FORECASTING OF THE MEAN MONTHLY RAINFALL OF GUSAU, ZAMFARA STATE FROM (1994-2015) USING TIME SERIES ANALYSIS

A PROJECT SUBMITTED TO THE DEPARTMENT OF MATHEMATICS, STATISTICS UNIT, FACULTY OF SCIENCE, USMANU DANFODIYO UNIVERSITY, SOKOTO

IN PARTIAL FULFILMENT FOR THE REQUIREMENT FOR THE AWARD OF BACHELOR OF SCIENCE, B.sc 
(Hons) DEGREE IN STATISTICS





OCTOBER, 2016


CERTIFICATION
This is to certify that the research work for the project and the subsequent preparation of this report by Mubarak Hassan Augie ADM.NO 1110306948 in the department of mathematics is fully adequate in scope and quality as a research for the degree of Bachelor of Science in statistics, Usmanu Danfodiyo University Sokoto.

   
 DR. YAKUBU MUSA                                                             Date
 Project supervisor




DR. A. DANBABA                             Date
Head of statistics unit




DR. UMAR USMAN                                                                 Date
Project coordinator                                                                   



DEDICATION
This work is dedicated to my parents Dr. Hassan Abubakar Augie, my Mother and my entire family in general.

















TABLE OF CONTENTS
Title page …………………………………………………………………………………………i
Certification ii
Dedication iii
Acknowledgements iv
Table of Contents v
LIST OF FIGURES…………………………………………………………………………………..VI
LIST OF TABLES……………………………………………………………………………………VII
abbreviations……………………………………………………………………………………vii
Abstract ..x
CHAPTER ONE: GENERAL INTRODUCTION
1.0 INTRODUCTION………………………………………………………………………………... 1
BACKGROUND OF THE STIUDY AREA…………………………………………………….4
AIM AND OBJECTIVE OF RESEARCH………………………………………………………5
STATEMENT OF THE PROBLEM…………………………………………………………….5
SCOPE AND LIMITATION OF THE STUDY…………………………………………………6
SIGNIFICANCE OF THE STUDY………………………………………………………………6
BASIC DEFINITIONS OF SOME TERMS……………………………………………………..6
CHAPTER TWO: LITERATURE REVIEW
2.0 INTRODUCTION…………………………………………………………………………………1
2.1SOME REVIEWED RESEARCH WORK ON RAINFALL……………………………………1

CHAPTER THREE: METHODOLOGY
3.0 INTRODUCTION………………………………………………………………….………...…….14
3.1 TIME SERIES MODELS……………………………………………………………….….….…..14
WHITE NOISE PROCESS……………………………………………………….…..……...14
AUTO REGRESSIVE PROCESS AR( p)…………………………………………...…..….14
MOVING AVERAGE PROCESS MA( q)……………………………………………...…..15
ARMA (p, q) PROCESS …………………………………………………………...…….…..15
ARIMA (p, d, q) PROCESS……………………………………………………………..…...16
SARIMA (p, d, q) (P, D, Q) PROCESS………………………………………………...……16
     STATIONARITY IN BOX – JENKINS MODEL…………………………………..………17
3.3         TYPES OF NON-STATIONARITY.……………………………………………………….. 18
3.3.1       NON-STATIONARITY IN THE MEAN………………………………………………..….18
3.3.2       NON STATIONARITY IN THE VARIANCE…………………………………………...…19
3.3.3       NON-STATIONARITY WITH A PERIODIC OR SEASONAL COMPONENT…….….19
3.4          MODELLING TECHNIQUE……………………………….………………………….…….20
3.4.1 MODEL IDENTIFICATION…………………………………………………………….…...20
3.4.2       UNIT ROOT TEST……………………………………………………………………………21
3.4.3       INFORMATION CRITERIA AND MODEL SELECTION…………………………...…..23
3.4.4       MODEL ESTIMATION………………………………………………………………………25
3.4.5       MODEL DIAGNOSTIC CHECKING……………………………….....................................26
4.4         FORECASTING………………………………………………………………………………..28
CHAPTER FOUR: DATA ANALYSIS
4.0 INTRODUCTION……………………………………………………………………………..31
4.1 THE DATA FOR THE ANALYSIS………………………………………………………….31
4.2         TIME SERIES PLOTS…………………………………………………………………………...32
4.2.1      TIME SERIES PLOT BEFORE DIFFERENCING ……………………………………………32
4.2.2 THE PLOT OF ACF AND PACF OF THE DATA BEFORE DIFFERENCING………........33 
4.2.3 TIME SERIES PLOT AFTER TAKING THE LOG…………………………………..……....34
4.2.4 UNIT ROOT TEST BEFORE DIFFERENCING……………………………………..…...…...35
4.2.5 TIME SERIES PLOT AFTER FIRST DIFFERENCE ……...…………………………......….36
4.2.6 THE PLOT OF ACF AND PACF OF THE DATA AFTER FIRST DIFFERENCE….….….37
4.2.7 UNIT ROOT TEST AFTER DIFFERENCING…………………………………………….......38
4.2.8 SEASONAL UNIT ROOT TEST BEFORE SEASONAL DIFFERENCING…………….….39
4.2.9      TIME SERIES PLOT AFTER SEASONAL DIFFERENCING ………………………………32
4.2.10 THE PLOT OF ACF AND PACF AFTER SEASONAL DIFFERENCING…….....................41
4.2.8 SEASONAL UNIT ROOT TEST AFTER SEASONAL DIFFERENCING……………….….42
4.3 MODEL IDENTIFICATION………………………………………………….……………....….42
4.4 MODEL ESTIMATION…………………………..……………………………………………....43
4.5 MODEL DIAGONISTIC CHECKING….…………………………………………………...…..45
4.5.1 LJUNG – BOX TEST……………………………………………………………………………...50
4.5.2 JAQUE BERA TEST……………………………………………………………………………....51
4.5.3 THE TEST STATISTIC FOR NORMALITY…………………………………………..….……53
4.6 FORECAST………………………………………………………………………………….……..56
4.7 FORECAST EVALUATION ……...……………………………………………………………..56
CHAPTER FIVE: SUMMARY, CONCLUSION AND RECOMMEDATION
5.0 SUMMARY…………………………………………………………………………….……63
5.1 CONCLUSION………………………………………………………………………….…..63
5.2 RECOMMENDATION………………………………………………………………….….64
REFERENCE…………………………………………………………………….....65
APPENDIX









LIST OF FIGURES
Fig 4.1: Time series plot before differencing
Fig 4.2: ACF and PACF plot before differencing.
Fig 4.3:            Time series plot after taking the log
Fig 4.4:            Time series plot after first difference
Fig 4.5:            ACF and PACF after first difference
Fig 4.6:            Time series plot after seasonal differencing
Fig 4.7:            ACF and PACF after seasonal differencing
Fig 4.8:            The ACF and PACF of SARIMA (1, 1, 2) (1, 1, 1)12
Fig 4.9:            The ACF and PACF of SARIMA (1, 1, 3) (1, 1, 1)12
Fig 4.10:          The ACF and PACF of SARIMA (2, 1, 1) (1, 1, 1)12
Fig 4.11:          Time series plot of SARIMA (1, 1, 2) (1, 1, 1)12
Fig 4.12: Time series plot of SARIMA (1, 1, 3) (1, 1, 1)12
Fig 4.13: Time series plot of SARIMA (2, 1, 1) (1, 1, 1)12
Fig 4.14: Test statistic for normality of SARIMA (1, 1, 2) (1, 1, 1)12
Fig 4.15: Test statistic for normality of SARIMA (1, 1, 3) (1, 1, 1)12
Fig 4.16: Test statistic for normality of SARIMA (2, 1, 1) (1, 1, 1)12
Fig 4.17 Forecast plot of rainfall for SARIMA (1, 1, 2) (1, 1, 1)12
Fig 4.18 Forecast plot of rainfall for SARIMA (1, 1, 3) (1, 1, 1)12
Fig 4.19 Forecast plot of rainfall for SARIMA (2, 1, 1) (1, 1, 1)12


LIST OF TABLES
Table 4.1:          Result of Philips - Perron (pp) Test
Table 4.2:          Result of KPSS test
Table 4.3:          Result of Philips - Perron (pp) Test
Table 4.4:          Result of KPSS test
Table 4.5:          Result of Hegy test before seasonal differencing
Table 4.6:          Result of Hegy test after seasonal differencing
Table 4.7:          Results of SARIMA model identification for rainfall
Table 4.8:          Results of SARIMA (1, 1, 2) (1, 1, 1)12 parameter estimates
Table 4.9:          Results of SARIMA (1, 1, 3) (1, 1, 1)12 parameter estimates
Table 4.10:        Results of SARIMA (2, 1, 1) (1, 1, 1)12 parameter estimates
Table 4.11:        Result of ljung Box Test
Table 4.12:        Result of Jaque-Bera Test
Table 4.13:        Results of SARIMA (1, 1, 2) (1, 1, 1)12 Forecast Values for 2 years
Table 4.14:       Results of SARIMA (1, 1, 3) (1, 1, 1)12 Forecast Values for 2 years
Table 4.15:       Results of SARIMA (2, 1, 1) (1, 1, 1)12 Forecast Values for 2 years
Table 4.16:       Results of forecast evaluation

ABBREVIATIONS
ACF                         Autocorrelation Function
PACF                      Partial Autocorrelation Function
AIC                         Akaike Information Criteria
SIC                         Schwert Information Criteria
ADF                        Augumented Dickey Fuller
KPSS                      Kwiatkowski Philips Schmidt Shin
ARMA                    Auto Regressive Moving Average Process
ARIMA                   Autoregressive Integrated Moving Average
SARIMA                Seasonal Autoregressive Integrated Moving Average








ABSTRACT

Time series analysis and forecasting has become a major tool in different applications in meteorological phenomena such as rainfall, humidity, temperature, draught and so on. Among the most effective approaches for analyzing time series data is the SARIMA (Seasonal Autoregressive Integrated Moving Average) model introduced by Box and Jenkins (1976). In this study, Box-Jenkins methodology was used to model monthly rainfall data taken from Nigerian meteorological Agency Gusau, Zamfara State for the period from 1994 to 2015 with a total of 264 readings. SARIMA (1, 1, 3)(1, 1, 1)12 model was developed. This model was used to forecast monthly rainfall for the upcoming 24 months (2 years) to help decision makers establish priorities in terms of water demand management and agriculture. Thus, SARIMA (1, 1,3)(1,1,1)  provides a good fit for the rainfall data of Gusau metropolis and is appropriate for short term forecast.








 CHAPTER ONE
INTRODUTION
Rainfall is a determinant factor of many natural occurrences. Vegetation distribution and types over land masses are as a result of rainfall. Animals breeding period synchronize with rainfall period. Rainfall events have been directly linked to sickness and diseases particularly those of waterborne and vector-borne types. Crop planting, yields and harvest are influenced by rainfall. Investments in agricultural produce and products are expected to be done in accordance with the knowledge of rainfall and other weather conditions.
 Naturally, rainfall variability is of spatial and temporal forms and within these variations if by time series analysis no significant trend is obtained then the rainfall is steady, otherwise, it has changed (Janhabi & Ramakar,  2013).
With regards to rainfall, studies have pointed to the fact that the climate is changing (Goswami et al, 2006). Specifically, the research of (Ragab & Prudhomme, 2002) discusses the variability and uncertainty of rainfall across the whole globe amidst global warming and among others it states that while North Africa witnesses rainfall decrease, the regime of rainfall in some parts of South Africa is increasing. On the other hand, no coherent trend has been recorded in some other regions, in the face of different significant trends of other regions in Sri Lanka (Jayawardene et al, 2005). However, (Odjugo, 2010) argues that not any change in the climate should be considered as climate change since climate fluctuates. And in his overview of climate change in Nigeria, he established that rainfall has decreased by 81mm within 105 years period. Thus, not to mistake climate fluctuation for climate change (Nsikan et al, 2011).

 The growth of population demands for increased domestic water supplies and at the same time results in higher consumption of water due to expansion in agriculture and industry. Mismanagement and lack of knowledge about existing water resources and changing climatic conditions have consequences of an imbalance of supply and demand of water. The problem is pronounced in semi-arid and arid areas where these resources are limited.
 Generally, the study of weather and climatic elements of a region is vital for sustainable development of agriculture and planning. Particularly, rainfall and temperature temporal analyses for trends, fluctuations and periodicities are deemed necessary as such can indirectly furnish with “health” status of an environment. (Afangideh et al, 2010).

Before the modeling of the rainfall forecast, preliminary statistical tests were carried out on the time series to test the presence of trend and stationarity in it. A simple linear regression analysis may provide a primary indication of the presence of trend in the time-series data. The Box-Jenkins method is univariate time series analysis. It is thus essential to analyse the presence of the unit root in the time series. The stationarity of the time series can be tested with phillips perron Test. This test is popularly used to check the presence of unit root in the time series (Mahalakshmi et al, 2014).
In a country like India where around 70% of the total population is dependent on agriculture directly or indirectly, prediction of rainfall plays a very major role. The Artificial Neural Network (ANN) based on the time series analysis was used to predict the summer monsoon rainfall. (Akashdeep et al, 2013).
Rainfall is one of the key climatic resources of Zamfara state (Zamfara State, Nigeria). Crops and animals derived their water resources largely from rainfall. It is considered as the main determinant of the types of crops that can be grown in the area and also the period of cultivation of such crops and the farming systems that can be practiced. Nigeria is a country with diverse ethnic groups practicing different cultures. And having variations in farming and religious beliefs. The country can primarily be divided into two major geographical zones namely the north and south, though that has historical underlying (obot et al, 2010). Furthermore, both the north and south are segmented into three regions each, making a total of six geopolitical zones. These six geopolitical zones in the country are: North east, North West, North central, South West, South East and South South. Policies, resources allocations sites of infrastructures and even political and other appointments are mostly considered by zoning in Nigeria. Gusau is in Zamfara state which is in North West geopolitical zone in Nigeria. Zamfara state came into being in 1 October 1996. It was created out of the old Sokoto State. The capital is Gusau. Its major towns include Gusau, Mafara, and Maru. The state has a population of 3,838,160 and has a total land area of 39,762 km2. It is bordered in the North by Niger republic, to the South by Kaduna State. In the east it is bordered by Katsina State and to the West by Sokoto and Niger States. The climate is semi-arid with a zone of savannah-type vegetation as part of the sub-Saharan Sudan belt of West Africa. Rainfall is concentrated in a short wet season, which extends from mid-May to mid-September whilst the dry season (with no single rain) last more than 7 months. Farming is the major occupation in the place. It is well known that crop production in semi-arid regions is largely determined by climatic and soil factors. The amount of rainfall are among the most important factors that affect agricultural systems. Rainfall is the limiting factor in the area. Rainfall governs the crop yields and determines the choice of the crops that can grow. These show that a detailed knowledge of rainfall regime is an important prerequisite for agricultural planning. The analysis of rainfall for agricultural purposes must include information concerning the trends or changes of precipitation, the start end length of the rainy season, the distribution of rainfall amounts through the year, and the risk of dry and wet spells. The purpose of this study is to characterize trend of total amount of rainfall in Gusau, Zamfara State, Nigeria.
BACKGROUND OF THE STUDY AREA
Geographically, Gusau metropolis lies on the geographical coordinates of 12°09′ North, 6°40′ East. The climate is largely influenced by the interactions between two air masses: the tropical maritime air mass from the Atlantic Ocean and the tropical continental air mass from Sahara Desert. The climate is characterized by rapid changes in temperature and humidity. Temperatures are generally high in the months of March-May, with highest recorded mean monthly temperature of about 40°C in April (Ayoade, 2004). The lowest temperatures occur in the months of December and January due to the influence of harmatan. Seasonal rainfall prediction by NIMET (2013) shows that the northern Nigeria region will have a range of mean annual rainfall that is between 300mm-1200m. The relative humidity is very high during rainy season reaching about 65 to 70%. The vegetation of the study area falls under the Sudan savannah type of classification, with the presence of grasses and shrubs less than one meter in height. The forest vegetation in some parts of the State comprises neem (Dogonyaro) and baobab trees (Kuka) (Adefolalu, 1990). Most part of the state lies within the Sudan Savanna zone where trees and short grass species of less than one meter co-dominate (Davis, 1982 p12). The study area lies on a sedimentary formation which is made up of unconsolidated sediments believed to have been formed or deposited in a syncline during the Cretaceous and Tertiary era (Davis, 1982). The relief of Zamfara State is generally plain though interrupted by spot of plateau, sandstone with resistant layer of lateritic material in some parts of the state. The Zamfara plain on the one hand form monotonous lowland derived from Softer sedimentary rock with an average height of 300m compared with 700m (Davis, 1982). The study area is characterized by loess and sandy soils.
AIM AND OBJECTIVES OF THE STUDY
The aim of this research work is to develop a SARIMA model for the mean monthly rainfall of Gusau metropolis from January 1994 to December 2015.The model will then be used to provide forecast for the subsequent years.
The objectives of this research work are:
To identify the nature of time plot of monthly rainfall of Gusau metropolis from January 1994 to December 2015.
To identify the correlogram of the data series through the use of Autocorrelation function (ACF) and partial Autocorrelation function (PACF) of the data series.
To test whether or not there is need for log transformation of the data series.
To detect whether or not there exist seasonality in the data.
To develop a SARIMA model that will be used to explain the time dependent structure of rainfall patterns of Gusau metropolis.
To forecast the average monthly rainfall of Gusau metropolis.
STATEMENT OF THE PROBLEM
Rainfall being a largely important resource to the crops and animals. It is considered as the main determinant of the types of crops that can be grown. Consequently, the shortage of rainfall lead to have devastating effects on communities and the surrounding environment. The amount of devastation depends on the strength of the drought and the length of time and area is considered to be in drought conditions. However higher rate of rainfall to the society lead to flooding which have numerous social, economic and environmental effects that can vary depending on the demographics of a population and the economic development of an area.
So with this problem at hand, the study intended to analyze and model the average monthly rainfall in Gusau metropolis in order to know how the average rainfall behave.
1.4 SCOPE AND LIMITATION OF THE STUDY
This project work/research is restricted to Nigeria Meteorological Agency in Zamfara State. The study covers a period of twenty two (22) years (1994-2015). It takes into consideration of the monthly amount of rainfall in Gusau city. As already stated in the purpose of study, this research was specifically carried out to analyses the monthly amount of rainfall in Gusau City, however the work is only limited to twenty two (22) year’s figures (1994-2015).
1.5 SIGNIFICANCE OF THE STUDY
The time series analysis will help us in understanding the past behavior of the amount of monthly rainfall in Gusau, Zamfara State and also the direction of fluctuation.
 The knowledge of the behavior of the amount of rainfall will enables us iron intra-year variation and
Study of past amount of rainfall will enhance the prediction of the feature rainfall in Gusau.
 BASIC DEFINITIONS OF SOME TERMS
Time series: - Time series can be defined as a sequence of observations ordered by time. In other words time series can be defined as a set of observations taken at specific times, usually at equal intervals.
Forecast: - Forecasting is the process of estimating future values of numerical parameters on the basis of the past. To do this, a model is created. This model is an artificial equation that captures the important features of the data. Forecasting is what the whole procedure is designed to accomplish. Once the model has been selected, estimated and checked, it is usually a straight forward task to compute forecasts. Of course, this is done by computer.
Stationary time series: - A stationary time series is a time series whose statistics do not change over time in the mean (no trend), variance and periodic (seasonal) variation of the series.
Non- stationary time series: - A time series is said to be non- stationary time series, if it does not have a constant mean and variance. In other words non- stationary time series is one that contains trend and seasonal variations.
Moving average: - According to Murray R. Speigel and Larry J. Stephen (1999), a moving average for time period is simply the arithmetic average of the values in that defined period.
Trend: - Trend is defined as long term change in the mean, i.e. it is a smooth upward and downward increment of a time series over a long period of time. 
Differencing: - This is defined as a mathematical technique useful for removing trend in time series. I.e. the series of changes from one period to the next.
De-seasonalised time series: - This is defined as a time series that has had the effect of season removed by dividing each original time series observation by corresponding seasonal factor.
The signal: - This is the component of the data that contains information, Say {Sn} this is the component of the time series that can be forecast.
The noise: - This is the randomness that is observed, which may be Due to numerous other variables affecting the signal, measurement Imperfections, etc. Because the noise is random, it cannot be forecast.
Backshift Operator: - A tool that enables complicated time series model to be written in a simple form, and also allows the models to be manipulated.
Auto covariance function: - This is second mixed central moment of a stochastic process, the function measures the similarity between two points on the same series observed at different times.
Variance Function: - This is the central moment of a stochastic process, the function measures the degree to which the process is spread out over the real line.
Model selection in the Box-Jenkins framework uses various graphs based on the transformed and differenced data to try to identify potential SARIMA processes which might provide a good fit to the data. Later developments have led to other model selection tools such as Akaike’s Information Criterion.
Parameter estimation means finding the values of the model coefficients which provide the best fit to the data. There are sophisticated computational algorithms designed to do this.
 Model checking involves testing the assumptions of the model to identify any areas where the model is inadequate. If the model is found to be inadequate, it is necessary to go back to Step 2 and try to identify a better model.







CHAPTER TWO: LITERATURE REVIEW
2.0 INTRODUCTION
Time series analysis refers to problems in which observations are collected at regular time intervals and there are correlations among successive observations. Applications cover virtually all areas of Statistics but some of the most important include economic and financial time series, and many areas of environmental or ecological data. In this chapter is going to focus on some review research work on rainfall.

2.1 SOME REVIEWED RESEARCH WORK ON RAINFALL
Several studies have taken place in the analysis of pattern and distribution of rainfall in various regions of the world. Different time series methods with different objectives are employed to analyze rainfall data in various literatures. In terms of using a formal time series model to forecast, the patterns and intensity of rainfall overtime, Adejuwon (2010) studied annual rainfall in Nigeria using power spectral analysis based on Benin, Sapele, Warri and Forcados Synoptic station in Edo and Delta States (formerly Mid-Western Nigeria) over 67 years and found that Benin synoptic station shows significant spectral peaks at 6.7, 4.6 and 3.7 years periodicities. The most pronounced peak at the station was found to be 3.7 years periodicity. In Sapele, the most pronounced periodicity of 5 years was observed. Although, the spectral peaks were significant at 4.6 and 3.7 years, respectively, at Warri, the most pronounced of these peaks was found to be 3.7 years. However, in the case of Forcados, a single significant spectral peak of 3.6 years cycle was prominent and it was then concluded that periodicities were evident with significant cycles of between 3 and 6 years.
Edwin and Martins (2014) examined the stochastic characteristics of the Ilorin monthly rainfall in Nigeria using four different modeling techniques (Decomposition, Square root transformation-deseasonalisation, Composite and Periodic Autoregressive) where they compared the results from the various methods employed.
Mahsin et al. (2012) use Box-Jenkins methodology to build seasonal ARIMA model for monthly rainfall data taken for Dhaka station, Bangladesh, for the period from 1981-2010. In their paper, ARIMA (0, 0, 1) (0, 1, 1)12 model was found adequate and the model is used for forecasting the monthly rainfall. Seyed et al.,(2011) use time series method to model weather parameter in Iran at Abadeh Station and recommended ARIMA(0,0,1)(1,1,1)12 as the best fit for monthly rainfall data and ARIMA(2,1,0)(2,1,0)12 for monthly average temperature for Abadeh station.
Amha (2010) studied the monthly rainfall in Tigray region based on Mekelle station. He employed univariate Box-Jenkins method to analyze rainfall in the region and found that SARIMA model is suitable for forecasting future value of monthly rainfall data and used this model to forecast 12-month rainfall pattern in the study area. Further he concluded that there is no tendency of decreasing or increasing pattern of monthly rainfall over the forecast period from January 2010 to September 2011.
A study made by Mersha (2002) on rainfall cyclicity over selected stations in Ethiopia also shows that there appears to be cyclic tendency in the annual rainfall data, particularly, Gode, Dire Dawa, Negelle, and Debre Zeit station. Seifu (2004) also show that the rainfall at Dire-Dawa, D/Markos and Jijiga has periodic tendency.
Alamerew and Eshetu (2009) assess local climate of Addis Ababa. They used time domain approach like ARIMA for modeling mean minimum temperature and frequency domain time series approach, particularly, spectral analysis for rainfall data. Using spectral analysis they determined periodicity of drought in Addis Ababa and found that the periodicity of 11.24 year is dominant cycle for the annual rainfall of Addis Ababa. Based on their result, they concluded that drought recurs in Addis Ababa region between 10 to 11 years.
Shamsuddin Shahid (2009) has analyzed Rainfall variability and the trends of wet and dry periods in Bangladesh over the time period 1958–2007 has been assessed using rainfall data recorded at 17 stations distributed over the country. The result shows a significant increase in the average annual and pre-monsoon rainfall of Bangladesh. The number of wet months is found to increase and the dry months to decrease in most parts of the country. Seasonal analysis of wet and dry months shows a significant decrease of dry months in monsoon and pre-monsoon
Al-Ansari et al. (2003) dealt with the statistical analysis of the rainfall measurements for three meteorological stations in Jordan: Amman Airport (central Jordan), Irbid (northern Jordan) and Mafraq (eastern Jordan). Normal statistical and power spectrum analyses as well as ARIMA model were performed on the long-term annual rainfall measurements at the three stations. The result shows that possible periodicities of the order of 2.3 - 3.45, 2.5 - 3.4 and 2.44-4.1 years for Amman, Irbid and Mafraq stations, respectively, were obtained. A time series model for each station was adjusted, processed, diagnostically checked and lastly an ARIMA model for each station is established with a 95% confidence interval and the model was used to forecast 5 years annual rainfall values for Amman, Irbid and Mafraq meteorological stations. Further result indicated that there is decreasing trend for forecasted rainfall results in all stations.
Alamerew and Eshetu (2009) assess local climate of Addis Ababa. They used time domain approach like ARIMA for modeling mean minimum temperature and frequency domain time series approach ,particularly, spectral analysis for rainfall data. Using spectral analysis they determined periodicity of drought in Addis Ababa and found that the periodicity of 11.24 year is dominant cycle for the annual rainfall of Addis Ababa. Based on their result, they concluded that drought recurs in Addis Ababa region between 10 to 11 years.
Nguyen and Pandey (1994) proposed a mathematical model to describe the probabi1ity distributions of temporal rainfall using data from seven rain gauge stations. The study considers multifractal multiplicative cascade model. The model provides adequate estimates of the hourly rainfall distribution and hence can be used in locations where these short-duration Rainfall data are not available.
Yilma et al. (1994) assessed the statistical link between annual rainfall of Addis Ababa and sun spot number series. They used transfer function plus noise model and spectral analysis using smoothed periodogram method to determine the periodic behavior in the sunspot number series and annual rainfall series of Addis Ababa for the period 1900-1991 and found that periodogram of Addis Ababa annual rainfall series show similar feature with sunspot series periodogram, i.e both series show similar dominant periodicity of 10.22 year. Based on their result, they concluded that an „Average‟ sun activity may be associated with a contemporaneous „Average‟ rainfall process, but the anomalous sun activities may later induce anomalies in rainfall (Drought and Flood) through Ocean-Atmospheric phenomenon.
Nicholson and Entekhabi (1986) conducted a detailed power spectrum analysis of African annual rainfall series using Blackman-Turkey and Fourier methods. Their analysis revealed that quasi-periodicities were clustered in four bands at 2.2–2.4, 2.6–2.8, 3.3–3.8 and 5.0-6.3 years, common throughout equatorial and southern Africa but only weakly evident in northern Africa.
Harvey et al., (1987) investigate how patterns of rainfall correlate with general weather conditions and frequency of the cycles of rainfall. They used rainfall data from Brazil for a particular region which often suffers from drought to assess the cyclical behavior of rainfall. They use a model that allows cyclical components to be modeled explicitly. They found that cyclical components are stochastic rather than deterministic, and the gains achieved from forecast by taking account of the cyclic component are small in the case of Brazil.
Winstanley (1973a, b) reported that monsoon rains from Africa to India decreased by more than 50% from 1957 to 1970 and predicted that the future monsoon seasonal rainfall, averaged over 5 to 10 years is likely to decrease to a minimum around 2030.
Stringer (1972) reported that at least 35 quasi-periods with more than one year in length have been discovered in records of pressure, temperature, precipitation, and extreme weather conditions over many parts of the earth surface. A very common quasi-periodic oscillation is the quasi-biennial oscillation (QBO), in which the climatic events recur every 2 to 2.5 years. After reviewing all these research work, it is now paramount that the study of rainfall of Gusau metropolis is of great importance.











CHAPTER THREE: METHODOLOGY
3.0 INTRODUCTION
This chapter is designed to explain the basic methodology of the research work. Hence the various materials and methods applied for the purpose of achieving the objectives of the study are going to be explained thoroughly below. In particular, the basic procedures in SARIMA model building are explained in details.
3.1 TIME SERIES MODELS
3.1.1 WHITE NOISE PROCESS: Newbold and Granger (1974) in their book on time series said a time series {} is called a white noise process denoted by, if the following conditions are satisfied:
( i.e. Zero mean)
(i.e. constant variance)
Cov { , } = 0 if  i.e. not serially correlated in this case we write
~.
3.1.2 AUTO REGRESSIVE PROCESS AR(p): The auto regressive process uses weighted time lagged (previous) values to generate new current values for the time series. A time series    {} is an AR (p) process if it has the following representation.

                         (3.1.1)
 For   where {}  is a series of independent identically distributed (i.i.d) random variables, and   is some constant.
3.1.3 MOVING AVERAGE PROCESS MA(q): The moving average technique is often used for linear fitting. A moving average process of order q denoted by MA(q) is a stationary time series process {  } by box and Jenkins (1976), if it has representation of the form

                                      (3.1.2)
3.1.4 AUTOREGRESSIVE MOVING AVERAGE PROCESS ARMA(p,q): According to pandit and wu (1983), an ARMA(p,q) process is defined from the combination of the  order autoregressive and  order moving average process. A time series {  } is an ARMA (p,q) process if it has a representative  form of
,                          (3.1.3)
Where {  },  is some constant, and the  and  are defined as for AR and MA models respectively and is a series of unknown random errors (white noise) which are assumed to follow the normal probability distribution. An ARMA process is stationary if the AR component of the series is stationary and invertible if the MA component is invertible.
3.1.5 AUTOREGRESSIVE INTEGRATED MOVING AVERAGE PROCESS ARIMA(p,d,q): This process was developed to help remove trends and uncover hidden patterns in non-stationary data because; ARMA process can only model stationary data. Although the theory behind ARIMA time series model was developed much earlier, the systematic procedure for applying the technique was documented in the landmark book by box and Jenkins (1970). Since the ARIMA forecasting and box and Jenkins forecasting usually refer to the same set of techniques.
Knowing that stationary time series is integrated of the order d and if by differencing the terms it becomes an ARIMA (p,d,q) process, then the difference process can have an ARIMA (p,d,q) representation. In this case the time series  can be expressed as
                          (3.1.4)
Where
3.1.6 SEASONAL AUTOREGRESSIVE INTEGRATED MOVING AVERAGE PROCESS SARIMA(p, d, q) (P, D, Q): Pandit and wu (1983) said a time series  (p, d, q) (P, D, Q) representation if it can be expressed as:-
                       (3.1.5)
Where the AR components are:-
                            (3.1.6)
And the MA components are:-
                     (3.1.7)
Where, the SARIMA (p, d, q) (P, D, Q) is denoted thus:-
p=denotes the number of autoregressive terms.
d=is an integer which denotes the number of time the series must be differenced to attain stationarity.
q=denotes the number of moving average terms.
 P=denotes the number of seasonal autoregressive terms.
D=denotes the number of seasonal difference required to attain stationarity.
Q=denote the number of seasonal moving average terms.
s=denote the seasonal period or the length of the season.
3.2 STATIONARITY IN BOX-JENKINS MODELS
The Box-Jenkins model assumes that the time series is stationary. A stationary series has: 1. Constant mean 2. Constant variance 3. Constant autocorrelation structure.
Stationarity can be assessed from a run sequence plot. The run sequence plot should show constant location and scale. It can also be detected from an autocorrelation plot. Specifically, non-stationarity is often indicated by an autocorrelation plot with very slow decay. Box and Jenkins recommend differencing non-stationary series one or more times to achieve stationarity
3.3 TYPES OF NON-STATIONARITY
Up till now, all time series considered have been assumed stationary. This assumption was crucial to the definitions of the autocorrelation function (ACF) and the partial autocorrelation function (PACF). However in practice, many time series are not stationary. Methods for identifying non-stationary series are considered. The three (3) types of non-stationarity are:-
Series that have a non-stationarity mean
Series that  have a non-stationarity variance
Series with a periodic or seasonal component.
3.3.1 NON-STATIONARITY IN THE MEAN
One common type of non-stationarity is non-stationary mean. Typically, the mean of the series tends to increase or fluctuate. This is easiest to identify by looking at a plot of the data. Sometime, the sample ACF may indicate a non-stationary mean if the terms take a long time to decay to zero.
If a dataset exhibits a non-stationary mean, the solution is take difference. That is, if a time series   is non-stationary in the mean, compute the difference   . Generally, this makes any time series with a non-stationary mean into a time series with a stationary mean. Occasionally, the differenced time series will also be non-stationary in the mean, and another set of differences will be needed. It is rare to ever need more than two set of differencing, as there may be danger of having an over-differenced time series.


3.3.2 NON-STATIONARITY IN VARIANCE                                                                         
A less common type of non-stationarity with climatic data is non-stationarity in the variance. A non-stationary variance is a common difficulty, in many business applications. Generally a series that is non-stationary in the variance, has a variance that gets larger over time (that is, as time progresses, the observation become more variable). In this case, usually taking logarithms of the time series will be of help. Another possible difficulty is that the time series contain negative values, in cases like these, adding a sufficiently large constant to the data (which won’t affect the variance) and then taking the logarithm is of essence. If the time series is non-stationary in the mean and the variance, logs should be taken before differences (to avoid taking logs of negative values).
3.3.3 NON-STATIONARITY WITH A PERIODIC OR SEASONAL COMPONENT
The most common type of non-stationarity is when time series exhibits a seasonal pattern. “Seasonal” does not necessarily have anything to do with the season of dry or raining seasons. It means that there is some kind of regular pattern in the data. This type of non-stationarity is very common in climatological and meteorological applications, where there is often an annual pattern evident in the data. Seasonal data in time series is data that shows regular fluctuation aligned usually with some natural time period (not just the actual season of raining and dry seasons).
The length of a season is the time period over which the pattern repeats. For example, monthly data might show an annual pattern with a season of length 12, as the data may have a pattern that repeats each year i.e. each twelve months. These patterns usually appear in the sample ACF and PACF.
3.4 MODELLING TECHNIQUE
To identify a perfect ARIMA model for a particular time series data, Box and Jenkins (1976) proposed a methodology that consists of four phases: (i) Model identification (ii) Estimation of model parameters (iii) Diagnostic checking for the identified model and (iv) Application of the model (i.e. forecasting).
3.4.1 MODEL IDENTIFICATION
In model identification, Chatfield C. (1980) wrote that model identification involves the use of the data and any available information to suggest a suitable model to describe how the data has been generated. Also the input series of the SARIMA model need to be stationary, that is it should have a constant mean, variance and autocorrelation through time. Therefore, in some cases the series’ first needs to be differenced before stationarity is achieved. However one should keep in mind that, some time series may require little or no differencing and that over differenced series produce less stable estimate. The first step in the identification of a SARIMA process is to examine the time plot to indicate and identify the presence of trend and seasonal variation. However the three commonly used graphical methods in identification in time series are:- the time plot of the series, the time plot of autocorrelation at various lags (ACF) and the time plot of the partial autocorrelation function (PACF).
GRAPHICAL ANALYSIS (Time plot of a series): This is a graphical representation of a time series of observed values  plotted against the observations. Such plots give initial clue about the likely nature of the time series.
AUTOCORRELATION FUNCTION (ACF): One simple test of stationarity is based on the so-called Autocorrelation Function.
The ACF at lag k, denoted by is defined as
                              (3.2.1)
Where covariance at lag k and variance of the process are defined as:


 is the covariance (or Auto covariance) at lag k, is the covariance between variances of ; if k = 1, is the covariance between two adjacent values of y.
PARTIAL AUTOCORRELATION FUNCTION (PACF):- the partial autocorrelation function (), measure correlation between (time series) observations that are k times period apart after controlling for correlations at intermediate lags (i.e. lags less than k). In other words, partial autocorrelation is the correlation between after removing the effect of the intermediate y’s.
3.4.2 UNIT ROOT TEST
In probability theory and statistics, a unit root is a feature of some stochastic processes that can cause problems in statistical inference involving time series models. A linear stochastic process has a unit root if 1 is a root of the process’s characteristics equation. Such process is non-stationarary but does not always have a trend. Among the most widely used test are the phillips perron test, Kwiatkowski-Phillips-Schmidt-Shin (KPSS) Test and Augmented Dickey Fuller (ADF) Test.
KPSS TEST:- Kwiatkowski, Phillips, Schmidt and shin (1992) proposed a test of the null hypothesis that an observable series is trend stationary (stationary around a deterministic trend). The integration properties of a series  may also be investigated by testing the null hypothesis that the series is stationary against a unit root. Assuming no linear trend term, the data generating process is given as:-
 Where a random is walk,  and is a stationary process. Kwiatkowski (1992) proposed the following test statistic
 KPSS Where With  and  an estimator of the long run variance of                        (3.3.1)
The null hypothesis of the test is against the alternative hypothesis  . Reject the null hypothesis if the test statistic is greater than the asymptotic critical values.
PHILLIPS – PERRON (PP) TEST
The Phillips–Perron test (named after Peter C. B. Phillips and Pierre Perron) is a unit root test. That is, it is used in time series analysis to test the null hypothesis that a time series is integrated of order 1. It builds on the Dickey–Fuller test of the null hypothesis  in
                      ,                                       (3.3.2)
 Where  is the first difference operator. The phillips perron test the null hypothesis of non stationarity against the alternative that a variable was generated by a stationary process.


HEGY TEST
HEGY (1990) introduced a factorization of the seasonal differencing polynomial  and developed a testing procedure for seasonal unit roots, which consists in estimating via OLS the following regression:
                          (3.3.3)
Where
    and  Notice that when (α=1) and  have unit roots only at θ = 0,  and , respectively. Hence, the unit root found at θ = 0 in  implies accepting the null that  is zero. Similarly, when  is zero, it indicates the existence of a unit root at  When both and  are zero, then we have a pair of complex unit roots at .
3.4.3 INFORMATION CRITERIA AND MODEL SELECTION
Akaike Information Criterion (AIC): Suppose that we have a statistical model of some data. Let L be the maximum value of the likelihood function for the model; let k be the number of estimated parameters in the model. Then the AIC value of the model is the following.
                                                   (3.4.1)
Given a set of candidate models for the data, the preferred model is the one with the minimum AIC value. Hence AIC rewards goodness of fit (as assessed by the likelihood function).
Hannan-Quinn Information Criterion (HQC): In statistics the Hannan-Quinn information criterion (HQC) is a criterion for model selection. It is an alternative to Akaike information criterion (AIC) and Bayesian information criterion (BIC). It is given as
                                              (3.4.2)
 Where is the log likelihood, k is the number of parameters, and n is the number of observations.
Schwartz Information Criterion (SIC):  In statistics, the Bayesian information criterion (BIC) or Schwarz criterion (also SBC, SBIC) is a criterion for model selection among a finite set of models; the model with the lowest BIC is preferred. It is based, in part, on the likelihood function and it is closely related to the Akaike information criterion (AIC). The BIC was developed by Gideon E. Schwarz and published in a 1978, where he gave a Bayesian argument for adopting it. The BIC is formally defined as
                                                (3.4.3)
Where
The maximized value of the likelihood function of the model i.e., where  are the parameter values that maximize the likelihood function;
The observed data;
 The parameters of the model;
The number of data points in the number of observations, or equivalently, the sample size;
The number of free parameters to be estimated. If the model under consideration is a linear regression, is the number of regressors, including the intercept;
3.4.4 MODEL ESTIMATION
After choosing the most appropriate model then, the model parameters are estimated by using several estimation procedures. The estimation-stage results will be used to check: (i) parameter estimates, (ii) the appropriateness of coefficient estimates which includes the statistical significance of estimated coefficient and standard error and correlation matrix.
Maximum Likelihood Estimation
In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of a statistical model given observations, by finding the parameter values that maximize the likelihood of making the observations given the parameters. The philosophy behind maximum likelihood estimates is to find a set of parameters which maximize the likelihood of observing the data to which the model is being fitted. To use the method of maximum likelihood, one first specifies the jointly density function for all observations. For an independent and identically distributed sample, the joint density function is
                   (3.5.1)
Looking at this function from a different perspective by considering the observed values  to be fixed parameters of this function, whereas will be the function’s variable and allowed to vary freely; this function will be called the likelihood:
() =                                  (3.5.2)
Where “;” denote a separation between the two input arguments: and the observations. In practice it is often more convenient to work with the natural logarithm of the likelihood function, called the log likelihood:
                                                                   (3.5.3)
In Time series analysis, there may be several adequate models that can be used to represent a given data set, and hence, numerous criteria for model comparison have been introduced in the literature. One of them is based on the so-called information criteria. The idea is to balance the risks of under fitting (selecting an order smaller than the true order) and over fitting (selecting an order larger than the true order).
3.4.5 MODEL DIAGNOSTIC CHECKING
At the diagnostic checking stage, we check to see if the tentative model is adequate for its purpose. If a model is rejected we repeat the circle of the identification, estimation and diagnostic checking in an attempt to find a better model. The model having been identified and the parameters estimated, diagnostic checks are then applied to the model, in other to discover the ways in which a model is adequate so as to suggest appropriate modification.
There are many statistical tests used for diagnostic checking of randomness such as: (1) The Ljung- Box Q statistic, (2) jaque – bera test
LJUNG-BOX Q (LBQ) STATISTIC: The box-ljung test (1978) is a diagnostic tool used to test the lack of fit of a time series model. The test is applied to the residuals of a time series after fitting an ARMA (p,q) model to the data. The test examines m autocorrelations of the residuals. If the autocorrelations are very small, we conclude that the model does not exhibit significant lack of fit. In general, the Box-ljung test is defined as:
H0: The model does not exhibit lack of fit.
Ha: The model exhibits lack of fit.
                                           (3.6.1)
Where  is the estimated autocorrelation of the series at is is lag, and is the number of lags being tested. The Box-Ljung test rejects the null hypothesis (indicating that the model has significant lack of fit) if 
               
Where  is the chi-square distribution table value with h degrees of freedom and significance level α. Because the test is applied to residuals, the degrees of freedom must account for the estimated model parameters so that h = m – p – q, where p and q indicate the number of parameters from the ARMA(p,q) model fit to the data.
JARQUE-BERA TEST: Jarque and Bera (1987) have proposed test for normality based on skewness and kurtosis of a distribution. The Jarque-Bera test is a two-sided goodness of fit test suitable when a fully-specified null distribution is unknown and its parameters must be estimated. The test statistic is:-
                                      (3.6.2)
Where n is the sample size, s is the sample skewness, and k is the sample kurtosis. The test checks the pairs of hypothesis; and That is, the distribution is symmetry and hence normal. And for the alternative hypothesis it implies that, the distribution is asymmetry and hence non-normal.
The null hypothesis is accepted if the test statistic is less then critical values, and rejected if the test statistic is greater than the critical values.
 NORMAL PROBABILITY PLOT
The normal probability plot (Chambers et al., 1983) is a graphical technique for assessing whether or not a data set is approximately normally distributed. The data are plotted against a theoretical normal distribution in such a way that the points should form an approximate straight line. Departures from this straight line indicate departures from normality.
3.4.6 FORECASTING
The last step in time series modeling is forecasting. A forecast is an estimate of a future event achieved by systematically combining and casting forward in predetermined way data about the past. It is simply a statement about the future. Good forecast can be quite valuable and would be worth a great deal. Long-run planning decisions require consideration of many factors: general economic conditions, industry trends, probable competitor’s actions, overall political climate, and so on. Forecasts are possible only when a history of data exists.
Forecasting has applications in a wide range of fields where estimates of future conditions are useful. Not everything can be forecasted reliably, if the factors that relate to what is being forecast are known and well understood and there is a significant amount of data that can be used very reliable forecasts can often be obtained. If this is not the case or if the actual outcome is effected by the forecasts, the reliability of the forecasts can be significantly lower.
Quantitative forecasting models are used to forecast future data as a function of past data. They are appropriate to use when past numerical data is available and when it is reasonable to assume that some of the patterns in the data are expected to continue into the future. These methods are usually applied to short- or intermediate-range decisions. Examples of quantitative forecasting methods are the simple and weighted N-Period moving averages, simple exponential smoothing, Poisson process model based forecasting and multiplicative seasonal indexes.
FORECAST ACCURACY MEASURES
Once forecasts are made they can be evaluated if the actual values of the series to be forecasted are observed. The forecast errors are the difference between the actual values in the test set and the forecasts produced using only the data in the training set. Some of the forecast accuracy measures are the (i) Root mean square error (RMSE) (ii) Mean squared error (MAE) (iii) Mean absolute error (MAE) and (iv) Mean absolute percentage error (MAPE).

Root Mean Squared Error (RMSE) =        (3.7.1)
Mean Square Error (MSE) =                                                   (3.7.2)
Mean Absolute Error (MAE) =          (3.7.3)                   
Mean Absolute Percentage Error (MAPE) =         (3.7.4)





















CHAPTER FOUR: DATA ANALYSIS
 4.0 INTRODUCTION
This chapter is going to focus on time series modeling and forecasting of the monthly rainfall of Gusau metropolis, using a dataset from the year 1994 to 2015. This research work uses the techniques of Seasonal Autoregressive Integrated Moving Average (SARIMA) methodology, developed by Box and Jenkins (1976), to find an appropriate model for the data.
The statistical software used for this time series analysis is gretl.
4.1 THE DATA FOR THE ANALYSIS
The data used in this research is the mean monthly rainfall data of Gusau metropolis, from January 1994 to December 2015. The Instrument for the measurement of Rainfall is the rain gauge, situated at the Nigerian Meteorological Agency Gusau.







4.2 Time series plots
To identify the model of any time series data, one most make a guess as to the data generation process. In doing this, one most begin by plotting the series.
4.2.1 TIME SERIES PLOT BEFORE DIFFERNCING: A time plot of the original series conducted below (fig 4.1) shows the time plot of the monthly rainfall of Gusau metropolis, for a period of 22 years (1994-2015).

Fig 4.1: Time series plot before differencing
From Fig 4.1 Examine the time series plot of the monthly rainfall of Gusau, the series is not stationary and trendy downward. It shows that the mean and variance are changing with time, meaning the series is not stationary.


4.2.2 ACF AND PACF BEFORE DIFFERENCING
In time series analysis using SARIMA, the identification of whether the data is stationary or not is not enough. One has to test further some other time series properties. For example, there is need to examine the correlogram of the time series data (i.e. the ACF and PACF of the data). Here is the autocorrelation function (ACF) and partial autocorrelation function (PACF) of the data series before differencing is performed.

Fig 4.2 ACF and PACF plot before differencing.
From the plots in Fig 4.2 above we note that there is strong presence of seasonal factors. This is confirmed by very high spikes at and around seasonal lags of period 12. The sinusoidal or periodic pattern in the ACF plot is suggesting that the series has a strong seasonal effect also, it can be seen in lag (10, 15, 20, 25, and 30) autocorrelation. Looking at the (PACF) on the other hand, it can be observed clearly that the model for the data is mixed one. That is to say the model for the data is mixed with both AR and MA component.
4.2.3 TIME SERIES PLOT AFTER TAKING THE LOG

Fig 4.3 Time series plot after taking the log
We can observe from the above that the log transformation did not have any effect on the variance. And this make us conclude that the original data do not need to be log transformed. Henceforth the original data will be used.



4.2.4 UNIT ROOT TEST BEFORE DIFFERENCING
Table 4.1: Result of Philips - Perron (pp) Test
Number of lags
1%
5%
10%
Test Statistic

4
-3.457964
-2.873186
-2.572924
-8.1587

5
-3.457964
-2.873186
-2.572924
-7.8905

6
-3.457964
-2.873186
-2.572924
-7.4632


H0: The series contains unit root
H1: The series is generated by stationary process
Decision: From table 4.1 above we can observe that the test statistic in PP test are less than critical values, at 1%, 5%, and 10% level of significance, we accept the null hypothesis (Ho) and conclude that the series contains unit root.
Table 4.2: Result of KPSS test
Number of lags
1%
5%
10%
Test Statistic

4
0.216
0.146
0.119
0.3529

5
0.216
0.146
0.119
0.3242

6
0.216
0.146
0.119
0.3170


H0: The series is stationary
H1: The series is not stationary
Decision: By observing the table above, we can notice that, the test statistics for the KPSS test is greater than asymptotic critical values at 1%, 5% and 10% level of significance, and therefore we reject the null hypothesis and conclude that the series is not stationary.

4.2.5 TIME SERIES PLOT AFTER FIRST DIFFERENCE

Fig 4.4: Time series plot after first difference
From fig 4.3 it shows that the level of variability of the series is very effective therefore from our first differencing we can observe that the data is stationary in both mean and variance.

4.2.6 ACF AND PACF AFTER FIRST DIFFERENCE

Fig 4.5: ACF and PACF after first difference
From fig 4.4: above we can observe that, the ACF of the first difference possess strong and persistent periodic nature even after differencing once. This clearly shows that the time series data has seasonal behavior.
Seasonal differencing was therefore carried out in order to remove the persistent periodicity found in the series. The plot of seasonal monthly rainfall, ACF and PACF was analyzed respectively.

4.2.7 UNIT ROOT TEST AFTER DIFFERENCING
Table 4.3: Result of Philips - Perron (pp) Test
Number of lags
1%
5%
10%
Test Statistic

4
-3.458064
-2.873231
-2.572947
-1.7701

5
-3.458064
-2.873231
-2.572947
-1.4566

6
-3.458064
-2.873231
-2.572947
-2.0372


H0: The series contains unit root
H1: The series is generated by stationary process
Decision: From table 4.1 above we can observe that the test statistic in PP test are less than critical values, at 1%, 5%, and 10% level of significance, we reject the null hypothesis (Ho) and conclude that the series is stationary.
Table 4.4: Result of KPSS test
Number of lags
1%
5%
10%
Test Statistic

4
0.216
0.146
0.119
0.0073

5
0.216
0.146
0.119
0.0082

6
0.216
0.146
0.119
0.0101


H0: The series is stationary
H1: The series is not stationary
Decision: By looking at the table above, we can observe that, the test statistics for the KPSS test is less than asymptotic critical values at 1%, 5% and 10% level of significance, and therefore we accept the null hypothesis (HO) and conclude that the series is stationary.
4.2.8 SEASONAL UNIT ROOT TEST BEFORE SEASONAL DIFFERENCING
The Hegy statistic test that a variable has a unit root. The null hypothesis of the HEGY test is that a unit root exists and the alternative is that the variable was generated by a stationary process and the decision rule is to reject the null hypothesis when the value of the test statistic is less than the critical value.
Table 4.5: Result of Hegy test before seasonal differencing
Number of lags
1%
5%
10%
Test Statistic

4
-2.54
-1.94 
-1.59   
-3.8397

5
-2.54
-1.94 
-1.59   
-3.8397

6
-2.54
-1.94 
-1.59   
-3.8397


H0: The series is not stationary
H1: The series is stationary
Decision: By looking at the table above, we can observe that, the test statistic for the Hegy test is less than asymptotic critical values at 1%, 5% and 10% level of significance, therefore we accept the null hypothesis and this makes us conclude that the series is not stationary




4.2.6 TIME SERIES PLOT AFTER SEASONAL DIFFERENCING

Fig 4.6: Time series plot after seasonal differencing
It can be seen that upon visual inspection of the time series plot after seasonal differencing we can observe that the series is stationary in both mean and variance.



4.2.10 ACF AND PACF AFTER SEASONAL DIFFERENCING

Fig 4.7: ACF and PACF after seasonal differencing
It can be seen that upon visual inspection of the ACF and PACF plot of the seasonal difference of monthly rainfall above that, the persistent trend and periodicity witnessed earlier has now been removed. This make us to conclude that the series is stationary at this point







4.2.11 SEASONALUNIT ROOT TEST AFTER SEASONAL DIFFERENCING
Table 4.6: Result of Hegy test after seasonal differencing
Number of lags
1%
5%
10%
Test Statistic

4
-2.54
-1.94 
-1.59   
-1.2790

5
-2.54
-1.94 
-1.59   
-1.1279

6
-2.54
-1.94 
-1.59   
-1.1291


H0: The series is stationary
H1: The series is not stationary
Decision: By looking at the table above, we can observe that, the test statistics for the Hegy test is greater than asymptotic critical values at 1%, 5% and 10% level of significance, therefore we reject the null hypothesis and conclude that the series is stationary after seasonal differencing once and hence the order of seasonal difference (D = 1).
4.3 MODEL IDENTIFICATION
The table below shows the selected models, so as to suggest the appropriate model for the data. You must identify the models before selection is being made, consider the following model identification. 
Table 4.7: Results of SARIMA model identification for rainfall
Model
AIC
SIC
    HQC

SARIMA(1,1,2)(1,1,1)12
1430.457
1454.792
1440.263

SARIMA(1,1,3)(1,1,1)12
1431.775
1459.587
1442.982

SARIMA(2,1,1)(1,1,1)12
1430.431
1454.767 
1440.238

From table 4.4 above we consider all the two models with minimum AIC, HQC and SIC for having satisfied the criteria of being white noise.
4.4 MODEL ESTIMATION
Table 4.8: Results of SARIMA (1, 1, 2) (1, 1, 1)12 parameter estimates
parameters
coefficient
std. error
     z
p-value

  const
−0.000706817
0.00148798
−0.4750
 0.6348 

  phi_1
  0.102140
0.327775
  0.3116
0.7553 

  Phi_1
−0.114142
0.0683435
−1.670
0.0049 ***

  theta_1
−0.922507
0.328687
−2.807
0.0050 ***

  theta_2
−0.0774927
0.327538
−0.2366
0.0017 *** 

  Theta_1
−0.999999
0.0858209
−11.65
2.24e-031 ***

KEY (1% ***) (5% **) (10% *)
INTERPRETATION
At 1% level of confidence, the p-values for the parameters Phi_1, theta_1, theta_2 and Theta_1 are found to be significant, therefore we reject the null hypothesis (H0) and conclude that the parameters are statistically significant. Also the constant term and the parameters Phi_1 are found to be statistically insignificant thereby accepting the null hypothesis.




Table 4.9: Results of SARIMA (1, 1, 3) (1, 1, 1)12 parameter estimates
parameters
coefficient
std. error
      z
p-value

  const
−0.000712551
0.00145934
−0.4883
0.6254 

  phi_1
−0.973264
0.0226225
−43.02
0.0000    ***

  Phi_1
−0.107451
0.0654365
−1.642
0.0006 *** 

  theta_1
0.164248
0.0713631
   2.302
0.0014  ***

  theta_2
−0.999991
0.0406333
−24.61
9.84e-134 ***

  theta_3
−0.164247
0.0614653
−2.672
0.0075   ***

  Theta_1
 −0.999992
0.0977529
−10.23
1.46e-024 ***

KEY (1% ***) (5% **) (10% *)
INTERPRETATION
At 1% level of significance the p-values for the parameters phi_1, Phi_1, theta_1, theta_2, theta_3 and Theta_1 are found to be significant, therefore we reject the null hypothesis (H0) and conclude that the parameters are statistically significant. Also the constant term is found to be statistically insignificant thereby accepting the null hypothesis.




Table 4.10: Results of SARIMA (2, 1, 1) (1, 1, 1)12 parameter estimates
Parameters
coefficient
std. error           
     z       
p-value

const
−0.000706769               
  0.00147772               
−0.4783
0.6324 

phi_1               
0.180404     
  0.0649937
  2.776   
0.0055 ***

phi_2                 
−0.0191843   
0.0654843   
−0.2930 
0.7696 

Phi_1               
−0.114925     
0.0684468   
−1.679   
0.0031 ***

theta_1             
−1.00000     
0.0280941   
−35.59   
1.69e-277 ***

Theta_1             
−0.999998     
0.0861305   
−11.61   
3.65e-031  ***

KEY (1% ***) (5% **) (10% *)
INTERPRETATION
At 1% level of significance the p-values for the parameters phi_1, Phi_1, theta_1 and Theta_1 are found to be significant, therefore we reject the null hypothesis (H0) and conclude that the parameters are statistically significant. Also the constant term and the parameter phi_2 are found to be statistically insignificant thereby accepting the null hypothesis.
4.5 MODEL DIAGONISTIC CHECKING
Before the interpretation and use of the model, we are to look at some tests/plots to check whether this volatility models have adequately captured all of the persistence in the variance of returns. And in as much as the model is adequate, then standard residuals are uncorrelated as specified in the results below.

Fig 4.8: The ACF and PACF of SARIMA (1, 1, 2) (1, 1, 1)12
The residual ACF and PACF of the model lie within the 90% confidence interval.  The model have passed the standard test criteria of being white noise.


Fig 4.9: The ACF and PACF of SARIMA (1, 1, 3) (1, 1, 1)12
The residual ACF and PACF of the model lie within the 90% confidence interval.  The models have passed the standard test criteria of being white noise.

Fig 4.10: The ACF and PACF of SARIMA (2, 1, 1) (1, 1, 1)12
The residual ACF and PACF of the model lie within the 90% confidence interval.  The models have passed the standard test criteria of being white noise.

Fig 4.11: Time series plot of SARIMA (1, 1, 2) (1, 1, 1)12
The time plots has a wave-like pattern; the plots showed that the series has constant mean and variance. The plots indicate the series is stationary.



Fig 4.12: Time series plot of SARIMA (1, 1, 3) (1, 1, 1)12
The time plots has a wave-like pattern; the plots showed that the series has constant mean and variance. The plots indicate the series is stationary.

Fig 4.13: Time series plot of SARIMA (2, 1, 1) (1, 1, 1)12
The time plots has a wave-like pattern; the plots showed that the series has constant mean and variance. The plots indicate the series is stationary.
4.5.1 LJUNG BOX TEST
The Box-Ljung test (1978) is a diagnostic tool used to test the lack of fit of a time series model. The test is applied to the residuals of a time series after fitting an appropriate model to the data. The test examines m autocorrelations of the residuals. If the autocorrelations are very small, we conclude that the model does not exhibit significant lack of fit. In general, the Box-Ljung test is defined as:
 Ho: The model does not exhibit lack of fit and
 H1: The model exhibits lack of fit.
The decision rule is to accept the null hypothesis when the p-value is greater than the level of significance, otherwise we reject the null hypothesis.
Table 4.11: Result of ljung Box Test
MODEL
TEST STATISTIC
P-VALUE

SARIMA(1,1,2)(1,1,1)12
9.3451
0.6732

SARIMA(1,1,3)(1,1,1)12
9.4130
0.6673

SARIMA(2,1,1)(1,1,1)12
9.2170
0.5219


INTERPRETATION
It can be seen that the p-value is greater than choosing alpha (α) at 5% level of significance in all the cases, hence we accept the null hypothesis Ho and conclude that the models are adequate and fit for forecast.
4.5.2 JAQUE-BERA TEST
The Jaque-Bera statistic tests the null hypothesis of normality against the alternative of non-normality and the decision rule is to accept the null hypothesis when the p-value is greater than the level of significance.


Table 4.12: Result of Jaque-Bera Test
MODEL
TEST STATISTICS
P-VALUE

SARIMA(1,1,2)(1,1,1)12
334.550
0.1517

SARIMA(1,1,3)(1,1,1)12
333.410
0.1718

SARIMA(2,1,1)(1,1,1)12
339.386
0.1688

INTERPRETATION
Since the p-value is greater than choosing alpha (α) at 5% level of significance, we accept the null hypothesis (Ho) and conclude that the residuals of the model are normally distributed.








4.5.3 THE TEST STATISTIC FOR NORMALITY

Fig 4.14: Test statistic for normality of SARIMA (1, 1, 2) (1, 1, 1)12
From the above histogram, it shows a bell shape, this can lead us to believe that the error term is normally distributed.



Fig 4.15: Test statistic for normality of SARIMA (1, 1, 3) (1, 1, 1)12
From the above histogram, it shows a bell shape, this can lead us to believe that the error term is normally distributed.


Fig 4.16: Test statistic for normality of SARIMA (2, 1, 1) (1, 1, 1)12
From the above histogram, it shows a bell shape, this can lead us to believe that the error term is normally distributed.




4.6 FORECAST
The ability of a good model fitted SARIMA model is in its ability to forecast the time series data is tested. This further testifies the validity of the model.
Table 4.13: Results of SARIMA (1, 1, 2) (1, 1, 1)12 Forecast Values for 2 years
Year/Month
Predictions
Std.error
95% interval

2016:01
-0.13
4.332
-8.62 - 8.36

2016:02
-0.01
4.332
-8.50 - 8.49

2016:03
0.49
4.332
-8.00 - 8.98

2016:04
3.77
4.332
-4.72 - 12.26

2016:05
11.20
4.332
2.71 - 19.70

2016:06
11.65
4.332
3.16 - 20.15

2016:07
15.74
4.332
7.24 - 24.23

2016:08
14.82
4.332
6.33 - 23.31

2016:09
12.51
4.332
4.02 - 21.00

2016:10
4.17
4.332
-4.32 - 12.67

2016:11
0.08
4.332
-8.42 - 8.57

2016:12
-0.44
4.332
-8.93 - 8.05

2017:01
-0.41
4.333
-8.90 - 8.08

2017:02
-0.28
4.333
-8.77 - 8.21

2017:03
0.22
4.333
-8.27 - 8.71

2017:04
3.54
4.333
-4.96 - 12.03

2017:05
10.97
4.333
2.48 - 19.47

2017:06
11.38
4.333
2.89 - 19.87

2017:07
15.38
4.333
6.89 - 23.87

2017:08
14.64
4.333
6.15 - 23.13

2017:09
12.26
4.333
3.77 - 20.76

2017:10
3.93
4.333
-4.56 - 12.42

2017:11
-0.20
4.333
-8.70 - 8.29

2017:12
-0.73
4.333
-9.22 - 7.76



Fig 4.17 forecast plot of rainfall for SARIMA (1, 1, 2) (1, 1, 1)12
From the table and figure above we have seen that the model shows a good result and it has accuracy of forecasting this lead us to conclude that the model suits the data well.

Table 4.14: Results of SARIMA (1, 1, 3) (1, 1, 1)12 Forecast Values for 2 years
Year/Month
Predictions
Std.error
95% interval

2016:01
0.00
4.317
-8.65 - 8.27

2016:02
0.00
4.317
-8.49 - 8.43

2016:03
0.43
4.317
-8.04 - 8.89

2016:04
3.76
4.318
-4.71 - 12.22

2016:05
11.17
4.318
2.71 - 19.63

2016:06
11.63
4.318
3.17 - 20.10

2016:07
15.66
4.319
7.20 - 24.13

2016:08
14.76
4.319
6.30 - 23.23

2016:09
12.54
4.319
4.08 - 21.01

2016:10
4.08
4.319
-4.39 - 12.54

2016:11
0.11
4.320
-8.36 - 8.57

2016:12
0.00
4.320
-9.03 - 7.91

2017:01
0.00
4.320
-8.89 - 8.05

2017:02
0.00
4.320
-8.86 - 8.07

2017:03
0.20
4.320
-8.27 - 8.67

2017:04
3.43
4.321
-5.04 - 11.90

2017:05
10.97
4.321
2.51 - 19.44

2017:06
11.27
4.321
2.80 - 19.74

2017:07
15.35
4.321
6.88 - 23.82

2017:08
14.49
4.321
6.02 - 22.96

2017:09
12.33
4.321
3.86 - 20.80

2017:10
3.75
4.321
-4.72 - 12.22

2017:11
0.00
4.322
-8.61 - 8.33

2017:12
0.00
4.322
-9.39 - 7.55



Fig 4.18 forecast plot of rainfall for SARIMA (1, 1, 3) (1, 1, 1)12
Similarly, from the table and figure above we have seen that the model shows a good result and it has accuracy of forecasting this lead us to conclude that the model suits the data well.


Table 4.15: Results of SARIMA (2, 1, 1) (1, 1, 1)12 Forecast Values for 2 years
Year/Month
    Predictions
         Std.error
     95% interval

2016:01
0.00
4.332
-8.62 - 8.36

2016:02
0.00
4.333
-8.50 - 8.48

2016:03
0.49
4.333
-8.01 - 8.98

2016:04
3.76
4.333
-4.73 - 12.26

2016:05
11.20
4.333
  2.71 -19.69

2016:06
11.65
4.333
 3.16 - 20.14

2016:07
15.74
4.333
 7.24 - 24.23

2016:08
14.82
4.333
 6.32 - 23.31

2016:09
12.51
4.333
4.02 - 21.00

2016:10
4.17
4.333
-4.32 - 12.66

2016:11
0.00
4.333
-8.42 - 8.56

2016:12
0.00
4.333
-8.93 - 8.05

2017:01
0.00
4.333
-8.90 - 8.08

2017:02
0.28
4.333
-8.78 - 8.21

2017:03
0.22
4.333
-8.28 - 8.71

2017:04
3.53
4.333
-4.96 - 12.03

2017:05
10.97
4.333
2.48 - 19.46

2017:06
11.38
4.333
2.88 - 19.87

2017:07
15.38
4.333
6.89 - 23.87

2017:08
14.64
4.333
6.14 - 23.13

2017:09
12.26
4.333
3.77 - 20.75

2017:10
0.00
4.333
-4.56 - 12.42

2017:11
0.00
4.333
-8.70 - 8.29



Fig 4.19 forecast plot of rainfall for SARIMA (2, 1, 1) (1, 1, 1)12
Similarly, from the table and figure above we have seen that the model shows a good result and it has accuracy of forecasting this lead us to conclude that the model suits the data well.
4.6.1 FORECAST EVALUATION
Table 4.16: Results of forecast evaluation
MODELS
MSE
RMSE
MAE

SARIMA(1,1,2)(1,1,1) 12
22.751
4.7698
2.9870

SARIMA(1,1,3)(1,1,1) 12
22.512
4.4657
2.0163

SARIMA(2,1,1)(1,1,1) 12
22.751
4.7698
2.9898


From the above forecast valuation we found that SARIMA(1,1,3)(1,1,1)12 to be the best model which give the best forecast, since it has the minimum mean square error (MSE), Root mean square error (RMSE) and mean absolute error (MAE).










CHAPTER FIVE: SUMMARY, CONCLUSION AND
RECOMMEDATION
5.1 SUMMARY
The objectives of this study is to forecast the mean monthly rainfall in Gusau Zamfara State. Having known the impact and useful of rainfall to the society, chapter one able to give the general description of rainfall, state the problem behind the shortage of rainfall, significance of rainfall, chapter two reviewed some relevant work on mean monthly rainfall. Chapter three provide the basic methodology of the research, chapter four analyzed and discussed the results properly which deduced that SARIMA (2, 1, 1) (1, 1, 1)12 found to be the best model and hence we made the forecast for two years ahead.
5.2 CONCLUSION
This project analyses the time series characteristics of rainfall data of Gusau for the period of 22 years with a view to understanding rainfall variation. Data for this study was obtained from the Nigerian Meteorological Agency (NIMET), the data was subject to time series test. The result showed a decreasing trend of rainfall trend in Gusau. Analysis of seasonality in monthly precipitation showed a concentration of rainfall in the months of July, August and September; however, the months of April, May, June and October do experience some showers of rainfall sometimes. Obviously, result of the seasonality analysis showed that January, February, March, April, November and December are dry months. This implies that growing season in Gusau do end around October. The implication is that farmers in the study area need to stream line their farming activities with a view to making effective use of the available rainfall. The project also suggests a need for building micro dams, developed underground water resources and or adopts conjunctive water management as part of drought management efforts.
5.3 recommendation
base on the study, we recommend the following:
The government is recommended to build micro dams, developed underground water resources and adopts conjunctive water management as part of drought management efforts.
The farmers are therefore recommended to stream line their farming activities with a view to make effective use of the available rainfall.
Emergency measures should be put in place for eventuality of heavy rainfall that can lead to flooding, which may also cause loss of lives and properties.









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