Fourier Series Modelling of Rainfall Data in Southeastern Nigeria

ABSTRACT Rainfall is a climate event that causes significant damage both in natural environment and human live. Rainfall forecast is an important issue in water resource planning. Due to seasonal behaviour of rainfall, Fourier series model was fitted to the mean monthly rainfall series inSouth-Eastern region of Nigeria, in which R2 = 0.74for both Enugu and Owerri inSouth-Eastern region of Nigeria. By examining the correlogram of residual associated with the model we found out that it shows no periodic oscillating nature of time series for both residual series associated after fitting fourier series model to the mean monthly rainfall. Also that there are no significant trend between residual and the dependent variables and no significant trend between residual and independent variables.Hence we use fourier series model fitted to obtain some forecast. The forecasted values and observation values are close.

TABLE OF CONTENTS

Title Page - - - - - - - - - - i

Abstract - - - - - - - - - - iii

Table of Contents - - - - - - - - - iv

CHAPTER ONE

1.1 Introduction - - - - - - - - - 1

1.2 Significance of Study - - - - - - - 2

1.3 Aims and Objective - - - - - - - - 2

1.4 Scope and Limitation of Work - - - - - - 3

CHAPTER TWO

Literature Review - - - - - - - - - 4

CHAPTER THREE

METHODOLOGY

3.1 Definition - - - - - - - - - 16

3.1.1 Time Series - - - - - - - - - 16

3.1.2 Fourier Series - - - - - - - - 16

3.1.3 Periodic Function - - - - - - - - 16

3.1.4 Principle of Parsimony - - - - - - - 16

3.1.5 Stochastic Process - - - - - - - - 17

3.1.6 Sample Mean- - - - - - - - - 17

3.1.7 Sample Variance - - - - - - - - 17

3.1.8 Autocovariance - - - - - - - - 17

3.1.9 Autocorrelation - - - - - - - - 18

3.1.10 Seasonal Function - - - - - - - - 18

3.1.11 R2 - - - - - - - - - - 19

3.2 A General Approach to the Time Series Modeling - - - 19

3.2.1 Testing for Significant Trend - - - - - - 20

3.2.2 Condition for the Use of Fourier series - - - - - 21

4

3.3 Goodness of the Fit Test and Method of Selection of Variables for a Model in

Fourier series Model - - - - - - - - 21

3.3.1 Testing for the Significance of Estimated Parameter - - - - 22

3.3.2 Approach to Model Selection in Time Series - - - - 23

3.4 Diagnostic Checking of fourier series model- - - - - 24

3.5 Estimating the Regression Parameters, a by the Method of Ordinary Least

Squares (OLS)- - - - - - - - - 27

3.6 Variance and Standard Error of the Estimates Parameter - - - 27

3.7 Confidence Interval for the Parameter - - - - - 29

CHAPTER FOUR

Analysis

4.1 Testing for the Significance of Trend - - - - - - 30

4.2 Fourier series model fitted for mean monthly Rainfall data of Enugu and Owerri 40

4.3 Fourier series model fitted using Significance Parameters for mean Monthly Rainfall

data of Enugu and Owerri - - - - - - -

40

4.4 Estimation of Parameter of Fourier Series Model by method of ordinary least square

of Mean Monthly rainfall of Enugu and Owerri series- - - -

41

4.5 Variance and Standard Error of estimated Parameter of Fourier function of mean

monthly rainfall series of Enugu and Owerri - - - - - 45

4.6 Testing for Adequacy of Fourier Series Model of Mean Monthly rainfall of Enuguand

Owerri - - - - - -- - - - - 47

4.7 Forecasting for Mean Monthly Rainfall of Enugu and Owerri - - 52

CHAPTER FIVE

SUMMARY, CONCLUSION AND RECOMMENDATION

5.1 Summary - - - - - - - - - 54

5.2 Conclusion - - - - - - - - - 55

5.3 Recommendation - - - - - - - - 55

References - - - - - - - - - 56

Appendix - - - - - - - - - 60