FORECASTING OF THE MONTHLY TEMPERATURE OF GUSAU METROPOLIS FROM (1991-2007) 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



SEPTEMBER, 2016


                                         CERTIFICATION
This is to certify that the research work for the project and the subsequent preparation of this report by MANSUR RABIU ADM NO: 1110306276. In the department of mathematics is fully adequate in scope and quality as a research for the degree of Bachelor of Science in statistics, Usman Danfodiyo University Sokoto.


 DR. YAKUBU MUSA                                                                Date
 Project supervisor




DR. ABUBAKAR D KOKO                                Date
Head of statistics unit




DR UMAR USMAN
Project coordinator
                                                                                                      Date 





DEDICATION
I humbly dedicate this research work to myself, my beloved parents for the love and understanding right from infancy, even till this present stage of life. Brother and sisters as well as my entire friend for their support and source of encouragement for my success. May Almighty Allah (s w a) continually shower his blessing on them all Amin.


                                                 







                                                     AKNOWLEDGEMENT
All praise and thanks be to Almighty Allah for sustaining my life to-date and his mercy in giving me the courage and patience to undertake this research project lucidly successful. My sincere and thoughtful gratitude goes to my ever meticulous and hardworking supervisor Dr. YAKUBU MUSA for his effort in patiently reading, criticizing and correcting the manuscripts throughout the period of study. His recommendations and suggestion made this research project highly authentic. May Almighty Allah bless him. My sincere appreciation and gratitude also goes to all other lecturers in the department of mathematics for their contribution to my success especially the head of department Dr. A D KOKO and head of statistics unit Dr. A D KOKO whose effort, support, suggestion, constructive critics and advice has been given to me with due respect and humility, I acknowledge your help through the course of my study I appreciate.
 I am highly grateful to my parents for their financial support, guidance, counseling and assistance throughout the course of the study .Finally, I wish to thanks Alh Aminu Moyi manager Unity bank Sokoto state  and my entire friends, school mates, relatives and all other associate.





                                 
                                                                    TABLE OF CONTENTS
Title page ………………………………………………I
Certification ii
Dedication iii
Acknowledgements iv
Table of Contents v
List of figures ix
List of tables x
Abbreviations xi
Abstract xii
CHAPTER ONE: GENERAL INTRODUCTION
1.0 INTRODUCTION------------------------------------------------------------------------------------1- 2
HISTORICAL BACKGROUND-------------------------------------------------------------------2 - 3
AIM AND OBJECTIVE OF RESEARCH---------------------------------------------------------3
SCOPE AND LIMITATION OF THE STUDY----------------------------------------------------3
STATEMENT OF PROBLEM------------------------------------------------------------------------4
BASIC DEFINITIONS OF SOME TERM--------------------------------------------------------5 - 6
CHAPTER TWO: LITERATURE REVIEW
2.0 INTRODUCTION-------------------------------------------------------------------------------------7
2.1 REVIEW OF RESEARCH WORK ON TEMPERATURE--------------------------------8 - 12
2.2 DEFINATION OF TIME SERIES--------------------------------------------------------------1
2.3 METHOD OF TIME SERIES ANALYSIS--------------------------------------------------13 - 14
CHAPTER THREE: METHODOLOGY
3.0 INTRODUCTION-------------------------------------------------------------------------------15
3.1 THE DATA FOR THE ANALYSIS-------------------------------------------------------15 - 16
3.2 METHOD OF DATA ANALYSIS---------------------------------------------------------------16
TIME SERIES MODELS----------------------------------------------------------------------17
3.3.1 WHITE NOISE PROCESS----------------------------------------------------------------17
3.3.2 AUTO REGRESSIVE PROCESS AR (P)------------------------------------------------17 - 18
MOVING AVERAGE PROCESS MA (q)-------------------------------------------------18
3.3.4 AUTOREGRESSIVE MOVING AVERAGE ARMA (p, q) PROCESS-----------------18 - 19
3.3.5 AUTOREGRESSIVE INTEGRATED MOVING AVERAGE ARIMA (p, d, q) PROCESS…..19
3.3.6 SEASONAL AUTOREGRESSIVE INTEGRATED MOVING AVERAGE SARIMA (p,d,q)(P,D,Q) PROCESS………………………………………………………………………………………..19 - 20
3.4 BUILDING SARIMA MODEL--------------------------------------------------------------20
3.4.1 MODEL IDENTIFICATION-----------------------------------------------------------------20 – 22
3.4.1.1 THE PLOT OF THE SERIES------------------------------------------------------------------21
3.4.1.2 THE ACF OF THE SERIES------------------------------------------------------------------21 – 22
3.4.1.3 THE PACF OF THE SERIES--------------------------------------------------------------------22
3.4.1.4 UNIT ROOT TEST--------------------------------------------------------------------------23 -25
3.4.1.5 SEASONAL UNIT ROOT TEST----------------------------------------------------------25 - 26
3.4.1.6 INFORMATION CRITERIA---------------------------------------------------------------26 - 27
3.4.2 MODEL ESTIMATION----------------------------------------------------------------------- 28
3.4.2.1 LEAST SQUARE ESTIMATE-------------------------------------------------------------28
3.4.3 MODEL VALIDATION OR DIAGNOSTIC CHECKING-------------------------------29
3.4.3.1 JARQUE-BERA TEST---------------------------------------------------------------------29 – 30
4.5 FORECASTING----------------------------------------------------------------------------------30 - 31
3.6 TYPES OF NON-STATIONARITY-----------------------------------------------------------31
3.6.1 NON-STATIONARITY IN THE MEAN---------------------------------------------------31 - 32
3.6.2 NON-STATIONARITY IN VARIANCE--------------------------------------------------32
3.6.3 NON-STATIONARITY WITH A PERIODIC OR SEASONAL COMPONENT---------------32 - 33

CHAPTER FOUR: DATA ANALYSIS
4.0 INTRODUCTION---------------------------------------------------------------------------------34
4.1 THE DATA FOR THE ANALYSIS-------------------------------------------------------------34
4.2 TIME PLOTS--------------------------------------------------------------------------------------34 - 35
4.3 THE PLOT OF ACF AND PACF OF THE DATA-------------------------------------------35 - 36
4.4 UNIT ROOT TEST------------------------------------------------------------------------------37 - 42
4.4.1 TEST FOR SEASONAL UNIT ROOT-----------------------------------------------------43 - 45
4.5 MODEL IDENTIFICATION---------------------------------------------------------------------45
4.6 PARAMETER ESTIMATES OF THE MODEL----------------------------------------------46 - 47
4.7 MODEL CHECKING----------------------------------------------------------------------------47 - 51
4.8 THE TEST STATISTIC FOR NORMALITY------------------------------------------------52 - 53
4.9 FORECASTING---------------------------------------------------------------------------54 - 63
CHAPTER FIVE: SUMMARY, CONCLUSION AND RECOMMEDATION
5.0 SUMMARY-----------------------------------------------------------------------------------64
5.1 CONCLUSION-------------------------------------------------------------------------------------64
5.2 RECOMMENDATION---------------------------------------------------------------------------65
5.3 REFERENCE----------------------------------------------------------------------------------65 - 66
5.4 APPENDIX-----------------------------------------------------------------------------------------67










LIST OF FIGURES               
Fig 4.1 Time plot of the monthly Temperature
Fig 4.2 the ACF and PACF of the monthly temperature
Fig 4.3 Time plot of the first difference of the original series   
 Fig 4.4 the ACP and PACF of first difference f monthly temperature   
Fig 4.5 plot of the seasonal difference of monthly temperature
Fig 4.6 the ACF and PACF of seasonal difference of monthly temperature
Fig 4.7 the ACF and PACF of SARIMA (2, 1, 2) (1, 1, 1)12
Fig 4.8 the ACF and PACF of SARIMA (2, 1, 2) (1, 1, 0)12
Fig 4.9 The residual plot of the SARIMA (2, 1, 2) (1, 1, 1)12
Fig 4.10The residual plot of the SARIMA (2, 1, 2) (1, 1, 0)12
Fig 4.11 Test statistic for normality of SARIMA (2, 0, 2) (1, 1, 1)12
Fig 4.12 Test statistic for normality of SARIMA (2, 0, 2) (1, 1, 0)12
Fig 4.13 Time plot of the five years forecast of SARIMA (2, 1, 2) (1, 1, 1)12
Fig 4.14 Time plot of the five years forecast of SARIMA (2, 1, 2) (1, 1, 0)12
                               
                                 LIST OF TABLES
Table 4.1: Results of the PP Test before differencing
Table 4.2: Results of the KPSS Test before differencing
Table 4.3: Results of the PP Test after differencing
Table 4.4: Results of the KPSS Test after differencing.
Table 4.5: Results of the HEGY test
Table 4.6 Results of SARIMA model identification for monthly temperature
Table 4.7 Results of SARIMA (2,1,2)(1,1,1)12 model estimation for monthly temperature
Table 4.8 Results of SARIMA(2,1,2)(1,1,0)12 model estimation for monthly temperature
Table 4.9 Results of JaqueBera test for temperature time series
Table 4.10 Results of SARIMA (2,1,2)(1,1,1)12 forecast values for five years.
Table 4.11 Results of SARIMA (2,1,2)(1,1,0)12 forecast values for five years.
Table 4.12: Results for the forecasting                                     
                                             








                                     


                              ABBREVIATION
ACF          Autocorrelation Function
PACF       Partial Autocorrelation Function
AIC           Akaike Information Criteria
PP             Phillips-Perron
KPSS         Schwert Information Criteria
(MA)         Moving Average Process
(AR)         Auto Regressive Process
ARMA       Autoregressive Moving Average
ARIMA     Autoregressive Integrated Moving Average
SARIMA    Seasonal Autoregressive Integrated Moving Average

       
                                                               



                                                 ABSTRACT
Temperature is very important factor of climate change. This study investigates the modeling of the monthly temperature of Gusau metropolis, using monthly data from January 1991 to December 2007. Instrument for the measurement for the temperature is a thermometer, situated at the Nigerian Meteorological Agency Gusau. The Box and Jenkins Seasonal Auto-regressive Integrated Moving Average technique was used for the analysis. The result of the analysis reveals that SARIMA(2, 1, 2) (1, 1, 1)12  is the best model for the data. The model was used for a five year forecast.









 CHAPTER ONE: GENERAL INTRODUTION
INTRODUCTION
Time series analysis and its application have become increasingly important in various field of research, such as business, economics, engineering, medicine, environometrics, social science, politics, and others. Since Box and Jenkins (1970,1976) published the seminal book time series Analysis: forecasting and control, a number of books and vast number of research papers have been published in this area, the goal of this book is to distill and integrate these research result into cohesive and comprehensible methodologies, and to provide a streamlined approach to time series analysis and forecasting. World-wide interest in global warming and climate changes has led to numerous trend detection studies. Most of climate change studies focus on changes in temperature and precipitation; however, temperature is very important as an environmental condition which influence the growth of plant, health, pollution etc.
Temperature is a physical property of matter that quantitatively express the common nations of hot and cold. Objective of low temperature are cold, while various degree of higher temperature are referred to as warm or hot. When a heat transfer path between them is open, heat spontaneously flows from bodies of a higher temperature to bodies of lower temperature. The flows rate increase with the temperature difference, while no heat will be exchanged between bodies of the same temperature, which are than said to be in “thermal equilibrium”. Serrin, j. (1986). Problems of temperature many physics and chemistry problems require the temperature to be an absolute temperature. This means the calculations require the temperature to be in the Kelvin temperature scale. More often than not, the data you take will be in Celsius. To convert Celsius to Kelvin require a simple formula:
K = ▫C + 273.15
Where
K is the temperature in Kelvin ▫C is the temperature in Celsius
1.1 HISTORICAL BACKGROUND
 The city of Gusau, home of  the most celebrated international farming and cultural festival is also the traditional home of the “HAUSAWA” ethnic group, as well as the headquarter of Gusau emirate and present Gusau local government area. Gusau is located in central of the state as a new settlement of hausawa and Fulani. It is interesting to note however that despite the prolonged hospitalities between the hausawa and Fulani which resulted in a series of battles, Gusau hadmiraculously remain inquired and independent until the arrival of the British colonialist. The means of the subsistence of the people of the area is mainly the famous river referred to as “GULBIN BAGA” the river and its plain stretching about 300 kilometer wide, provide ample opportunity for various horticultural activities which include cultivation of crops, fishing and cattle rearing. The famous Gusau farming festival has brought significance development to the town in term of social amenities and infrastructure. The climate of Gusau and its environment is of tropical continental types, essentially semi-arid in nature with gradually higher temperature throughout the year and marked seasonal rainfall. In general two season are recognize; the wet season and dry season (1997).The annual rainfall average is very slow is concentrated in a short season, which extent from mid may-mid October (ibid). the dry season last for more than 7 months being particular dry from November-march or April when drop of rainfall may fall. The harmattan persist from November-march during the normal vegetation growth ceases. Temperature remain very high in the day, but as a result of radiation due to cloudless sky, the night are very cold comes as it does from desert, the harmattan wind is not only dry but also dust-laden, such that stationary 0bject are quickly covered by dust particles bodily discomfort at this season is also considerable (Udo,1970).
1.2 AIM AND OBJECTIVES OF THE RESEARCH
The aim of this research work is to develop a SARIMA model for the monthly temperature of Gusau metropolis from January 1991 to December 2007.The model will then be used to provide forecast for the subsequent years. The following are the objectives of this research:
To identify the nature of time plot of monthly temperature of Gusau metropolis from January 1991 to December 2007.
To identify the correlogram of the data series through the use of Autocorrelation Function (ACF) and Partial Auto-correlation (PACF) 0f the data series.
To test whether or not there is need for Log Transformation of the data series, through the use of Range-mean plot.
To detect whether or not there exists periodicity in the data.
To identify the sarima model for monthly temperature of Zamfara state Gusau local government.
To forecast the future values of the expected monthly temperature.
  To evaluate the forecasted values using the data from 2010-2015
1.3 SCOPE AND LIMITATION OF THE STUDY
The project research was conducted base on SARIMA modeling approach of Box and Jenkins (1976), due to its power and flexibility in model fitting. The application in the research is limited to the monthly temperature records (%) Gusau local government of Zamfara state (1991-2007).


STATEMENT OF THE PROBLEM
A normal temperature is critical for good health. Low body temperature results when the body is unable to keep the body "thermostat" regulated within a safe range. Under normal circumstances, the body is able to generate and dissipate heat. The internal mechanisms can overcome most outside adversities of freezing cold or broiling heat. The ideal core temperature is considered to be around 98.6° Degree Fahrenheit or 37° degree Celsius. This temperature is however, the average body temperature and the overall normal temperature varies from a minimum of 97.7° Fahrenheit (36.5° Celsius) to a maximum of almost 99.5° Fahrenheit (37.5° Celsius).Normally, also this protective mechanism keeps your body temperature in the safe range in emergencies for maintaining life. Body temperature can fall due to numerous reasons .such as being exposed to cold weather or wearing soaked or wet clothing for a long time. On the other hand, abnormally low body temperature can also be a potential symptom of the following diseases and disorders:  Addison’s  Disease, Diabetes, Drug/alcohol abuse, Hypothyroidism, Infection, Kidney failure, Liver failure, Side effects of medications, Fast Breathing/Asthma, Stress, Cancer, etc. With this problems the project intended to model and accommodate the dynamics of conditional hetrocesdasticity of the monthly temperature, so as to find the best model to forecast the future of monthly temperature in the area in order to have a look on the outcome. 
.



1.5 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 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.
Autocovariance 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 INTRODUTION
Many aspects of temperature have been analyzed from various perspectives these include probability point estimation and the application of time series analysis. The following reference material was reviewed, each with information regarding temperature effects on data from tire/pavement noise measurements: sanberg 2002, sandberg 2004, Anfosso 2006, jabben 2006 and Bendtsen 2009. The measure temperature effect typically ranges from -0.028 t0 -0.056 dB per 1 ▫F. 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 monthly temperature. The history and development of time series analysis may be taught as having commenced in 1994 when Newton decomposed a light signal in to frequency compounds by passing the signal through a glass prison. Mathematical foundation began to be laid in the moods 1800s for the analysis of time series when goony represent white light as Fourier series. In the time period 1930-1950 substantial development in the area of time series analysis were provided by N. wiener. H. Crammer and A.N Kolmogorov. Box and Jenkins (1976) developed the time domain by collecting some of its major term and applying it to such important functions as forecasting and control. The range areas where time series analysis can be applied covers areas like Economics, Marketing, demography, meteorology, geophysics, agricultural, and medical science.

2.1 REVIEW OF RESEARCH WORK ON TEMPERETURE
Traditionally, published temperature criteria have relied on studies on resident and anadromous fish conducted in the Pacific Northwest, which may not directly apply to inland, resident rainbow trout in the southern portion of their range. The Big Creek study lies in the southern portion of the range of this species and contains inland resident rainbow trout. Recent research specific to California strains of rainbow trout and steelhead suggest that temperature tolerances of steelhead and rainbow trout in California may be higher than races in the Pacific Northwest. Temperature Preference and Growth the temperature criteria used in the PDEA primarily are based on recent studies focused on California rainbow trout stocks, which are summarized in a review by Myrick
and Cech (2001) that focused on California’s Central Valley O. mykiss populations.
 Myrick and Cech (2000) conducted studies with two strains of resident California rainbow trout (Eagle Lake and Mt. Shasta, both hatchery fish stocks are planted widely Attachment I Copyright 2007 by Southern California Edison Attachment I February 2007 in California) at temperatures of 10 - 25°C. They found that growth rates increased to a maximum near 19°C, and declined at temperatures greater than 19°C. This indicates that growth rates were greatest at 19°C and declined at temperatures above and below 19°C. Trout grew well at temperatures near 22°C, but growth rates declined rapidly as temperatures approached 25°C. These studies suggest if daily mean summer water temperatures are less than or equal to 20°C in the Project Area, conditions would be suitable for rainbow trout growth. These growth rates were consistently higher than those of American or Feather River steelhead. The Project area has been heavily stocked and rainbow trout introduced to almost every location in which they are now found (see Attachment H - Life History and Habitat Requirements of Fish Species in the Project Area). That the two hatchery strains had higher growth rates than the steelhead stocks suggests that studies utilizing hatchery rainbow trout strains may be more appropriate for determining temperature criteria in the Big Creek Project Area than studies that used steelhead, which are not present in the Project area. Of the two hatchery strains, the Mt. Shasta strain trout grew faster at the warmest temperatures than Eagle Lake trout. A large range of temperature preferences for rainbow trout/steelhead has been reported in the literature, with substantial regional variability. The preferred water temperatures for rearing juvenile steelhead on the American River are reported to range from 12.8°C to 15.6°C (CDFG 1991).
Bell (1986) reports a somewhat lower preferred range of 10 to 12.8°C for steelhead in the Pacific Northwest. In contrast, hatchery-reared, Central Valley steelhead consistently selected temperatures of 18 to 19°C, while wild Fish selected temperatures of about 17ºC (Myrick and Cech 2000). This lends support for the idea that O. mykiss populations in the southern portion of their range may have Higher temperature preferences than those in northern regions.
Myrick (1998) studied thermal preference of American River (Nimbus strain) steelhead over an 11 to 19°C range. The study did not reach an experimental acclimation temperature at which juvenile steelhead began to select cooler temperatures. Furthermore, selected temperatures closely matched the temperature at which highest growth rates were observed.
Myrick and Cech (2000) found hatchery Feather River steelhead acclimated to constant and cyclical thermal regimes had similar thermal preferences and selected temperatures in the 18 to 19°C range. They found that growth rate increased with ration level. Fish on a cyclical thermal regime (14 to 18°C) grew more slowly than under a constant regime, although the difference was not statistically significant. Stressful and Lethal Temperatures Based on available literature drawn largely from laboratory studies (Cherry et al. 1977, Raleigh et al. 1984, Currie et al. 1998, Coutant 1977) the upper incipient lethal temperature (UILT) for rainbow trout is within the range 25 to 30°C.
Eaton et al. (1995) identified upper temperature criteria for rainbow trout as 24.0°C. Myrick and Cech (2001) report critical thermal maximum (CTM) tolerances of 27.7 to 29.7°C for juvenile California steelhead, and as high as 32°C for Eagle Lake rainbow trout acclimated to Attachment I Copyright 2007 by Southern California Edison Attachment I February 2007 25°C. The temperatures that may be considered too warm or deleterious for a fish species depend upon the duration of exposure.
USEPA (1976) identified maximum weekly temperatures for survival for rainbow trout as 24°C. UILTs reported by USEPA (2001) for rainbow trout range from 24 to 26.9°C. For the purposes of the analysis in the PDEA, a daily maximum temperature of 24°C was selected as a temperature evaluation criterion for short-term high temperature exposure. The use of 24°C for short-term exposure may be considered conservative (overly protective) based on available information. The exposure considered in the evaluation criteria for daily maximum temperatures is generally one-hour or less. Since the temperatures cited above represent 24-hour to seven-day exposures, 24°C the lowest UILT identified, when used in comparison to an exposure of one hour or less, is very conservative.
Myrick and Cech (2000) found no strain-related (Eagle Lake and Mt. Shasta hatchery rainbow trout) differences in response to temperature of conversion efficiency, oxygen consumption rates, thermal tolerance or swimming performance, but the Mt. Shasta strain trout grew faster at the highest temperatures (22 to 25°C). Both strains were able to maintain weight at 25°C for 30 days, which the authors suggest may allow them to survive short (<1 month) periods of sublethal temperatures in California streams. To examine the effects of high temperatures frequently found in the lower portion of the San Joaquin River basin.
 Myrick and Cech (2002) tested effects of rearing temperatures of 19, 22 and 25°C on three California hatchery trout strains (Eagle Lake and Mount Shasta rainbow trout, and Kern River strain golden trout). As expected, growth rates were reduced at the higher temperatures, but the trout were capable of growing at temperatures 4 to 7°C higher than optimal temperatures, and they maintained their weight at temperatures close to incipient lethal temperatures. All strains were found to have a CTM between 29.6 to 32.0°C, although the authors recommend that the ILT values rather than CTM values should be used for determining upper temperature limits for water quality criteria. Myrick and Cech’s (2001) review indicated that steelhead can be expected to show significant mortality at chronic temperatures exceeding 25°C, although they tolerate temperatures as high as 29.6°C for short periods of time. However, the fish experience sub-lethal effects at temperatures below these limits.
The knowledge of temperature distribution is of basic interest for the understanding of the physical mechanism and for a selective manipulation of their properties. Hansan and James (1962). Warmth is one of the necessities of life. Vital activities are possible only between certain narrowly defined limits of temperature. Cold inhibits and excessive heat suspends them. Body heat is energy. It is employed not just in resisting cold, but also in accelerating cellular activities. Temperature, within certain narrow limits, is so absolutely essential to life that all functions are excited by any attempt at its variation. Animals are roughly divided into two major classes: warm-blooded and cold-blooded. This is according to whether they have means of producing and maintaining their own temperature or are dependent upon the surrounding medium (water or air) to provide it.  Robert A. (2005), in all changes of temperature outside the body, some compensation effort is required. But if our other relation are correct, the internal heat-regulating capacity of the body will be efficient. The maintenance of the heat-making mechanism of the body is an indispensable condition of health. Feeble and sick individuals who find it difficult or impossible to maintain normal temperature in a cold climate need to be kept warm. Chilling inhibits all functions of life and reduces their already greatly reduced stock of energy. The escape to warm climate is no mere luxury for such persons. Warmth of some degree is certainly a normal requisite of life. But experience and experiment have shown that when the temperature of the surrounding is out of all proportion to the need of the body and to its capacity to adjust itself, the body must and does suffer. The is not only discomfort, which normally causes us to seek relief from extreme of heat or cold, but there is some expenditure of energy in resisting extremes temperature. In lands where fogs, frost and darkness cramp the energies of man, as well as in regions where excessive and long-continued heat depresses his vital activities, life is handicapped. By means of clothing, housing and artificial heating arrangement, we are able to live in cold climates, By means of cooling system and a reduction of clothing, we live more comfortably in hot regions and seasons. But none of these arrangement are ideals. A warm climate serves man best; first-class habits of living enable him to live better in whatever climate he resides.Much of the world uses the Celsius scale (ºC) for most temperature measurement. It has the same incremental scaling as the Kelvin scale used by scientists but fixes its null points at 0ºC = 273.15k, approximately the freezing point of water ( at one atmosphere of pressure). The united state uses the Fahrenheit scale for common purposes, a scale on which water freezes at 32 ºF and boils at 212 0 F ( at one atmosphere of pressure ). Udo R. K (1970).


2.2 DEFINATION OF TIME SERIES
 A time series is a collection of observations of well-defined data items obtained through repeated
measurement overtime. For example measuring the value of retail sales each month of the year would comprises a time series. This is because sales revenue is well defined and consistently measured at equally spaced intervals. Data collected irregularly or only once are not time series.
Anderson T. W (1980), opined that time series is a sequences of observations ordered by time. Example include the noon temperature measured daily, the annual sales of passenger cars and the monthly average values of southern oscillation index (SOI), the number of people receiving unemployment benefit each month, and the number of bits of information sent through computer line per seconds. In each case, the observation are taking at regular time intervals.
A time series can be continuous or discrete depending on whether the observation is made continuously in time or only at specific time.
Most time series are stochastic in the sense that the future is only partly determined by past values.
2.3 METHOD OF TIME SERIES ANALYSIS
There are two main approaches used to analyze time series. Akaike, H.(1969).
These approaches are as follows:
Time domain approaches: analysis in the tine domain is most often used for stochastic observations. One common technique is the Box and Jenkeins ARIMA method which can be used univariate ( a single data set) or multivariate (comparing more two or more data set) analysis. The ARIMA technique used moving averages, detrending, and regression method to detect and remove autocorrelation in the data.
Frequency domain approaches: analysis in the frequency domain approaches is often used for
periodic and cyclical observations. Common techniques are spectral analysis and harmonic
analysis, and periodogram analysis. A special technique is past Fourier transform (FFT).
Mathematically, frequency domain techniques use fewer computation than time domain
techniques, thus for complex data, analysis frequency domain is most common.














                                             CHAPTER THREE: METHODOLOGY
3.0 INTRODUCTION
This chapter presents the method of research and procedure adopted for data collection and data analysis, methodology can be considered to include a multiple methods that can be applied in this research work. Therefore, this chapter covers method of research design, method of data collection and method of the data analysis.
3.1 THE DATA FOR THE ANALYSIS
The data used in this research is the monthly temperature data of Gusau local government in Zamfara state from January 1991 to December 2007. Instrument used for measuring temperature is a thermometer, situated at zamfara state meteorological inspectorate Gusau Air-strips. 
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. Time domain is usually parametric in nature and is based on direct modeling of the lagged relationships between a series and its past history, possibly to forecast its future values. The models for time series that are required to achieve optimal prediction and possible control are stochastic models. These models are used to calculate the probability of future values falling between upper and lower units specified. There are three primary stages in building a Box-Jenkins time series model: 1. Model Identification2. Model Estimation3. Model Validation
Box-Jenkins Model Identification
 The identification stage is the most important and also the most difficult: it consists to determine the adequate model from ARIMA family models. The most general Box-Jenkins model includes difference operators, autoregressive terms, moving average terms, seasonal difference operators, seasonal autoregressive terms, and seasonal moving average terms2.This phase is founded on the study of autocorrelation and partial autocorrelation.
The first step in developing a Box-Jenkins model is to determine if the series is stationary and if there is any significant seasonality that needs to be modeled.
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 stationary
3.2 METHOD OF DATA ANALYSIS
The method used to analyzed data in time series analysis involves the use of simple deterministic model such as the least square method or the use of complex stochastic model such as the moving average model (MA), Autoregressive model (RA), the Autoregressive moving average (ARMA), the Autoregressive integrated moving average (ARIMA). And seasonal Autoregressive integrated moving average (SARIMA). For the purpose of this research the seasonal Autoregressive integrated moving average was used, because it is the method that used as the basic for forecasting into the future. Ender’s and walters (2004).


3.3 TIME SERIES MODEL
A time series models is one which postulates a relationship amongst a number of sequence or time series, the time series model includes:
3.3.1 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
.
3.3.2 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.


 For   where {}  is a series of independent identically distributed (iid) random variables, and   is some constant.
The letter p denote the order of autoregressive model, defined how many time previous values the current value is related to. The model is called autoregressive because the series is regressed on to past values of itself.
Dickey and fuller (1979), stated that the autoregressive procedure is an extension of ordinary least square regression analysis specially designed for time series. In the absence of autocorrelated residuals, the linear regression procedure gives inaccurate estimate of how much of the series variability is accounted for the chosen predictors.
3.3.3 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


3.3.4 AUTOREGRESSIVE MOVING AVERAGE MODEL ARMA (p,q) PROCESS: 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
,
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.
3.3.5 AUTOREGRESSIVE INTEGRATED MOVING AVERAGE MODEL ARIMA (P,D,Q) PROCESS: 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

Where
3.3.6 SEASONAL AUTOREGRESSIVE INTEGRATED MOVING AVERAGE SARIMA (p,d,q) (P,D,Q) PROCESS: Pandit and wu (1983) said a time series  (p, d, q) (P, D, Q) representation if it can be expressed as :-

Where the AR components are:-

And the MA components are:-

Since the lag operator L is defined by:
Xt-1 = LXt
Where, the SARIMA (p,d,q) (P,D,Q)s 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.
3.4 BUILDING SARIMA MODEL
 The time series SARIMA building is a selection of the appropriate model for the data in achieving and iterative procedure based on the three fundament steps of Box and Jenkins (1976). These procedure are:
Model identification
Model estimation 
Model validation or Diagnostic checking.
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. If the data appears to be stationary no differencing is called for, and we identify d=0 where d is the order of difference of the series until its time plot appears to be stationary.
As mentioned earlier 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, usually the series’ first needs to be differenced until it is stationary i.e. the number of times the series needs to be differenced to achieve stationary. 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 commonly used tools in identification in time series are:-the time plot of the series, the time
Plot of autocorrelation at various lags (ACF), the time plot of the partial autocorrelation function (PACF), unit root test, seasonal unit root test and information criteria.
3.4.1.1 Time plot of a series:-This is define as a graph use to evaluate, pattern and behaviors in the data over time and also is a graphical representation of a time series of observed values  plotted against tine observations. It is a useful tool for Interpreting a set of autocorrelation co-efficient.
3.4.1.2 Time plot of Autocorrelation function:- Given a sample y0,y1,...,yT-1 of  T observations, we define the sample autocorrelation function to be the sequence of values
(1) rτ = cτ/c0, τ = 0,1,...,T − 1, wherein
(2)
Is the empirical autocovariance at lag τ and c0 is the sample variance. One should note that, as the value of the lag increases, the number of observations comprised in the empirical autocovariance diminishes until the final element cT−1 = T−1(y0 − y)(yT−1 – y) is reached which comprises only the first and last mean-adjusted observations. In plotting the sequence {rτ}, we shall omit the value of r0 which is invariably unity. Moreover, in interpreting the plot, one should be wary of giving too much credence to the empirical autocorrelations at lag values which are significantly high in relation to the size of the sample.

3.4.1.3 Time plot of partial Autocorrelation function:-The other analytical function that serves as a fundamental tool of Box–Jenkins time series analysis is the sample partial autocorrelation function (PACF). This partial autocorrelation function, used in conjunction with the autocorrelation function, can be used to distinguish a first-order from a higher order autoregressive process. It works in much the same way as a partial correlation. This function, when working at k lags, controls for the confounding autocorrelations in the intermediate lags. The effect is to partial out those autocorrelations, leaving only the autocorrelation between the current and kth observation. It is helpful to derive the partial autocorrelation function in order to understand its source and meaning. Consider the first-order autoregression process:
Yt = Φ1Yt-1 + et
YtYt-1 = Φ1Yt-1Yt-1 + etYt-1
E(YtYt-1) = Φ1E(Yt1Yt-1) + E(etYt-1).
With γ1 = autocovariance (ACV(Yt))
and γ0 = Variance (Yt), then
γ1 = Φ1γ0 .

3.4.1.4 UNIT ROOT TESTS
The detection of unit roots started to be studied in annual data (the so-called zero frequency). The extension of the resulting methodologies to consider seasonal frequencies occurred in two stages: first, the researchers studied the application to quarterly data —with the appearance of three additional frequencies; second, monthly data were considered, which imply eleven seasonal frequencies in addition to the usual one. As soon as the new methods were known alternative procedures were proposed. In this way, not only parametric tests but also semi parametric, nonparametric and Bayesian techniques were put forward. For each one of them the three stage process was a natural development. Furthermore, in each case there were different proposals concerning the form of the null and alternative hypotheses, not to mention the large number of different data generating processes which were considered. The consideration of broken trend alternative hypotheses added even more material to this huge amount of literature.
The history of (non-seasonal) unit root tests starts with Dickey and Fuller (1979) and the well known Augmented Dickey-Fuller (ADF) test with a non-stationary model as the null hypothesis. The first test for seasonal integration resembles a generalization of the ADF test for integration in annual data. Dickey, Hasza and Fuller (1984) (DHF from now on), following the methodology suggested by Dickey and Fuller (1979) for the zero-frequency unit-root case, propose a test of the hypothesis ρ = 1 against the alternative ρ < 1 in the model yt =ρyt−s +εt . The DHF test ⎯as well as similar ones proposed in the following years⎯ only allows for unit roots at all of the seasonal frequencies and has an alternative hypothesis which is considered rather restrictive, namely that all the roots have the same modulus. Trying to overcome these drawbacks Hylleberg et al. (1990) (from now on referred to as HEGY) propose a more general testing strategy that allows for unit roots at some (or even all) of the seasonal frequencies as well as the zero frequency. HEGY’s methodology allows to test for unit roots at some seasonal frequencies without maintaining that unit roots are present at all seasonal frequencies. Below are the list of unit root test use in the research.
a. 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.
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.
b.PHILLIPS-PERRON (p p) TEST:-phillips-perron (1988) test that a variable has a unit root. The null hypothesis is that the variable contains a unit root, and the alternative is that the variable was generated by a stationary process. phillips-perron uses Newey-west (1987) standard error to account for serial correlation where as the augumented Dickey-fuller test implemented

 The null hypothesis of unit root is accepted if the test statistic is greater than the critical values.
3.4.1.5 SEASONAL UNIT ROOT TEST
The HEGY (1990) procedure was extended for the case of monthly data in two different though similar directions. Franses (1991, 1991b) discusses a method to distinguish empirically between models (2) and (5) presented above.6 In his second paper this author shows that conventional autocorrelation checks cannot generally make this distinction because they are not discriminative. He also shows that considering a model like (5) or similar when (2) is more appropriate yields a deterioration of forecasting performance. Beaulieu and Miron (1993) (B&M from now on) use in a slightly different way the approach developed by HEGY to derive the mechanics of another procedure to test for seasonal unit roots using monthly data. These authors derive the asymptotics of HEGY’s procedure for monthly data and use Monte Carlo methods to compute the finite sample critical values of the associated test statistics. The main difference with Franses’ (1991a, 1991b) methodology is that B&M use mutually orthogonal regressors, obtaining a different somewhat more complicated test equation. Suppose that the series of interest (Xt) is generated by a general process like:
  φ(L) xt = α0 + α1t + αkDkt + εt
where εt is a white noise process and the deterministic terms include a constant, a linear trend and seasonal dummies. “We wish to know whether the polynomial in the backshift operator,
ϕ(L) , has roots equal to one in absolute value at the zero or seasonal frequencies. In particular, the goal is to test hypotheses about a particular unit root without taking a stand on whether other seasonal or zero frequency unit roots are present” (Beaulieu and Miron, 1993, page 307). The auxiliary regression model that allows to perform the test is given by the following equation:
φ(L)* Y13t = α0 + α1t + αkDkt + εt +πkYkt-1 + εt
where Yk k t ( = 1, 2,⋅⋅⋅,13) are auxiliary variables obtained by appropriately filtering the variable under study (Xt).7 The ϕ (L)* polynomial is a remainder with roots outside the unit circle which allows the augmentation necessary to whiten the errors in the estimation of the above equation. “In order to test
3.4.1.6 INFORMATION CRITERIA
a. The Akaike information criteria (AIC) is a measure of the relative quality of statistical models for a given set of data. Given a collection of models for the data, AIC estimate the quality of each model, relative to each of the other models.
Hence, AIC provides a means for model selection. AIC is founded on information theory: it offers a relative estimate of the information lost when a given model is used to represent the process that generates the data. In doing so, it deals with the trade-off between the goodness of fit of the model and the complexity of the model.
AIC does not provide a test of a model in the sense of testing a null hypothesis. i.e. AIC can tell nothing about the quality of the model in an absolute sense. If all the candidate models fit poorly, AIC will not give any warning of that
The general form for calculating AIC:
             AIC = 2k – 2in(L)
b. 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 formally define as
BIC = -2.inL + k .in(n).
Where
. L = the maximized value of the likelihood function of the model M.
c. The Hannan-Quinn information criterion (HQC) is criterion for model selection. It is an alternative to Akaike information criterion (AIC) and Bayesian information criterion (BIC). It is given as
BQC = -2Lmax + 2kin(in(n))
Where
Lmax is the log-likelihood, k is the number of parameters, and n is the number of observations.


3.4.2 MODEL ESTIMATION
  Anderson T.W (1980), in the estimation stage, parameter are been estimated for adequate model. The parameters of a SARIMA (p,d,q)(P,D,Q)s  model selected can be estimated consistently by least square or maximum likelihood estimation. The estimate of parameters used in the forecasting stage is to calculate new values of the series and the confidence interval for those predicted values. The estimation process is performed on transformed data before the forecast of the data is generated. Estimated coefficient are nearly correlated with one another.
3.4.2.1 LEAST SQUARES ESTIMATION
One of the assumptions underlying ordinary least squares (OLS) estimation is that the errors be uncorrelated. Of course, this assumption can easily be violated for time series data, since it is quite reasonable to think that a prediction that is (say) too high in June could also be too high in May and July. That kind of cyclical effect is indicative of positive autocorrelation, and it is quite common in time series data. But say we ignore this fact; why is it a problem to use OLS if the errors are autocorrelated? The following two tables can help to answer that. Consider a simple regression problem, and let be the first order autocorrelation of the errors  and be the first order autocorrelation of the predicting variable x (it’s likely that this, too, would exhibit autocorrelation; this is not a violation of any assumptions, but it can affect the properties of OLS estimators if there is also autocorrelation of errors). Further, assume the particular autocorrelation structure known as a first order autoregressive model (we’ll talk more about this a little later). The first problem with using OLS estimates in the context of autocorrelated errors is that they are inefficient; that is, they have higher variability (as estimates of the true parameters) than they should. The following table gives the efficiency of the OLS estimator of 1 compared to the best possible estimator (the efficiency is simply the ratio of variances):
3.4.3 MODEL VALIDATION AND 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 the applied to the model, in other to discover the ways in which a model is adequate so as to suggest appropriate modification. No model form ever represents the truth absolutely. It follows that given sufficient data; statistical test can discredit models which could nevertheless be entirely adequate for the purpose at hand. Therefore the best policy is to device the most sensitive statistical procedure possible. Model diagnostics for Box-Jenkins models is similar to model validation for nonlinear least squares fitting. Model diagnostics for Box-Jenkins models is similar to model validation for non-linear least squares fitting. That is, the error term t u is assumed to follow the assumptions for a stationary unvaried process. The residuals should be white noise (or independent when their distributions are normal) drawings from a fixed distribution with a constant mean and variance. If the Box-Jenkins model is a good model for the data, the residuals should satisfy these assumptions.
3.4.3,1 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:- 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 value.

3.5 FORECASTING
Forecasting involves basic definitions and assumptions. A decision needs to be made at current time t and the optimal decision depends on expected future value of a random variable, yt-h , the value being predicted or forecast. The number of time points forecast into the future forecast horizon is called the lead time, h. The value of the random variable for such a forecast is the value of yt-h . A forecaster would like to obtain a prediction as close as possible to the actual value of the variable in question at the concurrent or future temporal point of interest. As a rule, the more accurate the prediction, the less the cost of miscalculation. As the forecaster develops his model on the basis of the historical or estimation sample, he makes the first assumption that his model is a stable definition of the underlying data generation process. He conducts these tests on the validation period series. To do so, he extrapolates over the validation period and compares his predicted values to the actual values of the series. When he builds his model, he wishes to minimize the difference between his forecasts and the observed values of the process under examination during the validation period. This difference is known as the forecast error, et . One criterion for
measuring the precision of prediction is the sum of squared forecast errors. Amore commonly used criterion is the mean square forecast error (MSFE). This MSFE is the average difference between the true value and the predicted value,
Mean square forecast error: MSFEt (yt+h) =  2
where h = number of periods into the future horizon one wishes to forecast. This would be divided by T, but since T = 1, it is invisible. It would also be summed, but for one case, the sum is 1, so that is also invisible. It will, however, be shown that this can be estimated by the conditional expectation of yt h:
yt(h) = E(yt+h  yt , yt-1 , . . . , y1),

3.6 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.
3.6.1 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.
3.6.2 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).
3.6.3 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.












                                         

                                     

                                 CHAPTER FOUR: DATA ANALYSIS
4.0 INTRODUCTION
  This chapter is going to focus on time series modeling and forecasting the monthly temperature of Zamfara state from January 1991 to December 2007. Seasonal Autoregressive integrated moving average model of Box and Jankins was use to find and appropriate for data. The software Gretl  was use for the analysis.
4.1 THE DATA FOR THE ANALYSIS
The data used in this research is the monthly temperature data of Gusau, metropolis, from January 1991 to December 2007. Instrument for the measurement of temperature is a thermometer, situated at the Nigerian Meteorological Agency Gusau
4.2 TIME PLOT: A time plot of the original series conducted below (fig 4.2) shows the time plot of the monthly temperature of Gusau Metropolis, for a period of 17 years (1991-2007)






Fig 4.1 Time plot of the monthly Temperature
The observed time plot shows clearly that the mean is not constant but the variance is constant.   Furthermore we proceed to take the first difference of the original series to stabilize the mean.
4.3 THE PLOT OF ACF AND PACF OF THE DATA
The idea in this chapter is to examine the correlation structure of the data using the ACF and PACF. We proceed to examine the correlogram i.e the autocorrelation function (ACF) and partial auto correlation function in Fig 4.2Fig 4.2 ACF and PACF of the monthly temperature.
By observing the ACF of the series this shows clearly that there is persistence correlation up to lag 40  and the ACF clearly shows seasonal behavior because it is in a circle form or cyclic form, the lag are in oscillation form, it can be seen in lag(48). The seasonality is coming from the autocorrelation behavior. Looking at the (PACF) on the other hand, the correlation is decaying as the lags goes on but did not decay up to lag 72 even beyond lag 72, but there is damping effect even up to lag 75.


4.4 UNIT ROOT TEST
The Philips-perron (p p) test statistic that a variable has a unit root. The null hypothesis is that the variable contain a unit root, and the alternative is that the variable was generated by stationary process and decision role is to reject the null hypothesis when the value of test statistic is less than the critical value. The KPSS statistic tests the null hypothesis of stationarity against the alternative of non stationarity and the decision rule is to accept the null hypothesis when the value of the test statistic is less than the critical value. The results of the KPSS and PP tests are in Table 4.1 and 4.2
                  Table 4.1: Results of the PP Test before differencing
Number of Lag
1%                5%                    10%
Test Statistic

4
-3.4639        -2.8759       -2.5743
-0.6573

5
-3.4639        -2.8759       -2.5743
-1.3044

6
-3.4639        -2.8759       -2.5743
-0.2286

DICISION: from the table above the test statistic is greater than critical value at 1%,5% and 10% level of significance which make me to accept the null hypothesis and conclude that the series is not stationary before differencing.



  Table 4.2: Results of the KPSS Test before differencing
Number of Lag
 TEST
T-STATISTIC
10%                    5%           1%


           5
KPSS with trend
   0.512325
0.120                 0.148        0.217


KPSS without trend
   0.859467
0.348                 0.462        0.739

 
           4
KPSS with trend
   0.3110986
0.120                 0.148        0.217


KPSS without trend
   0.9219048
0.348                0.462         0.739

DECISION: The result of the KPSS test verify that at 10%, 5%, and 1% the series is not stationary because the test statistics is greater than the critical value. 
               










   Table 4.3: Results of the PP Test after differencing
Number of Lag
1%                5%                    10%
Test Statistic

4
-3.4641        -2.8760       -2.5744
-10.7628

5
-3.4641        -2.8760       -2.5744
-11.2313

6
-3.4641        -2.8760       -2.5744
-11.2127

DICISION: from the table above the test statistic is less than critical value at 1%,5% and 10% level of significance which make me to reject the null hypothesis and conclude that the series is  stationary before differencing.


Table 4.4: Results of the KPSS Test after differencing.
Number of Lag
 TEST
T-STATISTIC
10%                    5%                1%


           5
KPSS with trend
   0.012325
0.120                 0.148        0.217


KPSS without trend
   0.259467
0.348                 0.462        0.739

 
           4
KPSS with trend
   0.0110986
0.120                 0.148        0.217


KPSS without trend
   0.0219048
0.348                0.462         0.739

DECISION: The result of the KPSS test verify that at 10%, 5%, and 1% the series is stationary because the test statistics is less than the critical value. 










         

     



Fig 4.3 below shows the time series plot of the first difference of the original series
       Fig 4.3 Time plot of the first difference of the original series
By observing the time plot of the first difference above, it shows clearly that the mean and variance are constant after taking the first difference of the original series.
Fig 4.4 the ACP and PACF of first difference f monthly temperature
It can be observed upon visual inspection that the ACF and PACF of the first difference it possess strong and persistent periodic nature even after difference once. This clearly shows that the time series data has seasonal behavior.
Seasonal difference was therefore carriedout in order to removed the persistent pereiodicity found in the series. The plot of seasonal monthly temperature, ACF and PACF was analysed respectively.


4.4.1 TEST FOR SEASONAL UNIT ROOT
 The HEGY statistic that a variable has a unit root. The null hypothesis of the HEGY test is that a unit root exists and alternative hypothesis is that the variable has generated by a stationary process and the decision rule is to accept the null hypothesis when the value of test statistic is less than critical value.
                     Table 4.5: Results of the HEGY test
Number of lag
1%             5%                10%
Test statistic

4
-2.26         -1.95            -1.60
-2.294

5
-2.26          -2.01            -1.60
-2.386

6
-2.26           -1.99            -1.62
-2.286

DECISION: By observation the table of HEGY test above the test statistic is less than critical value at 1%, 5% and 10% level of significance, therefore we accept the null hypothesis which make us to conclude that the series is stationary and seasonality is remove at first seasonal differencing, and hence our order of seasonal differencing to be 1.
 Fig 4.5 plot of the seasonal difference of monthly temperature.
It can be seen upon visual inspection of the ACF and PACF of the seasonal difference of monthly temperature blow that, the persistent trend and periodicity witnessed earlier has now been removed. This make me to conclude that the data is stationary at this point.
              Fig 4.6 the ACF and PACF of seasonal difference of monthly temperature
4.5 MODEL IDENTIFICATION
   Table 4.6 Results of SARIMA model identification for monthly temperature
MODEL
AIC
BIC
HQ
LIKELIHOOD

SARIMA(2,1,2)(1,1,1 )12
714.2283
740.2465
724.7669
-349.1141

SARIMA(2,1,2)(1,1,0)12
748.8521
771.6180
758.0734
-367.4261

The table above shows the model that satisfied both stationarity and uncorrelated criteria 

4.6 PARAMETER ESTIMATES OF THE MODEL
Table 4.7 Results of SARIMA (2,1,2)(1,1,1)12 model estimation for monthly temperature
Variable
Coefficient
Std. Error
     Z
P-Value

Constants
Phi_1
Phi_2
Phi_1
theta_1
theta_2
theta_1
 -0.000842623
 −0.676540
  0.298117
  0.0301124
  -0.0421266
  -0.957871
  -0.99994
0.00105437
0.0852372
0.0706286
0.0746507
0.0552896
0.056845
0.137437
-0.7992
−7.937
4.221
0.4034
-0.7619
-16.85
-7.276
0.4242
2.07e-015 ***
2.43e-0.5 ***
0.6867
 0.4461
1.02 e-063 ***
3.44 e- 013***

The model parameters were found to be significant by comparing the choosing alpha (α) at 5% with respective p-value. It can be seen that the p-value is less than alpha (α) at 5% in almost all the estimation. This confirms that the model parameters are significant.








Table 4.8 Results of SARIMA(2,1,2)(1,1,0)12 model estimation for monthly temperature
Variable
Coefficient
Std. Error
Z
p-value

Constants
Phi_1
Phi_2
Phi_1
theta_1

theta_2
−0.00119648
−0.671009
0.291024
−0.497805
−1.81090e-08
−1.00000
0.00215107
0.0707199
0.0713257
0.0654806
0.0225588
0.0225588
−0.5562
−9.488
4.080
−7.602
−8.027e-07
−44.33
0.5781
2.35e-021 ***
4.50e-05  ***
2.91e-014 ***
1.0000
0.0000    ***

The model parameters were found to be significant by comparing the choosing alpha (α) at 5% with respective p-value. It can be seen that the p-value is less than alpha (α) at 5% in almost all the estimation. This confirms that the model parameters are significant.
4.7 MODEL 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.
4.7 The ACF and PACF of SARIMA (2, 1, 2) (1, 1, 1)12             
Fig 4.7 the ACF and PACF of SARIMA (2, 1, 2) (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 4.8 the ACF and PACF of SARIMA (2, 1, 2) (1, 1, 0)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.







TIME PLOT OF RESIDUAL

Fig 4.9 The residual plot of the SARIMA (2, 1, 2)(1, 1, 1)12
The plot shows that the time plot has a wave-like pattern; the plot showed that the series has constant mean and variance. The plots indicate the series is stationary

Fig 4.10The residual plot of the SARIMA (2, 1, 2)(1, 1, 0)12
The plot shows that the time plot has a wave-like pattern; the plot showed that the series has constant mean and variance. The plots indicate the series is stationary








4.9 THE TEST STATISTIC FOR NORMALITY
Table 4.9 Results of JaqueBera test for humidity time series
MODELS
TEST STATISTICS
P-VALUE

SARIMA(2,1,2)(1,1,1)12

SARIMA(2,1,2)(1,1,0)12

6.63513

1.272
0.036241

0.52948


DECISION: Since from the table above p-value is greater than alpha (ɑ) then we accept null hypothesis and conclude that the error term are normally distributed.
 Fig 4.11 Test statistic for normality of SARIMA (2, 0, 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. Otherwise the error term is non- normally.


          Fig 4.12 Test statistic for normality of SARIMA (2, 0, 2) (1, 1, 0)12
From the above histogram, it shows a bell shape, this can lead us to believe that the error term is normally distributed. Otherwise the error term is non- normally.
4.9 FORECASTING
Here, the five year of the model i.e. SARIMA (2, 1, 2) (1, 1, 1)12 will be forecast for the subsequent years to come. Forecasting (i.e. predicting the future value on the basic of the past).The SARIMA (2, 1, 2) (1, 1, 1)12 use appears to have a minimum standard error, therefore, the model can be used for future predictions. 
Table 4.10 Results of SARIMA (2,1,2)(1,1,1)forecast values for five years..                             
                                 Pridiction          std err              95% C I


 2010:01                       31.0                1.42            28.3 -      33.8

 2010:02                       33.7                1.42            30.9 -      36.5

 2010:03                       37.0                1.42            34.3 -      39.8

 2010:04                       38.6                1.42            35.8 -      41.4

 2010:05                       36.3                1.42            33.5 -      39.1

 2010:06                       32.8                1.42            30.0 -    35.6

 2010:07                       30.3                1.42            27.5 -    33.1

 2010:08                       29.3                1.42           26.5 -    32.1

 2010:09                       31.0                1.42           28.2 -      33.8

 2010:10                       33.3                1.42           30.5 -      36.1

 2010:11                       33.5                1.42           30.7 -      36.2

 2010:12                       31.8                1.42           29.0 -      34.5

 2011:01                       30.8                1.42           28.0 -    33.6

 2011:02                       33.4                1.42           30.6 -    36.2

 2011:03                       36.8                1.42           34.0 -    39.5

 2011:04                       38.3                1.42           35.5 -      41.1

 2011:05                       36.1                1.42           33.3 -      38.8

 2011:06                       32.5                1.42           29.8 -      35.3

 2011:07                       30.0                1.42           27.2 -    32.8

 2011:08                       29.0                1.42           26.2 -      31.8

 2011:09                       30.7               1.42           27.9 -      33.5

 2011:10                       33.0               1.42           30.2 -      35.8

 2011:11                       33.2               1.42           30.4 -      36.0

 2011:12                       31.5               1.42           28.7 -      34.2

 2012:01                       30.5              1.42           27.7 -      33.3

 2012:02                       33.1              1.42           30.4 -      35.9


 2012:04                       38.0             1.42           35.2 -      40.8

 2012:05                       35.8             1.42           33.0 -      38.5

 2012:06                       32.2             1.42           29.5 -            35.0

 2012:07                       29.7             1.42           26.9 -    32.5

 2012:08                       28.7             1.42           25.9 -      31.5

 2012:09                       30.4             1.42           27.6 -      33.2

 2012:10                       32.7             1.42           29.9 -      35.5

 2012:11                       32.9             1.42           30.1 -      35.7

 2012:12                       31.2             1.42           28.4 -      33.9

 2013:01                       30.2             1.42           27.4 -      33.0

 2013:02                       32.8             1.42           30.0 -      35.6

 2013:03                       36.2             1.42           33.4 -      39.0

 2013:04                       37.7             1.42           34.9 -      40.5

 2013:05                       35.5             1.42           32.7 -      38.2

 2013:06                       31.9             1.42           29.1 -      34.7

 2013:07                       29.4             1.42            26.6 -           32.2

 2013:08                       28.4             1.42           25.6 -      31.2

 2013:09                       30.1             1.42           27.3 -            32.9

 2013:10                       32.4             1.42           29.6 -             35.2

 2013:11                       32.6             1.42           29.8 -             35.4

 2013:12                       30.8             1.42           28.1 -             33.6

 2014:01                       29.9             1.42            27.1 -      32.7

 2014:02                       32.5             1.42            29.7 -             35.3

 2014:03                       35.9             1.42          33.1 -      38.7

 2014:04                       37.4             1.42          34.6 -      40.2

 2014:05                       35.2             1.42          32.4 -    37.9

 2014:06                       31.6            1.42             28.8 -               34.4


 2014:07                       29.1            1.42        26.3 -      31.9

 2014:08                       28.1           1.42        25.3 -      30.9

 2014:09                       29.8           1.42        27.0 -      32.6

 2014:10                       32.1           1.42        29.3 -      34.9

 2014:11                       32.3           1.42        29.5 -    35.1

 2014:12                       30.5            1.42        27.7 -    33.3

 2015:01                       29.6            1.42        26.8 -      32.4

 2015:02                       32.2            1.42        29.4 -      35.0

 2015:03                       35.6            1.42        32.8 -    38.3

 2015:04                       37.1            1.42         34.3 -      39.8

 2015:05                       34.8           1.42          32.1 -      37.6

 2015:06                       31.3           1.42        28.5 -      34.1

 2015:07                       28.8           1.42          26.0 -      31.6

 2015:08                       27.7           1.42        25.0 -      30.5

 2015:09                       29.5           1.42          26.7 -    32.3

 2015:10                       31.7           1.42          29.0 -    34.5

 2015:11                       31.9           1.42          29.2 -      34.7

 2015:12                       30.2           1.42          27.4 -      33.0

As we can see from the predicted values in March and April has the highest prediction which confirms that our model is good and valid.

Fig 4.13 Time plot of the two years forecast of SARIMA (2, 1, 2) (1, 1, 1)12
As it shown in the figure above the difference between actual series and forecasted series is small, that is to say the model has good prediction power.

Table 4.11 Results of SARIMA (2,1,2)(1,1,0)forecast values for five years..
OBSERVATION
Pridiction
std err
95% C I

2010:01
31.0
2.24
26.6 -     35.3

 2010:02
34.7
2.28
30.2 -     39.2

 2010:03
38.2
2.28
33.7 -     42.6

 2010:04
39.4
2.28
34.9 -     43.8

 2010:05
35.9
2.28
31.4 -     40.4

 2010:06
33.9
2.28
29.4 -     38.3

 2010:07
31.5
2.28
27.0 -     36.0

 2010:08
28.8
2.28
24.4 -     33.3

 2010:09
30.0
2.28
25.5 -     34.5

 2010:10
33.1
2.28
28.7 -     37.6

 2010:11
34.6
2.28
30.1 -     39.0

 2010:12
31.6
2.28
27.1 -     36.1

 2011:01
30.5
2.48
25.6 -     35.3

 2011:02
34.3
2.51
29.4 -     39.3

 2011:03
38.0
2.51
33.1 -     43.0

 2011:04
39.3
2.51
34.4 -     44.2

 2011:05
35.8
2.51
30.9 -     40.7

 2011:06
33.7
2.51
28.8 -     38.6

 2011:07
31.4
2.51
26.4 -     36.3

 2011:08
28.7
2.51         23.8 -     33.6

 2011:09
29.8
2.51         24.9 -     34.7

 2011:10
33.0
2.51         28.1 -     38.0

 2011:11
34.6
2.52         29.7 -     39.6

 2011:12
31.6
2.52         26.7 -     36.5

 2012:01
30.5
2.73         25.2 -     35.9

 2012:02
34.3
2.76         28.9 -     39.7

 2012:03
37.9
2.76         32.5 -     43.3

 2012:04
39.1
2.76         33.7 -     44.6

 2012:05
35.7
2.76         30.2 -     41.1

 2012:06
33.6
2.76         28.2 -     39.0

 2012:07
31.2
2.77         25.8 -     36.7

 2012:08
28.5
2.77         23.1 -     34.0

 2012:09
29.7
2.77         24.3 -     35.1

 2012:10
32.9
2.77         27.4 -     38.3

 2012:11
34.4
2.77         29.0 -     39.8

 2012:12
31.4
2.77         25.9 -     36.8

 2013:01
30.3
2.95         24.5 -     36.1

 2013:02
34.1
2.97         28.3 -     39.9

 2013:03
37.8
2.97         31.9 -     43.6

 2013:04
39.0
2.98         33.2 -     44.8

 2013:05
35.5
2.98         29.7 -     41.3

 2013:06
33.4                   
2.78      23.5-34.5

 2013:07
31.1
2.98
25.3 -     36.9

 2013:08
28.4
2.98
22.5 -     34.2

 2013:09
29.5
2.98
23.7 -     35.4

 2013:10
32.7
2.98
26.9 -     38.5

 2013:11
34.3
2.98
28.5 -     40.1

 2013:12
31.2
2.98
25.4 -     37.1

 2014:01
30.2
3.15
24.0 -     36.4

 2014:02
34.0
3.18
27.7 -     40.2

 2014:03
37.6
3.18
31.4 -     43.8

 2014:04
38.8
3.18
32.6 -     45.1

 2014:05
35.4
3.18
29.1 -     41.6

 2014:06
33.2
3.18
27.0 -     39.5

 2014:07
30.9
3.18
24.7 -     37.2

 2014:08
28.2
3.18
22.0 -     34.4

 2014:09
29.4
3.18
23.1 -     35.6

 2014:10
32.5
3.19
26.3 -     38.8

 2014:11
34.1
3.19
27.9 -     40.3

 2014:12
31.0
3.19
24.8 -     37.3

 2015:01
30.0

3.35
23.4 -     36.5

 2015:02
33.8
3.37
27.2 -     40.4

 2015:03
37.4
3.37
30.8 -     44.0

 2015:04
38.6
3.37
32.0 -     45.2

 2015:05
35.2
3.37
28.6 -     41.8

 2015:06                       33.1                 3.37         26.4 -     39.7

 2015:07                       30.7                 3.37         24.1 -     37.4

 2015:08                       28.0                 3.37         21.4 -     34.6

 2015:09                       29.2                  3.37         22.6 -     35.8

 2015:10                       32.3                  3.37         25.7 -     38.9

 2015:11                       33.9                   3.38         27.3 -    40.5

 2015:12                       30.9                   3.38         24.2 -    37.5

As we can see from the predicted values in March and April has the highest prediction which confirms that our model is good and valid.

Fig 4.14 Time plot of the five years forecast of SARIMA (2, 1, 2) (1, 1, 0)12
As it shown in the figure above the difference between actual series and forecasted series is small, that is to say the model has good prediction power.








4.9.1 FORECAST EVALUATION
         Table 4.12: Results for the forecasting
Model
RMSE
MSE

SARIMA(2,1,2)(1,1,1)12
1.5486
2.3982

SARIMA(2,1,2)(1,1,0)12
1.6794
2.8203

The SARIMA(2,1,2)(1,1,1)12  model was chosen to be the best because it has the least value of root mean square error (RMSE) and mean square error (MSE).












CHAPTER FIVE: SUMMARY, CONCLUSION, RECOMMEDATION AND RETERENCE
5.0 SUMMARY
   This project work has five chapters; chapter one comprises of the introduction, historical background, aim and objectives of research, scope and limitation of the study, significance of the study and basic definitions of some terms, data collection.
Chapter two is made up of application of time series in many field, some reviewed research work on monthly temperature and the use of time series analysis.
Chapter three comprise the method of data analysis, model of time series sarima model building and statistical test in time series analysis.
The basic procedure in SARIMA model building were explained as contained in chapter four and gretl software were used in the analysis.
5.1 CONCLUSION
One of the most important advantages of time series forecasting is that it helps to determining in advance what will happen in the near future.
    SARIMA (2, 1, 2) (1, 1, 1)12 was chosen as our best model for the series, because it has the lowest AIC and log likelihood.



5.2 RECOMMENDATION
From the discussion made so far in the review of the related literature and also the conclusions drawn from the result of the analysis carried out in chapter four. We would like to recommend the following.
SARIMA methodology pioneered by Box and Jenkins (1979) should be maintained as the most effective tool for analyzing time series data.
Adequate awareness and information about changes in the weather pattern should be made available to the general public, especially sensitive groups like farmers, fishermen etc.
Finally, further research work should be done on monthly temperature in order to study its ever changing pattern.   







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