EVALUATION OF STATISTICAL MODELS FOR CLIMATIC CHARACTERIZATION OF GKVK STATION BHOOMIKA RAJ, R. PALB 3153

EVALUATION OF STATISTICAL MODELS FOR CLIMATIC CHARACTERIZATION OF GKVK STATION BHOOMIKA RAJ, R. PALB 3153 DEPARTMENT OF AGRICULTURAL STATISTICS, APPLIED MATHEMATICS AND COMPUTER SCIENCE UNIVERSITY OF AGRICULTURAL SCIENCES GKVK, BENGALURU 2015

EVALUATION OF STATISTICAL MODELS FOR CLIMATIC CHARACTERIZATION OF GKVK STATION BHOOMIKA RAJ, R. PALB 3153 Thesis submitted to the University of Agricultural Sciences, Bengaluru In partial fulfilment of the requirements For the award of the degree of Master of Science (Agriculture) In Agricultural Statistics BENGALURU JULY, 2015

AFFECTIONATELY DEDICATED TO MY BELOVED PARENTS, MY SISTER, MY FRIENDS & MY TEACHERS

DEPARTMENT OF AGRICULTURAL STATISTICS, APPLIED MATHEMATICS AND COMPUTER SCIENCES UNIVERSITY OF AGRICULTURAL SCIENCES BENGALURU-560065 CERTIFICATE This is to certify that the thesis titled Evaluation of statistical models for climatic characterization of GKVK station. Submitted by Ms. BHOOMIKA RAJ, R., ID NO. PALB 3153 in partial fulfilment of the requirement for the degree of Master of Science (Agriculture) in Agricultural Statistics to the University of Agricultural Sciences, Bengaluru, is a record of bonafide research work done by her during the period of her study in this University under my guidance and supervision and that no part of the thesis has previously formed the basis for the award of any degree, diploma, associate ship, fellowship or other similar titles. Bengaluru, July, 2015 Mr. K.N. Krishnamurthy (MAJOR ADVISOR) Approved by: Chairman: Mr. K.N. Krishnamurthy Members: 1. Dr. M. Gopinath Rao 2. Dr. M.B. Rajegowda 3. Dr. G. B. Mallikarjuna

ACKNOWLEDGEMENT Success of any scientific research is not dependent on an individual effort. It is a harvest of seeds sown by group of like-minded people. It is a great juncture to recall all those faces and spirits in the form of teachers, friends and dear ones. I place on record my deep sense of gratitude with at most sincerity and heartfelt respects to the esteemed chairman of my Advisory Committee Mr. K. N. Krishnamurthy, Associate Professor, Dept. of Agricultural Statistics, GKVK, Bengaluru, for his excellent guidance, constant support, close counsel and valuable suggestions throughout the period of my study. I admit that it has been a great fortune for me to be associated from him during my degree programme. I am very much grateful and thankful to my advisory committee Dr. M. Gopinath Rao, Professor and Head, Dept. of Agricultural Statistics,UAS, GKVK, Bengaluru, Dr. M.B. Raje Gowda, The Registrar, UAS, GKVK, Bengaluru and Dr. G.B. Mallikarjuna, Associate Professor, Dept. of Agricultural Statistics, UAS, GKVK, Bengaluru, for their constant supervision, invaluable guidance and all the facilities extended during the course of this investigation. I gratefully record my indebtedness to the staff members Dr. D.M. Gowda, Professor and University Head, Mr. H..S.S Surendra, Associate Professor, Mr. V. Manjunath, Associate Professor, Mr. R Munirajappa,, Associate Professor, Dr. H. Chandrashekhar, Professor, Dr. S..S..S. Surpur, Professor, Mr. K. B. Murthy, Associate Professor, Dr. S. S Patil., Associate Prfessor for their timely help rendered during the course of my research. On my pesronel note, I humbly place before my parents Rajanna,T and Asha,T, my most sincere gratitude. Their blessings have renewed me every day, all the way on

the journey through my life. I am also thankful to my sister Yashaswini Raj, R for her constant support. I am fortunate to have a myriad of friends here. I am thankful for the emotional support from my friends Amrutha,T., Bhavana,D., Chethan,A. J., Nishitha,K., Pavithra, H.K, Priyanka Basavaraddi, Seema Raj, Shashi Kumar,N., Sitara Makam, Shivani Dechamma, Supriya,D., S Swapna Shetty, who encouraged me in each and every step of my post graduation. My special thanks to Aditya K. S., Bhavyashree,S., Golive Prashanthi, Navya, H. S., for their valuable suggestions, moral support and cooperation during the degree programme. It gives me great pleasure to thank all my classmates Deepti K.A., Kavitha D.S., Madhushree C.S., Pavithra R., Satish Y., Shilpashree G.S., Sowmya S Desai, and all juniors in the Department of Agricultural Statistics for their affection and cooperation rendered to me in all aspects. Last but not the least, I would like to thank non-teaching faculty Miss. Jayasudha, Mr. Bayanna and Mr. Umesh, Department of Agricultural Statistics for their kind co-operation and encouragement during my study and research. I owe all my success to my beloved father, mother, sister and teachers for their inspiration and support that saw me through all the trials and tribulations. BENGALURU JULY 2015 Bhoomika Raj,R.

EVALUATION OF STATISTICAL MODELS FOR CLIMATIC CHARACTERIZATION OF GKVK STATION BHOOMIKA RAJ, R. ABSTRACT Agro-climatic characters play an important role in deciding the cropping pattern of a region. The distribution of rainfall is one such climatic character essential to plan farm activities in a given region. The present study was conducted to know the climatic characterization of GKVK station. The secondary data of weather parameters over a period of 38 years (1976-2013) and Finger millet (GPU 28) crop yield (qtl/ha) data during kharif season for a period of 16 years (1998-2013) was collected from AICRP on Agro Meteorology and AICSMIP, UAS, GKVK, Bengaluru respectively. Among the weather parameters, amount of maximum daily rainfall (mm) was considered to fit appropriate probability distributions. The probability distributions viz., Normal, Log- normal, Gamma (1P, 2P, 3P), Generalized Extreme Value (GEV), Weibull (1P, 2P, 3P), Gumbel and Pareto were used to evaluate the best fit for maximum daily rainfall (mm). Kolmogorov-Smirnov test for the goodness of fit of the probability distributions showed that for majority of the data sets on rainfall at different study periods, Weibull (3P) distribution was found to be the best fit. However, all the data sets were scale dominated which indicated large variation in the distribution of rainfall. Simple and multiple linear regression analysis showed that none of the weather parameters influenced the finger millet crop yield significantly. The path coefficient analysis indicated that among all the weather parameters, direct effect of PET on rainfall was found to be highly negative, while evaporation had the highest positive direct effect on rainfall. Further, vapor pressure and evaporation had positive and negative indirect contributions through the other weather parameters respectively. Bengaluru July 2015 K.N. Krishnamurthy Major Advisor

fpé«pé ದ ದ ಹ ನ ರ ಪ ಯ ಅ ಅ ಶಗಳ ದ ಗಳ ಲ ಪನ sàæ«äpá gáeï Dgï ಶ ಕ -ಹ ನ ತ ಗಳ ಒ ದ ಪ ಶದ ಕ ದ ಯನ ಧ ಸ ವ ಮಹತ ದ ತ ವನ ವ ಸ ತ. ಅ ತಹ ಹ ನ ಲ ಣಗಳ, ಮ ತರ ಯ MAzÀÄ ಪ ಶದ ಕ ಚಟ ವ ಗಳ ಜ ಯ ಅತ ಗತ ತ ವನ ªÀ9» ÀÄತ zé. ಪ ಸ ತ ಅಧ ಯನªÀÅ fpé«pé ದ ದ ಹ ನ ತ ಯಲ ನ ಸ ತ. 16 ವಷ ಗಳ (1998-3013) ಲ ಮ ರ ಋತ ನ ( ಯ 28) ಇಳ ವ ( / ) ಮತ 38 ವಷ ಗಳ (1976-2013) ಅವ ಯ ಹ ನ ಯ ಕಗಳ ಯಕ ಯನ JL¹Dgï ಕ ಹ ನ ಮತ JL¹JªÀiïL, AiÀÄÄJJ ï, fpé«pé, ಗಳ j AzÀ ಕ ಮ ಪ ಯ ತ. ಹ ನ ಯ ಕಗಳ ಗ ಷ ನ ನ ಮ ( ) ಪ ಣವನ ಸ ಕ ಸ ಭವ ಯ ತರ ಗಳ ಳ ಲ ಪ ಗ ಸ. ಇ ಪ ಗ ¹gÀĪÀ ಸ ಭವ ಯ ತರ ಗ¼ÉAzÀgÉ, ನ, - ನ, (1P, 2P, 3P), ರಣ ದ ಎ ಲ (fe«), ಬ (1P, 2P, 3P), ಗ ಬ ಮತ. ಸ ಭವ ಯ ಹ ಗಳ, ಧ ಅಧ ಯನ ಲಗಳ ಮ ದ ಶ ಗ ಪ ಗಳ ಬಹ ಕ ಬ (3P) ತರ ಯ ಅತ ತ ಮ JA ÄzÀÄ - ಪ AiÀÄ ªÀÄÆ PÀ ಕ ಡ ಬ ತ. ಆ ಗ, ಎ UÀÄA ÀÄ ಪ ಣದ ಮ ಹ ಯ ವ ಸಗಳ ಸ ಸ ವ ಪ ವ ರ ತ. ವ ಹ ನ ಯ ಕಗ¼ÀÄ ಇಳ ವ ಗಮ ಹ ಪ ವ ÃgÀĪÀÅ ಲ JA ÄzÀÄ ಸರಳ ಮತ ಹಲ ರ ತ ಕ ಜ ತ ಷ ÄAzÀ w½zàä A vàä. ಗ ಗ ಕ ಷ AiÀÄ ಪ ರ, ಎ ಹ ನ ಯ ಕಗಳ, ಮ ಯ ಇ AiÀÄÄ ಚ ಗಮ ಹ ನ ತ ಕ ರ ಪ ಮªÀನ ಗ ಕರಣªÀÅ ಚ ಗಮ ಹ ಸ ತ ಕ ರ ಪ ಮªÀನ ಸ zàªàå. ಇದಲ, ಆ ಯ ಒತ ಡ ಮತ ಕರಣUÀ¼ÀÄ ಕ ಮ ಇತರ ಹ ನದ ಯ ಕಗಳ ಮ ಲಕ ಸ ತ ಕ ಮತ ನ ತ ಕ ಪ ಡ ಯನ ದ ªÀÅ. ಗಳ ರ ಜ 2015 ಕ ಷ ಮ,. ಎ. ಪ ಮ ಖ ಸಲ ರ

CONTENTS CHAPTER TITLE PAGE No. I INTRODUCTION 1-7 II REVIEW OF LITERATURE 8-20 III MATERIAL AND METHODS 21-35 IV RESULTS 36-67 V DISCUSSION 68-76 VI SUMMARY 77-79 VII REFERENCES 80-84

LIST OF TABLES TABLE No. TITLE PAGE No. 3. 1 Description of various probability distribution functions. 25 4. 1. 1 Summary of Statistics for Maximum daily rainfall (mm). 37 4. 1. 2 Study period wise probability distributions using goodness of fit test. 4. 1. 3 Parameters of the best fit probability distributions for Maximum daily rainfall. 50-51 52 4. 2. 1 Summary of Statistics for weather parameters. 54 4. 2. 2 Estimates of the simple linear regression model relating the crop yield with weather parameters. 4. 2. 3 Multiple linear regression of weather parameters on the yield of finger millet. 4. 2. 4 Observed and predicted yield of finger millet crop over a period of 16 years (1998-2013). 56 57 58 4. 3. 1 Correlation matrix between weather parameters and rainfall. 62 4. 3. 2 Path coefficients showing direct and indirect effects of weather parameters on rainfall. 4. 3. 3 Direct and indirect partitioning of correlation between rainfall and weather parameters. 63 67

LIST OF FIGURES FIGURE No. TITLE PAGE No. 1 Agro-climatic zones of Karnataka. 6-7 2 Conceptual regimes described in parameter space with the shape parameter on the x-axis and the scale parameter on the y-axis. 26 3 Cause and effect relationship. 33 4 Variation in Annual maximum daily rainfall (mm). 37-38 5 Variation in Seasonal maximum daily rainfall (mm). 37-38 6 7 8 9 10 Plot of best fitted distribution functions for Maximum daily rainfall (mm). Effect of weather parameters on the finger millet crop yield. Plot of PRESS residuals against predicted finger millet yield. Normal Probability Plot for finger millet yield against weather parameters. Path diagram for the direct and indirect effects of weather parameters on rainfall. 53-54 57-58 59-60 59-60 67-68

LIST OF ABBREVIATIONS SL. No. ABBREVATIONS 1 SMW - Standard Meteorological Week 2 K-S - Kolmogorov-Smirnov 3 Gamma (1P) - One parameter Gamma distribution 4 Gamma (2P) - Two parameters Gamma distribution 5 Gamma (3P) - Three parameters Gamma distribution 6 Weibull (1P) - One parameter Weibull distribution 7 Weibull (2P) - Two parameters Weibull distribution 8 Weibull (3P) - Three parameters Weibull distribution 9 GEV- Generalized Extreme Value 10 PET- Potential Evapotranspiration

I INTRODUCTION Climate is a measure of average pattern of variation in temperature, humidity, atmospheric pressure, wind speed, precipitation, atmospheric particle count and other meteorological variables in a given region over long periods of time. Climate is different from weather, in that weather only describes the short-term conditions of these variables in a given region. The Intergovernmental Panel on Climate Change (IPCC) (Anon., 2007) defines climate change as, changes in the mean and/or the variability of its properties that persists for an extended period, typically decades or longer. A region's climate is generated by the climate system, which has five components: atmosphere, hydrosphere, cryosphere, lithosphere, and biosphere. Climate is usually defined as the "average weather" or as the statistical description in terms of the mean and variability of relevant quantities over a period ranging from months to thousands or millions of years. The classical period is 30 years, as defined by the World Meteorological Organization (WMO). These quantities are most often surface variables such as temperature, precipitation, and wind. Climate in a wider sense is the state, including a statistical description, of the climate system. The difference between climate and weather is usefully summarized by the popular phrase Climate is what we expect, weather is what we get. Climate change brings about changes in the weather conditions. There is a reason to believe that climate change could affect agricultural productivity, and cause increased health hazards and submergence of lands due to rise in the sea level to name a few. Climate change is the net result of many factors caused by continuous evolution of Planet Earth through many geological eras. However, there is growing concern about manmade developments causing, even if partially or insignificantly, the climate change outcomes. The industrialization that started from the late 17 th century is believed to have accelerated the process of climate change by emissions of Greenhouse Gases (GHGs) to the atmosphere. The observed levels of GHGs have perhaps nearly crossed tolerance levels in the atmosphere so that the survival for many animal and human species is at stake, while developmental needs of human race are contributing to factors like deforestation, urbanization etc., that can speed up the process of climate change. Climatic variability is the major factor influencing the agriculture productivity. Global climate change and its impacts on agriculture have become an important issue. Agriculture production is highly dependent on climate and it is also adversely affected by increasing climatic variability. Climate change is one of the most important global environmental challenges facing humanity with implications for food production, natural ecosystems, freshwater supply, health, etc. Climate classification The Koppen Climate Classification System is the most widely used system for classifying the world's climates. Its categories are based on the annual and monthly Evaluation of Statistical Models for Climatic Characterization of GKVK Station 1

averages of temperature and precipitation. The Koppen system recognizes five major climatic types, each type being designated by a capital letter. A- Tropical Moist Climates: all months have average temperatures above 18 Celsius. B - Dry Climates: with deficient precipitation during most of the year. C - Moist Mid-latitude Climates with Mild Winters. D - Moist Mid-Latitude Climates with Cold Winters. E - Polar Climates: with extremely cold winters and summers. Climate of India India is home to an extraordinary variety of climatic regions, ranging from tropical in the south to temperate and alpine in the Himalayan north, where elevated regions receive sustained winter snowfall. The Himalayas and the Thar Desert strongly influence the nation s climate. The Himalayas act as a barrier to the frigid katabatic winds flowing down from Central Asia keeping the bulk of the Indian subcontinent warmer than most locations at similar latitudes. Land areas in the north of the country have a continental climate with severe summer conditions that alternates with cold winters when temperatures plunge to freezing point. In contrast are the coastal regions of the country, where the warmth is unvarying and the rains are frequent. The country is influenced by two seasons of rains, accompanied by seasonal reversal of winds from January to July. During the winters, dry and cold air blowing from the northerly latitudes from a north-easterly direction prevails over the Indian region. Consequent to the intense heat of the summer months, the northern Indian landmass becomes hot and draws moist winds over the oceans causing a reversal of the winds over the region which is called the summer or the South-west monsoon. This is the most important feature controlling the Indian climate because about 75 per cent of the annual rainfall is received during a short span of four months (June to September). Variability in the onset, withdrawal and quantum of rainfall during the monsoon season has profound impacts on water resources, power generation, agriculture, economics and ecosystems in the country. The variation in climate is perhaps greater than any other area of similar size in the world. There is a large variation in the amounts of rainfall received at different locations. The rainfall pattern roughly reflects the different climate regimes of the country, which vary from humid in the north east (about 180 days rainfall in a year), to arid in Rajasthan (20 days rainfall in a year). So significant is the monsoon season to the Indian climate, that the remaining season are often referred relative to the monsoon. The rainfall over India has large spatial as well as temporal variability. A homogeneous data series has been constructed for the period 1901-2003 based on the uniform network of 1476 stations and analyzed the variability and trends of rainfall. Normal monsoon rainfall more than 150cm is being observed over most parts of northeast India, Konkan and Goa. Normal monsoon rainfall is more than 400cm over major parts of Meghalaya. Annual rainfall is more than 200 cm over these regions. 2 Bhoomika Raj, R., M.Sc. (Ag. Stat.), 2015

For the country as a whole, mean monthly rainfall during July (286.5 mm) is highest and contributes about 24.2 per cent of annual rainfall (1182.8 mm). The mean rainfall during August is slightly lower and contributes about 21.2 per cent of annual rainfall. June and September rainfall are almost similar and contribute 13.8 per cent and 14.2 per cent of annual rainfall, respectively. The mean South-west monsoon (June, July, August & September) rainfall (877.2 mm) contributes 74.2 per cent of annual rainfall (1182.8 mm). Contribution of pre-monsoon (March, April & May) rainfall and postmonsoon (October, November & December) rainfall in annual rainfall is mostly the same (11 per cent). Coefficient of variation is higher during the months of November, December, January and February. India is characterized by strong temperature variations in different seasons ranging from mean temperature of about 10 C in winter to about 32 C in summer season. There are four major types of climate found in India. They are presented as below: 1. The cold weather season (December to February) 2. The hot weather season (March to May) 3. The rainy season (June to September) 4. The Season of Retreating South-west Monsoon (October-November). 1) Cold Weather Season January and February are the coldest months of this season. The temperature is between 10 C to 15 C in Northern India and about 25 C in Southern India. 2) Hot Weather Season The North Indian region experiences a well defined hot weather season during the month of April and May. The temperature starts rising by the middle of March and by mid May, mercury touches 41 to 42 C. Temperature even exceeds 45 C in areas of central and north-west India. 3) The Rainy Season The inflow of South-westerly monsoon in India brings the season of rain in India. The monsoon may burst in the first week of June or even earlier in the coastal areas, while in the interior it may be delayed to the first week of July. With the onset of rains, temperature starts falling. The Indian sub-continent receives the bulk of its rainfall during the south-west monsoon period. 4) The Season of Retreating Monsoon The South-west monsoon begins to retreat from northern India by the second week of September. The pattern of retreat also shows interesting regional variations. The Evaluation of Statistical Models for Climatic Characterization of GKVK Station 3

weather during the season is characterized by high day temperatures, but nights are pleasant with the mean minimum temperature going down to 20 C or even lower. Climate of Karnataka Climate of Karnataka presents an exceptional diversity. While the hilly and plateau regions demonstrate a different climatic behavior, the plain presents comparatively a warmer atmosphere. The varying geographic and physiographic conditions of the State is responsible for the climatic variation in the State from arid to semi-arid in the plateau region, sub humid to humid tropical in the Ghats and humid tropical monsoon type in the west coast plains. For meteorological purposes, the State has been divided into three subdivisions: Coastal Karnataka, which includes, Dakshina Kannada and Uttara Kannada districts. North Interior Karnataka, which includes, Belagavi, Bidar, Vijapura, Dharwad, Kalaburgi and Raichur districts. South Interior Karnataka, which includes the remaining districts of Bengaluru Rural, Begaluru urban, Ballari, Chikkamamagaluru, Chitradurga, Kodagu, Hassan, Kolar, Mysuru, Mandya, Shivamogga and Tumakuru districts. As per Koppen's classification, the State witnesses three climatic types. The tropical monsoon covers the entire coastal belt and the adjoining areas. The southern half of the State, outside the coastal belt experiences hot, seasonally dry tropical savanna climate. The remaining regions of the southern half of the State experiences hot, semiarid, tropical steppe type of climate. According to the Thronthwaite's classification, the coastal and Malnad regions are pre-humid i.e. those having moisture index of 100 per cent and above. The interior regions are semi-arid (moisture index of minus 66.7 per cent to minus 33.3 per cent). Moist sub-humid and dry-humid zones (moisture index of minus 33.3 to plus 20 per cent) are the transition zones covering the region between the Malnad and the coast. The arid zone in the State is confined to the east of Bellary district, most of Raichur district, east of Chitradurga district and the adjoining Pavagada taluk of Tumakuru district with small areas in Vijapura and the adjoining north-eastern Belagavi district. Very dry areas with moisture indices less than minus 60 per cent occur in Chitradurga, Bellary, Raichur and Vijapura districts, west and south of Kalaburgi district and north Tumakuru district. Semi- arid regions with moisture indices of less than 50 per cent occur in Bidar district in the north, Bengaluru district and adjoining areas of Tumakuru and Mysuru districts in the south. The sub- humid zone in the State exists as a narrow belt east of Western Ghats from Belagavi in the north to the west of Heggadadevanakote taluk of Mysuru district in the south. Adjoining to this in the west is a narrow strip of humid and a wider strip of pre-humid zones. About 77 per cent of the total geographical area of the State, covering interior Karnataka is arid or semi-arid with the State contributing 15 per cent of the total semi-arid or 3 per cent of the total arid areas of the country. 4 Bhoomika Raj, R., M.Sc. (Ag. Stat.), 2015

Annual Rainfall Coastal Karnataka: 3456mm North Interior Karnataka: 731mm South Interior Karnataka: 1126mm Summer The average weather of Karnataka is dry and warm over different regions and summers starting from the month of April which last till the month of May. These months are the hottest months in Karnataka; somewhere the humidity percentage is comparatively low, but as the month of June starts to pervade humidity in the air could make you uncomfortable as the monsoon is reaching the state soon. The average temperature remains around 34 C with 75 per cent humidity. Monsoon Monsoon season starts from June and lasts till September, as prominent downfalls in temperature are noted, but at this time the percentage of humidity gets a little higher in the atmosphere. The Tropical Monsoon climate covers the entire coastal belt and adjoining areas. This area experiences heavy rainfall 3456 mm annually, while the North interior Karnataka and its adjoining areas; Vijapura, Bagalkote, Belagavi, Haveri, Gadag, Dharwad, Kalaburgi, Ballari, Koppal and Raichur districts experience mediocre rainfall of 731mm per annum. On the other side, the South interior Karnataka receives a blissful shower of monsoon annually. Winter Winters start from January and last till February. During winters, the climate of Karnataka remains pleasant. Every region of Karnataka experiences a cold winter - wave and average temperature falls between 32 C to 20 C or even below that. Karnataka receives a delightful shower during months of October and November. Agro-Climatic Zones 1. North Eastern Transition Zone 2. North Eastern Dry Zone 3. Northern Dry Zone 4. Central Dry Zone 5. Eastern Dry Zone 6. Southern Dry Zone 7. Southern Transition Zone 8. Northern Transition Zone Evaluation of Statistical Models for Climatic Characterization of GKVK Station 5

9. Hill Zone 10. Coastal Zone Need for the study Agro climatic characters play an important role in deciding the ways of life, crop of the region and other developmental plans. The study of climate at different scales and its variations with the topography and the nature of soils can play a fundamental role in energy and/or environmental conscious land use planning. Rainfall is one of the most important natural input resources to crop production. Its occurrence and distribution is erratic, and shows temporal and spatial variations in nature. Hence, there is a need to study the distribution both on long term (annually or seasonally) as well as short term (monthly or weekly). Most of the hydrological events occurring as natural phenomena are observed only once. One of the important problems in hydrology deals with the interpreting past records of hydrological event in terms of future probabilities of occurrence. The average rainfall of the region is generally considered as the basis for deciding irrigation management and cropping pattern. Long term analysis is vital for firm planning and execution of crop cultivation. Rainfall is considered as principle source of water. The success or failure of crops particularly under rainfed conditions is closely linked with the rainfall patterns. The important characteristics of rainfall influencing production from rainfed farming are the duration of wet spells and durations of intervening dry spells. Rainfall during the monsoon is not uniform. The annual and seasonal analysis of rainfall will give general idea about the rainfall pattern of the region, whereas the monthly analysis of rainfall will be of much use as far as agricultural planning is concerned. The spatial and temporal variability of rainfall and its uneven and inadequate distribution is based on other weather parameters such as, maximum and minimum temperatures, relative humidity, evapo-transpiration, sunshine hours, etc. This determines the success or failure of crops especially in drought prone areas. Knowledge of the distribution of rainfall is essential for successful farming. It is also important to know the chances of occurrence during the critical stages of the crops for deciding the sowing date, cropping pattern and planning for protective irrigation and intercultural operations. With this view in mind, the study was conducted for GKVK station with the following specific objectives: 1. To fit appropriate Statistical distributions for rainfall. 2. To study the impact of weather parameters on the crop yield. 3. To study direct and indirect effects of weather parameters on rainfall. 6 Bhoomika Raj, R., M.Sc. (Ag. Stat.), 2015

Fig.1: Agro-climatic zones of Karnataka Evaluation of Statistical Models for Climatic Characterization of GKVK Station 7

II REVIEW OF LITERATURE This chapter deals with the reviews of related past studies. Keeping in view of the objectives of the study, the reviews have been presented under the following headings: 2.1 To fit appropriate Statistical distributions for rainfall. 2.2 To study the impact of weather parameters on the crop yield. 2.3.To study direct and indirect effects of weather parameters on rainfall. 2.1 To fit appropriate Statistical distributions for rainfall. Jinfan et al. (1998) compared the general two-parameter Weibull distribution with other widely used distributions for describing the distribution of daily precipitation event sizes at 99 sites from the U.S. Pacific Northwest. They found that a one-parameter Weibull model of a wet day precipitation amount based on the Weibull distribution provided a better fit to historical daily precipitation data for eastern U.S. sites than other one-parameter models. The fit of the single-parameter Weibull to the 99 stations included in this study was significantly better than other single-parameter models tested, and performed as well as the widely endorsed, more cumbersome, two-parameter gamma model. Both the one and two parameter Weibull distributions were shown to have L- moments that were consistent with historical precipitation data, while the ratio of L-skew and L-variance in the gamma model was inconsistent with the historical record by this measure. In addition, it was found that the two-parameter gamma distribution was better fit using the method of moment estimators than maximum likelihood estimates. Gregory et al. (2007) demonstrated the feasibility of fitting cell-by-cell probability distributions to grids of monthly interpolated, Continent-wise data. They applied these grids to improve satellite-remote sensing of drought and interpretation of probabilistic climate outlook for forecasts. They showed that the gamma distribution was well suited for these applications because it was capable of representing a variety of distribution shapes. This study tested the goodness of fit using the Kolmogorov Smirnov (KS) test, and compared these results against another distribution commonly used in rainfall events, the Weibull distribution. The gamma distribution was suitable for roughly 98 per cent of the locations over all months. The techniques and results presented in this study provided a foundation for the use of the gamma distribution to generate drivers for various rain related models. These models were used as decision support tools for the management of water and agricultural resources as well as food reserves by providing decision makers with ways to evaluate the likelihood of various rainfall accumulations and assess different scenarios in Africa. Evaluation of Statistical Models for Climatic Characterization of GKVK Station 7

Suhaila and Jemain (2007) have tested Gamma, Weibull, Kappa and Mixed exponential distributions to fit the daily amount of rainfall in Peninsular Malaysia. They estimated the parameters for each distribution using maximum likelihood method. The selected model was chosen based on the minimum error produced by seven goodness of fit (GOF) tests, namely, the Median of Absolute Difference (MAD) between the empirical and hypothesized distributions, the traditional Empirical Distribution Function (EDF) statistics which included Kolmogorov- Smirnov, Anderson Darling statistic and Cramer-Von-Mises statistic and the new method of EDF statistics based on likelihood statistics. Based on this goodness of fit tests, the Mixed Exponential was found to be the most appropriate distribution for describing the daily amount of rainfall in Peninsular Malaysia. Lars and Vogel (2008) studied the distribution of wet-day daily rainfall and identified that the 2-parameter Gamma (2P) distribution as the most likely candidate distribution based on traditional goodness of fit tests. They used probability plot correlation coefficient test statistics and L-moment diagrams to examine the complete series and wet-day series of daily precipitation records at 237 U.S. stations. The analysis indicated that the Pearson Type-III (P3) distribution fits the full record of daily precipitation data remarkably well, while the Kappa (KAP) distribution best described the observed distribution of wet-day daily rainfall. It was also shown that the Gamma (2P) distribution performed poorly in comparison to either the P3 or KAP distributions. Deka et al. (2009) determined the best fitting distribution to describe the annual series of maximum daily rainfall data for the period 1966 to 2007 of nine distantly located stations in North East India. Five extreme value distributions viz., generalized extreme value distribution; generalized logistic distribution, generalized Pareto distribution, lognormal distribution and Pearson distribution were fitted for this purpose using the method of L-moment and LQ-moment. The performance of the distributions was evaluated by using three goodness of fit tests namely, relative root mean square error, relative mean absolute error and probability plot correlation coefficient. Further, L- moment ratio diagram was also used to confirm the goodness of fit for the above five distributions. Finally, goodness of fit test results was compared and generalized logistic distribution was empirically proved to be the most appropriate distribution for describing the annual maximum rainfall series for the majority of the stations in North East India. Mohita and Singh (2010) studied the annual daily maximum rainfall of Pantnagar and showed that it received rainfall ranging from 49.32mm (minimum) to 229.40mm (maximum) indicating a very large range of fluctuation during the period of study. It was observed that the best probability distributions obtained for the maximum daily rainfall for different data sets are different. The lognormal and gamma distribution were found as the best fit probability distribution for the annual and monsoon season period of study, respectively. Generalized extreme value distribution was observed in most of the weekly period as best fit probability distribution. The best fit probability distribution of monthly 8 Bhoomika Raj, R., M.Sc. (Ag. Stat.), 2015

data was found to be different for each month. The scientific results clearly established that the analytical procedure devised and tested in this study may be suitably applied for the identification of the best fit probability distribution of weather parameters. Manikandan et al. (2011) analyzed daily rainfall data of 37 years and annual one day maximum rainfall was sorted to estimate the probable one day maximum rainfall for different return periods by using the probability distribution function at Tamil Nadu Agricultural University (TNAU) Campus. The mean value of annual one day maximum rainfall was found to be 77.79 mm with standard deviation, coefficient of variation and skewness of 28.56, 0.37 and 0.71 respectively. October month received the highest amount of one day maximum rainfall and the highest number of one day maximum rainy days. Five probability distributions such as Normal, Log Normal, Gumbel, Pearson Type- III and Log Pearson Type-III distributions were tested to determine the best fit probability distribution that described the annual one day maximum rainfall by comparing with the Chi-square value. The results revealed that the log-normal distribution was the best fit probability distribution for annual one day maximum rainfall. Based on the best fit probability distribution, the minimum rainfall of 33.99 mm in a day can be expected to occur with 99 per cent probability and one year return period and maximum of 173.8 mm rainfall can be received with one per cent probability and 100 year return period. The results from the study could be used as a rough guide by design engineers and hydrologists for appropriate planning and design of small soil and water conservation structures, irrigation and drainage systems. Bhim Singh et al. (2012) analyzed the daily rainfall data of 39 years (1973-2011) to determine the annual one day maximum rainfall of the Jhalarapatan area of Rajasthan, India. The observed values were estimated by Weibull's plotting position and expected values were estimated by four well known probability distribution functions viz., normal, log-normal, log-pearson type-iii and Gumbel. The expected values were compared with the observed values and goodness of fit was determined by Chi-square (χ 2 ) test. The results showed that the log-pearson type-iii distribution was the best fit probability distribution to forecast annual one day maximum rainfall for different return periods. Based on the best fit probability distribution, the minimum rainfall of 44.74 mm in a day can be expected to occur with 99 per cent probability and one year return period and maximum of 252.98 mm rainfall can be received with one per cent probability and 100 year return period. And it was observed that July month received the highest amount of one day maximum rainfall (46 per cent) followed by August (38 per cent) and September (13 per cent). Oseni and Femi (2012) studied several types of statistical distributions to describe the rainfall distribution in the Ibadan metropolis over a period of 30 years. The exponential, gamma, normal and Poisson distributions were compared to identify the optimal model for a daily rainfall amount based on data recorded at rain gauge station at the Forestry Research Institute of Nigeria, Jericho, and Ibadan (FRIN). The models were Evaluation of Statistical Models for Climatic Characterization of GKVK Station 9

evaluated based on chi- square and Kolmogorov-Smirnov tests. Overall, this study showed that the exponential distribution was the best model followed by normal and Poisson model that had the same estimated rainfall amount for describing the daily rainfall in Ibadan metropolis. Lala et al. (2013) analyzed daily rainfall data for 28 years (1983-2010) of Central Meghalaya, Nongstoin station for estimating maximum daily rainfall. The annual maximum daily rainfall data was fitted to five different probability distribution functions i.e. Normal, Log-normal, Pearson Type-III, Log Pearson Type-III and Gumbel Type-I extreme. The probable rainfall value for different return periods was estimated. These estimated values were compared with the values obtained by Weibull s Method. The analysis indicated that, the Gumbel distribution gave the closest fit to the observed data. Hence, Gumbel distribution may be used to predict maximum rainfall, which will be a great importance for economic planning and design of small and medium hydraulic structures. Majid et al.(2013) determined the best probability distribution of the rainfall data in the Schoeckelbach basin, which is situated at the Northern Graz in Austria using L- moments method (parameter estimation technique). They used Chi- square, Kolmogorov- Smirnov and Root mean square error (RMSE) tests to test the goodness of fit. They compared between four commonly used rainfall frequency distribution, such as Generalized Extreme Value (GEV), Gumbel, Log- Pearson type II I(LP III) and 3 parameter Log normal (LN III) and showed that Gumbel was the best probability distribution for Schoeckelbach basin. The best probability distribution was used to determine the Intensity- Duration- Frequency (IDF) relation for the Schoeckelbach basin. Mohsen et al. (2013) investigated the spatial and temporal characteristics of rainfall in the United Arab Emirates (UAE). The region is divided into four climate zones (East Coast, Mountains, Gravel Plains and Desert Foreland) of distinguished rainfall distribution. The rainfall patterns, rainfall probability of occurrences, rainfall intensityduration-frequency (IDF) relationship, probable maximum precipitation (PMP) and drought scenarios were investigated. Daily rainfall data from a network of stations across the UAE were used. Standard statistical techniques were applied for data analyses. The Gumbel, log Pearson, generalized extreme value, log normal, Wake by and Weibull probability distributions were tested to fit extreme rainfalls. Both Gumbel and Weibull distributions were found adequate. Measures of dispersion and symmetry of rainfall patterns were found relatively high. The estimated PMP values were found highest in the East Coast region and lowest in the Gravel Plains region. The estimated drought severity index showed that the regions have similar trends of drought patterns over the years. The study is useful for sustainable water resources planning and management in the region. Taofik et al. (2013) carried out a study to evaluate the utility of Weibull probability distribution in the estimation of the distribution of rainfall data of South- 10 Bhoomika Raj, R., M.Sc. (Ag. Stat.), 2015

Western Nigeria. The study was executed using the rainfall data of 4 stations (Ibadan, Ikenne, Ilora and Orin Ekiti) of the Institute of Agricultural Research and Training, Ibadan. The data obtained from the database were subjected to Weibull distribution analysis using Peak Fit program and the parameters were estimated using maximum likelihood estimation. The results showed that the parameter estimates ranged between 0.32907 for the Ibadan rainfall and 0.37024 for the Ilora rainfall while ranged from 0.8712 for rainfall at Ibadan to 0.9913 for rainfall at Ikenne. The statistical properties of this distribution were that the estimated mode of the rainfall was zero because α and < 1 and the mean ranged between 5.2272 (Ibadan) and 5.9478 (Ikenne) while the estimated mean for Ilora was 5.8975 and Orin Ekiti was 5.8411. Both the estimated and actual means followed the same trend, indicating the plausibility of the distribution function. Deka (2014) demonstrated the application of the method of Trimmed L-moment (TL-moment) to determine the best fitting distribution for annual series of maximum daily rainfall data of nine distantly located stations of North- East India. For this purpose, three extreme value distributions viz. Generalized Extreme Value distribution (GEV), Generalized Logistic distribution (GLD), Generalized Pareto distribution (GPD) were considered and the parameters for each of the distributions were estimated using the method of TL-moment. The performance of the distributions was evaluated by using three goodness of fit tests, namely, relative root mean square error, relative mean absolute error and probability plot correlation coefficient. Goodness of fit test results were compared and Generalized Extreme Value distribution was empirically proved to be the most appropriate distribution for describing the annual maximum rainfall data for all the stations considered when the parameters were estimated by using TL-moment method. 2.2 To study the impact of weather parameters on the crop yield. Jain et al. (1980) have made an attempt to study the forecasting of rice yield on climatic variables at different stages of crop growth in Raipur District using the yield data of 25 years and weekly weather variables viz., maximum temperature, relative humidity, total rainfall and number of rainy days. Based on the principal component analysis, weather variables were obtained and used as independent variables in the regression equation. Among the different models, two models were found suitable. In the first, weighted averages of weekly weather variables and their interactions using the power of the week as weights were used. The respective correlation coefficients with yield in place of week number were taken in the second model. The study revealed that forecasting of rice yield is possible by weekly climatic variables, 2 ½ months after sowing for crop of 5 months duration. Using the second model, the study showed that the weighted weather index/ principal component could be used in place of weather variables. Kulkarni et al. (2004) studied the crop yield weather relations using a nonparametric approach based on the concept of two-way contingency tables that had been proposed to account for the associations of the combined effect of rainfall pattern on crop Evaluation of Statistical Models for Climatic Characterization of GKVK Station 11

yield. The approach identified both the different rainfall patterns and the levels of crop yields that were likely to occur during the crop season and then measured their associations. The approach was applied for studying the weather (rainfall) effect on the three crops viz., Jowar, Groundnut and Cotton by using 40 years crop yield-rainfall data (1961-2000) of Anantapur district of Andhra Pradesh. The analysis provided useful information about the yield response range corresponding to the different rainfall patterns that were likely to occur during the crop growth season. Das et al. (2007) carried out a preliminary study to test a crop simulator model, ORYZA 2000 in the baro seasons during 1999-2000 and 2000-2001 in West Bengal. The model predicts the drastic changes in the yield by -10 per cent, -45.8 per cent and 72.1 per cent with change in temperature, +1 0 C, +2 0 C and +3 0 C respectively. They also revealed that even with warmer climate up to 1 0 C, production may increase by 10 per cent in an atmosphere with doubled CO 2 concentration. Chunqiang et al. (2008) reported that long term changes of reference evapotranspiration and crop water requirement can have great effect on agricultural production, hydrological cycle as well as water resources management and analyzed the effect of climate change and variability on reference evapo-transpiration and crop water requirements. The results showed that reference evapo-transpiration and water requirement had decreased with time. Geethalakshmi et al. (2008) used Info Crop model to evaluate the impact of climate change on crop yield for baseline data and daily weather data created for 2020, 2050 and 2080. The CO 2 concentration used in the model for the baseline, 2020, 2050 and 2080 runs were 376,414.522 and 682 ppm, respectively as projected in the IPCC (2001) report. Simulated rice production indicated that, generally rice production showed a declining phase in Tamil Nadu by 8.7, 23.6 and 42.2 per cent from the production level of the year 2000 in 2020, 2050 and 2080 respectively. Guruswamy et al. (2008) have assessed the mean annual and seasonal rainfall behaviour and also have identified the influence of weather parameters on the coconut productivity. South West Monsoon and winter rainfall seasons had negative correlation with coconut productivity, whereas summer and North East Monsoons had positive correlation with coconut productivity. The percentage of barren nut production in coconut had positive correlation with summer rainfall and negative correlation with winter, south west and north east monsoons. The weather variables namely maximum temperature, minimum temperature, relative humidity, evaporation and rainfall had positive correlation with coconut productivity. But there was no significant influence on the productivity of the palms. Jadhav et al. (2008) conducted an experiment for 5 years on sorghum by using four different sowing windows to study the relationship between weather parameters and yield 12 Bhoomika Raj, R., M.Sc. (Ag. Stat.), 2015

in Rabi season. Rabi sorghum sown at MW 39 (24-30 Sept) produced a maximum grain yield (850 kg/ha), fodder yield (29.06q/ha) and total monetary returns (Rs.13781/ha). However, the crop sown at MW 40 was also produced the on par yields with crop sown at MW 39 and of high degree of sustainability. Weather parameter showed a significant positive association with grain yield. The stepwise multiple regressions were obtained for different finishes wise weather parameters with a yield of Rabi sorghum, which explained 78 per cent variation in the yield. Kersebaum et al. (2008) studied the impact of projected climate change for winter wheat production which was simulated for 9 sites across Germany using the dynamic agro ecosystem model HERMES and downscaled climate change scenarios of the GCM ECHAM5 output for SRESM emission scenario A1B until 2050. Yield reductions between 2 and 11 per cent were estimated for 8 sites for the period 2031-2050. Decreasing summer precipitation led to a reduced efficiency of nitrogen fertilizers. Khadtare et al. (2008) conducted a study for 5 years on sunflower by using 4 different sowing windows to know the relationship between weather parameters and yield in Rabi season. The minimum temperature, relative humidity and bright sunshine had a highly significant positive association with grain yield at all the phonological phases. Significant negative association with grain yield by pan evaporation indicated that, at early growth stages Rabi sunflower did not favour the moisture stress condition. A significant positive association with grain yield at all stages of growth by minimizing temperature indicates Rabi sunflower responds well to the low temperature condition throughout the growth period. The weather parameter influences their contribution and performance of Rabi sunflower crop sown at different dates were assessed and model on combined effect was developed using stepwise multiple regression for predicting grain yield. Chaudhari et al. (2009) developed the regional models to know the yield response to temperature (minimum, maximum and its diurnal range) and precipitation for meteorological (met) sub-divisions of India and to study the impact of future climate change on major food crops viz. wheat, rice, potato and rapeseed-mustard. The area weighted averages of district-wise crop yield data were computed at met sub-division level for 1977 2007 for 9 major wheat (Triticum aestivum L.) producing met subdivisions, 16 major rice (Oryza sativa L.) producing met subdivisions, 6 major potato (Solanum tuberosum L.) growing sub-divisions of and 8 major rapeseed-mustard (Brassica spp.) growing subdivisions. Fortnightly correlation weighted weather parameters like minimum and maximum temperature and precipitation for the respective met sub-division and periods of the crop season were used to develop the empirical relationships. The study showed that, there was a clear negative response of yields to increase minimum temperatures for all three Rabi crops such as wheat, potato and rapeseed-mustard while the mixed response was observed for kharif rice. The reduction impact was high for Rabi crop as compared to kharif rice. The crop yield also showed Evaluation of Statistical Models for Climatic Characterization of GKVK Station 13

negative response to increase in maximum temperature. The crops like potato and rapeseed-mustard showed positive response to increased maximum temperature. This might be due to its strong positive correlations with diurnal temperature range (DTR). The mixed impact (increase and decrease both) was observed on rice yield for the increased precipitation. The estimated impacts of diurnal temperature range (DTR) changes on yields were generally less for wheat and rice crops while more for potato and rapeseed-mustard crops with the unit increase in DTR. Rojalin et al. (2009) assessed the impact of temperature and CO 2 on the productivity of the four major cereal food crops (wheat, rice, maize and pearl millet) taking New Delhi as study area. This study also aimed at identifying the most sensitive growth phase of these crops to the rising temperature. A soil-plant growth simulator CropSyst model was used for this purpose. Their study concluded that Rabi (winter) crops are (wheat) more sensitive to high temperature stress than the kharif crops. The positive effect of CO 2 was more in C4 crops (maize) as compared to C3 crops. The present study also indicated that adverse effect of rising temperature could be mitigated through the doubled CO 2 up to 10 0 C in pearl millet, up to 20 0 C in rice and wheat, up to 30 0 C in maize. However, the rise in temperature beyond that temperature could cause significant reduction in yield of all cereal crops under the present management environment. Hence, further research is required to find out the appropriate management practices to sustain the present level yield in a changed climate condition. Ratna and Amrender (2011) studied the application of Neural Networks (NNs) for crop yields (rice, wheat and sugar cane), by forecasting using Multilayer Perceptron (MLP) architecture with different learning algorithm has been attempted. For development of neural network based forecast models, yields of the crop at district level (Uttar Pradesh state, India) was considered as an output variable and indices of weather variables viz. maximum and minimum temperatures, rainfall and morning relative humidity were considered as input variables. Forecasts based on MLP architecture using a Conjugate gradient descent algorithm for learning have been found to be close to the observed ones in most of the cases. The findings of the study substantiate that neural networks possess potential for prediction purposes. Parekh and Suryanarayana (2012) carried out a study to determine the predominance of various meteorological data on the yield of wheat. The meteorological data for the Rabi season are collected and correlated with yield of wheat in Vallabh Vidyanagar for the period 1981-1999 using neural network fitting. Then, the model was re-trained until the best coefficient of correlation was obtained and this corresponding model was considered as the best model and this was further used to validate the dataset from 2000 to 2006. This whole procedure was repeated for three different Alternatives. In Alternative 1, only maximum and minimum temperatures were correlated with yield data. In Alternative 2, maximum and minimum temperature and relative humidity were correlated with yield data. In Alternative 3, Maximum and minimum temperatures, 14 Bhoomika Raj, R., M.Sc. (Ag. Stat.), 2015

sunshine hours and relative humidity were correlated with yield. The correlation between maximum and minimum temperatures & relative humidity and yield, the co-efficient of correlation for training, i.e. 1.00 validations, i.e. 0.97 and testing, i.e. 0.95 were best for 70 per cent, whereas in 30 per cent dataset, R came out to be 0.62, which is good. They evidently concluded from the study that considering three variables, the correlation was achieved as the best. This revealed that the yield of a crop was very much depended on maximum and minimum temperatures and relative humidity. Govinda (2013) examined the effect of precipitation and temperature variation on the yield of major cereals (e.g., rice, wheat, maize, millet and barley) in Dadeldhura district of Nepal. He used the time series information of yield and seasonal meteorological data (e.g., precipitation and temperature) to assess the effect using the exponential growth regression methods. The findings of this study confirmed that maximum temperature and minimum precipitation have had adverse effects on the yield of rice and maize. Temperature is statistically significant for the yield of all cereals whereas precipitation is statistically significant for the yield of maize and barley. Finally, precipitation and temperature have a statistically significant effect on rice and maize yield. The low rainfall and high temperature during the crops growing seasons had severely affected the yield of major cereals in Dadeldhura district. Elbariki et al. (2014) carried out a study to examine the trends of important weather parameters and their effect on the production of cereal crops in Solan district of Himachal Pradesh. The descriptive statistics and regression analysis revealed no significant trend in all the selected weather parameters in annual, seasonal and monthly basis for the period (1984 2011). In regression analysis cubic and quadratic functions (non significant) were found to be the best fit. Sen s estimation analysis indicated increasing trend of maximum temperature by 2.95 o C, minimum temperature by 0.50 o C and decreasing trend of rainfall, relative humidity and sunshine hours. Analysis of relationship of crop yields to time in regression analysis did not show any significant trend. However, Sen s estimate showed increasing trends for selected crops. There was no significant correlation of individual weather parameters with crop yields both in case of multiple regression analysis some effects were observed, indicating thereby that crop yields are influenced by combinations of weather parameters. 2.3 To study direct and indirect effects of weather parameters on rainfall. Kaushik et al.(2003) evaluated thirty-seven genotypes of American cotton at Agricultural Research Station, Sriganganagar during kharif 2001. Randomized block design was followed with two replications. A significant positive association was observed for a number of bolls per plant, sympodial branches per plant, plant height and monopodium branches per plant with seed cotton yield per plant. Besides, these components had positive inter-relationships among themselves. Path-coefficient analysis at genotypic level revealed that sympodia per plant, monopodia per plant and boll weight Evaluation of Statistical Models for Climatic Characterization of GKVK Station 15

(g) had positive direct effect on seed cotton yield. Selection based on these characters may contribute considerably to the improvement in seed yield. Sharma et al. (2004) studied the effect of meteorological factors on the population build up of green leaf hopper Nephotettix virescens Dist (Cicadellidae, Hemiptera), plant hoppers Cofana spectra Dist (Delphacidae, Hemiptera) and C. Yasumatsui Young (Kolla mimica, Hemiptera) and rice gundhi bug Leptocoriza acuta Thunberg (Alydidae, Hemiptera) in rice growing season (July to November) through light trap collection during ten years (1988 1997). Maximum populations of Nephotettix virescens Dist (Cicadellidae, Hemiptera) and C. Yasumatsui Young (Kolla mimica, Hemiptera) were recorded in the third week of October during all the years. Cofana spectra Dist had maintained peak activity in respect of population in the last week of September and third week of October. Leptocoriza acuta Thunberg (Alydidae, Hemiptera) had a maximum population in second and third weeks of October during the aforesaid period. No meteorological factors had a significant effect on the population build up of Nephotettix virescens Dist, Cofana spectra Dist and C. yasumatsui Young in the month of October. In the case of Leptocoriza acuta Thunberg, no other factor, but rainfall had a positive correlation of order 0.857 with population build up in the fourth week of September. Anwar et al. (2009) conducted a study at Wheat Research Institute, Faisalabad during 2006-2008 where significant genotypic differences existed among the genotypes. Correlation coefficients were computed for grain yield plant-1, tillers plant-1, spikelet s spike-1, 1000 grain weight, spike length, days to heading, days to maturity and plant height from the F1 crosses developed from four lines and three testers including their parents. The results showed that grain yield plant-1 was positively and significantly correlated with number of tillers plant-1 and days to maturity at genotypic level but non significantly correlated at phenotypic level. Days to maturity had positive genotypic correlation with grain yield plant-1, number of tillers plant-1 and 1000-grain weight. Days to maturity and tillers plant-1 had a positive direct effect on grain yield plant-1 also. Therefore, more days to maturity and more tillers plant-1 would be important selection criteria for improved grain yield plant-1 in the breeding material studied. Ramazan (2009) conducted an in 1998-1999 and 1999-2000 growing seasons with 7 different durum wheat genotypes under West Anatolia conditions. The correlations among plant height, grain number per spike, grain weight per spike, 1000 grain weight, test weight and grain yield as well as direct and indirect effects of those traits on the grain yield were investigated using Path analysis. Grain number per spike, 1000-grain weight, plant height and test weight had significant direct effects on grain yield. It was concluded that these characteristics could be important selection criteria in durum wheat breeding studies. Er et al. (2009) conducted a study to determine the relationships between climatic parameters and sugar beet produced in Konya (Cumra) plain. There were statistically 16 Bhoomika Raj, R., M.Sc. (Ag. Stat.), 2015

significant relationships between the monthly mean sugar content of sugar beet and the climatic factors; rainfall, temperature, cloudiness mean and wind velocity (r = 0.630*, r = - 0.898**, r = - 0.900**, r = -0.700* in 1999; r = 0.528, r =-0.980**, r = 0.673, r = - 0.545 in 2000, respectively). According to path analysis results, temperature, cloudiness mean and rainfall had more direct and indirect effects than other climatic factors on sugar content of sugar beet. Javare Gowda et al. (2009) installed a light trap in Teak plantation to study the effect of various weather parameters. The path analysis of five associated weather parameters indicated that, on the outbreak of moth rainfall had a positive direct effect on the emergence of male and female moth during 2003-04 and 2004-05 (0.7079 r = 0.7920; 0.7308 r = 0.7904 and 0.4608 r = 0.4342; 0.5555 r = 0.5018, respectively). Indirectly relative humidity in the evening through rainfall influenced the outbreak of male moth (0.1947 r= 0.4549) and through maximum temperature it influenced the emergence of the female moth (0.2110 r = 0.4175) during 2003-04. Similarly, relative humidity in the morning through the relative humidity in the evening influenced the outbreak of male and female moth (0.8190 r = 0.1402; 0.7988 r= 0.1688, respectively) during 2004-05. Diogo et al.(2013) conducted a study to evaluate the relationships between morphological characteristics and fresh matter yield of cactus pear and identify their direct and indirect effects. Nineteen accessions were evaluated for the following traits: number (NCl), thickness (ThCl), length (LCl) and width (WCl) of cladodes, plant height (PH) and plant width (PW), and green matter yield (GMY), dry matter yield (DMY) and dry matter percentage (DM). The correlations were estimated, and path analysis was performed by the method proposed by Wright. GMY was strongly correlated with DMY, allowing indirect selection for this trait. NCl and ThCl had a direct effect on GMY and can be used for indirect selection or as secondary traits in the selection process. Given the lack of significant correlations between MS and DMY, it is possible to select a palm variety with high DMY and DM. Nevzat et al. (2010) conducted a research to determine characters effecting grain yield in fifty bread wheat (Triticum aestivum L.) cultivars and advanced lines by using simple correlation coefficient and path analysis under 2 locations (high rainfall and low rainfall; 745 and 506 mm, respectively). A total of 50 genotypes, 25 for each independent experiment, were tested for grain yield, test weight, 1000-kernel weight, Zeleny sedimentation, protein content and plant height. Grain yield was significantly correlated with plant height in high rainfall condition. It was significantly correlated with all components except Zeleny sedimentation and protein content in low rainfall condition. Results suggest that plant height and test weight are the primary selection criteria for improving grain yield in bread wheat in high and low rainfall conditions. Hassan (2011) carried out an investigation using 13 alfalfa accessions during 2009 Evaluation of Statistical Models for Climatic Characterization of GKVK Station 17

to 2010 and located on the experimental field of East Azarbaijan Agriculture and Natural Resources Research Center (AZARAN), Iran. The main objective of his research was to evaluate positive effect and reliability of yield and quality traits as selection criteria in alfalfa breeding. Significant differences were observed for most of the yield and quality components. Variability coefficients were high for yield components, while quality traits showed relatively low variation. Plant height (PH), number of stems (NS), number of nodes (NN) and leaf size (LS) were positively correlated with plant yield. Crude protein (CP) content was correlated directly with acid detergent fiber (ADF) and natural detergent fiber (NDF) while the correlation with crude fiber (CF) was inverse. The direct effect of the number of stems on yield had the highest value (0.698, P < 0.01). Direct effect of number of nodes on yield was positive (0.508, P < 0.01). Only the plant height had a lower direct effect than the indirect effect on yield. The direct effects of independent traits on CP were significant (P < 0.05), except for trait leaf dry weight (LDW). The direct effect of ADF had the highest value (2.440, P < 0.01), which was positive and significant. LDW trait had a negative direct effect on the CP (- 0.248, P < 0.05), while the indirect effect on the NDF trait on CP was neither high nor justifiable. ADF, CF, NDF and LDW traits had the highest indirect effects on CP trait via correlation with in vitro dry matter digestibility (IVDMD). XueMei et al. (2011) collected the data of daily mean air temperature, precipitation and runoff during the period of 1958 2007 in the Kaidu River watershed, this paper analyzed the changes in air temperature, precipitation and runoff and revealed the direct and indirect impacts of daily air temperature and precipitation on daily runoff by path analysis. The results showed that mean temperature, time series of the annual, summer and autumn had a significant fluctuant increase during the last 50 years (P < 0.05). Only winter precipitation increased significantly (P < 0.05) with a rate of 1.337 mm/10a. The annual and winter runoff depths in the last 50 years significantly increased with the rates of 7.11 mm/10a and 1.85 mm/10a, respectively. The driving function of both daily temperature and precipitation on daily runoff in annual and seasonal levels is significant in the Kaidu River watershed by correlation analysis. The result of path analysis showed that the positive effect of daily air temperature on daily runoff depth is much higher than that of daily precipitation in annual, spring, autumn and winter, however, the trend is opposite in the summer. Shweta et al. (2012) conducted correlation and path coefficient analyses in order to identify potential traits determining yield under terminal heat stress during grain filling stages and to study character associations among 25 morphological and physiological traits in a set of genetically divergent 36 bread wheat genotypes under normal and late sown conditions. Grain yield depicted positive association with grain weight (GW), grains/spike (G/S), biological yield (BY), harvest index (HI) under normal (NS) and late sown (LS) conditions, GGR-3 (grain growth rate at 28 days after anthesis) in LS and negative association with DM (days to maturity), HU (heat units) and PTU (photo 18 Bhoomika Raj, R., M.Sc. (Ag. Stat.), 2015

thermal unit) in NS and with ChA-1 (CHB a at anthesis), CHB a -1 (total CHB at anthesis) and CHB a -2 (total CHB at 28 days after anthesis) in LS. Path coefficient analysis revealed that, out of eight characters significantly related to yield under normal sown conditions, PTU, harvest index, biological yield and grains per spike directly affected the grain yield in a positive direction. On the basis of strong association with yield and marked direct influence on yield, the numbers of grains were considered to be first order yield components and ought to have top priority in selection under normal sown conditions. Under late sown conditions BY, harvest index (HI), CHB a -1 and grain weight had a direct positive effect, but the grains per spike exhibited negative direct effect despite of the fact that it possessed a positive significant correlation with grain yield. Based on the results, it suggested that higher numbers of grains and high grain weight should be given priority for selection of high yielding genotypes in NS and LS, respectively. Khan and Naqvi (2012) conducted a research with the objectives to identify the utmost traits that may be beneficial for the higher productivity of the grains under stress environment. They used thirteen genotypes which were obtained from different sources as research material. An experiment was carried out in randomized complete block in a split plot design. Water regimes (irrigated and non-irrigated) were allocated to the main plots and genotypes to the subplots. Path coefficient could be used as an important tool to bring about appropriate cause and effect relationship between yield and yield components. According to obtained results the selection on the basis of number of spikes, number of spikelet s and number of grains in this material would likely to be most useful for increasing grain yield because of their direct positive contribution to grain yield under irrigated condition. However a number of spikes, spikelet numbers, spike length and grains number may be used as an effective selection criterion for increasing grain yield of wheat under different irrigation levels. Therefore, they concluded that these traits could be selected for the different stress environments and it would be beneficial for the yield. Sileshi and Temesgen (2013) determined the factors that influence stem radial growth of juvenile Eucalyptus hybrids grown in the east coast of South Africa. Measurement of stem radius was conducted using dendrometers on sampled trees of two Eucalyptus hybrid clones (Eucalyptus grandis Eucalyptus urophylla, GU and E. grandis Eucalyptus camaldulensis, GC). Daily averages of climatic data (temperature, solar radiation, relative humidity and wind speed) were simultaneously collected with total rainfall from the site. They employed path analysis for this study. The joint effect of the climatic variables as well as the direct effect of each climatic variable was studied. Bootstrap estimation procedures, which relax the distributional assumption of the maximum likelihood estimation method, were used. It is found that all variables had a positive effect on stem radial growth. The study showed that tree age is the most important determinant of radial measure. Saied et al. (2013) carried out a study on the relationship among some agronomic Evaluation of Statistical Models for Climatic Characterization of GKVK Station 19

and morphological attributes with protein yield of bread wheat cultivars in randomized complete block design. Relationships among measured traits were assessed by the phenotypic correlation coefficient, step-wise regression and path analysis. Correlation analysis showed the significant relation of seed and protein yield with all the traits except correlation of grain filling duration and peduncle length with seed yield as well as spike yield and peduncle length with protein yield. Regression analysis by using step-wise method showed that 83.8 percent of total variation existed in protein yield accounted for by traits entered into the regression model, namely; grain filling rate, grain filling duration, peduncle length and spike harvest index. Grain filling rate accounted for the highest amount of variation about 46.5 per cent. Other traits accounted for 20, 11.3 and 6 percent of variation of protein yield, respectively. Because of that, these traits are recommended as the main components of protein yield in bread wheat cultivars. Path analysis showed that the traits grain filling rate and grain filling duration had the highest and positive direct effects on protein yield. On the other hand, peduncle length and spike harvest index showed low positive and negative direct effect on protein yield. Overall, grain filling rate and grain filling duration were recognized as the best indirect selection criteria to improve protein yield in bread wheat cultivars via selection especially from preliminary generations of breeding programs. Yogesh and Preeti (2014) studied the abundance of the ladybird beetles (LBB), Coccinella septumpunctata L. (Coleoptera- Coccinellideae) in relation to climatic factor in two consecutive cropping seasons 2005-06 and 2006-07. The study revealed that, the cotton LBB was first recorded in the 27 th SMW i.e. the first week of July during both the year of study and remained up and remained active till 50th SMW (2 nd week of December). The peak population was observed (9.76 / 5 plant) during 37 th SMW i.e. 3 rd week of September. The correlation studies between Population of LBB and weather factors showed that the LBB population had a significant positive correlation with maximum temperature (0.542) & minimum temperature (0.560). The multiple coefficient value between the LBB population and group of variable clearly indicated that 79.50 per cent change in LBB population was affected by maximum temperature, minimum temperature, morning relative humidity, evening relative humidity, sunshine hours, wind velocity, rainfall and rainy days respectively. The data also revealed that 20.50 per cent variation caused by inexplicable reason or due to error beyond the control of the experiment or due to factors not included in the investigation. Path coefficient analysis of LBB population and abiotic factors revealed that the minimum temperature had positive and high direct effect (1.6592) followed by morning relative humidity (0.1972), rainfall (0.1535), and sunshine hours (0.1519) and evening relative humidity (0.016) respectively. 20 Bhoomika Raj, R., M.Sc. (Ag. Stat.), 2015

III MATERIAL AND METHODS This chapter deals with a brief description of the materials which provide the necessary database for the study along with the statistical tools employed in the analysis. The methodology is presented under the following headings: 3.1. Description of the study area. 3.2. Data description. 3.3. Analytical tools and techniques applied. 3.4. Descriptive statistics 3.5. Fitting of probability distributions 3.6. Regression analysis 3.7. Path analysis 3.1 DESCRIPTION OF THE STUDY AREA The present study was conducted to know the climatic characterization of Gandhi Krishi Vignana Kendra (GKVK) station. The station GKVK is located at Bengaluru Urban District of Bengaluru North Taluk. It belongs to the Eastern Dry zone (Zone-V). The geographical co-ordinates of this station are 77 o 35 longitude, 12 o 58 latitude and 930 amsl altitude. The Eastern dry zone includes Kolar, Tumakuru, Bengaluru (Urban), Bengaluru (Rural), Chikkaballapur and Ramanagara. The zone has an average annual rainfall of 944 mm (1976-2013). Major crops grown in this zone are ragi, groundnut, rice and maize. 3.2 DATA DESCRIPTION The present study was based on the secondary data on weather parameters over a period of 38 years (1976-2013) which was collected from AICRP on Agro Meteorology, University of Agricultural Sciences, GKVK, Bengaluru. Weather parameters: 1. Temperature ( o C) 2. Relative Humidity (%) 3. Rainfall (mm) 4. Potential Evapotranspiration (mm) 5. Number of rainy days Evaluation of Statistical Models for Climatic Characterization of GKVK Station 21

6. Sunshine hours (hrs/day) 7. Vapor Pressure (pa) 8. Wind Speed (km/hr) 9. Evaporation (mm) 10. Cloud Amount (Oktas/day) 11. Soil temperature ( o C) 1.Temperature ( o C) Temperature is one of the important weather parameters which regulate the growth of a crop in any region. The mean atmospheric temperature is derived using maximum and minimum temperatures observed during 24 hours. Daily maximum and minimum temperatures were recorded using thermometer. 2. Relative Humidity (%) Relative humidity is the moisture content in the atmosphere which is indirectly calculated using the dry bulb and wet bulb thermometer readings recorded at 7.00 A.M. IST (RH I) and 2.00 P.M. IST (RH II). Average of these two time periods was considered in the present study. 3. Rainfall (mm) The available daily rainfall data was used to compute weekly, monthly, seasonal and annual rainfall. 4. Potential Evapotranspiration (mm) It is the maximum amount of soil moisture that soil and a crop canopy jointly evaporate and transpire under no limiting availability of soil moisture. 5. Number of rainy days If 2.5 mm or more rainfall is received in an area in 24 hours, then such a day is considered as rainy day. The number of rainy days in each year was recorded. 6.Sunshine hours(hrs/day) It is the duration of sunshine in a given period for a given location which is expressed as an average of several years. 22 Bhoomika Raj, R., M.Sc. (Ag. Stat.), 2015

7.Vapor Pressure (pa) It is an indication of a liquid's evaporation rate and relates to the tendency of particles to escape from the liquid (or a solid). 8. Wind Speed (km/hr) It is an average velocity of the atmosphere recorded at 10 feet height above the ground level. 9. Evaporation (mm) Evaporation is a type of vaporization of a liquid that occurs from the surface of a liquid into a gaseous phase that is not saturated with the evaporating substance. 10. Cloud Amount (Oktas/day) It is the extent of cover by clouds in the portion of the sky visible from a particular location. 11. Soil temperature ( o C) The temperature measured at a given soil depth, typically 5cm, 10cm and 15 cm at 7.00 A.M. (Soil temperature I) and 2.00 P.M. (Soil temperature II). It is the measurement of the warmth of the soil. CROP DATA Finger millet (GPU 28) crop yield (qtl/ha) data during kharif season for a period of 16 years from 1998-2013 was collected from ZARS, UAS, Bengaluru. This data was used to know the influence of weather parameters on the crop yield. 3.3 ANALYTICAL TOOLS AND TECHNIQUES APPLIED To assess the climatic characterization of the GKVK station, following statistical tools were used. 3.3.1 Descriptive Statistics The Descriptive Statistics such as minimum, maximum, mean, standard deviation, skewness and coefficient of variation (CV) were calculated for the weather parameters. Standard formulae were used to compare the above measures. Evaluation of Statistical Models for Climatic Characterization of GKVK Station 23

3.3.2 Fitting of probability distributions Rainfall is highly variable in a given period. Hence, there is a need to study the distribution both on long term (annually or seasonally) as well as short term (monthly or weekly) basis. Among the weather parameters, amount of daily maximum rainfall (mm) was considered to fit appropriate probability distributions. The probability distributions viz. normal, log normal, Gamma (1P, 2P, 3P), generalized extreme value (GEV), Weibull (1P, 2P, 3P), Gumbel and Pareto were used to evaluate the best fit probability distribution for rainfall. Description of parameters: Shape parameter A shape parameter is any parameter of a probability distribution that is neither a location parameter nor a scale parameter (nor a function of either or both of these only, such as a rate parameter). Shape parameters allow a distribution to take on a variety of shapes, depending on the value of the shape parameter. These distributions are particularly useful in modeling applications since they are flexible enough to model a variety of data sets. Examples of shape parameters are skewness and kurtosis. Scale parameter In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions. The larger the scale parameter, the more spread out the distribution. The scale parameter of a distribution determines the scale of the distribution function. The scale is either estimated from the data or specified based on historical process knowledge. In general, a scale parameter stretches or squeezes a graph. The examples of scale parameters include variance and standard deviation. Location parameter The location parameter determines the position of central tendency of the distribution along the x-axis. The location is either estimated from the data or specified based on historical process knowledge. A location family is a set of probability distributions where µ is the location parameter. The location parameter defines the shift of the data. A positive location value shifts the distribution to the right, while a negative location value shifts the data distribution to the left. Examples of location parameters include the mean, median, and the mode. 24 Bhoomika Raj, R., M.Sc. (Ag. Stat.), 2015

Table 3.3.2: Description of various probability distribution functions. Distribution Probability density function Range Parameters Gamma (1P) 1 k 1 f ( x) = x exp( x) 0 x <+ Γ( k) k >0 Gamma (2P) and β > 0 k 1 ( k µ ) / β f ( x) = ( x) exp k Gamma (3P) GEV f ( x) = ( x µ ) k 1 1 exp β f ( x) = 1 exp β exp β τ ( k) ( k µ ) / β k β Γ( k) 1 ( 1+ kz) k ( 1+ kz) [ z exp( z) ] 1 1/ k k 0 k = 0 1 + k z > 0 for k 0 - < x <+ for k=0 where, ( µ) z = x β k = Shape parameter β = Scale parameter µ = Location parameter Γ = Gamma function k = Shape parameter β = Scale parameter µ = Location parameter Normal Log- normal Gumbel Pareto f ( x ) f ( x) f ( x) Where, f 1 1 2 2 ( x µ ) = exp 2 σ 2π 1 exp xσ 2π 2σ 2 ( ln( x) µ ) = 2 1 = exp β z = x µ β kβ β k ( x) = k + 1 Weibull (1P) β β f x = β x 1 ( ) exp( x ) Weibull (2P) Weibull (3P) f ( x) = f ( x) = k x β β k 1 k x µ β β 2σ z ( z + e ) x exp β k 1 k x µ exp β k - < x <+ - < µ <+ σ > 0 µ = Mean σ = Standard deviation 0 < x < + σ = Scale parameter µ = Location parameter β >0 - < x <+ 1 x + k, β > 0 x 1 x > 0 β > 0 0 x <+ k, β, > 0 β = Scale parameter µ = Location parameter k = Shape parameter β = Scale parameter β = Shape parameter k = Shape parameter β = Scale parameter µ = Location parameter ( µ 0 yields the two parameter Weibull distribution) Evaluation of Statistical Models for Climatic Characterization of GKVK Station 25

A conceptual understanding of the parameters The shape parameter describes the form of the curve. Distributions with a low shape parameter are positively skewed, and as the shape value increases the distribution curve becomes more symmetrical. The mapped parameters provide some spatial context to the rainfall values and distributions. The term shape-dominated refer to the distributions with a larger shape parameter, and scale dominated refer to the distributions with a larger scale parameter. Figure 2 shows a conceptual graphic with the shape and scale parameters on the x- axis and y-axis, respectively. The axes on this graphic are not numerated as the concept of small and large values may vary in time and space. This graphic provides an idea of how the parameters describe regimes. Areas in the low rainfall region of the plot describe study periods that are typically arid during the interval of analysis. The empty area in the graph indicates that areas with at least a minimal amount of rainfall are in either the shape-dominated or scale dominated category. A shape-dominated regime describes a pattern where the rainfall tends to be symmetrically distributed, indicating that drier-than-average events are as common as wetter-than-average events. Scale-dominated rainfall describes locations where the variance is quite large in comparison to the mean. Scale - dominated Scale Small Large Variable rain More extreme events Low rainfall Average is unable to sustain agriculture Shape dominated Consistent rain Fewer extreme events Small Shape Large Fig.2: Conceptual regimes described in parameter space with the shape parameter on the x-axis and the scale parameter on the y-axis. 26 Bhoomika Raj, R., M.Sc. (Ag. Stat.), 2015

Testing for goodness of fit: The goodness of fit test measures the discrepancy between observed values and the expected values. Kolmogorov- Smirnov test was used to test for the goodness of fit. In the present investigation, the goodness of fit test was conducted at α = 0.05 level of significance. It was applied for testing the following hypothesis: H 0 : The maximum daily rainfall data follows a specified distribution. H 1 : The maximum daily rainfall data does not follow a specified distribution. Kolmogorov- Smirnov test ( K-S test) This test was used to decide whether a sample comes from a hypothesized continuous PDF. The KS test compares the cumulative distribution functions of the theoretical distribution the distribution described by the estimated shape and scale parameters with the observed values and returns the maximum difference between these two cumulative distributions (Wilks, 1995). This maximum difference in cumulative distribution functions is frequently referred to as the KS-statistic. It is based on the empirical distribution function i.e., on the largest vertical difference between the theoretical and empirical cumulative distribution functions which is given as: i 1 i D = max F i, 1 i n n n ( X ) F( X ) Where, X i = Random sample, i= 1, 2,, n. i CDF 1 = Fn ( X ) = [Number of observations x] n Plot of the best fitted distribution functions for maximum daily rainfall (mm) The frequency density at different levels of rainfall was calculated using the following formulae. Relative Frequency = Class frequency Total frequency Frequency Density = Relative frequency Class width The above analysis will be carried out using XLSTAT software. Evaluation of Statistical Models for Climatic Characterization of GKVK Station 27

3.3.3 REGRESSION ANALYSIS The influence of weather parameters on the crop yield was assessed separately by relating finger millet crop yield with the weather parameters through linear functional forms. For this, regression models were used. Regression Data Finger millet crop yield was taken as the dependent variable (Y) and weather parameters (X 1, X 2, X 3, X 4, X 5, X 6, X 7, and X 8 ) were considered as independent variables. X 1 : Minimum temperature (Min Temp.) X 2 : Maximum temperature (Max Temp.) X 3 : Relative Humidity (RH) X 4 : Rainfall X 5 : Number of rainy days X 6 : Potential Evapotranspiration (PET) X 7 : Sunshine hours (SSH) X 8 : Soil temperature (Soil Temp.) Simple Linear Regression Analysis The simple linear regression models were fitted using crop yield as dependent variable (Y) and weather parameters as independent variables (X 1, X 2,, X 8 ) one at a time. Y β β X + e where, i = 1,2,,8. i = o + 1 i i Multiple Linear Regression Analysis The multiple linear regression model for the crop yield (Y) and the weather parameters (X i ) is given as, Y = Xβ + ε Where, Y = ( Y Y,..., ) finger millet. is the vector of values of the dependent variable, crop yield of 1, 2 Y n 28 Bhoomika Raj, R., M.Sc. (Ag. Stat.), 2015

X = 1 1... 1 X X... X 11 21 n 1.................. X X X 1 k 2 k... nk, a (n x p) matrix = ( β β ) is the vector of the parameters and ε = ( ε ε,..., ) β,..., 1, 2 β k is the error 2 vector. The error vector ε is assumed to be normally distributed with N [ 0, σ ]. The least square estimator of the parameter vector β is 1 ( X X ) X Y ˆ β = b = And its sampling variance is V 2 ( b) = ( X X ) 1 σ The fitted equation is Y ˆ = Xb The variance of Ŷ at the given values of explanatory variables is ( ˆ 2 1 0 ) = σ X 0( X X ) X 0 V Y Where, = ( x, x x ) x0 01 02,..., 0k 1, 2 is the vector of given values of the predictor variables. An unbiased estimator of ˆ σ 2 σ ( y y ) n 2 2 1 ˆ i i = ( Y Y b X Y ) ( n k) = MSres = i= 1 n k Where, k is the number of parameters to be estimated. Testing significance of multiple linear regression model The significance of multiple linear regression models was tested using F- test. Test Hypothesis: H : β i = β i j 0 j ε k Evaluation of Statistical Models for Climatic Characterization of GKVK Station 29

H 1 : β β i j Level of significance: α = 5% Test statistic: F = MS MS Reg Res ANOVA Table Source of variation Degrees of freedom Sum of Squares Mean Sum of Squares F Regression k SS Reg MS Reg MS MS Reg Res Error n-k-1 SS Res MS Res Total n-1 SS T SS Re g = ˆ β X Y ( Yi ) n 2 SS s = Y Y βˆ X Y Re SS T = Y Y ( Yi ) n 2 MS MS Re g = Re s SS Re g k SS Re s = n k 1 Critical region: Reject H 0 at α level of significance if F cal F( k, n k 1) Testing significance of regression coefficient The significance of regression coefficients was tested using t- test. 30 Bhoomika Raj, R., M.Sc. (Ag. Stat.), 2015

Test Hypothesis: H : β 0 j = 1,2,..., k 0 j = H 1 : β j 0 Level of significance: α = 0.05 Test statistic: ˆ β j t = ~ tα ( n k 2 ) df SE ( ˆ β j ) Where, SE( ˆ β ) = j MS S Re s xx S xx ( X ) 2 = i X Model Adequacy Checking Adequacy of a model depends on the validity of assumptions underlying the model. The assumptions made in a linear regression model are, apart from lineardependence of Y on predictors, independence and identical distribution (normal) of the predictors with zero mean. Gross violations of the assumptions may yield an unstable model in the sense that a different sample could lead to a totally different model with opposite conclusions. We cannot detect departures from the underlying assumption by examination of the summary statistics such as t or F statistic or R 2 (Draper and Smith, 1981). These are Global model properties and as such they do not ensure model adequacy. Hence, diagnostic methods, primarily based on the study of the model residuals are used. The diagnostics checks for the present study used are for randomness and normality of residuals. Coefficient of Determination (R 2 ) The statistic R 2 indicates the percentage of variation in the dependent variable explained by the independent variables included in the regression model. R 2 is computed using the uncorrected sum of squares. R n 2 i= 1 = n ( yˆ y) ( yi y) i= 1 Where, 0 R 2 1 i 2 2 Evaluation of Statistical Models for Climatic Characterization of GKVK Station 31

Assumptions of error term An important assumption of multiple regression models is that the residual, ε follows normal distribution. This assumption is required to test the hypothesis about the regression coefficients. This assumption is verified using, i. Randomness of error terms ii. Normal probability plots. Randomness of error terms (Runs test) Randomness of residuals is tested using the non-parametric one sample runs test. A run is defined as the succession of identical symbols which are followed and preceded by different symbols or no symbols at all. The null and alternative hypothesis is given by H 0 : The sequence of residuals is random. H 1 : The sequence of residuals is not random. Let n 1 be the number of elements of one kind and n 2 be the number of elements of the other kind in a sequence of N = n 1 + n 2 binary events. If both n 1 and n 2 are less than or equal to 20, and if the number of runs (r) falls between the critical values provided in the tables, we cannot reject the null hypothesis. Normal probability plot Suppose e 1, e 2,,e n are the residuals ranked in increasing order as e 1 < e 2 < < e n. If we plot e i against the cumulative probabilities P i =(i-1/2)/n, i=1,2,,n, in the normal probability plot. The residuals can be assumed to be normally distributed if the resulting points lie approximately on a straight line. 3.3.3 PATH ANALYSIS Path analysis quantifies the direct and indirect effect of each variable and helps to visualize correlation in its totality to determine selection indices. It is an extension of the regression model, which researchers use to test the fit of a correlation matrix with a causal model that has been tested. Path analysis is a standardized partial regression and a technique to elucidate the contribution of each variable to the total correlation coefficient. The aim of the path analysis is to provide estimates of the magnitude and significance of the hypothesized causal connections among sets of variables displayed through the use of path diagrams. 32 Bhoomika Raj, R., M.Sc. (Ag. Stat.), 2015

Path analysis was used to examine the influence of weather parameters on the rainfall. The direct and indirect effects of weather parameters on the rainfall were identified considering the following weather variables: X 1 : Minimum temperature (Min Temp.) X 2 : Maximum temperature (Max Temp.) X 3 : Potential Evapotranspiration (PET) X 4 : Vapor Pressure (VP) X 5 : Relative Humidity (RH) X 6 : Wind Speed (WS) X 7 : Sunshine hours (SSH) X 8 : Cloud Amount (CL AMT) X 9 : Evaporation (EVP) PATH DIAGRAM The graphical or diagrammatical representation of path analysis is known as path diagram. Generally, arrows are used to show the relations. A single headed arrow is meant for showing cause to effect. In path diagram, there are two kinds of variables, independent variables known as the causal variables and the dependent variables known as endogenous variables. If Y is a character, which is linearly determined by correlated variables, X 1, X 2 and X 3 because of the closed system, a path diagram can be formulated as in Fig.3. X 1 a r 12 Y b X 2 r 13 c r 23 Residual X 3 Fig.3: Cause and effect relationship Evaluation of Statistical Models for Climatic Characterization of GKVK Station 33

Symbols a, b and c represent path coefficients from causes X 1, X 2 and X 3 respectively. r 12, r 23 and r 13 are the correlation coefficients and R is the variation in Y that is due to the unknown agencies (residuals). A character Y is determined by three correlated variables, viz. X 1, X 2 and X 3. The relationship can be expressed in the form of a partial regression equation and is given by, Y µ + β X + β X + X + R = 1 1 2 2 β 3 3 Where, β 1, β2 and β 3 are the partial regression coefficients. β 1 is the partial regression coefficient of Y on X 1, which measures the amount of change that can be brought about in Y due to one unit change in X 1, when X 2 and X 3 are held constant. β 2 and β 3 have similar meanings. R is the residual component. When the variables are expressed as deviation from their respective means and also in terms standardized units, their means will be zero and variance will be equal to unity. Thus, if we denote ( X 1 X 1 )/ σ Y by a lower case letter X 1, mean of X 1 will be zero and variance will be unity. Replacing the expression β 1σ X 1 / σ Y by term a which is a standardized partial regression coefficient and it is called as the path coefficient. It should be pointed out that the path coefficients have a direction similar to the regression coefficients, i.e., they may bear a positive or negative sign. Unlike the correlation coefficients, the path coefficients could be greater than or lesser than unity like regression coefficients. A set of three simultaneous equations is then formulated from the path diagram as: r = a + br + cr YX 1 X 1 X 2 X 1 X 2 r = ar + b + cr YX 2 X 2 X X 1 2 X 3 ryx = ar + br + c 3 X 2 X1 X 3 X 2 r = Estimates of simple correlation coefficient between X 1 and X 2. X 1 X 2 r = Estimates of simple correlation coefficient between X 2 and X 3. X 2 X 3 r = Estimates of simple correlation coefficient between X 1 and X 3. X 1 X 3 a = Direct effect of variable X 1. b = Direct effect of variable X 2. c = Direct effect of variable X 3. Y = Dependent variable. Information provided in the above equation can be arranged in the matrix form as: 34 Bhoomika Raj, R., M.Sc. (Ag. Stat.), 2015

1 rx 1 r X1 X X 2 3 r X X r 1 1 3 2 X X 2 r X X r 1 2 1 3 X X 3 a b c = r r r YX YX YX 1 2 3 Which is expressed as A B = C Here the matrix A is symmetrical about the diagonal elements. The diagonal elements in this matrix represent the direct effects of X 1, X 2 and X 3, respectively on Y. All the direct effects have been replaced by unity in this matrix. Elements of the column matrix B specify the path coefficients to be estimated. The column matrix C represents the estimated values of correlation coefficients between dependent variable Y, and the component variables, i.e., X 1, X 2 and X 3. Estimates of unknown are calculated as: B = A -1 C Where, A -1 = Inverse of matrix A. C = Estimates of simple correlation coefficient between dependent variables. The indirect effects are calculated from the three simultaneous equations which provide the estimated path coefficients. This method can permit us to identify relatively important components of a dependent variable, on the basis of their direct and indirect influences. Besides the direct effect of the exogenous variables (X i ) on the endogenous (Y), there is an indirect effect of X i on Y via other X s by virtue of its relationship with others. A change in variable, say X 1, will affect its linear correlation r 12, with another variable, X 2, which invariably affects Y. Let P i be the direct effect of X i on Y, and change is only partial and proportional to r 12. That is, r ij P i is an indirect effect of X i via X j. The percentage direct contributions of any variable and its residual effect are worked out as follows: Direct percentage contribution = (P i ) 2 100. Where, P i is the direct effect of X i on Y. Once the path coefficient values corresponding to different independent variables are worked out, the next problem lies in estimating the residual (R) i.e., the unexplained part of the model. The residual is given by: Where, R = 1 - r ij P i r ij = Correlation between i th independent variables and the response variable. P i = Direct effect. The regression analysis and path analysis will be carried out using MS Excel. Evaluation of Statistical Models for Climatic Characterization of GKVK Station 35

IV RESULTS Keeping in view the specific objectives of the present investigation, the data on the weather parameters and the finger millet crop was subjected to statistical analysis using appropriate techniques. The results of the analysis are presented under the following headings: 4.1 Fitting appropriate probability distributions for rainfall. 4.2 Impact of weather parameters on the yield of finger millet crop. 4.3 Direct and indirect effects of weather parameters on rainfall. 4.1 Fitting appropriate probability distributions for rainfall. Rainfall data was classified into 28 sets viz., 1 annual, 1 seasonal, 5 seasonal (June to October) months and 21 seasonal Standard Meteorological Weeks (23 rd SMW- 43 rd SMW) to study the distribution pattern of rainfall at different levels. The best fit probability distributions for maximum daily rainfall on different sets of data were identified based on the criteria of Kolmogorov-Smirnov test. 4.1.1 Descriptive statistics The summary of statistics such as maximum, minimum, mean, standard deviation, skewness and coefficient of variation values of maximum daily rainfall are presented in Table 4.1.1. The results show that both annual and seasonal maximum rainfall was observed to be 200 mm. Monthly maximum rainfall during monsoon season ranged from 98.6 mm to 200 mm while weekly maximum rainfall was between 43.2 mm to 200 mm. It was also observed that the minimum rainfall was 56.8 mm annually, 43.6 mm seasonally and it ranged from 6.4 mm to 16 mm monthly. For all the weeks, the minimum rainfall was found to be 0.0 mm. Mean annual rainfall was found to be 94.6 mm, whereas for overall seasonal months, it was 90.6 mm. During the seasonal months the mean rainfall ranged from 32.1mm to 67.6 mm while for weekly, it varied from 5.2 mm to 33.5 mm. The value of coefficient of variation annually was observed to be 39.2 per cent which was the lowest among all the data sets. Seasonally, the rainfall varied by about 43.6 per cent whereas monthly, the variation ranged from 58.6 to 66.8 per cent. For Standard Meteorological Weeks, the rainfall variation was found to be very high ranging from 86.9 to 201.9 per cent. Evaluation of Statistical Models for Climatic Characterization of GKVK Station 36

Table 4.1.1: Summary of statistics for maximum daily rainfall (mm) Study Period Range Mean Standard deviation Parameters CV (%) Skewness Maximum Minimum Annual 1Jan 31 Dec 94.6 37.1 39.2 1.0 200.0 56.8 Seasonal 1 June- 28 Oct 90.6 39.4 43.6 1.0 200.0 43.6 June 1 June-30 June 32.1 20.9 65.0 1.1 98.6 6.4 July 1 July-31 July 40.5 27.0 66.8 1.8 139.5 10.2 August 1 Aug-31 Aug 43.1 25.9 60.1 0.6 98.8 10.2 September 1 Sept-30 Sept 67.6 39.6 58.6 0.8 158.4 15.4 October 1 Oct- 28 Oct 66.8 43.3 64.8 1.4 200.0 16.0 23 rd SMW 4 June-10 June 18.0 18.5 102.8 0.7 58.8 0.0 24 th SMW 11 June-17 June 9.3 11.6 124.6 1.7 49.4 0.0 25 th SMW 18 June-24 June 9.3 14.3 153.3 2.3 56.2 0.0 26 th SMW 25 June-1 July 5.2 7.7 148.8 3.6 43.2 0.0 27 th SMW 2 July-8 July 16.3 21.7 133.7 2.4 98.6 0.0 28 th SMW 9 July-15 July 14.5 23.1 159.3 4.5 139.5 0.0 29 th SMW 16 July-22 July 16.7 14.5 86.9 0.7 47.2 0.0 30 th SMW 23 July-29 July 11.5 11.6 101.0 1.5 47.2 0.0 31 st SMW 30 July-5 Aug 18.5 20.8 112.4 2.3 97.8 0.0 32 nd SMW 6 Aug-12 Aug 14.2 14.6 103.0 1.5 62.2 0.0 33 rd SMW 13 Aug-19 Aug 17.2 18.6 108.5 1.9 79.6 0.6 34 th SMW 20 Aug-26 Aug 21.6 26.3 121.5 1.8 98.8 0.0 35 th SMW 27 Aug-2 Sept 22.0 22.6 102.6 1.2 73.6 0.0 36 th SMW 3 Sept-9 Sept 29.0 36.4 125.7 1.9 151.8 0.0 37 th SMW 10 Sept-16 Sept 32.9 36.7 111.6 1.5 136.0 0.0 38 th SMW 17 Sept-23 Sept 23.4 23.6 100.8 1.1 89.5 0.0 39 th SMW 24 Sept-30 Sept 33.5 39.2 116.8 1.6 158.4 0.0 40 th SMW 1 Oct-7 Oct 26.4 22.7 85.8 1.0 84.0 0.0 41 st SMW 8 Oct-14 Oct 24.8 34.8 140.3 3.6 200.0 0.0 42 nd SMW 15 Oct-21 Oct 18.8 19.4 103.1 1.0 64.4 0.0 43 rd SMW 22 Oct-28 Oct 18.1 36.6 201.9 3.1 157.6 0.0 37 Bhoomika Raj, R., M.Sc. (Ag. Stat.), 2015

250 Annual Maximum rainfall (mm) 200 150 100 50 0 Year Fig. 4: Variation in Annual maximum daily rainfall (mm) Seasonal Maximum rainfall (mm) 250 200 150 100 50 0 Year Fig. 5: Variation in Seasonal maximum daily rainfall (mm) Evaluation of Statistical Models for Climatic Characterization of GKVK Station 38

The coefficient of skewness measures the asymmetry of the frequency distribution of the data. For the present study, the coefficient of skewness for all the data sets ranged from 0.6 to 4.5 indicating positive skewness. From this result, we can conclude that in all the data sets the average rainfall for the period exceeded the modal value. Figure 4 shows the year wise variation in annual maximum daily rainfall from 1976-2013. It ranged from 56.8 mm (in 2006) to 200 mm (in 1997). The minimum of the maximum daily rainfall, i.e., 56.8 mm occurred during 10 th SMW (5 th Mar-11 th Mar) and the maximum of the maximum daily rainfall of 200 mm occurred during the 41 st SMW (8 th Oct-14 th Oct). Figure 5 represents the year wise variation in seasonal maximum daily rainfall from 1976-2013 and it ranged from 43.6 mm (in 2009) to 200 mm (in 1997). The minimum of the maximum daily rainfall of 43.6 mm occurred during 30 th SMW (23 rd Jul- 29 th July) and the maximum daily rainfall occurred during the 41 st SMW (8 th Oct-14 th Oct). From both the figures it reveals that the variation in rainfall is stationary over the years as well as during the seasons. 4.1.2 Study period wise fitting of probability distributions The probability distributions used to evaluate the best fit for maximum daily rainfall were Normal, Log normal, Gamma (1P, 2P, 3P), Generalized Extreme Value (GEV), Weibull (1P, 2P, 3P), Gumbel and Pareto. The goodness of fit for different probability distributions was tested using Kolmogorov- Smirnov test (K-S test). The test statistic D along with the p-values for each data set was computed for 11 probability distributions. Table 4.1.2 represents the study period wise probability distributions and their goodness of fit statistics. Annual For the data set on annual maximum daily rainfall, the calculated K-S test statistic value was found to be non-significant for the probability distributions such as, Gamma (2P), Log- normal and Weibull (2P, 3P). This means that the null hypothesis which states that the rainfall data follow specified distributions cannot be rejected and hence it fits the above mentioned probability distributions. The test statistic value for Gamma (2P) distribution was 0.1448 and the p-value was found to be 0.3847 while log- normal distribution had a test statistic value of 0.1480 and a p-value of 0.3484. The test statistic value for Weibull (2P) was found to be 0.1630 and the p-value was 0.2415 whereas Weibull (3P) had a test statistic value of 0.0856 and a p-value of 0.9374. Among the four probability distributions, Weibull (3P) had the lowest test statistic value and the highest p-value. Seasonal (June-October) During the seasonal maximum daily rainfall, the distributions which fitted well were Gamma (2P), Log- normal and Weibull (2P, 3P). Gamma (2P) distribution had a test statistic value of 0.1650 and a p-value of 0.2404 while the test statistic value for lognormal distribution was found to be 0.1573 with a p-value of 0.2759. The test statistic 39 Bhoomika Raj, R., M.Sc. (Ag. Stat.), 2015

value and p-value for Weibull (2P) distribution was observed to be 0.1807 and 0.1617 respectively, whereas for Weibull (3P) distribution, the test statistic value and p-value were found to be 0.1252 and 0.5516 respectively. Weibull (3P) had the lowest test statistic value and the highest p-value among all the four probability distributions. June For the seasonal month of June, Gamma (2P), Generalized Extreme Value (GEV), log- normal and Weibull (2P, 3P) distributions fitted well. The test statistic value and p- value for Gamma (2P) distribution was observed to be 0.0908 and 0.9055 respectively. GEV distribution had a test statistic value of 0.1220 and p-value of 0.5947 while the test statistic value for log- normal distribution was observed to be 0.0989 with a p-value of 0.8405. The test statistic value and p-value for Weibull (2P) distribution was observed to be 0.0851 and 0.9357 respectively whereas Weibull (3P) distribution had a test statistic value of 0.0802 with a p-value of 0.9600. Among these five probability distributions, Weibull (3P) had the lowest test statistic value and the highest p-value. July It was observed that for the seasonal month of July, the distributions which fitted well were Gamma (2P), log- normal, Pareto and Weibull (2P, 3P). Gamma (2P) distribution had a test statistic value and p-value of 0.1184 and 0.6341 respectively while it was observed to be 0.0874 and 0.9242 respectively for log-normal distribution and for Pareto distribution, the test statistic value and p-value were found to be 0.2067 and 0.0670 respectively. The test statistic value for Weibull (2P) was observed to be 0.1401 with a p-value of 0.4163 whereas Weibull (3P) had a test statistic value of 0.0856 with a p-value of 0.6519. Among these five probability distributions, log-normal had the lowest test statistic value and the highest p-value. August For the seasonal month of August, Gamma (2P), Generalized Extreme Value (GEV), log- normal and Weibull (2P, 3P) were found to be the best fits. The test statistic value and p-value for Gamma (2P) distribution was observed to be 0.1426 and 0.4972 respectively. GEV distribution had a test statistic value of 0.1819 with a p-value of 0.1439 while the test statistic value for log- normal distribution was observed to be 0.1453 with a p-value of 0.3712. The test statistic value and p-value for Weibull (2P) distribution was observed to be 0.1286 and 0.5268 respectively whereas Weibull (3P) distribution had a test statistic value of 0.1315 and p-value of 0.4972. Among these five probability distributions, Weibull (2P) had the lowest test statistic value and the highest p-value. September During the seasonal month of September, the distributions which fitted well were Gamma (2P), GEV, log- normal, Pareto and Weibull (2P, 3P). Gamma (2P) distribution Evaluation of Statistical Models for Climatic Characterization of GKVK Station 40

had a test statistic value of 0.0804 and p-value of 0.9593 respectively, while it was observed to be 0.0849 and 0.9369 respectively for GEV distribution. For log-normal distribution, the test statistic value and p-value were found to be 0.1171and 0.6467 respectively and it was found to be 0.1517 and 0.3198 for Pareto distribution respectively. The test statistic value for Weibull (2P) was observed to be 0.0847 and the p-value was 0.9377 whereas Weibull (3P) had a test statistic value of 0.0853 and a p- value of 0.9344. Among these six probability distributions, Gamma (2P) had the lowest test statistic value and the highest p-value. Maximum number of distributions fitted well for the study period September when compared to all the other study periods. October It was found that for the seasonal month of October, the distributions which fitted well were Gamma (2P), log- normal, Pareto and Weibull (2P, 3P). Gamma (2P) distribution had a test statistic value and p-value of 0.0965 and 0.8369 respectively while it was observed to be 0.0698 and 0.9860 respectively for log-normal and Pareto distribution, the test statistic value and p-value were found to be 0.1630 and 0.2373 respectively. The test statistic value for Weibull (2P) was observed to be 0.1230 with a p- value of 0.5711 whereas Weibull (3P) had a test statistic value of 0.0853 and a p-value of 0.8942. Among these five probability distributions, log-normal had the lowest test statistic value and the highest p-value. 23 rd SMW (4 June-10 June) For this Standard Meteorological Week, Gamma (2P) was the only distribution which fitted well with a K-S test statistic value of 0.2105 and a p-value of 0.0591. 24 th SMW (11 June-17 June) For the 24 th Standard Meteorological Week, the distributions which fitted well were Gamma (2P) and Weibull (3P). The K-S test statistic value for Gamma (2P) was observed to be 0.1315 and the p-value was 0.4857, while for Weibull (3P) distribution, test statistic value was found to be 0.0784 and the p-value was observed to be 0.9590. Weibull (3P) had the lowest test statistic value and the highest p-value among the two best fitted distributions. 25 th SMW (18 June-24 June) During the 25 th Standard Meteorological Week, Gamma (2P) and Weibull (3P) distributions were found to be the best fits. Gamma (2P) had a test statistic value of 0.1842 and a p-value of 0.1349. The K-S test statistic value and p-value for Weibull (3P) distribution was observed to be 0.0945 and 0.8694 respectively. Among the two best fitted distributions, Weibull (3P) had the lowest test statistic value and the highest p- value. 41 Bhoomika Raj, R., M.Sc. (Ag. Stat.), 2015

26 th SMW (25 June-1 July) For the 26 th Standard Meteorological Week, the distributions which fitted well were Pareto and Weibull (3P). The test statistic value for Pareto distribution was observed to be 0.1627 and the p-value was 0.2433, while for Weibull (3P) distribution, test statistic value was found to be 0.1520 and the p-value was observed to be 0.3177. Weibull (3P) had the lowest test statistic value and the highest p-value among the two best fitted distributions. 27 th SMW (2 July-8 July) It was observed that for the 27 th Standard Meteorological Week, the distributions such as Gamma (2P) and Weibull (3P) fitted well. Gamma (2P) had a test statistic value of 0.1052 and a p-value of 0.7709. The test statistic value and p-value for Weibull (3P) distribution was observed to be 0.0734 and 0.9828 respectively. Among the two best fitted distributions, Weibull (3P) had the lowest test statistic value and the highest p- value. 28 th SMW (9 July-15 July) For the 28 th Standard Meteorological Week, Weibull (3P) was the only distribution which fitted well with a K-S test statistic value of 0.1290 and a p-value of 0.5225. 29 th SMW (16 July-22 July) During the 29 th Standard Meteorological Week, the distributions which fitted well were Gamma (2P) and Weibull (3P). The K-S test statistic value for Gamma (2P) was observed to be 0.1180 and the p-value was 0.6376, while for Weibull (3P) distribution, test statistic value was found to be and the p-value was observed to be 0.0799. Weibull (3P) had the lowest test statistic value and the highest p-value among the two best fitted distributions. 30 th SMW (23 July-29 th July) It was found that Gamma (2P) and Weibull (3P) distributions fitted well for the 30 th Standard Meteorological Week. Gamma (2P) hada test statistic value of 0.1393 and a p-value of 0.4235. The test statistic value and p-value for Weibull (3P) distribution was observed to be 0.1163 and 0.6716 respectively. Among the two best fitted distributions, Weibull (3P) had the lowest test statistic value and the highest p-value. 31 st SMW (30 July-5 Aug) For the 31 st Standard Meteorological Week, the distributions which fitted well were Gamma (2P) and Weibull (3P). Gamma (2P) had a test statistic value of 0.1417 and a p- value of 0.4016. The test statistic value and p-value for Weibull (3P) distribution was Evaluation of Statistical Models for Climatic Characterization of GKVK Station 42

observed to be 0.1344 and 0.4859 respectively. Weibull (3P) had the lowest test statistic value and the highest p-value among the two best fitted distributions. 32 nd SMW (6 Aug-12 Aug) It was observed that the 32 nd Standard Meteorological Week, the distributions such as Gamma (2P) and Weibull (3P) fitted well. The K-S test statistic value for Gamma (2P) was observed to be 0.0879 and the p-value was 0.9186 while for Weibull (3P) distribution, test statistic value was found to be 0.0889 and the p-value was observed to be 0.9119. Gamma(2P) distribution had the lowest test statistic value and the highest p- value among the two best fitted distributions. 33 rd SMW (13 Aug-19 Aug) During the 33 rd Standard Meteorological Week, the distributions which fitted well were Gamma (2P) and Weibull (3P). The K-S test statistic value for Gamma (2P) was observed to be 0.1017 and the p-value was 0.8055, while for Weibull (3P) distribution, test statistic value was found to be 0.1175 and the p-value was observed to be 0.6426. Among the two best fitted distributions, Gamma(2P) distribution had the lowest test statistic value and the highest p-value. 34 th SMW (20 Aug-26 Aug) For the 34 th Standard Meteorological Week, Gamma (2P) and Weibull (3P) distributions were found to the best fits. The K-S test statistic value for Gamma (2P) was found to be 0.1167 and the p-value was 0.6507 while for Weibull (3P) distribution, test statistic value was observed to be 0.1409 and the p-value was found to be 0.4092. Gamma (2P) distribution had the lowest test statistic value and the highest p-value among the two best fitted distributions. 35 th SMW (27 Aug-2 Sept) It was found that for the 35 th Standard Meteorological Week, the distributions which fitted well were Gamma (2P) and Weibull (3P). The K-S test statistic value for Gamma (2P) was observed to be 0.0908 and the p-value was 0.8981 while for Weibull (3P) distribution, test statistic value was found to be 0.0802 and the p-value was observed to be 0.9602. Weibull (3P) had the lowest test statistic value and the highest p-value among the two best fitted distributions. 36 th SMW (3 Sept-9 Sept) During the 36 th Standard Meteorological Week, Gamma (2P) and Weibull (3P) distributions were found to be the best fits. The K-S test statistic value for Gamma (2P) was observed to be 0.1315 and the p-value was 0.4973 while for Weibull (3P) distribution, test statistic value was found to be 0.0665 and the p-value was observed to 43 Bhoomika Raj, R., M.Sc. (Ag. Stat.), 2015

be 0.9946. Among the two best fitted distributions, Weibull (3P) had the lowest test statistic value and the highest p-value. 37 th SMW (10 Sept-16 Sept) For the 37 th Standard Meteorological Week, the distributions which fitted well were Gamma (2P) and Weibull (3P). The K-S test statistic value for Gamma (2P) was observed to be 0.0872 and the p-value was 0.9230 while for Weibull (3P) distribution, test statistic value was found to be 0.0824 and the p-value was observed to be 0.9498. Weibull (3P) had the lowest test statistic value and the highest p-value among the two best fitted distributions. 38 th SMW (17 Sept-23 Sept) It was observed that for the 38 th Standard Meteorological Week, Gamma (2P) and Weibull (3P) distributions were found to be the best fits. The K-S test statistic value for Gamma (2P) was observed to be 0.1315 and the p-value was 0.4973 while for Weibull (3P) distribution, test statistic value was found to be 0.0676 and the p-value was observed to be 0.9933. Among the two best fitted distributions, Weibull (3P) had the lowest test statistic value and the highest p-value. 39 th SMW (24 Sept-30 Sept) For the 39 th Standard Meteorological Week, the distributions which fitted well were Gamma (2P) and Weibull (3P). The K-S test statistic value for Gamma (2P) was observed to be 0.1735 and the p-value was 0.1827 while for Weibull (3P) distribution, test statistic value was found to be 0.1503 and the p-value was observed to be 0.3301. Weibull (3P) had the lowest test statistic value and the highest p-value among the two best fitted distributions. 40 th SMW (1 Oct-7 Oct) During the 40 th Standard Meteorological Week, Gamma (2P), GEV and Weibull (3P) distributions were found to be the best fits. The K-S test statistic value for Gamma (2P) was observed to be 0.1243 and the p-value was 0.5572 while for GEV distribution, test statistic value was found to be 0.1295 and the p-value was observed to be 0.5051. The test statistic value for Weibull (3P) distribution was found to be 0.0839 and the p- value was 0.9311. Among the three best fitted distributions, Weibull (3P) had the lowest test statistic value and the highest p-value. 41 st SMW (8 Oct-14 Oct) It was found that for the 41 st Standard Meteorological Week, Gamma (2P) and Weibull (3P) distributions were found to be the best fits. The K-S test statistic value for Gamma (2P) was observed to be 0.1842 and the p-value was 0.1334 while for Weibull (3P) distribution, test statistic value was found to be 0.1004 and the p-value was observed Evaluation of Statistical Models for Climatic Characterization of GKVK Station 44

to be 0.8013. Weibull (3P) had the lowest test statistic value and the highest p-value among the two best fitted distributions. 42 nd SMW (15 Oct-21 Oct) During the 42 nd Standard Meteorological Week, the distributions which fitted well were Gamma (2P) and Weibull (3P). The K-S test statistic value for Gamma (2P) was observed to be 0.1578 and the p-value was 0.2701 while for Weibull (3P) distribution, test statistic value was found to be 0.0830 and the p-value was observed to be 0.9361. Among the two best fitted distributions, Weibull (3P) had the lowest test statistic value and the highest p-value. 43 rd SMW (22 Oct-28 Oct) For the 42 nd Standard Meteorological Week, Pareto and Weibull (3P) distributions were found to be the best fits. The K-S test statistic value for Pareto was observed to be 0.2116 and the p-value was 0.0568 while for Weibull (3P) distribution, test statistic value was found to be 0.1662 and the p-value was observed to be 0.2187. Weibull (3P) had the lowest test statistic value and the highest p-value among the two best fitted distributions. The above results indicated that Weibull (3P) distribution fitted well for majority of the study periods. 4.1.3 Parameters of the best fit probability distributions The best fit probability distribution for each data set was found out based on the K- S test statistic and the p- value. Among all the fitted probability distributions for maximum daily rainfall, the distribution with lowest test statistic value and highest p- value was considered as the best fit. The best fit probability distribution for each data set along with the parameters is presented in the Table 4.1.3. Annual For the annual maximum daily rainfall, Weibull (3P) was found to be the best fit probability distribution with shape parameter of 0.8125, scale parameter of 37.1637 and location parameter of 56.3503. The shape of the distribution is shown in Fig.6(A). The distribution followed reverse J- shaped curve on linear axes which approached each of the orthogonal axes asymptotically. Since the distribution was scale dominated, there was large variation in the distribution of rainfall. Seasonal For the seasonal maximum daily rainfall, Weibull (3P) was found to be the best fit probability distribution with shape parameter of 2.1501, scale parameter of 74.9242 and location parameter of 12.0745. Fig.6(B) shows the shape of the distribution. The distribution approximated normal with consistency in the distribution of rainfall. 45 Bhoomika Raj, R., M.Sc. (Ag. Stat.), 2015

June For the data set on the seasonal month of June, Weibull (3P) was found to be the best fit probability distribution with shape parameter of 1.2689, scale parameter of 29.7402 and location parameter of 4.8815. The shape of the distribution is shown in Fig.6 (C). The distribution was observed to be positively skewed where the average rainfall exceeded the modal rainfall. Since the distribution was scale dominated, there was large variation in the distribution of rainfall. July During the seasonal month of July, log-normal was found to be the best fit probability distribution with shape parameter of 3.5139 and scale parameter of 0.6204. Fig.6 (D) shows the shape of the distribution in which positive skewness was observed with the average rainfall exceeding the modal rainfall. Since the distribution was shape dominated, there was not much variation in the distribution of rainfall. August For the seasonal month of August, Weibull (2P) was found to be the best fit probability distribution with shape parameter of 1.7800 and scale parameter of 48.7270. The shape of the distribution is shown in Fig.6 (E). The distribution showed slight positive skewness with the average rainfall exceeding the modal rainfall. There was large variation in the distribution of rainfall since it was found to be scale-dominated. September During the seasonal month of September, Gamma (2P) was found to be the best fit probability distribution with shape parameter of 2.9149 and scale parameter of 23.1780. Fig.6 (E) shows the shape of the distribution. The shape of the distribution approximated normal and there was consistency in the distribution of rainfall since the distribution was shape dominated. October For the data set on the seasonal month of October, log-normal was found to be the best fit probability distribution with shape parameter of 4.0168 and scale parameter of 0.6175. The shape of the distribution is shown in Fig.6 (F) where the distribution was positively skewed with the average rainfall exceeding the modal rainfall. There was consistency in the distribution of rainfall since it was found to be shape-dominated. 23 rd SMW (4 June-10 June) During this Standard Meteorological Week, Gamma (2P) was found to be the best fit probability distribution with shape parameter of 0.9467 and scale parameter of 19.0531. Fig.6 (a) shows the shape of the distribution where the curve got stretched out to Evaluation of Statistical Models for Climatic Characterization of GKVK Station 46

the right and its height has decreased. Since the distribution was scale dominated, large variation was observed in the distribution of rainfall. 24 th SMW (11 June-17 June) During 24 th Standard Meteorological Week, Weibull (3P) was found to be the best fit probability distribution with shape parameter of 0.72, scale parameter of 8.0669 and location parameter of - 0.1998. The shape of the distribution is shown in Fig.6 (b) where a vertical asymptote near the origin was observed. There was large variation in the distribution of rainfall as it was dominated by the scale parameter. 25 th SMW (18 June-24 June) For the data set on 25 th Standard Meteorological Week, Weibull (3P) was found to be the best fit probability distribution with shape parameter of 0.7201, scale parameter of 7.3989 and location parameter of - 0.2997. Fig.6 (c) shows the shape of the distribution. The distribution got stretched out to the right and its height has decreased. Since the distribution was scale dominated, there was large variation in the distribution of rainfall. 26 th SMW (25 June-1 July) During the 26 th Standard Meteorological Week Weibull (3P) was found to be the best fit probability distribution with shape parameter of 0.8702, scale parameter of 5.2704 and location parameter of - 0.4514. The shape of the distribution is shown in Fig.6 (d) in which the distribution got stretched out to the right and its height has decreased. There was large variation in the distribution of rainfall as it was dominated by the scale parameter. 27 th SMW (2 July-8 July) For the 27 th Standard Meteorological Week Weibull (3P) was observed to be the best fit probability distribution with shape parameter of 0.8197, scale parameter of 14.9306 and location parameter of - 0.4496. Fig.6 (e) shows the shape of the distribution where the distribution got stretched out to the right and its height has decreased. There was large variation in the distribution of rainfall as it was dominated by the scale parameter. 28 th SMW (9 July-15 July) For the data set on the 28th Standard Meteorological Week, Weibull (3P) was found to be the best fit probability distribution with shape parameter 0.8214, scale parameter 13.0598 and location parameter - 0.4496. The shape of the distribution is shown in Fig.6 (f). The curve had a vertical asymptote near the origin. Since the distribution was scale dominated, there was large variation in the distribution of rainfall. 47 Bhoomika Raj, R., M.Sc. (Ag. Stat.), 2015

29 th SMW (16 July-22 July) During the 29 th Standard Meteorological Week Weibull (3P) was found to be the best fit probability distribution with shape parameter of 1.0279, scale parameter of 17.9583 and location parameter of - 0.4506. The shape of the distribution is shown in Fig.6 (g). The distribution showed positive skewnes with the shape parameter value more than 1. There was large variation in the distribution of rainfall as it was dominated by the scale parameter. 30 th SMW (23 July-29 th July) For the data set on 30 th Standard Meteorological Week, Weibull (3P) was observed to be the best fit probability distribution with shape parameter of 1.1748, scale parameter of 14.1673 and location parameter of - 1.5595. The shape of the distribution is shown in Fig.6 (h) in which positive skewness was observed with shape parameter value more than 1. Since the distribution was dominated by the scale parameter, there was large variation in the distribution of rainfall. 31 st SMW (30 July-5 Aug) During this week, Weibull (3P) was found to be the best fit probability distribution with shape parameter of 1.1494, scale parameter of 19.7428 and location parameter of - 0.6940.The shape of the distribution is shown in Fig.6 (i). The distribution showed positive skewness with the shape parameter value more than 1. There was large variation in the distribution of rainfall as it was dominated by the scale parameter. 32 nd SMW (6 Aug-12 Aug) For the 32 nd Standard Meteorological Week, Gamma (2P) was found to be the best fit probability distribution with shape parameter of 0.9427 and scale parameter 15.0176. The shape of the distribution is shown in Fig.4 (j). It was observed that the distribution stretched out towards right. Since the distribution was dominated by the scale parameter, there was large variation in the distribution of rainfall. 33 rd SMW (13 Aug-19 Aug) During the 33 rd Standard Meteorological Week, Gamma (2P) was found to be the best fit probability distribution with shape parameter of 0.8501 and scale parameter 20.1691. Fig.6 (k) shows the shape of the distribution where a vertical asymptote near the origin was observed. There was large variation in the distribution of rainfall as it was dominated by the scale parameter. Evaluation of Statistical Models for Climatic Characterization of GKVK Station 48

34 th SMW (20 Aug-26 Aug) For the data set on 34 th Standard Meteorological Week, Gamma (2P) was found to be the best fit probability distribution with shape parameter of 0.6770 and scale parameter of 31.9156. Fig.6 (l) shows the shape of the distribution where a vertical asymptote near the origin was observed. Since the distribution was dominated by the scale parameter, there was large variation in the distribution of rainfall. 35 th SMW (27 Aug-2 Sept) During the 35 th Standard Meteorological Week, Weibull (3P) was found to be the best fit probability distribution with shape parameter of 0.9380, scale parameter of 22.4655 and location parameter of - 0.6777. Fig.6(m) shows the shape of the distribution. The distribution got stretched out to the right and its height has decreased as the shape parameter approached a value 1. There was large variation in the distribution of rainfall as it was dominated by the scale parameter. 36 th SMW (3 Sept-9 Sept) For the data set on 36 th Standard Meteorological Week, Weibull (3P) was observed to be the best fit probability distribution with shape parameter of 0.6683, scale parameter of 24.6470 and location parameter of - 0.4496. Fig.6 (n) shows the shape of the distribution. Since the shape parameter was near to a value 0.5, the distribution showed vertical asymptote near the origin. There was large variation in the distribution of rainfall as it was dominated by the scale parameter. 37 th SMW (10 Sept-16 Sept) During this week, Weibull (3P) was found to be the best fit probability distribution with shape parameter of 0.8899, scale parameter of 31.4378 and location parameter of - 0.4496. Fig.6 (o) shows the shape of the distribution where the curve stretched out towards right. There was large variation in the distribution of rainfall as it was dominated by the scale parameter. 38 th SMW (17 Sept-23 Sept) For the 38 th Standard Meteorological Week, Weibull (3P) was observed to be the best fit probability distribution with shape parameter of 0.8559, scale parameter of 24.0877 and location parameter of - 1.0237. Fig.6 (p) shows the shape of the distribution where the curve stretched out towards right. Large variation was observed in the distribution of rainfall as it was dominated by the scale parameter. 49 Bhoomika Raj, R., M.Sc. (Ag. Stat.), 2015

39 th SMW (24 Sept-30 Sept) During the 39 th Standard Meteorological Week, Weibull (3P) was found to be the best fit probability distribution with shape parameter of 0.6439, scale parameter of 24.7259 and location parameter of - 0.1998. Fig.6 (q) shows the shape of the distribution. The distribution showed vertical asymptote near the origin. As the distribution was dominated by the scale parameter, there was large variation in the distribution of rainfall. 40 th SMW (1 Oct-7 Oct) For the data set on 40 th Standard Meteorological Week, Weibull (3P) was observed to be the best fit probability distribution with shape parameter of 1.4546, scale parameter of 34.9827 and location parameter of - 5.1230. Fig.6 (r) shows the shape of the distribution. The distribution was observed to be positively skewed where the average rainfall exceeded the modal rainfall. There was large variation in the distribution of rainfall as it was dominated by the scale parameter. 41 st SMW (8 Oct-14 Oct) During this Standard Meteorological Week, Weibull (3P) was found to be the best fit probability distribution with shape parameter of 0.8737, scale parameter of 24.8221 and location parameter of - 1.5266. Fig.6(s) shows the shape of the distribution. The distribution showed vertical asymptote near the origin. There was large variation in the distribution of rainfall as it was dominated by the scale parameter. 42 nd SMW (15 Oct-21 Oct) For the data set on 42 nd Standard Meteorological Week, Weibull (3P) was observed to be the best fit probability distribution with shape parameter of 0.8635, scale parameter of 19.5985 and location parameter of - 1.0179. Fig.6 (t) shows the shape of the distribution. Since the shape parameter approached a value 1, the distribution got stretched out to the right and its height has decreased. There was large variation in the distribution of rainfall as it was dominated by the scale parameter. 43 rd SMW (22 Oct-28 Oct) During this week, Weibull (3P) was found to be the best fit probability distribution with shape parameter of 0.5184, scale parameter of 9.9949 and location parameter of - 0.1998. Fig.6 (u) shows the shape of the distribution. The distribution showed vertical asymptote near the origin. The distribution was dominated by the scale parameter which showed large variation in the distribution of rainfall. Evaluation of Statistical Models for Climatic Characterization of GKVK Station 50

Table 4.1.2: Study period wise probability distributions using goodness of fit test. Study Range Kolmogorov Smirnov Period Distribution Statistic p- value Gamma (2P) 0.1448 0.3847 Annual 1Jan 31 Dec Log- normal 0.1480 0.3484 Weibull (2P) 0.1630 0.2415 Weibull (3P) 0.0856 0.9374 Gamma (2P) 0.1650 0.2404 Seasonal 1 June- 28 Oct Log- normal 0.1573 0.2759 Weibull (2P) 0.1807 0.1617 Weibull (3P) 0.1252 0.5516 Gamma (2P) 0.0908 0.9055 GEV 0.1220 0.5947 June 1 June-30 June Log- normal 0.0989 0.8405 Weibull (2P) 0.0851 0.9357 Weibull (3P) 0.0802 0.9600 Gamma (2P) 0.1184 0.6341 Log- normal 0.0874 0.9242 July 1 July-31 July Pareto 0.2067 0.0670 Weibull (2P) 0.1401 0.4163 Weibull (3P) 0.1166 0.6519 Gamma (2P) 0.1426 0.4972 GEV 0.1819 0.1439 August 1 Aug-31 Aug Log- normal 0.1453 0.3712 Weibull (2P) 0.1286 0.5268 Weibull (3P) 0.1315 0.4972 Gamma (2P) 0.0804 0.9593 GEV 0.0849 0.9369 September 1 Sept-30 Sept Log- normal 0.1171 0.6467 Pareto 0.1517 0.3198 Weibull (2P) 0.0847 0.9377 Weibull (3P) 0.0853 0.9344 Gamma (2P) 0.0965 0.8369 Log- normal 0.0698 0.9860 October 1 Oct- 28 Oct Pareto 0.1630 0.2373 Weibull (2P) 0.1230 0.5711 Weibull (3P) 0.0895 0.8942 23 rd SMW 4 June-10 June Gamma (2P) 0.2105 0.0591 24 th SMW 11 June-17 June Gamma (2P) 0.1315 0.4857 Weibull (3P) 0.0784 0.9590 Contd. 51 Bhoomika Raj, R., M.Sc. (Ag. Stat.), 2015

Table 4.1.2: Study period wise probability distributions using goodness of fit test. Study Kolmogorov Smirnov Range Period Distribution Statistic p- value 25 th SMW Gamma (2P) 0.1842 0.1349 18 June-24 June Weibull (3P) 0.0945 0.8694 26 th SMW 25 June-1 July Pareto 0.1627 0.2433 Weibull (3P) 0.1520 0.3177 27 th SMW Gamma (2P) 0.1052 0.7709 2 July-8 July Weibull (3P) 0.0734 0.9828 28 th SMW 9 July-15 July Weibull (3P) 0.1290 0.5225 29 th SMW Gamma (2P) 0.1180 0.6376 16 July-22 July Weibull (3P) 0.0799 0.9615 30 th SMW Gamma (2P) 0.1393 0.4235 23 July-29 July Weibull (3P) 0.1163 0.6716 31 st SMW Gamma (2P) 0.1417 0.4016 30 July-5 Aug Weibull (3P) 0.1344 0.4859 32 nd SMW Gamma (2P) 0.0879 0.9186 6 Aug-12 Aug Weibull (3P) 0.0889 0.9119 33 rd SMW Gamma (2P) 0.1017 0.8055 13 Aug-19 Aug Weibull (3P) 0.1175 0.6426 34 th SMW Gamma (2P) 0.1167 0.6507 20 Aug-26 Aug Weibull (3P) 0.1409 0.4092 35 th SMW Gamma (2P) 0.0908 0.8981 27 Aug-2 Sept Weibull (3P) 0.0802 0.9602 36 th SMW Gamma (2P) 0.1315 0.4973 3 Sept-9 Sept Weibull (3P) 0.0665 0.9946 37 th SMW Gamma (2P) 0.0872 0.9230 10 Sept-16 Sept Weibull (3P) 0.0824 0.9498 38 th SMW 17 Sept-23 Sept Gamma (2P) 0.1315 0.4973 Weibull (3P) 0.0676 0.9933 39 th SMW 24 Sept-30 Sept Gamma (2P) 0.1735 0.1827 Weibull (3P) 0.1503 0.3301 Gamma (2P) 0.1243 0.5572 40 th SMW 1 Oct-7 Oct GEV 0.1295 0.5051 Weibull (3P) 0.0839 0.9311 41 st SMW 8 Oct-14 Oct Gamma (2P) 0.1842 0.1334 Weibull (3P) 0.1004 0.8013 42 nd SMW 15 Oct-21 Oct Gamma (2P) 0.1578 0.2701 Weibull (3P) 0.0830 0.9361 43 rd SMW 22 Oct-28 Oct Pareto 0.2116 0.0568 Weibull (3P) 0.1662 0.2187 Evaluation of Statistical Models for Climatic Characterization of GKVK Station 52

Table 4.1.3: Parameters of the best fit probability distributions for Maximum daily rainfall. Study Period Range Best fit Shape paramete r (k) Parameters Scale paramete r (β) Location paramet er (µ) Annual 1Jan 31 Dec Weibull (3P) 0.8125 37.1637 56.3503 Seasonal 1 June- 28 Oct Weibull (3P) 2.1501 74.9242 12.0745 June 1 June-30 June Weibull (3P) 1.2689 29.7402 4.8815 July 1 July-31 July Log-normal 3.5139 0.6204 August 1 Aug-31 Aug Weibull (2P) 1.7800 48.7270 September 1 Sept-30 Sept Gamma (2P) 2.9149 23.1780 October 1 Oct- 28 Oct Log-normal 4.0168 0.6175 23 rd SMW 4 June-10 June Gamma (2P) 0.2105 19.0531 24 th SMW 11 June-17 June Weibull (3P) 0.7229 8.0669-0.1998 25 th SMW 18 June-24 June Weibull (3P) 0.7201 7.3989-0.2997 26 th SMW 25 June-1 July Weibull (3P) 0.8702 5.2704-0.4514 27 th SMW 2 July-8 July Weibull (3P) 0.8197 14.9306-0.4496 28 th SMW 9 July-15 July Weibull (3P) 0.8214 13.0598-0.4496 29 th SMW 16 July-22 July Weibull (3P) 1.0279 17.9583-0.4506 30 th SMW 23 July-29 July Weibull (3P) 1.1748 14.1673-1.5595 31 st SMW 30 July-5 Aug Weibull (3P) 1.1494 19.7428-0.6940 32 nd SMW 6 Aug-12 Aug Gamma (2P) 0.9427 15.0176 33 rd SMW 13 Aug-19 Aug Gamma (2P) 0.8501 20.1691 34 th SMW 20 Aug-26 Aug Gamma (2P) 0.6770 31.9156 35 th SMW 27 Aug-2 Sept Weibull (3P) 0.9380 22.4655-0.6777 36 th SMW 3 Sept-9 Sept Weibull (3P) 0.6683 24.6470-0.4496 37 th SMW 10 Sept-16 Sept Weibull (3P) 0.8899 31.4378-0.4496 38 th SMW 17 Sept-23 Sept Weibull (3P) 0.8559 24.0877-1.0237 39 th SMW 24 Sept-30 Sept Weibull (3P) 0.6439 24.7259-0.1998 40 th SMW 1 Oct-7 Oct Weibull (3P) 1.4546 34.9827-5.1230 41 st SMW 8 Oct-14 Oct Weibull (3P) 0.8737 24.8221-1.5266 42 nd SMW 15 Oct-21 Oct Weibull (3P) 0.8635 19.5985-1.0179 43 rd SMW 22 Oct-28 Oct Weibull (3P) 0.5184 9.9949-0.1998 53 Bhoomika Raj, R., M.Sc. (Ag. Stat.), 2015

4.2 Impact of weather parameters on the yield of finger millet crop 4.2.1 Descriptive statistics Descriptive statistics of different weather parameters is given in the Table 4.2.1. It was observed that minimum temperature had a mean value of 19.30 0 C whereas standard deviation was found to be 0.40 0 C. The coefficient of variation was found to be 2.08 per cent and negative skewness was observed. Minimum temperature ranged from 18.10 0 C to 19.77 0 C. The mean of the maximum temperature was found to be 28.32 0 C and standard deviation was 0.51 0 C. It had a lesser coefficient of variation of 1.79 per cent and the skewness was observed to be positive. The maximum temperature ranged from 27.36 0 C to 29.28 0 C. Relative humidity had a mean value of 71.73 per cent and standard deviation was observed to be 2.07 per cent. The coefficient of variation of the relative humidity was observed to be 2.88 per cent and it was negatively skewed. The relative humidity ranged from 67.55 per cent to 74.78 per cent. The mean value of daily rainfall was found to be 4.08 mm while the standard deviation was 1.32 mm. It had a higher value of coefficient of variation i.e., 32.18 per cent and skewness was found to be positive. The daily rainfall ranged from 2.10mm to 6.28 mm. Number of rainy days had a mean value of 30 and standard deviation was observed to be 7 days. The coefficient of variation was found to be 23.11 per cent and the skewness was negative. The number of rainy days ranged from 19 to 40. The mean value of PET was observed to be 39.62 mm and standard deviation was 7.63 mm. It was observed that the coefficient of variation was 19.25 per cent and PET was found to be positively skewed. PET ranged from 24.02 mm to 57.35 mm. It was observed that mean value of the sunshine hours was 4.95 hrs/day while it had a standard deviation of 0.72 hrs/day. The coefficient of variation was observed to be 14.54 per cent and it was negatively skewed. The sunshine hours ranged from 3.56 hrs/day to 6.18 hrs/day. Soil temperature had a mean value of 26 0 C whereas the standard deviation was observed to be 1.63 0 C. The coefficient of variation was observed to be 6.23 per cent and soil temperature was negatively skewed. The soil temperature ranged from 23.24 0 C to 28.57 0 C. Evaluation of Statistical Models for Climatic Characterization of GKVK Station 54

Frequency Density 0.25 0.2 0.15 0.1 0.05 0 0 50 100 150 200 250 Frequency Density 0.018 0.016 0.014 0.012 0.01 0.008 0.006 0.004 0.002 0 0 50 100 150 200 Frequency Density 0.03 0.025 0.02 0.015 0.01 0.005 0 0 20 40 60 80 100 Rainfall (mm) Rainfall (mm) Rainfall (mm) Weibull (3)(0.813,37.164,56.350) Weibull (3)(2.150,74.924,12.075) Weibull (3)(1.269,29.740,4.882) Fig.6(A): Distribution of Annual rainfall Fig.6(B): Distribution for Seasonal rainfall Fig.6(C): Distribution of rainfall during June 0.025 0.03 0.014 Frequency Density 0.02 0.015 0.01 0.005 Frequency Density 0.025 0.02 0.015 0.01 0.005 Frequency Density 0.012 0.01 0.008 0.006 0.004 0.002 0 0 50 100 150 Rainfall (mm) Log-normal(3.514,0.620) 0 0 20 40 60 80 100 Rainfall (mm) Weibull (2)(1.780,48.727) 0 0 50 100 150 200 Rainfall (mm) Gamma (2)(2.915,23.178) Fig.6(D): Distribution of rainfall during July Fig.6(E): Distribution of rainfall during August Fig.6(F): Distribution of rainfall during September Fig.6: Plot of the best fitted distribution functions for maximum daily rainfall (mm) Contd.

Frequency Density 0.016 0.014 0.012 0.01 0.008 0.006 0.004 0.002 0 0 50 100 150 200 250 Rainfall (mm) Frequency Density 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 0 20 40 60 Rainfall (mm) Frequency Density 2.5 2 1.5 1 0.5 0 0 20 40 60 Rainfall (mm) Log-normal(4.017,0.618) Gamma (2)(0.947,19.053) Weibull (3)(0.723,8.067,-0.200) Fig.6(G): Distribution of rainfall during October Fig.6(a): Distribution of rainfall during 23 rd SMW Fig.6(b): Distribution of rainfall during 24 th SMW Frequency Density 2.5 2 1.5 1 0.5 0 0 20 40 60 Rainfall (mm) Frequency Density 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10 20 30 40 50 Rainfall (mm) Frequency Density 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0 20 40 60 80 100 Rainfall (mm) Weibull (3)(0.720,7.399,-0.300) Weibull (3)(0.870,5.270,-0.451) Weibull (3)(0.820,14.931,-0.450) Fig.6(c): Distribution of rainfall during 25 th SMW Fig.6(d): Distribution of rainfall during 26 th SMW Fig.6(e): Distribution of rainfall during 27 th SMW Fig.6: Plot of the best fitted distribution functions for maximum daily rainfall (mm) Contd.

Frequency Density 0.6 0.5 0.4 0.3 0.2 0.1 0 0 50 100 150 Rainfall (mm) Frequency Density 0.06 0.05 0.04 0.03 0.02 0.01 0 0 10 20 30 40 50 Rainfall (mm) Frequency Density 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 0 10 20 30 40 50 Rainfall (mm) Weibull (3)(0.821,13.060,-0.450) Weibull (3)(1.028,17.958,-0.451) Weibull (3)(1.175,14.167,-1.560) Fig.6(f): Distribution of rainfall during 28 th SMW Fig.6(g): Distribution of rainfall during 29 th SMW Fig.6(h): Distribution of rainfall during 30 th SMW Frequency Density 0.045 0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 0 20 40 60 80 100 Frequency Density 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0 20 40 60 Frequency Density 0.3 0.25 0.2 0.15 0.1 0.05 0 0 20 40 60 80 Rainfall (mm) Rainfall (mm) Rainfall (mm) Weibull (3)(1.149,19.743,-0.694) Gamma (2)(0.943,15.018) Gamma (2)(0.850,20.169) Fig.6(i): Distribution of rainfall during 31 st SMW Fig.6(j): Distribution of rainfall during 32 nd SMW Fig.6(k): Distribution of rainfall during 33 rd SMW Fig.6: Plot of the best fitted distribution functions for maximum daily rainfall (mm) Contd.

Frequency Density 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 0 20 40 60 80 Frequency Density 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 20 40 60 80 100 Frequency Density 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 50 100 150 200 Rainfall (mm) Rainfall (mm) Rainfall (mm) Weibull (3)(0.938,22.466,-0.678) Gamma (2)(0.677,31.916) Weibull (3)(0.668,24.647,-0.450) Fig.6(l): Distribution of rainfall during 34 th SMW Fig.6(m): Distribution of rainfall during 35 th SMW Fig.6(n): Distribution of rainfall during 36 th SMW 0.12 0.25 2.5 Frequency Density 0.1 0.08 0.06 0.04 0.02 Frequency Density 0.2 0.15 0.1 0.05 Frequency Density 2 1.5 1 0.5 0 0 50 100 150 Rainfall (mm) 0 0 20 40 60 80 100 Rainfall (mm) 0 0 50 100 150 200 Rainfall (mm) Weibull (3)(0.890,31.438,-0.450) Weibull (3)(0.856,24.088,-1.024) Weibull (3)(0.644,24.726,-0.200) Fig.6(o): Distribution of rainfall during 37 th SMW Fig.6(p): Distribution of rainfall during 38 th SMW Fig.6(q): Distribution of rainfall during 39 th SMW Fig.6: Plot of the best fitted distribution functions for maximum daily rainfall (mm) Contd.

Frequency Density 0.025 0.02 0.015 0.01 0.005 0 0 20 40 60 80 Frequency Density 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0 50 100 150 200 250 Frequency Density 0.25 0.2 0.15 0.1 0.05 0 0 20 40 60 Rainfall (mm) Rainfall (mm) Rainfall (mm) Weibull (3)(1.455,34.983,-5.123) Weibull (3)(0.874,24.822,-1.527) Weibull (3)(0.864,19.599,-1.018) Fig.6(r): Distribution of rainfall during 40 th SMW Fig.6(s): Distribution of rainfall during 41 st SMW Fig.6(t): Distribution of rainfall during 42 nd SMW 14 Frequency Density 12 10 8 6 4 2 0 0 50 100 150 200 Rainfall (mm) Weibull (3)(0.518,9.995,-0.200) Fig. 6(u): Distribution of rainfall during 43 rd SMW Fig.6: Plot of the best fitted distribution functions for maximum daily rainfall (mm)

Table 4.2.1: Summary of statistics for weather parameters (Period: 1998-2013) Weather parameters Mean Standard deviation CV (%) Skewness Maximum Minimum Minimum temperature Maximum temperature Relative Humidity 19.30 0.40 2.08-1.83 19.77 18.10 28.32 0.51 1.79 0.08 29.28 27.36 71.73 2.07 2.88-0.46 74.78 67.55 Rainfall 4.08 1.32 32.18 0.03 6.28 2.10 No. of rainy days 30.38 7.02 23.11-0.33 40.00 19.00 PET 39.62 7.63 19.25 0.13 57.35 24.02 SSH 4.95 0.72 14.54-0.18 6.18 3.56 Soil temperature 26.00 1.63 6.23-0.33 28.57 23.24

4.2.2 Simple Linear Regression Analysis Simple linear regression analysis was carried out separately for each climatic factor to know the effect of weather parameters on the finger millet crop yield and the results are presented in Table 4.2.2. All the regression coefficients pertaining to the weather parameters were found to be non- significant. The result also revealed that there was a positive influence of minimum temperature, relative humidity, rainfall and number of rainy days on the yield of finger millet whereas maximum temperature, PET, sunshine hours and soil temperature had a negative influence on the yield of finger millet. The fitting of the regression equation of individual weather parameter on the yield are depicted in Fig.7. 4.2.3 Multiple Linear Regression Analysis Multiple linear regression analysis was carried out initially involving all the eight weather parameters viz., minimum temperature, maximum temperature, relative humidity, rainfall, number of rainy days, PET, soil temperature and sunshine hours to know the overall effect of weather parameters on the finger millet crop yield. However, there was a problem of multicollinearity in the data set (with a value of VIF exceeding 10) showing high correlation between predictors namely, sunshine hours and maximum temperature. Hence, the predictor sunshine hours was removed from the data set and revised regression analysis was carried out with the remaining seven weather parameters. The results are presented in the Table 4.2.3. The results showed that minimum temperature, relative humidity and number of rainy days showed positive influence on finger millet where as maximum temperature, rainfall, PET and soil temperature showed negative influence on finger millet crop yield. None of the weather parameters had significant influence on finger millet crop yield. The model F ratio (0.662) was not significant with R 2 value of 0.367. The model was free from multicollinearity with all the Variance Inflation Factor (VIF) showing a value below 10.

Table 4.2.2: Estimates of the simple linear regression model relating the crop yield with weather parameters Weather parameters Minimum temperature Maximum temperature Relative Humidity Intercept Slope t- value SE R 2 23.1182 0.8552 0.1691 NS 5.0550 0.0020 117.2400-2.7408-0.6946 NS 3.9453 0.0333-13.1600 0.7358 0.7610 NS 0.9669 0.0397 Rainfall 38.0127 0.3942 0.2549 NS 1.5458 0.0046 No. of rainy days 27.0195 0.4149 1.5463 NS 0.2682 0.1459 PET 64.6818-6.1016-1.4253 NS 4.2807 0.1267 SSH 51.1680-2.3323-0.8463 NS 2.7556 0.0487 Soil temperature 89.3051-1.9112-1.6688 NS 1.1452 0.1659 NS Non-significant

39 Bhoomika Raj, R., M.Sc. (Ag. Stat.), 2015 Table 4.2.3: Multiple linear regression of weather parameters on the yield of finger millet Regression Min. Temp Max. Temp RH Rainfall NS Non-significant No. of rainy days PET Soil Temp. Coefficients 2.522-1.107 0.521-4.001 0.864-2.249-0.181 SE 5.897 5.327 2.393 4.498 0.735 6.468 2.849 t- value 0.428 NS -0.208 NS 0.218 NS -0.889 NS 1.175 NS -0.348 NS 0.064 NS VIF 1.226 1.591 5.308 7.606 5.786 1.800 4.658 F R 2 0.662 NS 0.367

Yield (qtl/ha) 70 60 50 40 30 20 10 0 18 19 20 Minimum temperature ( 0 C) Yield(qtl/ha) 70 60 50 40 30 20 10 0 27 27.5 28 28.5 29 29.5 Maximum temperature ( 0 C) Yield(qtl/ha) Y = - 13.16 + 0.7359X 70 60 50 40 30 20 10 0 66 68 70 72 74 76 Relative Humidity (%) Yield (qtl/ha) 70 60 50 40 30 20 10 0 0 2 4 6 8 Rainfall (mm) Fig.7: Effect of weather parameters on the finger millet crop yield (qtl/ha) Evaluation of Statistical Models for Climatic Characterization of GKVK Station 40

Contd. 70 70 60 60 Yield (qtl/ha) 50 40 30 20 Yield (qtl/ha) 50 40 30 20 10 10 0 15 25 35 45 Number of Rainy days 0 3 3.5 4 4.5 5 PET (mm) Yield (qtl/ha) 70 60 50 40 30 20 10 0 3 4 5 6 7 Sunshine hours (hrs/day) Y = 89.305-1.9112 X 70 60 50 40 30 20 10 0 22 24 26 28 30 Soil Temperature ( 0 C) Fig.7: Effect of weather parameters on the finger millet crop yield (qtl/ha) 41 Bhoomika Raj, R., M.Sc. (Ag. Stat.), 2015 Yield (qtl/ha)

4.2.4 Test for Randomness and Normality of error terms In order to test the adequacy of the model, randomness of errors was verified using runs test. The results are shown in Table 4.2.3.1. From the analysis, it was confirmed that the error terms were randomly distributed (with the total number of runs being 10) for the model. The distribution of predicted residuals against the response variable (yield) indicated a horizontal pattern confirming that the data is free from model defects (as shown in Fig.8). Further, the normality of error terms was checked by using normal probability plot (Fig.9). The plot indicated near normal type with points scattered on either side of the line. Table 4.2.4: Observed and predicted yield of finger millet crop over a period of 16 years (1998-2013). Year Y Y ) = ( Y Y ) e i ) 1998 41.28 39.56 1.72 1999 40.71 32.73 7.98 2000 27.6 39.95-12.35 2001 36.9 40.32-3.42 2002 38.2 36.14 2.06 2003 24.02 36.59-12.57 2004 40.76 40.96-0.20 2005 38.95 36.03 2.92 2006 36.31 36.64-0.33 2007 57.35 46.50 10.85 2008 38.5 39.64-1.14 2009 35.1 37.70-2.60 2010 44.39 49.42-5.03 2011 48.26 48.33-0.07 2012 44.4 41.43 2.97 2013 41.2 43.28-2.08 Run test statistic No. of runs (r) =10 NS, Critical values: r 4, r = 13 L = U Evaluation of Statistical Models for Climatic Characterization of GKVK Station 42

4.3 Direct and indirect effects of weather parameters on rainfall. Path coefficient analysis was carried out to know the direct and indirect effects of weather parameters on rainfall. The detailed results are presented below: 4.3.1 Correlation between weather parameters and rainfall. Correlations between weather parameters and rainfall were useful because they can indicate a predictive relationship that can be exploited in practice. Weather parameters are believed as the main factors that influence the rainfall. Therefore, it becomes important to know the correlation between the weather parameters and rainfall. The weather parameters considered were Maximum temperature (Max temp.), Minimum temperature (Min temp.), Potential Evapotranspiration (PET), Vapor Pressure (VP), Relative Humidity (RH), Wind Speed (WS), Sunshine hours (SSH), Cloud amount (CL AMT), and Evaporation (EVP). Table 4.3.1 represents the Correlation matrix between weather parameters and rainfall. 4.3.1.1 Maximum temperature Maximum temperature showed a positive significant correlation (0.3575) with evaporation while non-significant positive correlation was observed with minimum temperature (0.1654), PET (0.0856), sunshine hours (0.2325) and rainfall (0.1676). On the other hand, it showed a significant negative correlation with relative humidity (- 0.4974) and non-significant negative correlation with vapor pressure (- 0.0201), wind speed (- 0.1850) and cloud amount (- 0.0642). 4.3.1.2 Minimum temperature Minimum temperature had positive non-significant correlation with maximum temperature (0.1654), vapor pressure (0.2306) and rainfall (0.3196) and positive significant correlation was not observed with any of the weather parameters. The negative significant correlation was observed with sunshine hours (- 0.3961) while it showed negative non-significant correlation with PET (- 0.0822), wind speed (- 0.2199), cloud amount (- 0.1119) and evaporation (- 0.1947). 4.3.1.3 Potential Evapotranspiration (PET) Potential Evapotranspiration showed positive significant correlation with wind speed (0.8040), sunshine hours (0.5551), cloud amount (0.3677) and evaporation (0.6429) while a positive non-significant correlation was observed with maximum temperature (0.0856). PET had negative significant correlation with vapor pressure (-0.7904), relative humidity (- 0.5344) and rainfall (- 0.6584) whereas a negative nonsignificant correlation was seen with minimum temperature (- 0.0822). 43 Bhoomika Raj, R., M.Sc. (Ag. Stat.), 2015

Fig. 8:Plot of PRESS residuals against predicted finger millet yield Fig.9: Normal Probability Plot for finger millet yield against weather parameters Evaluation of Statistical Models for Climatic Characterization of GKVK Station 44