Statistical Models for Forecasting Mango and Banana Yield of Karnataka, India

Transcription

1 J. Agr. Sci. Tech. (2018) Vol.20: Statistical Models for Forecastig Mago ad Baaa Yield of Karataka, Idia Dowloaded from jast.modares.ac.ir at 21:59 IRST o Friday October 19th 2018 S. Rathod 1 *, ad G. C. Mishra 2 ABSTRACT Horticulture sector plays a promiet role i ecoomic growth for most of the developig coutries. Idia is the largest producer of fruits ad vegetables i the world ext oly to Chia. Amog the horticultural crops, fruit crops are cultivated i majority of the area. Fruit crops play a sigificat role i the ecoomic developmet, utritioal security, employmet geeratio, ad overall growth of a coutry. Amog fruit crops, mago ad baaa are largest producig fruits of Idia. Geerally, Karataka is called as the horticultural state of Idia. I Karataka, mago ad baaa are highest producig fruit crops. With these prospective, yield of mago ad baaa of Karataka have bee chose as study variables. Forecastig is a primary aspect of developig ecoomy so that proper plaig ca be udertake for sustaiable growth of the coutry. I this study, classes of liear ad oliear, parametric ad o-parametric statistical models have bee employed to forecast yield of mago ad baaa of Karataka. The major drawback of liear models is the presumed liear form of the model. I most of the cases, the time series are ot purely liear or oliear as they cotai both liear ad oliear compoets. To overcome this problem a hybrid model has bee proposed which cosists of liear ad oliear models. The hybrid model with the combiatio of Autoregressive Itegrated Movig Average (ARIMA) ad Support Vector Regressio model performed better i both model buildig as well as i model validatio as compared to other models. Keywords: ARIMA, Hybrid models, NLSVR, Regressio model, Time Series, TDNN. INTRODUCTION Agriculture is backboe of Idia ecoomy accoutig for 14 percet of the atio s Gross Domestic Product (GDP) ad about 11 percet of the coutry s exports. Nearly about 65 percet of coutry s populatio still depeds o agriculture for employmet ad livelihood. Though the cotributio of agricultural sector to GDP is decreasig from last decade, but a sigificat chage i the compositio of agriculture, showig shiftig from croppig towards horticulture, livestock, ad fisheries, is oticeable. The horticulture sector cotributed 30 percet of agricultural GDP, while the cotributio of livestock sector is 4 percet. Idia witessed the shift i area from food grais towards horticultural crops over the last five years ( to ) (NHB data base, ). The area uder horticultural crops has bee icreased about 18 percet but the augmetatio of area of food grais is oly 5 percet durig this period. Sice last decade, area uder horticultural crops is icreasig by 3 percet per year ad the aual productio is also icreasig by 7 percet. Amog the horticultural crops, fruit crops are cultivated i a area of 7, ( 000 ha) with productio of 88, ( 000 tos) (NHB data base, ). Fruit crops play a sigificat role i the ecoomic developmet, utritioal security, employmet geeratio, ad overall 1 ICAR-Idia Agricultural Statistics Research Istitute, New Delhi, Idia *Correspodig author; satosha.rathod@icar.gov.i 2 Istitute of Agricultural Scieces, Baaras Hidu Uiversity, Varaasi, UP, Idia

2 Rathod ad Mishra Dowloaded from jast.modares.ac.ir at 21:59 IRST o Friday October 19th 2018 growth of the coutry. Fruit crops have climatic specificity ad excellet fruits havig delicacy, utritive value, ad good market acceptability are grow widely i temperate, tropical, ad subtropical parts of the coutry. A large size of the populatio i Idia is egaged i fruit productio, distributio ad marketig (Yadav ad Padey, 2016). Fruits beig the major source of vitamis ad mierals are aptly called protective foods ad are idispesable part of huma diet. Although Idia cotributes percet of total fruits of the world, the availability of fruits i the coutry has bee estimated to be oly 182 grams per day per perso, amoutig to the deficit of 48 grams per day per perso (Aoymous, 2015a). Hece, there exists a great gap to icrease the yield of fruit crops. Idia is the largest producer of fruits ext to Chia. The aual productio of fruits has bee estimated to be MT from a area of Millio ha. Over the decades, icrease i area ad productio accouts to aroud percet ad percet, respectively (Aoymous, 2015). With availability of diversified climatic ad soil coditios, it is possible to grow a assorted rage of tropical, subtropical, temperate, ad arid zoe fruit crops i the coutry (Radha ad Mathew, 2007). Idia has emerged as leader i productio of several horticultural crops viz., mago, baaa, cashew ut, areca ut, potato, papaya, okra, etc. Amog the fruit crops, Mago is cultivated i a area of 2, ( 000 ha) with a productio of 18,002 ( 000 tos) with average productivity of 7.3 MT ha -1. Baaa cotributes a area of ( 000 ha) with a productio of 26, tos ( 000 tos) (NHB data base, ) with average productivity of 37 MT ha -1. Amog fruit crops, mago ad baaa are largest producig fruits of Idia. O the other had, Karataka is called as the horticultural state of Idia. I Karataka, baaa is secod i area (101,532 ha) ad first i productio (2,581,752 MT) with average productivity of MT ha -1. Mago is first i area (173,080 ha) ad secod i productio (1,641,165 MT) with average productivity of 9.48 MT ha -1 ) (Aoymous, 2015b). The Mago ad baaa are cultivated i almost all districts of Karataka. With these prospective, yields of mago ad baaa of Karataka have bee chose as study variables. Statistical forecastig is used to provide assistace i decisio makig ad plaig the future more effectively ad efficietly. Forecastig is a primary aspect of developig ecoomy so that proper plaig ca be udertake for sustaiable growth of the coutry. Maily there are two approaches of forecastig viz., (i) Predictio of preset series based o behavior of past series over a period of time called as the extrapolatio method, (ii) Estimatio of future pheomeo by cosiderig the factors which ifluece the future pheomeo, i.e., the explaatory method (Diebold ad Lopez, 1996). Statistical forecastig is the likelihood approximatio of a evet takig place i future. (Box ad Jekis, 1970) Cosiderig the above metioed facts, a study was coducted to model ad forecast the yield of mago ad baaa i Karataka. Most commoly used classical liear time series models are ARIMA ad liear regressio models. Rathod et al. (2011), Narayaaswamy et al. (2012a), Narayaaswamy et al. (2012b), ad Pardhi et al. (2016) applied regressio aalysis to study the effect of agricultural iputs ad weather parameters o agricultural ad horticultural crops. Naveea et al. (2014) used differet time series models to forecast the cocout productio of Idia. Kha et al. (2008) ad Qureshi (2014) forecasted mago productio of Pakista usig differet statistical models. Omar et al. (2014) carried out price forecastig ad spatial co-itegratio of baaa i Bagladesh. Soares et al. (2014) compared differet techiques for forecastig yield of baaa plats. Olse ad Goodwi (2005) carried out a statistical survey o Orego hazelut productio. Peiris et al. (2008) predicted cocout productio i Sri Laka usig seasoal climate iformatio. Mayer ad Stepheso (2016) carried out statistical forecastig of Australia macadamia crop. The major drawback of regressio ad AutoRegressive Itegrated Movig Average (ARIMA) models is the presumed liear form of 804

3 Models for Forecastig Mago ad Baaa Yield Dowloaded from jast.modares.ac.ir at 21:59 IRST o Friday October 19th 2018 the model, i.e. a liear correlatio patter is assumed amog the time series, ad hece o oliear patters ca be captured by these models. The time series which cotai both liear ad oliear compoets, rarely are pure liear or oliear. Uder such coditio Neither ARIMA or Artificial Neural Network (ANN) ad Noliear Support Vector Regressio (NLSVR) models are adequate i modelig the series which cotais both liear ad oliear patters. To overcome this difficulty, hybrid time series method was evolved. Applicatios of hybrid methods i the literature (Zhag, 2003; Che ad Wag, 2007; Jha ad Siha, 2014; Kumar ad Prajeshu, 2015; Ray et al., 2016; Naveea et al., 2017a; Naveea et al., 2017b; Rathod et al., 2017) shows that amalgamatig differet methods ca be a efficiet ad effective way to improve time series forecastig. With these motivatios, attempt has bee made to develop hybrid forecastig models for forecastig yield of mago ad baaa i Karataka by combiig ARIMA with ANN ad ARIMA with NLSVR models. The details Table 1. Variables cosidered for regressio aalysis. methodology is explaied i subsequet sectios. MATERIALS AND METHODS Data Descriptio Yearly data o yield (MT ha -1 ) of mago ad baaa were collected from data base of Natioal Horticulture Board (NHB) ad Daily data o weather variables viz., maximum temperature ( 0 C), miimum temperature ( 0 C), relative humidity (fractio), precipitatio (mm) ad wid speed (miles per secod) ad solar radiatio (mega Joules per square meter) were obtaied from a secodary website of Idia meteorological departmet ad from Aual data o other exogeous variables (Table 1) were collected from Agricultural Statistics at a Glace published Sl No Variables Uits 1 Mago yield MT ha -1 2 Mago area Hectares 3 Mago productio Millio tos 4 Baaa yield MT ha -1 5 Baaa area Hectares 6 Baaa productio Millio Tos 7 Maximum temperature (Idex 1) Degree Celsius ( 0 C) 8 Miimum temperature (Idex 1) Degree Celsius ( 0 C) 9 Relative humidity (Idex 1) Fractio 10 Precipitatio (Idex 1) Millimeter (mm) 11 Wid speed (Idex 1) Miles per secod (mps) 12 Solar radiatio (Idex 1) Megajoules per square meter (MJ m -2 ) 13 Maximum temperature (Idex 2) Degree Celsius ( 0 C) 14 Miimum temperature (Idex 2) Degree Celsius ( 0 C) 15 Relative humidity (Idex 2) Fractio 16 Precipitatio (Idex 2) Millimeter (mm) 17 Wid speed (Idex 2) Miles per secod (mps) 18 Solar radiatio (Idex 2) Mega Joules per square meter (MJ m -2 ) 19 Avg. size of operatioal holdigs Hectares 20 Area sow Hectares 21 Net area irrigated Hectares 22 Fertilizer distributio Tos 23 Argil. credit cooperative societies Numbers 24 Regulated markets Numbers 25 Rural road legth Kilo meters (Kms) 26 No of IP sets Numbers 805

4 Rathod ad Mishra Dowloaded from jast.modares.ac.ir at 21:59 IRST o Friday October 19th 2018 by Departmet of Ecoomics ad Statistics, Karataka. For regressio aalysis i.e. weather based forecastig, data o yield (MT ha -1 ), area (ha) ad productio (MT) of mago ad baaa from 1985 to 2011 were used for model buildig ad data from 2012 to 2014 were used to check the forecastig performace of the model. The iformatio o weather variables (daily data) ad other agricultural variables (aual data) were also used from 1985 to I this study, iformatio o exogeous variables are ot available for loger period of time, however, time series models yield better results whe we cosider data o loger period. To overcome this costrait, oly uivariate data o yield of mago ad baaa has bee cosidered. For mago ( to ) ad baaa ( to ), yearly data o yield (MT ha -1 ), area (ha), ad productio (MT) were collected from data base of Natioal Horticulture Board (NHB) ad To forecast yield of mago of Karataka, data from 1980 to 2011 were used for model buildig ad 2012 to 2014 were used to check the forecastig performace of the models. I the case of baaa of Karataka, data from 1954 to 2011 were used for model buildig ad 2012 to 2014 were used for model validatio. The list of variables cosidered for regressio based forecastig is listed i Table 1. Statistical Methodologies I this work, umber of statistical techiques viz., regressio model, weather idices, ARIMA, ANN, NLSVR ad proposed hybrid methodology are used to forecast the yield of mago ad baaa of Karataka, Idia. Regressio Model (1) Where, Y is the depedet (respose) variable, X are idepedet (predictor or stimulus) variables, β 0, β 1,..., β p are the regressio coefficiets ad ε is the error term. A importat issue i regressio modelig is the selectio of explaatory variables which are really ifluecig the depedet variable. There are may methods for selectio; stepwise regressio aalysis is frequetly used variable selectio algorithm i regressio aalysis (Motgomery et al., 2003). I this work, step wise regressio aalysis has bee used because umber of exogeous variables is more. Weather Idices For the daily data (d) o p variables, ew weather variables ad iteractio compoets ca be geerated with respect to each of the weather variables usig the below metioed procedure (Agrawal et al., 2001). I order to study the idividual effect of each weather variable, two ew variables from each variable ca be geerated as follows: Let be the value of the i th weather variable at the d th day, is the simple correlatio coefficiet betwee weather variable at the d th day ad yield Y i over a period of years, which is expressed as follows; r id [ d 1 X 2 ( d 1 d 1 XY d 1 X 2 X ) ] [ d 1 d 1 Y Y 2 ( d 1 Y ) (2) The geerated variables are give as follows; (3) j rid xid For j= 0, we have d 1 Z ij, j 0,1 uweighted j rid geerated variable d 1 as: Z d ij 1 x id (4) ad weighted geerated variables as: 2 ] 806

5 Models for Forecastig Mago ad Baaa Yield Dowloaded from jast.modares.ac.ir at 21:59 IRST o Friday October 19th 2018 Z i1 r id d 1 r x id d 1 id (5) Weather idices were costructed usig daily weather variables. After calculatig these idices [weighted (Idex 2) as well as uweighted (Idex 1) idices], they were used as idepedet variables i regressio models [Equatio (1)]. AutoRegressive Itegrated Movig Average (ARIMA) Model Oe of the most importat ad widely used classical time series models is the AutoRegressive Itegrated Movig Average (ARIMA) model. The popularity of the ARIMA model is due to its liear statistical properties as well as the popular Box- Jekis methodology (Box ad Jekis, 1970) for model buildig procedure. Ofte, the time series are o-statioary i ature. To obtai the statioary time series, we eed to itroduce the differecig term d. to make the o-statioary series to statioary oe. The, the geeral form of ARMA model is represeted as ARIMA (p, d, q). p is order of autoregressive term, d is the order of differecig term ad q is the order of movig average term i ARIMA(p,d,q) model. The process Y t is said to follow itegrated ARMA model if. The ARIMA model is expressed as follows: (6) Where, ad is the white oise. The Box-Jekis ARIMA model buildig cosists of three steps viz., idetificatio, estimatio, ad diagostic checkig. First step i model buildig is to idetify the model i.e. to determie the model order. Secod step is to estimate the parameters of model based o idetified model order. Fially, the third step is diagostic checkig of residuals. Time Delay Neural Network (TDNN) The ANN for time series aalysis is termed as Time Delay Neural Network (TDNN). The time series pheomeo ca be mathematically modelled usig eural etwork with implicit fuctioal represetatio of time, whereas static eural etwork like multilayer perceptro is preseted with dyamic properties (Hayki, 1999). Oe simple way of buildig artificial eural etwork for time series is the use of time delay also called as time lags. These time lags ca be cosidered i the iput layer of the ANN. The TDNN is the class of such architecture. Followig is the geeral expressio for the fial output Y t of a multilayer feed forward time delay eural etwork. Where, (7) ad are the model parameters, also called as coectio weights, p is the umber of iput odes, q is the umber of hidde odes ad is the activatio fuctio. The architecture of eural etwork is represeted i Figure 1. Noliear Support Vector Regressio (NLSVR) Model Support Vector Machie (SVM) is a supervised machie learig techique which was origially developed for liear classificatio problems. Later i the year 1997, the support vector machie for regressio problems were developed by Vapik by itroducig ε-isesitive loss fuctio (Vapik et al., 1997) ad it has bee exteded to the oliear regressio estimatio problems. Modelig of such problems is called as NoLiear Support Vector Regressio (NLSVR) model. The basic priciple ivolved i NLSVR is to trasform the origial iput time series ito 807

6 Rathod ad Mishra Dowloaded from jast.modares.ac.ir at 21:59 IRST o Friday October 19th 2018 a high dimesioal feature space ad the build the regressio model i a ew feature space. Let us cosider a vector of data set where is the iput vector, y i is the scalar output ad N is the size of data set. The geeral equatio of Noliear Support Vector Regressio estimatio fuctio is give as follows: (8) Where, (.) : R R h is a oliear mappig fuctio which maps the origial iput space ito a higher dimesioal feature space vector. W R h is weight vector, b is bias term ad superscript T deotes the traspose. The Proposed Hybrid Methodology The hybrid method cosiders the time series as a combiatio of both liear ad o-liear compoets. This approach follows the Zhag s (2003) hybrid approach. Accordigly, the relatioship betwee liear ad oliear compoets ca be expressed as follows Figure 1. Neural etwork structure. (9) Where, ad represet the liear ad oliear compoet, respectively. I this work, the liear part is modeled usig ARIMA model ad o-lier part by TDNN ad NLSVR. The methodology cosists of three steps. Firstly, a ARIMA model is employed to fit the liear compoet. Let the predictio series provided by ARIMA model be deoted as. I the secod step, the residuals ( ) obtaied from ARIMA model are tested for o-liearity by usig BDS test (Brock et al., 1996) Oce the residuals cofirm the o-liearity, the, they are modelled ad predicted usig TDNN ad NLSVR. Fially, the forecasted liear ad oliear compoets are combied to geerate aggregate forecast. y ˆ ˆ ˆt L t N t (10) Where, ad represet the predicted liear ad oliear compoet, respectively. The graphical represetatio of hybrid methodology is expressed i Figure 2. Fially, the performace of the models uder cosideratio is compared by usig Mea Absolute Percetage Error (MAPE). RESULTS AND DISCUSSION Regressio Aalysis Regressio aalysis has bee carried out to kow the factors ifluecig yield of mago ad baaa i Karataka. Regressio model was fitted for yield of mago ad baaa. The depedet variables i the study are yield of mago ad baaa, whereas idepedet variables iclude weather variables ad some other socio-ecoomic 808

7 Models for Forecastig Mago ad Baaa Yield Dowloaded from jast.modares.ac.ir at 21:59 IRST o Friday October 19th 2018 Figure 2. Schematic represetatio of hybrid methodology. ad agricultural variables listed i Table 1. Data o these variables were cosidered from to Data from to were used for model buildig ad data from to were used for model validatio. The summary statistics of yield of mago ad baaa time series is preseted i Table 2, which shows that baaa yield time series are highly heterogeeous as Coefficiet of Variatio (CV) is very high. To begi with regressio aalysis, multiple liear regressio aalysis was carried out for all the four data set separately. As explaied i methodology sectio, weather idices have bee developed ad these idices are cosidered as idepedet variables i MLR model. For oe idepedet variable, two weather idices viz., weighted ad uweighted idices has bee developed as Table 2. Descriptive Statistics of time series uder cosideratio. Statistics Mago yield Baaa yield Mea Media Mode Stadard deviatio Miimum Maximum Skewess Kurtosis Coefficiet of variatio (%) explaied i methodology sectio. (weighted ad u-weighted), therefore, total umber of idepedet variables becomes 22 i this study. The Multiple Liear Regressio (MLR) aalysis was carried out by cosiderig all the idepedet variables (Table 1). It was foud that the coefficiet of determiatio (R 2 ) of MLR model is very high (Table 3). Although the R 2 value of MLR model is very high, most of the variables i the models are o-sigificat ad Variace Iflatig Factor (VIF) is also very high. This clearly idicates that there is a multi-colliearity problem amog the idepedet variables. To overcome the multi-colliearity problem, oe of the measures is to drop the uimportat variables which are explaiig less variatio i depedet variables i the model. The droppig of variable ca be doe through the stepwise regressio aalysis (Gujarati et al., 2013). Hece, stepwise regressio aalysis was carried out to fit the model (Table 4 ad 5). I stepwise regressio aalysis, the value of R 2 was decreased ad multicolliearity problem was also reduced as Variace Iflatig Factor (VIF< 10) was less ad most importatly umber of sigificat variables were icreased. For stepwise regressio aalysis of time series o mago yield (Table 4) ifer that as area ad productio icreases, the mago yield also icreases. As the variables viz. et irrigated area ad umber of agricultural credit cooperative 809

8 Rathod ad Mishra Dowloaded from jast.modares.ac.ir at 21:59 IRST o Friday October 19th 2018 Table 3. MLR Model iformatio. Regressio model/depedet variables Adj R 2 Number of sigificat variables Mago yield Nil 19 Baaa yield Table 4. Stepwise regressio aalysis of mago yield time series. Number of variables havig VIF> 10 Variable Coefficiet Std error t Test Probability VIF R 2 Itercept Area < Net area irrigated E < Productio Agricultural credit cooperative societies Relative humidity societies icreased, the mago yield also icreased. However, it has egative relatio with relative humidity. As discussed earlier, the data set from to were used for model buildig ad data from to were used for validatio. Performace of stepwise regressio model i both traiig ad testig data set is give i Tables 12 ad 13, respectively. Results of ARIMA Model A ARIMA model was built usig SAS 9.4 software available at ICAR-Idia Agricultural Statistics Research Istitute, New Delhi. The ARIMA model for both mago ad baaa yield time series was built separately. The Box-Jekis methodology of model buildig was followed. After idetificatio of cadidate model order, maximum likelihood method was used for parameter estimatio (Tables 6 ad 7). Based o the probability of residuals obtaied, oe ca say that the residuals are o-correlated. Sice, the model satisfies Box-Jekis methodology of model buildig viz., model idetificatio, statioarity, parameter estimatio ad diagostic checkig, the, oe ca go for forecastig task. The forecastig performace of mago ad baaa yield time series i both traiig ad testig data sets are give i Tables 12 ad 13. Table 5. Stepwise regressio aalysis of Baaa Yield time series. Variable Coefficiet Std error t Test Probability VIF Adj R 2 Costat Productio Area Solar radiatio Relative humidity No of regulated markets Table 6. Parameter estimatio of ARIMA (0 1 1) for mago yield time series. Parameter Estimate Stadard error t Value Approx Pr> t Lag P(Resi.) at 6 Lag Costat MA

9 Models for Forecastig Mago ad Baaa Yield Dowloaded from jast.modares.ac.ir at 21:59 IRST o Friday October 19th 2018 Results of TDNN Model A feed forward time delay eural etwork was fitted for both mago ad baaa yield time series separately usig R software with the help of package forecast (Hydma, 2017). The Leveberg-Marquardt (LM) back propagatio algorithm was used for TDNN model buildig ad based o repetitive experimetatio, the learig rate ad mometum term for all TDNN models were set as 0.02 ad 0.001, respectively. Sigmoidal ad liear fuctios were used as activatio fuctio i hidde ad output layers, respectively. More tha 80 percet of the observatios i data set were used for model traiig ad remaiig observatios were used for testig or validatio. Differet umbers of eural etwork models were tried before arrivig at the fial structure of the model ad, fially, the adequate model give i Table 8 was obtaied. After model buildig, forecastig the time series was doe both for traiig ad testig data set. The forecast values of TDNN model are give i Tables 12 ad 13. Results of NLSVR Model The oliear support vector regressio model was employed for all four time series separately usig R software with the help of package e1071 (David, M. (2017). The parameter specificatios, cross validatio error are give i Table 9. The performace of NLSVR model strogly depeds o the Table 7. Parameter estimatio of ARIMA (0 1 2) for baaa yield time series. kerel fuctio ad set of hyper-parameters. The RBF kerel fuctio i NLSVR requires optimizatio of two hyperparameters, i.e. the regularizatio parameter C, which balaces the complexity ad approximatio accuracy of the model ad the kerel badwidth parameter, which defies the variace of RBF kerel fuctio. These tuig parameters viz., C ad, are user defied parameters. NLSVR tuig parameters (Table 9) were defied to miimize the traiig testig error. As i TDNN, the two lag delay was used as model iput variables. Also, the traiig ad testig data ratio was followed the same as i TDNN i.e. more tha 80 percet of data set was used for traiig the model ad remaiig for model testig. The forecastig performace of NLSVR model i both traiig ad testig data set are give i Tables 12 ad 13. Results of Hybrid Time Series Models As discussed i methodology sectio, the first step i hybrid time series modelig was to test the oliearity of the residuls. The BDS test was applied to test the oliearity of the residulas. The BDS test result (Tables 10 ad 11) shows that the residuals obtaied from regressio models were lier ad osigificat ad residuals of ARIMA models are oliear ad sigificat. Therfore, hybrid models were built oly with ARIMA model. Oce the residual series is foud to be oliear, it ca be modelled ad Parameter Estimate Stadard error t Value Approx Pr> t Lag P (Resi.) at 6 Lag MU MA < MA < Table 8. TDNN Model Specificatios. Time series Activatio fuctio Time No of hidde Total No of Hidde layer Output layer delay odes parameters Mago yield Sigmoidal Liear Baaa yield Sigmoidal Liear

10 Rathod ad Mishra Dowloaded from jast.modares.ac.ir at 21:59 IRST o Friday October 19th 2018 Table 9. NLSVR Model specificatios. Time series Kerel fuctio No of SVs C K Fold cross validatio (K) Cross validatio error Mago yield RBF Baaa yield RBF Table 10. Noliearity testig of Regressio residuals by BDS test. Time series Parameter Dimesio (m= 2) Dimesio (m= 3) Statistic Probability Statistic Probability Mago yield Baaa yield Table 11. Noliearity testig of ARIMA residuals by BDS test. Time series Parameter Dimesio (m= 2) Dimesio (m= 3) Statistic Probability Statistic Probability Mago yield < Baaa yield < < predicted usig oliear models. The oliear models, amely, TDNN ad NLSVR were used for modelig ad forecastig of ARIMA residuals i this study. After the cofirmatio of oliearity of ARIMA residuls, the same residuls were modelled ad forecasted usig TDNN ad NLSVR models. Further, the forecasted residuals were combied with the forecasts obtaied from origial ARIMA model. ARIMA-TDNN Hybrid Time Series Model After the cofirmatio of oliearity of ARIMA residuls, the same residuls were modelled ad forecasted usig TDNN model. Further, the forecasted residuals were combied with the forecasts obtaied from origial ARIMA model. These modelig procedure is called ARIMA-TDNN Hybrid time series model. Fially, the forecastig performace of ARIMA-TDNN hybrid model i both traiig ad testig data sets are give i Tables 12 ad 13, respectively. ARIMA-NLSVR Hybrid Time Series Model The same procedure was follwed as described i ARIMA-TDNN hybrid time series model. After the cofirmatio of oliearity of ARIMA residuls, the same residuls were modelled ad forecasted usig NLSVR model. Further, the forecasted residuals were combied with the forecasts obtaied from origial ARIMA model. These modelig procedure is called ARIMA-NLSVR Hybrid time series model. Fially, the forecastig performace of Table 12. Compariso of forecastig performace of all models i traiig data set. Criteria Stepwise regressio ARIMA TDNN NLSVR ARIMA- TDNN ARIMA- NLSVR Mago yield MAPE Baaa yield MAPE

11 Models for Forecastig Mago ad Baaa Yield Dowloaded from jast.modares.ac.ir at 21:59 IRST o Friday October 19th 2018 Table 13. Compariso of forecastig performace of all models i testig data set. Year Actual Stepwise regressio Forecast ARIMA TDNN NLSVR ARIMA- TDNN ARIMA- NLSVR Mago yield MAPE Baaa yield MAPE ARIMA-NLSVR hybrid model i both traiig ad testig data sets are give i Tables 12 ad 13, respectively. Forecastig Performace of Models uder Cosideratio The models viz. stepwise regressio, ARIMA, TDNN, NLSVR, ARIMA-TDNN ad ARIMA-SVR were studied for forecastig mago ad baaa yield time series of Karataka, Idia. Forecastig performace of each model uder traiig ad testig data set was compared. Eve though we cosidered may exogeous variables i regressio model, the uivariate ARIMA, TDNN ad NLSVR performed better as compared to regressio models i both mago ad baaa yield data. The performace of machie itelligece techiques like TDNN ad NLSVR is better as compared to liear time series models uder both traiig ad testig data set. Eve though we cosidered may exogeous variables i stepwise regressio model, the also the result of uivariate models, specially the machie learig models, performed better compared to regressio model. Eve the coefficiet of variatio time series were very high, the TDNN, NLSVR ad hybrid models performed better. The reaso could be the oliear machie learig techiques which ca capture the heterogeeous tred i the data set ad, therefore, performed well as compared to regressio ad ARIMA model. The TDNN ad NLSVR models performed well over liear models like stepwise regressio model ad ARIMA model due to their superior predictive ability i oliear ad heterogoous data set. Amog the machies, itelligece techiques like TDNN ad NLSVR, the NLSVR performed better i both traiig ad testig the data set. As discussed i hybrid methodology sectio, the hybrid models have their ow advatage over sigle models. Based o the lowest MAPE values of all models obtaied for both traiig (Table 12) ad testig data set (Table 13) cosidered, oe ca ifer that hybrid model cosistig of ARIMA ad NLSVR i.e. ARIMA-NLSVR model, outperformed all remaiig models. Both hybrid models viz., ARIMA-TDNN ad ARIMA-NLSVR, outperformed the sigle model viz., ARIMA, TDNN ad NLSVR. Fially, amog all models uder study, ARIMA-NLSVR model s performace was the best. Hybrid methodology cosiders both liearity ad oliearity of the data set, hece, the performace of ARIMA-NLSVR model is superior as compared to all other models uder both traiig ad testig data set for modelig ad forecastig mago ad baaa yield time series of Karataka. 813

12 Rathod ad Mishra Dowloaded from jast.modares.ac.ir at 21:59 IRST o Friday October 19th 2018 CONCLUSIONS Based o the results obtaied i this work, oe ca coclude that machie itelligece techiques like time delay eural etwork ad oliear support vector regressio perform better as compared to classical time series models uder heteroscedastic ad oisy time series data. The mai fidig of this study is the performace of hybrid time series model which is better as compared to sigle models. I this study, sice the data set cosisted of both liear ad oliear patter, the hybrid model performed better as compared to sigle time series or machie learig techiques for modelig ad forecastig mago ad baaa yield time series of Karataka. Amog the hybrid models, the ARIMA with Noliear Support Vector Regressio i.e. ARIMA- NLSVR model performed superior as compared to all other models uder both traiig ad testig data set. The stepwise regressio aalysis shows that some variables which strogly ifluece the yield of mago ad baaa, the govermet or policy makers should emphasize focus o such factors, for overall developmet of croppig patter uder cosideratio. Based o the results obtaied, oe ca coclude that the farmers or policy makers ivolved i mago ad baaa crop productio ca pla well i advace to further icrease the productivity of crops by suitable maagemet of the iputs ad weather variable which obtaied sigificat i this study. The hybrid approach ca be further exteded usig some other machie learig techiques for varyig autoregressive ad movig average orders so that practical validity of the model ca be well established. The validity of hybrid models ca be geeralized by applyig this approach to other horticultural ad agricultural data. REFERENCES 1. Agrawal, R., Jai, R. C. ad Mehta, S. C Yield Forecast Based o Weather Variables ad Agricultural Iputs o Agro- Climatic Zoe Basis. Id. J. Agric. Sci., 71 (7): Aoymous. 2015a. Horticultural Statistics at a Glace. Horticulture Statistics Divisio Departmet of Agriculture, Cooperatio ad Farmers Welfare Miistry of Agriculture ad Farmers Welfare Govermet of Idia. 3. Aoymous. 2015b. Karataka at a Glace. Departmet of Ecoomics ad Statistics, Govermet of Karataka, Idia. 4. Brock, W. A., Dechert, W. D., Scheikma, J. A. ad Lebaro, B A Test for Idepedece Based o the Correlatio Dimesio. Eco. Rev., 15: Che, K. Y. ad Wag, C. H Support Vector Regressio with Geetic Algorithm i Forecastig Tourism Demad. Tour. Maage., 28: David, M E1071: Misc Fuctios of the Departmet of Statistics, Probability Theory Group. R Package Versio 1.6-8, 7. Diebold, F. X. ad Lopez, J. A Forecast Evaluatio ad Combiatio: Hadbook of Statistics 14. Elsevier Sciece, Amsterdam. 8. Gujarati, D. N., Porter, D. C. ad Guasekar, S Basic Ecoometrics. Fifth Editio, Tata McGraw-Hill Educatio Pvt. Ltd, ISBN 10: /ISBN 13: Hayki, S Neural Networks: A Comprehesive Foudatio. New York. Macmilla, ISBN Hydma, R. J Forecast: Forecastig Fuctios for Time Series ad Liear Models. R Package Versio 8.1., Jha, G. K. ad Siha, K Time-Delay Neural Networks for Time Series Predictio: A Applicatio to the Mothly Wholesale Price of Oilseeds i Idia. Neural Comput. Appl., 24(3): Kha, M., Mustafa, K., Shah, M., Kha, N. ad Kha, J. Z Forecastig Mago Productio i Pakista a Ecoometric Model Approach. Sarhad J. Agri., 24(2): Kumar, T. L. M. ad Prajeshu, Developmet of Hybrid Models for Forecastig Time-Series Data Usig Noliear SVR Ehaced by PSO. J. Stat. Theor. Pract., 9(4):

13 Models for Forecastig Mago ad Baaa Yield Dowloaded from jast.modares.ac.ir at 21:59 IRST o Friday October 19th Mayer, D. G. ad Stepheso, R. A Statistical Forecastig of the Australia Macadamia Crop. Acta Hortic., 1109: doi: /ActaHortic Motgomery, D. C., Peck, E. A. ad Viig, G Itroductio to Liear Regressio Aalysis. 3rd Editio, Joh Wiley ad Sos (Asia) Pte. Ltd. 16. Narayaaswamy, T., Suredra, H. S. ad Rathod, S. 2012a. Multiple Stepwise Regressio Aalysis to Estimate Root Legth, Seed Yield per Plat ad Number of Capsules per Plat i Sesame (Seasamum idicum L.). Mysore J. Agricu. Sci., 46 (3): Narayaaswamy, T., Suredra, H. S ad Rathod, S. 2012b. Fittig of Statistical Models for Growth Patters of Root ad Shoot Morphological Traits i Sesame (Seasamum idicum L.). Eviro. Ecol., 30(4): Natioal Horticultural Board (NHB) Data Base Curret Sceario of Horticulture i Idia Naveea, K., Rathod, S., Shukla, G. ad Yogish, K. J Forecastig of Cocout Productio i Idia: A Suitable Time Series Model. It. J. Agric. Eg., 7(1): Naveea, K., Sigh, S., Rathod, S. ad Sigh, A. 2017a. Hybrid ARIMA-ANN Modellig for Forecastig the Price of Robusta Coffee i Idia. It. J. Curr. Microbiol. Appl. Sci., 6(7): Naveea, K., Sigh, S., Rathod, S., ad Sigh, A. 2017b. Hybrid Time Series Modellig for Forecastig the Price of Washed Coffee (Arabica Platatio Coffee) i Idia. It. J. Agric. Sci., 9(10): Olse, J. ad Goodwi, J The Methods ad Results of the Orego Agricultural Statistics Service: Aual Objective Yield Survey of Orego Hazelut Productio. Acta Hortic., 686: Omar, M. I., Dewa, M. F. ad Hoq, M. S Aalysis of Price Forecastig ad Spatial Co-Itegratio of Baaa i Bagladesh, Eur. J. Busiess Maage., 6(7): Pardhi, R., Sigh, R., Rathod, S. ad Sigh, P. K Effect of Price of Other Seasoal Fruits o Mago Price i Uttar Pradesh. Eco. Affairs, 61(4): Peiris, T. S. G., Hase, J. W. ad Zubair, L Use of Seasoal Climate Iformatio to Predict Cocout Productio i Sri Laka. It. J. Climatol., 28: doi: /joc Qureshi, M. N Modellig o Mago Productio i Pakista. Sci. It., (Lahore), 26(3): Radha, T. ad Mathew, L Fruit Crops. New Idia Publ. Agecy. 28. Rathod, S. Suredra, H. S., Muirajappa, R. ad Chadrashekar. H Statistical Assessmet o the Factor Ifluecig Agricultural Diversificatio i Differet Districts of Karataka. Eviro. Ecol., 30 (3A): Rathod, S., Sigh, K, N., Paul, R. K., Meher, R. K., Mishra, G. C., Gurug, B., Ray, M. ad Siha, K A Improved ARFIMA Model usig Maximum Overlap Discrete Wavelet Trasform (MODWT) ad ANN for Forecastig Agricultural Commodity Price. J. Id. Soc. Agric. Stat., 71(2): Ray, M., Rai, A., Ramasubramaia, V. ad Sigh, K. N ARIMA-WNN Hybrid Model for Forecastig Wheat Yield Time- Series Data. J. Id. Soc. Agric. Stat., 70(1): Soares, J. D. R., Pasqual, M., Lacerda, W. S., Silva, S. O. ad Doato, S. L. R Compariso of Techiques Used i the Predictio of Yield i Baaa Plats. Scietia Hortic., 167: Vapik, V., Golowich, S. ad Smola, A Support Vector Method for Fuctio Approximatio, Regressio Estimatio, ad Sigal Processig. I: Advaces i Neural Iformatio Processig Systems, (Eds.): Mozer, M., Jorda, M ad Petsche, T. MIT Press, Cambridge, MA, 9: , 33. Yadav, A. S. ad Padey, D. C Geographical Perspectives of Mago Productio i Idia. Imperial J. Iterdiscipliary Res., 2(4): Zhag, G. P Time Series Forecastig Usig a Hybrid ARIMA ad Neural Network Model. Neurocomputig, 50:

14 Rathod ad Mishra مدل های آماری برای پیش بینی عملکرد موز و مانگو در ایالت کارناتاکا هندوستان Dowloaded from jast.modares.ac.ir at 21:59 IRST o Friday October 19th 2018 س. رتود و ج. س. میشرا چکیده تخش تاغثاوی وقش تارسی در رشذ اقتصادی تیشتز کش ر ای در حال ت سع تاسی می کىذ. ىذيستان تعذ اس چیه تشرگتزیه ت لیذ کىىذ می ي سثشی در ج ان است. در میان گیا ان تاغثاوی تیشتز مساحت ت درختان می اختصاص دارد. درختان می در ت سع اقتصادی امىیت غذایی ایجاد شغل ي رشذ عم می کش ر وقش عمذ ای دارد. در میان می ا م س ي ماوگ تیشتزیه می ای ت لیذی ىذ ستىذ. ت ط ر کلی کارواتاکا ایالت تاغثاوی ىذ شىاخت می ش د. در ایه ایالت م س ي ماوگ تیشتزیه ت لیذ کىىذ می میثاشىذ. تا ایه تص یز عملکزد ماوگ ي م س کارواتاکا ت عى ان متغیز ای مطالع حاضز اوتخاب شذوذ. گفتىی است ک پیش تیىی کزدن یک جىث اساسی در اقتصاد ای در حال ت سع است تا تت ان تزوام ریشی تزای رشذ پایذار کش ر را ت گ و ای مىاسة اوجام داد. در ایه پژي ش تزای پیش تیىی عملکزد م س ي ماوگ در کارواتاکا مذل ای آماری در گزي ای خطی غیز خطی پارامتزیک ي غیز پارامتزی ت کار رفت. عمذ تزیه ایزاد مذل ای خطی میه فزض خطی ت دن مذل است سیزا در تیشتز م ارد سزی ای سماوی ت ط ر خالض)کامل( خطی یا غیز خطی ویستىذ چ ن آن ا ز دي جشء خطی ي غیز خطی را داروذ. تزای رفع ایه مسل یک مذل دي رگ ) یثزیذ( پیشى اد شذ ک حايی مذل ای خطی ي غیز خطی است. مذل یثزیذی تزکیثی اس مذل Autoregressive Itegrated (ARIMA) Movig Average ي Support Vector Regressio ت د ي در مقایس تا دیگز مذل ا در مزحل ساخت مذل ي مزحل راستی آسمایی آن وتیج ت تزی داشت. 816