TIME SERIES ANALYSIS AND FORECASTING OF MONTHLY AIR TEMPERATURE CHANGES IN NAIROBI KENYA

Transcription

1 TIME SERIES ANALYSIS AND FORECASTING OF MONTHLY AIR TEMPERATURE CHANGES IN NAIROBI KENYA OSCAR OCHIENG OCHANDA I56/74805/2014 A RESEARCH PROJECT SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENT FOR THE AWARD OF DEGREE OF MASTER OF SCIENCE IN SOCIAL STATISTICS OF THE UNIVERSITY OF NAIROBI 2016

2 DECLARATION I, the undersigned, declare that this project is my original work and to the best of my knowledge has not been presented for the award of the degree in any other University. Signature:.. Date:. Name: Oscar Ochieng Ochanda Registration Number: I56/74805/2014 This project has been submitted with my approval as University supervisor. Signature:.. Date:. Dr. John Ndiritu School of Mathematics i

3 Abstract Climate change has for a long time been the biggest debate among many people all over the world. Temperature is a key element that can be used to detect climate change. Time series analysis and forecasting is one of the major tools used by scientists in meteorological and environmental fields to study phenomena like temperature, rainfall and humidity. The aim of this research is to build a time series SARIMA model and use this model to analyze and forecast the maximum and minimum air temperature of Nairobi City in order to inform stakeholders who depend directly or indirectly on it to plan in advance. The appropriate orders of models are picked based on the results of ACF and PACF plots and evaluated using the AIC criterion. The best forecasting SARIMA model for maximum temperature is (0, 0, 2) (0, 1, 1) 12 and that for minimum temperature is (1, 0, 0) (0, 1, 1) 12. The results show that the minimum temperature is gradually increasing over years supporting the fact that global warming is real. ii

4 Dedication I dedicate this work to my entire family and friends for tirelessly supporting and encouraging me during my study period. iii

5 Acknowledgement I thank the Almighty God for his guidance through my study period. Secondly, special thanks go to my supervisor Dr. Ndiritu for the great insight, encouragement, tireless unreserved support and guidance throughout the research process. I appreciate the University of Nairobi administrators and staff at large for regular guidance on the whole research process. I also sincerely thank my friends and family for supporting, advising and encouraging me throughout the time I have worked for my project. iv

6 Abbreviations RMSE MAE MA AR ARIMA ARMA AIC SARIMA Root Mean Square Error Mean Absolute Error Moving Average Autoregressive Autoregressive integrated moving Autoregressive moving average Akaike Information Criterion Seasonal Autoregressive Integrated Moving Average ACF Autocorrelation Function PACF Partial Autocorrelation Function ADF Augmented Dickey Fuller MAPE Mean Absolutes Percent Error AIC Akaike's Information Criterion ACF Autocorrelation Function IPCC Intergovernmental Panel on Climate Change KMD Kenya Meteorological Department KPSS Kwiatkowski Phillips Schmidt Shin v

7 Table of Content Declaration...i Abstract... ii Dedication...iii Acknowledgement...iv Abbreviations... v List of Figures...x List of Tables...xi Chapter Introduction 1.1 Background study Statement of the problem General objectives Specific Objectives Justification Scope of the study Limitations of the study Outline 6 Chapter Literature review 7 vi

8 2.1 Introduction to General review Kenyan scenario Types of time series modeling methods...11 Chapter METHODOLOGY Data Overview Time Series Definition Time Series Components Methods The Box-Jenkins Method Conceptual Framework Types of Models The Autoregressive Models Moving Average Models Autoregressive Moving Average Models Autoregressive Integrated Moving Average Models Seasonal ARIMA Model Model Identification..19 vii

9 3.6.1 Stationarity Analysis ACF and PACF Augmented Dickey Fuller test Kwiatkowski Phillips Schmidt Shin (KPSS) test Estimation of parameters Model selection criterion Akaike's Information Criterion (AIC) Model Diagnostic Box-Ljung Test Forecasting Mean Absolute Error Mean Square Error Chapter Data analysis Time plots of temperature changes Descriptive Analysis Decomposition of time series data Stationarity checks using the ACF, PACF, KPSS and Dickey-Fuller.31 viii

10 4.5 Seasonal differencing Model building for monthly temperature series Model Identification Statistics for tentative models Parameter Estimation Diagnostic Analysis Diagnostic analysis for Maximum temperature model Diagnostic analysis for Minimum temperature model Model validation Forecasting Results Chapter Conclusion and recommendations..57 References.. 59 Appendices 62 Appendix 1: Decomposition of Forecasted Minimum Temperatures 62 Appendix 1: Decomposition of Forecasted Maximum Temperatures...63 Appendix 1: Residual Plots for Minimum Temperature 64 Appendix 1: Residual Plots for Maximum Temperature...65 ix

11 List of Figures Figure 1: Box- Jenkins ARIMA Model...14 Figure 2: Time series plots of Maximum and Minimum Temperatures Figure 3: Decomposition of Maximum temperature.30 Figure 4: Decomposition of Minimum temperature...31 Figure 5: ACF and PACF of Maximum temperature Figure 6: ACF and PACF of Minimum temperature.32 Figure 7: Time series plot for seasonal differenced Maximum and Minimum temperatures 35 Figure 8: ACF and PAF for seasonal differenced Maximum temperature series...36 Figure 9: ACF and PAF for seasonal differenced Minimum temperature series..36 Figure 10: Residuals QQ plot of Maximum Temperature Model..43 Figure 11: Residual Plots of maximum temperature model...44 Figure 12: Residuals QQ plot of Minimum Temperature Model..45 Figure 13: Residual Plots of minimum temperature model...46 Figure 14: Observed and fitted values of Maximum temperature series...47 Figure 15: Observed and fitted values of Minimum temperature series 48 x

12 List of Tables Table 1: Behavior of the ACF and PACF for ARMA Models...21 Table 2: Behavior of the ACF and PACF for Pure Seasonal ARMA Models...22 Table 3: Summary statistics of maximum and minimum temperatures 29 Table 4: Augmented Dickey Fuller Test.33 Table 5: KPSS Test 34 Table 6: ADF test after seasonal differencing...37 Table 7: KPSS test after seasonal differencing...38 Table 8: Statistics for tentative Maximum time series models...40 Table 9: Statistics for tentative Minimum time series models...41 Table 10: Parameter estimates for ARIMA (0, 0, 2) (0, 1, 2) Table 11: Parameter estimates for ARIMA (1, 0, 2) (0, 1, 2) Table 12: Observed and fitted values of Maximum temperature series ( )...48 Table 13: Observed and fitted values of Maximum temperature series ( )...50 Table 14: Forecasting Accuracy Statistic...52 Table 15: Forecasted Monthly Maximum temperature.53 Table 16: Forecasted Monthly Minimum temperature...55 xi

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14 CHAPTER 1 INTRODUCTION A Time Series is a data set which describes a function or property that varies or changes over time. Time Series Analysis examines this changing data, often with the objective of predicting the future occurrences (David Corliss, 2009). Analyzing time series data and forecasting its future values are among the most significant challenges in many fields like agriculture, meteorology, finance, economics, engineering, industrial production, and various environmental studies. 1.1 Background study Currently the entire world s population is believed to be suffering from harsh climatic conditions. Climatic changes have the ability to influence nature and thus threaten humans in different aspects of life economically, socially and politically. Despite the fact that the climatic changes pose a threat to the entire globe, many people believe that developing countries (like Kenya) in the tropical regions of the world will be impacted more severely than developed ones. The most influential factors in the climate are temperature and moisture. According to a study by Tanusree and Kishore (2012), climate change seems to be one of the most important issues in the recent two decades and temperature has been identified as one of the key elements that can indicate climate change. The gradual rise in the mean temperature of the Earth s atmosphere and its oceans is referred to as Global warming. It is widely believed that the changing temperature due to global warming is permanently changing the entire Earth s climate. For a long time the biggest debate in a number of local and international forums worldwide has been whether global warming is real. Some 1

15 people think that global warming is not real. However several climate scientists have carried out researches and have come to a conclusion that the globe is gradually warming. People perceive the impacts of global warming differently with some taking the necessary precautions to help reduce the rates of the rising temperatures. In the past century alone, studies have shown that the globe s mean temperature has risen by between 0.4 C and 0.8 C. According to a study by IPCC (2007), the temperatures could rise by between 1.4 C and 5.8 C by the end of the 21 st century. This increase in temperature may seem to be minute but the impacts are great. Increase in temperatures are likely to lead to a global increase in drought conditions, decreased water supplies due to evapotranspiration and an increase in urban and agricultural demand. In Kenya the observed mean annual temperatures have risen by 1.0 C since It is estimated that this increase occurs at a rate of 0.2 C per decade and it is predicted to rise up to 2.5 C by the year 2050 and to 4 C by Vital sectors of the Kenyan economy like Agriculture greatly rely on climate. Plants can grow only within certain limits of temperature. Each plant species has an optimal temperature limit for its different stages of growth and functions. They also have an upper and lower lethal limits between which they can properly grow. Temperature determines which species can survive in a particular region. Several farmers are however unaware of the changing climate and are also ignorant of the adverse impacts it will have on their livelihoods. High temperatures causes prolonged droughts, affects the amount of water in the soil, affects rainfall patterns and reduces water catchment areas. The increased temperatures can also cause an outbreak of pests and diseases that affects plants, animals and humans. Farmers who are aware of the changing climate are also helpless and unaware of what to do. They continue with poor agricultural 2

16 practices like burning of wastes and poor disposal of unused fertilizers that worsen the situation by releasing green house gasses to the atmosphere. Studying temperature changes is thus vital for the Kenyan economy as Agriculture which is the country s largest source revenue is directly affected by the rising temperatures. The Kenyan government derives nearly 45% of its revenue from the agricultural sector. As the largest employer in the economy, the Agricultural sector accounts for about 60% of the country s employment. In addition more than 80% of Kenyan population living in rural areas depends on agricultural related activities for their daily livelihoods. Climatic studies on temperature are therefore vital for the survival of the Agricultural sector as the key source of revenue to the government of Kenya. Nairobi is one of the few capital cities in the world with a national park. The animals in this park have however continued to diminish both as a result of climate change and other factors. Increase in temperatures affects the lives of the animals directly as they face shortage of food and water. This has a direct impact on the tourism sector which is one of our foreign exchange earners. Kenya is ranked 33 rd on vulnerability to climate related disasters and 155 th on readiness. This means that the country is rather vulnerable to, yet highly unready to combat climate change effects. The Kenyan economy also greatly depends on Agriculture and tourism which are directly affected by the rising temperatures. Nairobi, the capital city of Kenya is the major industrial town and has the highest number of vehicles in the country. Factories and motor vehicles are known to be the leading emitters of green house gases like carbon dioxide, methane and nitrous oxide. These gases thicken the ozone layer making the planet warmer. 3

17 1.2 Statement of the problem Climate change is among the greatest environmental threats to food production, forest biodiversity, availability of water and livelihoods in Kenya. Kenya s economy greatly depends on agricultural production which is directly affected by the climate change. Temperature is one of the key elements of climate and it is important to various sectors of the economy like Agriculture and tourism. Temperature affects water sources, pests that attack plants, animals and human diseases. Unlike other towns in Kenya, Nairobi has the highest number of factories and motor vehicles. It is therefore the leading town in air pollution (emission of green house gasses) in Kenya. These gases thicken the ozone layer making the planet warmer and affecting the general climate system thus leading to climate related disasters. Despite the increasing climate changes, majority of Kenyan citizens are still not well informed. Analyzing and forecasting of temperature changes will thus help various stakeholders and government to plan in advance in order to counter climate related disasters. 1.3 General objectives The objective of this research is to build a time series model and use this model to analyze and forecast the variation in maximum and minimum air temperature in Nairobi City in order to inform stakeholders who depend directly or indirectly on it to plan in advance. 4

18 1.4 Specific Objectives 1. To analyze temperature variations and fit a Seasonal ARIMA model for monthly minimum and maximum temperatures. 2. To carry out short-term prediction for the temperatures using the selected Seasonal ARIMA models. 1.5 Justification Climate change is currently among the important points of debate and temperature is one of its main components. Understanding the nature and scale of possible climate changes in Kenya using the temperature data from meteorological unit of Nairobi is of importance to agricultural production and human health. The main aim of forecasting temperature is to inform the various sectors of the economy that depend directly or indirectly on it to plan in advance for any eventualities. Numerous hydro-meteorological applications use Time series analysis and forecasting in studying trends and variations in variables like temperature, humidity, rainfall, stream flow and many other environmental parameters. Forecasting the variations in temperature is thus important since very high temperatures may predispose plants, livestock and human to heat related diseases. 1.6 Scope of the study Nairobi is located in the south-central part of Kenya at about 5,524 feet (1,684 meters) above the sea level. Its geographic coordinates are S latitude and E. Besides being the country s capital city, it is also the major industrial town with several factories that manufacture beverages and food staffs. It is also the most populated town in the country with the 5

19 highest number of vehicles. It is the busiest town in Central and East Africa acting as a major link to Mombasa port which is a gateway to various exports and imports. Major cooperates and international organizations operating in East Africa also have their headquarters in Nairobi. 1.7 Limitations of the study The availability of temperature data is difficult to come by in Kenya. The data was obtained at a high price yet such data are supposed to be available online for the public use. The project is restricted to the objectives of the research. The research work is characterized by some other setbacks like time constraints and the difficulties in obtaining relevant materials on the topic. 1.8 Outline This project is divided into five sections. Chapter one describes the background, objectives, justification, scope and the limitations of the study. Chapter two explores the past literature of similar studies that have been done in the past. In the third chapter the methodologies that are used in the data analysis are explained in details using their various mathematical equations. In chapter four the data is analyzed and the results presented in form of graphs and tables. In chapter five the conclusions and recommendations of the study are given. 6

20 CHAPTER 2 LITERATURE REVIEW 2.1 Introduction to General review One of the challenging subjects in climate research is Global warming and several studies have shown that during the 20 th century the average global air temperature has increased by C. A higher percentage of this increase has happened within the last three decades (Syeda, 2012). According to Oluwafemi et al (2010), climate change can influence all natural systems and thus threaten the human development and their social, political and economical survival. Temperature is among the key elements that affect the climatic system and it has been studied by a number of researchers in the recent decades. Forecasting of temperature assists stakeholders who directly depend on it to prepare in advance for any eventualities. Numerous hydro-meteorological applications use time series analysis and forecasting to study trends and variations in variables like temperature, humidity, rainfall, stream flow and many other environmental parameters (Nury et al, 2013). Time series analysis can be done using different approaches, however concentration is needed since the events are evolved through time and have a relation successively. Giving the thought about the future events by analyzing the lagged events is the main objective of the time series analysis. Nigar and Mehdi (2015) acknowledges that various methods are available for time series analysis of temperature, however one of the most effective and popular approach is Box-Jenkins ARIMA model. In their study to forecast minimum and maximum temperature in Bengal they apply the Seasonal ARIMA model. They described their process of model building and forecasting as follows: Differencing the series so as to achieve Stationarity; Identifying the 7

21 model to be tentatively entertained; estimating the parameters of the tentative models; Diagnostic Analysis, (the adequacy of the chosen model); Using the model for forecasting. Nigar and Mehdi tests for Stationarity using Augmented Dickey Fuller (ADF) and autocorrelation using the Ljung- Box test. They also used the chow breakpoint test to affirm the stability of their model and found out that their model was stable to forecast. Their outcomes indicated an increasing pattern of temperature in upcoming days which is a sign of increasing global warming in terms of temperature. Nury et al (2013) uses Seasonal ARIMA models to obtain short term forecasts of monthly minimum and maximum temperatures in the Moulvibazar and Sylhet districts in north-east Bangladesh. For the maximum and minimum temperatures at Sylhet station ARIMA (1,1,1) (1,1,1) 12 and ARIMA (1,1,1) (0,1,1) 12, respectively, are obtained, whereas ARIMA (1, 1, 0) (1, 1, 1) 12 and ARIMA (0, 1, 1) (1, 1, 1) 12 are the suitable models for Moulvibazar station. They concluded that the temperature time series fitted by the SARIMA model for the two stations can be used for estimating missing temperature values and for forecasting thus helping decision makers to establish better strategies and to set up priorities for arming themselves against upcoming weather changes which may have effects on the water resources in the Sylhet division. Syeda (2012) carries out an investigation in Bangladesh to study the variability and trend for seasonal and annual mean maximum temperature for six meteorological stations: Khulna, Dhaka, Sylhet, Rajshahi, Chittagong and Barisal. He applies univariate Box-Jenkin s ARIMA modelling technique to predict the average monthly maximum temperature from for these stations. In his study he encounters the problem of missing data and uses the median of the corresponding years to fill them. He tested the Stationarity of the residuals for average maximum temperature using ACF and PACF and checked for normality by normal probability plot. He 8

22 found out that the average maximum temperature is varying indicating that the climate of Bangladesh is gradually changing. He points out at the fact that these changes will impact the agricultural sector of the county and recommends that proper planning is required to sustain the development of the sector. Tanja (2010) analyzed and forecasted temperature changes in Belgium using an ARIMA model in order to show that it is a valuable tool in the study of climatic changes. He proves that in deed climatic change can be confirmed using mathematical theories but advises that his results must be treated as tentative because as we advance in time, the uncertainty about the predictions grows. Uchechukwu et al (2014) carry out a study to analyze the trend and forecast maximum monthly temperature changes in South Eastern Nigeria. Due to the seasonal variations in climatic data they opt to apply the full Seasonal ARIMA model. They found this SARIMA model to be a useful technique which can assist various decision makers to come up with better policies and to set up priorities for equipping themselves against future weather changes. Based on their best suited SARIMA model, the forecasted five years maximum temperature seems to be slightly stable from that of the reference period, with maximum values not exceeding the period s maximum. Their approach showed dependable results and they recommended that careful understudy of various mathematical models like the ARIMA could help track future rise in monthly temperature although for a relatively short time intervals. In a research to study temporal variation in temperature over Dibrugarh city in India, Tanusree and Kishore (2012) fit the traditional seasonal ARIMA model using Box-Jenkins approach. Using their most adequate SARIMA model, the forecasted temperature data showed reliable values compared to the actual recorded data. Their results support the effectiveness of the SARIMA model in predicting the average monthly temperature. 9

23 Stein and Lloret (2001) carry out a study to forecast temperatures of water and air for fishery purposes in Northwest Atlantic. They apply Seasonal ARIMA models that counted for 92% of the total variability in the monthly means of air temperature. Their forecasted values showed good agreement with the actual observed values of temperature. They concluded that for highly variable time series, ARIMA models yield better forecasts than the simple models which are only based on means of previous observations. Amirpooya et al (2011) uses time series SARIMA model to predict average monthly temperature in Ahwaz station. He found out that the average temperatures are rising over time indicating the fact that the globe is warming. 2.2 Kenyan Scenario A report by Conor on behalf of IPCC (2015) shows that over the last 50 years, the seasonal average temperature has been gradually increasing in many regions of Kenya. From this report, the rate of the temperature rise has been high during the recent decades. This indicates that the frequency of dry seasons is increasing. Lukoye et al (2010) conducted a research to determine the trend in mean maximum and mean minimum air temperature in Nairobi. They employed linear regression model on temperature data from 1966 to Their results showed that the mean annual minimum temperature in Nairobi has an increasing trend. Even though this study is on the same region a different model will be employed (SARIMA). This study also uses monthly minimum and maximum temperatures from 1985 to The conclusions will therefore be based on both minimum and maximum temperatures using the most current data. 10

24 2.3 Types of time series modeling methods a) Univariate modeling method: This is a type of modeling which generally uses only time as an input variable with no other outside explanatory variable. In univariate time series ARMA models from which the SARIMA models are derived, stationarity is a mandatory and sufficient condition. This is achieved through either seasonal or non seasonal differencing. b) Multivariate modeling method: Here, two or more variables are used to measure a person, place or thing. The variables may or may not be dependent on each other. 11

25 CHAPTER 3 METHODOLOGY 3.1 Data Overview Average Maximum and Minimum Monthly temperature data covering Nairobi region has been collected from the Kenya Meteorological Department (KMD). This data was recorded at Wilson Airport Weather Station in Nairobi and was available in monthly basis covering a 29 year period from January 1985 to October In this study the temperature data from Wilson Airport Weather Station is taken as a classical representative of Nairobi region. The temperatures are measured in degrees Celsius. The temperature data is a continuous univariate time series as it contains a single variable (temperature) which is measured at every instant of time. However, this data was merged into monthly intervals transforming it to a discrete univariate time series. 3.2 Time Series Definition This is a data set which describes a function or property that varies or changes over time. It is mathematically defined as a set of vectors x(t), t = 0, 1, 2... Where t represents the time elapsed. The variable x(t) is treated as a random variable. The observations taken during an event in a time series are organized in a proper chronological order (Ratnadip and Agrawal, 2013) Univariate time series contain measurements of a single variable while multivariate time series contain more than one variable. Time series can also be continuous or discrete depending on the instance when the measurements are taken. In discrete the measurements are taken at discrete (distinct) points of time, for example weekly, monthly, yearly etc while for continuous they are taken at every instant of time. 12

26 3.3 Time Series Components Time series can be affected by four main components: Trend, Cyclical, Seasonal and Irregular components. Trend is the series component that reflects its evolution in the long term. Cyclical component explains the periodic changes in the series. They are caused by circumstances which are not regular but repeat in cycles. Seasonal component describe the fluctuations within a year during the season while irregular component gathers the erratic fluctuations due to unforeseeable phenomenon and do not repeat in any particular pattern. 3.4 Methods The Box-Jenkins Method This study follows the Box-Jenkins methodology for modeling. The methodology involves four step processes which include: a) Model identification- In this step, the appropriate model structure of AR, MA or ARIMA and order of model is identified. Models can be identified by studying the plots of the ACF and PACF (Box and Jenkins 1976). b) Estimation of the model parameters - Non-linear least-squares estimation or Maximum likelihood estimation methods are employed to estimate the coefficients of the models. A more complicated iteration procedure is required when estimating the parameters of MA and ARMA models (Box and Jenkins, 1976; Chatfield, 2004). c) Diagnostic checking is important in ensuring the adequacy of the model. The residuals of the model have to be a white noise and that estimated parameters must also be statistically significant. According to Anderson (1977) misspecification can be identified by observing; the plots of the residual means and variance over time; plots of the 13

27 autocorrelation function and partial autocorrelation function of the residuals or performing a Box-Ljung test. d) Forecasting- Here the appropriate model is estimated to obtain the forecasted values. These four steps are used to form the conceptual framework of this study Conceptual Framework Conceptual framework is a scheme of concepts which the researcher operationalizes in order to achieve the set objectives. The following conceptual framework proposed by Box- Jenkins (1976) is considered in this study. Model Identification Estimation of Parameters Diagnostic Checking (Is the Model Adequate?) No Yes Figure 1: Box- Jenkins ARIMA Model Forecasting 14

28 3.5.0 Types of Models The Autoregressive Models An autoregressive model, AR (p), is a model in which a linear combination of previous measurements of the variable and a random error term with a constant term are used to forecasts the variable of interest. An autoregressive model of order p, AR (p) is mathematically represented as: (3.1) Where ε t is the source of randomness and is referred to as the white noise. are unknown parameters relating yt to yt 1, yt 2, yt p and must be estimated from sample data. Introducing a lag operator B to equation (3.1) it becomes (3.2) Moving Average Models A MA (q) model uses past errors to predict the variable of interest. The general algebraic representation of a moving average of order q, MA (q) is specified as: (3.3) ε, ε,... Where are the parameters of the model and t t 1 are white noise error terms. The residuals are assumed to follow a normal distribution. Thus a MA model is a linear regression of the current values of the time series against the residuals of one or more prior observations (Ratnadip and Agrawal, 2013). If a lag operator B is introduced to equation (3.3) it becomes (3.4) 15

29 3.5.3 Autoregressive Moving Average Models The AR model includes the lagged terms on the time series itself while the MA model includes lagged terms on the noise or residuals. If the AR and MA models are effectively combined together we form the ARMA model. Thus ARMA (p, q), where p is the autoregressive order and q the moving average order is generally defined as (3.5) It is important to note that the ARMA models can only be used when time series data is stationary Autoregressive Integrated Moving Average Models In practice, many time series are always non-stationary. ARMA models are therefore inadequate to effectively describe non-stationary time series which are more frequently encountered in actual practice. Box and Jenkins (1976) proposed the ARIMA model which is a generalization of an ARMA model to include the case of non- Stationarity. When using the ARIMA model, finite differencing is applied to the data to remove non- Stationarity. The non seasonal differencing is expressed as (3.6) The model is referred to as an ARIMA (p, d, q) and is represented algebraically as: ɛ t ~ ( 0, 2) WN s Where: i. WN stands for white noise. ii. p represents non-seasonal AR order, d represents non seasonal differencing and q represents non seasonal MA order, 16

30 Generally d 1is enough in most cases. If d 0, the model reduces to an ARMA (p, q) model. In case a series has both seasonal and non seasonal behaviors then the ARIMA model may mislead to the selection of a wrong order for non-seasonal component because it may not be able to capture the behavior along the seasonal part of the series Seasonal ARIMA Model Xier Li (2009) alludes that most natural factors like temperature have strong seasonal components. It is therefore necessary to use autoregressive and moving average polynomials that identify with the seasonal lags. One such model is the SARIMA model. SARIMA model is an extension of ARIMA model and it is applied when the series contains both seasonal and nonseasonal behavior. SARIMA model is sometimes called the multiplicative seasonal autoregressive integrated moving average and is denoted by SARIMA ( p, d, q)( P, D, Q) S The Seasonal AR can be written as: (3.8) Or (3.9) The Seasonal MA can be written as (3.10) Or (3.11) 17

31 The seasonal differencing is expressed as (3.12) Combining equations 2, 4, 5, 7, 9 and 10 we get SARIMA (3.13) Where the constant equals (3.14) Where; p represents non-seasonal AR order, d represents non seasonal differencing, q represents non seasonal MA order, P represents seasonal AR order, D represents seasonal differencing, Q represents seasonal MA order, S represents seasonal order (for monthly data S = 12 ) y t represents time series data at period t, k B is the backward shift operator ( y = y ) B t t- k and ɛ t is the random shock (white noise error). 18

32 In this model non- Stationarity is removed from the series using appropriate order of seasonal differencing. A first order seasonal difference is the difference between an observation and the corresponding observation from the previous year and is calculated as: = -. (For monthly time series, S=12). Z t yt yt- s Non-seasonal differencing is also necessary if trend is present in the data. Often a first nonseasonal difference will detrend the data (Penstate, 2012) Model Identification Stationarity Analysis Stationarity is achieved when a time series has a constant mean, variance and autocorrelation over time. Stationarity is a necessary and sufficient condition for ARIMA models before performing any analysis. Plotting the series and its autocorrelation is the standard way to check for non Stationarity. The time series graph can be examined through time to determine whether it has any trend or variability over time. For a non stationary series, the autocorrelation function decays slowly. For a series characterized by trend, seasonality or any other non stationary patterns, we analyze the series after differencing. In order to obtain a stationary data from a first order nonstationarity, we first sieve the observations with ARIMA models by differencing them d times, using Δ d y instead of y t as the time series. t 19

33 This is normally done with the transformation Δ t t - t = (3.15) y y y - 1 For the non-seasonal part the above equation results to the values d = 0,1, 2... and for the seasonal part D = 0,1,2.... Sometimes a series might need differencing more than once or to be differenced at lags which are greater than one period. In case of a second order non-stationarity, a simple transformation like the log transformation could be helpful Autocorrelation and Partial Autocorrelation Functions (ACF and PACF) When using the ARIMA models, Model specification and selection is a crucial step of the analysis process. A proper model for the series is identified by analyzing the ACF and PACF. They reflect how the observations in a time series are related to each other. It is useful that the ACF and PACF are plotted against consecutive time lags for the purposes of modeling and forecasting. The order of the AR and MA are determined by these plots. For a time series, the autocovariance function ACVF at lag k is defined as: c k n k 1 ( xt μ)( xt k μ) n (3.16) t 1 If x t is a stationary process with mean μ, the autocorrelation of order k is simply the relation between x t and x t-k. The ACF estimate for the sample at lag k is thus defined as k E ( x )( x E t t k ( x ) t 2 (3.17) 20

34 The PACF of a stationary process, x t, denoted hh is (3.18) And h 2 (3.19) and are not correlated with xt 1,..., xt h 1 The ACF and PACF plots are used to identify the terms of the SARIMA model. The nonseasonal terms are identified from the early lags 1, 2, 3,. Non-seasonal MA terms are indicated by spikes in the ACF at low lags while non-seasonal AR terms are indicated by spikes in the PACF at low lags. The seasonal terms are examined from the patterns across lags that are multiples of S. For monthly data, we look at lags 12, 24, 36, and so on (probably won t need to look at much more than the first two or three seasonal multiples). The ACF and PACF are judged at the seasonal lags in the same way it is done for the earlier lags. Initial terms of p, q, P and Q can be selected using the guide of table 1 and table 2 below. Since we are using estimates, in most cases we will not be sure whether the sample autocorrelation function or partial autocorrelation function is cutting off or tailing off (Shumway and Stoffer, 2006). Models that look different can also be very similar. Precision should therefore not be a major concern at this stage of model fitting. AR(p) MA(q) ARMA(p, q) ACF Tails off Cuts off after lag q Tails off PACF Cuts off after lag p Tails off Tails off Table 1: Behavior of the ACF and PACF for ARMA Models 21

35 AR(P)s MA(Q)s ARMA(P,Q)s ACF Tails off at lags ks, k=1, 2, Cuts off after lag Qs Tails off at lags ks PACF Cuts off after lag Ps Tails off at lags ks k=1, 2, Tails off at lags ks Table 2: Behavior of the ACF and PACF for Pure Seasonal ARMA Models Augmented Dickey Fuller test Stationarity can also be checked using ADF test. Its fundamental aim is to test the null hypothesis that ɸ =1 in: (3.20) Against the one-sided alternative ɸ <1 H 0: that the series has a unit root vs. H 1 : that the series is stationary Kwiatkowski Phillips Schmidt Shin test (KPSS) It tests the null hypothesis of trend-stationarity verses an alternative of a unit root Estimation of parameters Maximum likelihood or non-linear least-squares estimation methods can be used to estimate the coefficients of the models. According to Box and Jenkins, (1976) and Chatfield, (2004), the 22

36 estimation of parameters of a MA and an ARMA models normally requires a more complicated iteration procedure. Let X 1, X 2,, X n be drawn from a Gaussian ARMA (p, q) process with mean zero. Then the likelihood parameters are defined as the density of X= (X 1, X 2,, X n )' under the Gaussian model with those parameters: (3.21) Where A denotes the determinant of a matrix A, and Γ n is the variance/covariance matrix of X with the given parameter values. The maximum likelihood estimator identifies the values of parameters that increase the probability of obtaining fitted values that are close to the observed values Model selection criterion Even though the Autocorrelation and Partial Autocorrelation Functions help in determining the model s order, it only hints on where model building can begin from (Aidoo, 2010). Several models can therefore be considered from this case. However the final model is chosen using a penalty function statistics such as Akaike Information Criterion or Bayesian Information Criterion. Burnham & Anderson (1998) states that the motivation behind the selection criteria of the model is to identify the best model that neither under-fits nor over-fits the data Akaike's Information Criterion (AIC) The goodness of fit of estimated statistical models can be measured by the AIC. The competing models are ranked according to their values of AIC. The model which attains the lowest value of 23

37 information criterion is considered to be the best. The AIC evaluates a given model depending on the closeness of its fitted values to the observed values. It selects the simplest model that best explains the given data with minimum number of parameters and penalizes the complex model for having more model parameters. Penalizing the model with more parameters discourages over fitting. The AIC takes the form AIC 2k n log RSS n (3.22) Or AIC 2k 2log( L) (3.23) Where k= the number of parameters in the statistical model, (p+q+p+q+1) L= the maximized value of the likelihood function for the estimated model. RSS= the residual sum of squares of the estimated model. n= is the number of observation, or equivalently, the sample size In case two or more different models have the same AIC then according to the principle of parsimony the simplest model with fewer parameters is selected Model Diagnostic Here, each selected model is assessed to determine how well it fits the temperature data. For a model that fits the data well, the standardized residuals estimated from it should be independently and identically distributed with zero mean and constant variance. Such a sequence 24

38 is referred to as white noise. According to Chatfield (1996), the residuals of a well fitted model, should be randomly distributed. Several diagnostic statistics like normality QQ plots, standardized time residuals, ACF and PACF of the residuals are used in determining the goodness of fit of the selected model Box-Ljung Test The Box- Ljung test is defined as: H 0: the model does not exhibit lack of fit H 1 : the model exhibit lack of fit Given a time series Y of length n, the test statistic is defined as: (3.26) Q r k is the estimated autocorrelation of the series at lag k m is the number of lags being tested n is the sample size m k ( 2) r n n k 1 n 2 k The statistic follows a. For a level of significance α, the critical region for rejection of the hypothesis is where m is the degrees of freedom. 25

39 3.9 Forecasting Forecasting is important in decision making process. The chosen model should therefore produce accurate forecasts. The selected model does not always necessarily provide the best forecasting therefore it is important to apply other tests such as MAE, MSE and MAPE to confirm the forecasting accuracy of the model. Forecasting an ARMA process with mean, m-step-ahead forecasts can be defined as (3.27) The precision of the forecast is assessed with a prediction interval of the form (3.28) Where is identified such that the desired degree of confidence is achieved. Suppose it is Gaussian process, then having will yields approximately 95% prediction interval for Mean Absolute Error MAE is defined as (3.29) Where e t = X t - F t is the error term 26

40 X t is the actual observation for time period t, F t is the forecast value for period t and n is the number of forecasting values (Spyros et al., 1998) Mean Square Error MSE is defined as (3.30) Where e t = X t - F t is the error term and X t is the actual observation for time period t, F t is the forecast value for period t and n is the number of forecasting values (Spyros et al., 1998). 27

41 Temp (ºC) CHAPTER 4 DATA ANALYSIS The statistical software packages used the analysis is R and SPSS. 4.1 Time plots of temperature changes Time Figure 2: Time plot of Maximum and Minimum temperatures There is no evidence systematic variation about the mean on the time series plot for maximum temperature. 28

42 For minimum temperature there is a clear upward trend indicating that the series is non stationary. However both series exhibit seasonality which is evident from the strong yearly cycles. 4.2 Descriptive Analysis Summary of temperature records (January, October, 2014) Temperature ( C) Maximum Minimum Range Minimum Value Maximum Value Mean Statistic Standard Error Standard Deviation Variance Table 3: Summary statistics of maximum and minimum temperatures The highest maximum temperature was recorded in February 2006 while the lowest maximum temperature was recorded in July The lowest minimum temperature was recorded in August 1996 while the highest minimum temperature was recorded in April The maximum temperature is more varying (standard deviation=1.7632) compared to the minimum temperature (standard deviation=1.4192). The months of February and March recorded the highest mean maximum temperatures while the lowest mean maximum temperatures were recorded in June, July and August. The lowest mean minimum temperatures were recorded in July and August while the highest mean minimum temperatures were recorded in March and April. 29

43 Both maximum and minimum temperatures seem to be unstable throughout the year. However, from April both maximum and minimum temperatures drop significantly till July and start rising again till October. The lowest temperatures are experienced in June, July and August with July being the coldest month of the year. High temperatures are experienced in the months of February and March Decomposition of time series data Figure 3: Decomposition of Maximum temperature Ordinarily, time series data exhibit trend, seasonal, cyclical and random components. From figure 3 above and figure 4 below, it is evident that both maximum and minimum temperature 30

44 series have seasonal, random and trend components. The upward trend is clearly evident for the minimum temperature series. Figure 4: Decomposition of Minimum temperature 4.4 Stationarity checks using the ACF, PACF, KPSS and Dickey-Fuller Figures 5 and 6 below show the plots of the ACFs and the PACFs for the monthly maximum and minimum temperature series. The plots show strong seasonal wave patterns that decline moderately. The non seasonal lags decay rapidly. This confirms the presence of seasonality behavior and thus the time series is non stationary. 31

45 Therefore, from the time series and autocorrelation plots, it is obvious that both maximum and minimum temperature series have seasonal variation. To make the series stationary, seasonal differencing is required. Figure 5: Autocorrelation and Partial autocorrelation function of Maximum temperature 32

46 Figure 6: Autocorrelation and Partial autocorrelation function of Minimum temperature Formal tests of Stationarity are performed next to confirm the conclusions from visual inspection of seasonal and non-seasonal Stationarity. Unit root test Here we test: H 0 : temperature series has a unit root (there is non-stationarity and non-seasonality) H 1 : the series is stationary Augmented Dickey Fuller Test Dickey Fuller Lag Order P Value Maximum Temp Minimum Temp

47 Table 4: Augmented Dickey Fuller Test According to the ADF tests results for both maximum and minimum temperature series shown in table 4 above, we do not reject the null hypothesis and conclude that the two series are not stationary. This is because the more negative the Dickey Fuller is, the stronger the rejection of the null hypothesis which is not the case here. Table 5 below shows the test results of the KPSS tests. It tests the null hypothesis that a series is trend-stationary verses an alternative of non stationarity. It is important to note that any absence of a unit root in a KPSS test is not enough proof of general stationarity but of trend stationarity. From the results, for maximum temperature series we do not reject the null hypothesis because the p-value of at 5% level of significance. Thus the maximum temperature is trend stationary. For minimum temperature series we reject the null hypothesis that it is trend stationary since the p-value of at 5% level of significance and conclude that minimum temperature is non stationary. KPSS Test KPSS level Lag Parameter P Value Maximum Temp Minimum Temp Table 5: KPSS Test 34

48 The above results confirm that there is non-stationarity in the original temperature series and thus differencing is necessary for Stationarity to be achieved. 4.5 Seasonal differencing Figure 7 below shows the time series plot for maximum and minimum temperatures after taking a seasonal differencing. The pattern does not show any systematic trend, visible seasonality or predictable cycles. This indicates that the series are now stationary. Figure 7: Time series plot for seasonal differenced Maximum and Minimum temperatures 35

49 Figure 8: ACF and PAF for seasonal differenced Maximum temperature series Figure 9: ACF and PAF for seasonal differenced Minimum temperature series 36

50 ACF and PACF sequence of residuals is examined to know if stationarity has been achieved by differencing or by removal of the trend (Janacek and Swift, 1993). As the lag value increases, the autocorrelation functions for a stationary series converges to zero quite rapidly. Following this view, ACF and PACF plots shown in Figures 8 and 9 above are in support of monthly maximum and minimum temperature series being stationary after having a seasonal differencing. ADF and KPSS tests are carried out to confirm Stationarity of the two series. According to the ADF tests results for both maximum and minimum temperature series shown in table 6 below, the null hypothesis are rejected and thus the two time series data are stationary. Augmented Dickey Fuller Test Dickey-Fuller Lag Order P-Value Maximum Temp Minimum Temp Table 6: ADF test after seasonal differencing From the results of the KPSS test for both maximum and minimum temperature series shown in table 7 below, the null hypothesis is not rejected because the p values of at 5% level of significance. This confirms that both series have achieved trend Stationarity. 37

51 KPSS Test KPSS level Lag Parameter P-Value Maximum Temp Minimum Temp Table 7: KPSS test after seasonal differencing 4.6 Model building for monthly temperature series According to Takele (2012), the process of model fitting involves data plotting, data transformation if necessary, identification of dependence order, estimation of parameter, diagnostic analysis and choosing appropriate model. In this section, a univariate SARIMA methodology is used to model maximum and minimum monthly temperatures of Nairobi. 4.7 Model Identification ACF and PACF plots are used in the identification of the values p, q, P and Q. For the nonseasonal part, spikes of the ACF at low lags are used to identify the value of q while the value of p is identified by observing the spikes at low lags of the PACF. For the seasonal part the value of Q is observed from the ACF at lags that are multiples of S while for P, the PACF is observed at lags that are multiples of S. Looking at the ACF plots and PACF plots for maximum and minimum differenced time series, the following models are suggested; 38

52 Suggested models for Maximum temperature SARIMA (0, 0, 2) (1, 1, 0) 12 SARIMA (0, 0, 3) (0, 1, 1) 12 SARIMA (0, 0, 2) (0, 1, 1) 12 SARIMA (1, 0, 2) (0, 1, 1) 12 SARIMA (0, 0, 3) (1, 1, 0) 12 SARIMA (1, 0, 3) (0, 1, 1) 12 Suggested models for Minimum temperature SARIMA (1, 0, 2) (0, 1, 1) 12 SARIMA (1, 0, 0) (0, 1, 1) 12 SARIMA (2, 0, 1) (0, 1, 1) 12 SARIMA (2, 0, 0) (0, 1, 1) 12 SARIMA (1, 0, 1) (0, 1, 1) 12 SARIMA (0, 0, 1) (1, 1, 0) 12 39

53 4.8 Statistics for tentative models Maximum temperature Model p-value Chi-square Df AIC SARIMA (0, 0, 2) (1, 1, 0) SARIMA (0, 0, 3) (0, 1, 1) SARIMA (0, 0, 2) (0, 1, 1) SARIMA (1, 0, 2) (0, 1, 1) SARIMA (0, 0, 3) (1, 1, 0) SARIMA (1, 0, 3) (0, 1, 1) , Table 8: Statistics for tentative Maximum time series models The AIC evaluates a given model depending on the closeness of its fitted values to the observed values. It selects the simplest model that best fits the given data using the minimum number of parameters and penalizes the complex model for having more model parameters. Penalizing the model with more parameters discourages over fitting. The best model is the one with the lowest value of AIC. The best model for maximum temperature is SARIMA (0, 0, 2) (0, 1, 2) 12 while for minimum temperature is SARIMA (1, 0, 2) (0, 1, 2)

54 Minimum temperature Model p-value Chi-square Df AIC SARIMA (1, 0, 2) (0, 1, 1) SARIMA (1, 0, 0) (0, 1, 1) SARIMA (2, 0, 1) (0, 1, 1) SARIMA (2, 0, 0) (0, 1, 1) SARIMA (1, 0, 1) (0, 1, 1) SARIMA (0, 0, 1) (1, 1, 0) Table 9: Statistics for tentative Minimum time series models 4.9 Parameter Estimation Maximum temperature time series-sarima (0, 0, 2) (0, 1, 1) 12 Parameter Estimate Standard error ma ma sma Table 10: Parameter estimates for SARIMA (0, 0, 2) (0, 1, 1) 12 41

55 SARIMA (0, 0, 2) (0, 1, 1) 12 has an estimated variance of with a log likelihood of Its AIC was The Box Ljung test yielded a chi square of with a p value equal to From Box Ljung test, the p value of > 0.05 and this confirms that SARIMA (0, 0, 2) (0, 1, 1) is adequate for forecasting. Minimum temperature time series-sarima (1, 0, 0) (0, 1, 1) 12 Parameter Estimate Standard error ar sma Table 11: Parameter estimates for SARIMA (1, 0, 0) (0, 1, 1) 12 SARIMA (1, 0, 0) (0, 1, 1) 12 has an estimated variance of with a log likelihood of Its AIC was The Box Ljung test yielded a chi square of with a p value of From the Box Ljung test, the p value of > 0.05 and this confirms that SARIMA (1, 0, 0) (0, 1, 1) is adequate for forecasting Diagnostic Analysis For a well fitted model, the standardized residuals estimated from the model should behave as an independent identically distributed sequence with zero mean and constant variance. 42

56 4.1.1 Diagnostic analysis for Maximum temperature model A normal probability plot or a Q-Q plot can help in identifying departures from normality. From figure 10 below the residuals are approximately normal distributed with zero mean. Figure 10: Residuals of Maximum Temperature Model Figure 11 shows the value of the Q-statistic at lag 1 through 10, the plots of ACF of the residuals and the standardized residuals. 43

57 p value ACF Standardized Residuals Time ACF of Residuals Lag p values for Ljung-Box statistic lag Figure 11: Residual Plots of maximum temperature model From the time plot of the residuals against time, we can see that there is no obvious pattern in the plot except for a possible outlier, and it looks like an independently and identically distributed sequence of mean zero with a constant variance. The plots of the ACF of the residuals lack enough evidence of significant spikes which clearly shows that the residuals are white noise. The results also showed that the residuals are non - significant with Box Ljung test value of and a p value of

58 From the above tests, it is clear that the fitted model is adequate since the residuals are white noise. That is, SARIMA (0, 0, 2) x (0, 1, 1) 12 is adequate for modeling the log transformed monthly maximum temperature series in Nairobi Diagnostic analysis for Minimum temperature model From Figure 12 below the residuals are approximately normal distributed with zero mean. Figure 12: Residuals of Minimum Temperature Model Figure 13 shows the value of the Q-statistic at lag 1 through 10, the plots of ACF of the residuals and the standardized residuals. 45

59 Figure 13: Residual Plots of minimum temperature model From the time plot of the residuals against time, we can see that there is no obvious pattern in the plot except for a possible outlier, and it looks like an independently and identically distributed sequence of mean zero with a constant variance. The plots of the ACF of the residuals lack enough evidence of significant spikes which clearly shows that the residuals are white noise. The results also showed that the residuals are non - significant with Box Ljung test statistic of and a p-value of

60 From the above tests, it is clear that the fitted model is adequate since the residuals are white noise. That is, SARIMA (1, 0, 0) (0, 1, 1) 12 is adequate for modeling the log transformed monthly minimum temperature series in Nairobi Model validation In order to test the adequacy and predictive ability of the chosen models, the actual data sets, predicted values, lower and upper limits are plotted and displayed in Figure 14 and Figure 15 below. The graphs show that the predicted values are well-fitted through the original data with the lower and upper limits containing majorities of the original data. This indicates that the models chosen for maximum and minimum temperature series are the best fitted ones for the data sets. Figure 14: Observed and fitted values of Maximum temperature series 47

61 Figure 15: Observed and fitted values of Minimum temperature series Table 12 and Table 13 below also shows the observed values verses the predicted values that affirms the adequacy of the chosen maximum and minimum time series models Table 12: Observed and fitted values of Maximum temperature series ( ) Year Observed Forecast LCL UCL Noise residual 2005M M M M M M M M M M M M M M M M M