CHAPTER ONE

INTRODUCTION

1.1 BACKGROUND OF STUDY

Climate is the average atmospheric condition of an area over a considerable time. The term

weather should not be confused with climate, though they are closely related to each in the study

of Meteorological and climatology. It is obvious that people do not say that the climate of the

day is warm or cold, but we do talk of warm water, a cold morning, a sunny afternoon, a rainy

day or a chilly night. Any causal remarks about the atmospheric condition of a certain place at a

certain time are about weather. Weather is never stable and cannot be generalized.

Rainfall is a primary driving variable in nearly all physical and biological processes. The

fluctuations of rainfall every year has brought some treats to farmers and other sectors that make

use of rainfall to promote their production. One of the recent suggestions on how to boost food

production and raw material for agro-based industries is sectionalizing agricultural production.

That is it allow areas produce those crops suitable for their land. Hence, to determine the type of

crop suitable for any particular area, the knowledge of its rainfall variation is required. Countries

have always be taken unawares by drought disaster and this has cause some billion of naira and

dollars, just because of adequate statistical climatologically information. This problems has

become enormous especially in Nigeria, who loses their crops almost every year by planting

either too early or two late, since they lack the knowledge of rainfall variation.

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It is our believe here that information from this study will go a long way in helping the people to

know the rainfall variation. So that more will engage in more meaningful agricultural activities

that will enable them climb few steps in their economic ladder.

1.2 AIMS AND OBJECTIVES

1. To fit an appropriate time series model for the monthly rainfall.

2. To make forecast of the monthly rainfall for twelve steps ahead.

1.3 SCOPE OF STUDY

This study is restricted to the monthly rainfall values that were recorded at the Nigeria

Meteorological Station, Benin.

1.4 LITERATURE REVIEW

Many works have been done on rainfall over the years to lay emphasis on the fluctuations of

rainfall. One of such is;

Chukwukere (1983) carried out a statistical study of the monthly rainfall at Isujaba in Imo State

from 1972-1982, He came out with the result that rainfall of Isunjaba had fairly increasing linear

trend for the period considered and showed a regular cyclical movement.

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Hutchmson and Sam (1984) illustrate the patchy nature of daily rainfall in Gambia, particularly

at the start of the season in June. They point out that after the first rain, (day with 12.7mm being

used as a threshold). The second heavy rain may be expected 10 days later each succeeding rain

occurs at ever decreasing intervals.

Anarado (1993) who carried out an analysis on the rainfall in Nsukka, Enugu State from 1972-

1992. He found that the data has no significant trend value, it was neither increasing nor

decreasing over the period.

Chapman (1994) in an article on stochastic models for daily rainfall collected a long sequence of

daily rainfall and compares the performance of a range of models which have been proposed for

the stochastic generation for daily rainfall data using historical data from a set of rainfall sites.

Maximum likelihood estimators are used throughout for parameter estimation, and the Akaike

information criterion was used as a guide to parsimony in the number of parameters required.

The result showed that it is difficult to make substantial reductions in the number of parameters

used in an existing model developed.

Jimoh and Webster (1996) in a study on optimum order of a markov chain model for daily

rainfall in Nigeria noted that although models of Order 1 have been successfully employed, there

remains uncertainty concerning the optimum order for such models. This paper is concerned with

estimation of the optimum order of Markov chains and, in particular, the use of objective criteria

of the Akaike and Bayesian Information Criteria (AIC and BIC, respectively). Using daily

rainfall series for five stations in Nigeria, it has been found that the AIC and BIC estimates vary

with month as well as the value of the rainfall threshold used to define a wet day. There is no

apparent system to this variation, although AIC estimates are consistently greater than or equal to

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BIC estimates, with values of the latter limited to zero or unity. The optimum order is also

investigated through generation of synthetic sequences of wet and dry days using the transition

matrices of zero-, first- and second-order Markov chains. It was found that the first-order model

is superior to the zero-order model in representing the characteristics of the historical sequence

as judged using frequency duration curves. There was no discernible difference between the

model performances for first- and second-order models. There was no seasonal variation in the

model performance, which contrasts with the optimum models identified using AIC and BIC

estimates. It is concluded that caution is needed with the use of objective criteria for determining

the optimum order of the Markov model and that the use of frequency duration curves can

provide a robust alternative method of model identification. Comments are also made on the

importance of record length and non-stationary for model identification.

Ejekwummadu (1998) carried his analysis on the monthly rainfall of Owerri from 1987 to

1997.He came out with the result that rainfall of Owerri for the period under consideration had

fairly increasing linear trend. The months of wet seasons was shown to be between March and

October with its peaks at August and September. He found out that, there was no evidence of

cyclical movement in the data. He forecasted the rainfall values for 1998 using his data and

found out that the forecast values and the original data value were very close and concluded that

his analysis was good.

Nnaji (2001) in his study forecasting seasonal rainfall for agricultural decision-making in

northern Nigeria used the Least Absolute Deviation (LAD) multiple regression equation as a

model for forecasting rainfall in the savanna agro-ecological region of northern Nigeria. The

model is employed in forecasting seasonal rainfall, in order to assess the climatological success

range for improving the production of three staple cereals grown in the region. Historic climate

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data spanning 35 years are used to generate a computer-based statistical model, which utilizes

November, December and January (NDJ) values of identified climate controlling variables in the

region, and the previous year’s rainfall as predictors. Nineteen synoptic stations in the study area

are selected, and because of high inter-annual variability of rainfall in the region, grouped into

three sub-regions using cluster analysis. Seasonal rainfall was forecasted in a probabilistic

fashion for respective sub-regions and a measure of the anticipated variability about the

forecasted value established using confidence bounds. The model was tested in recommending

rain-fed potentiality for growing corn, sorghum and millet, during 1991–1995, by deriving

probabilistic estimates of receiving critical rainfall threshold for growing each cereal. Result

indicated with high probability, 0.98 and 0.91, that millet and sorghum, respectively, can be

grown without irrigation under rain-fed conditions, while corn which required more seasonal

rainfall, had the lowest probability at 0.60.

Olawepo (2002) carried out a research on time series analysis monthly rainfall of Enugu urban

from (1981-2006). He noted in his work that the data was not stationary and also has seasonal

variation. After removing the trend and the seasonal variation, making the data to be stationary,

he identified seasonal autoregressive integrated moving average process of order (1,2,1) as being

the appropriate model for the series. He validated his model by comparing the original values of

the data with the forecast value which are very close to each other.

Asogwa (2004) studies the time series analysis of mean daily temperature and rainfall of Nsukka

from January 2001 to December 2002. Her concern was to examine the relationship between the

temperature and rainfall and also to fit an appropriated model for the data. She employed

autoregressive integrated moving average (ARIMA) approach and applied Box and Ljung (1978)

test for diagnostic. She concluded that ARIMA (0,1,1) was the model suitable for both data and

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that there exists a relationship between temperature and rainfall, but only 2.2% of rainfall were

explained by temperature and 98.8% were due to other factors that influence rainfall.

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CHAPTER TWO

DATA PRESENTATION

2.1 SOURCE OF DATA

The data for this study were collected from the Department of the Nigeria Meteorological

Station, Benin City Edo State Capital. The data are recorded rainfall values from January 1981 to

December 2000 making up two-hundred and forty data points.

The Meteorological Station records monthly rainfall value measured using rain-gauge. The

centre has several stations that keep the record of daily rainfall, monthly rainfall, average

rainfall, daily temperature, monthly temperature, wind relative humidity etc. Rainfall is

measured with graduated cylinder that has 3.8cm (11

2 inch) diameter. The reading is done at eye

level and to an accuracy of 0.25mm (0.01 inch) for greater accuracy. For Meteorological

readings, a rainy day is reckoned as a period of 24 hours with at least 0.25mm (0.01 inch) or

more rain being recorded. If the amount exceeds 1mm (0.04 inch) it is considered a wet day. The

rain gauge must be examined everyday to make accurate records.

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Table 1: monthly rainfall of Benin City, Edo State.

Year Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

1981 5.6 6.8 108.4 119.8 277.2 180.6 276.1 230 394.8 178.1 12.9 1.1

1982 101.9 111.4 98.2 211.4 146 174.1 243.9 66.8 391 381.9 42.4 20

1983 4.9 49.4 39 76.8 267.4 277.2 166.5 142.8 425.2 50.3 29.8 22.8

1984 6.4 45.8 87.3 59.3 120.4 134.1 223.8 181.8 235.1 154.8 4.4 2.5

1985 8.7 16 110.5 33.2 173.4 202.5 244.3 305.2 197.9 147.5 106.1 50

1986 5.6 45.1 109.4 52.3 162.7 65.1 214.2 117.7 231.3 166.5 67.1 30.2

1987 0.8 74.5 100.2 112.5 157.4 217 269.5 722.5 348.1 299.8 39.9 28.5

1988 7 71.1 154.8 136.5 168 227 393 191.2 445.4 273.2 23.3 58.6

1989 10 25.8 66.1 152 140.4 340 279.2 427.8 157.3 365.1 14.2 25.1

1990 19.6 18.8 55 256.8 181.4 204.1 353.4 614.5 296.9 289.9 33.7 68.6

1991 18.9 58.1 123.5 386.3 196.7 207.2 656.2 382.6 268 267.9 39.2 11.9

1992 0.3 0.2 41.4 222.7 240 335.8 515.9 76.4 256.3 292.2 35.6 16.1

1993 5.1 9.6 135 95.4 198.2 208.8 191.4 433.9 257.6 174.2 108.1 48.6

1994 27.5 14.6 111.4 149.8 327.9 351.3 444.4 461.2 391.8 204.5 43 21.8

1995 13 50.6 165.4 217.9 226.9 256.2 383.8 580.8 382.2 240.2 124.7 9.1

1996 7 92.6 188.2 298.2 322.2 281.1 182.3 392.3 476 292.7 5.6 1.1

1997 75.3 80 104 230.9 305.3 203.3 285 258 300 285 15.6 2.1

1998 44.1 1.8 104.6 104.8 214.6 214.4 506.1 95.6 387.9 244 58.8 42

1999 86.3 64.4 98.3 119.6 161.7 256.8 412.3 232 369 472.5 97.8 9.4

2000 4 73 60.8 170 191.8 413.7 294.7 237.9 345 351.2 49 48.7

Table 1 shows the monthly rainfall of Benin, Edo State.

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Table 2: Descriptive Statistics for Rainfall

Statistics Value

N

Range

Min

Max

Mean

Std. deviation

Skewness

Kurtosis

240

722.30

0.20

722.50

1.716

144.93845

0.952

0.691

Since skewness is positive, the data are positively skewed or skewed right, showing that the right

tail of the distribution is longer than the left. The distribution is moderately skewed since 0.5 <

Skd (0.952) < 1.

Since kurtosis (0.691) < 3, hence the distribution is platykurtic i.e less sharply peaked than the

normal distribution.

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The standard deviation of 144.93845 shows that the monthly rainfall are widely spread which

indicates that the rainfall values of Benin city are not clustered. Since the skewedness is positive,

it shows that the frequency polygon of the data has a longer tail to the right, which also tells us

that majority of the data points are less than the mean.

Seasonal Decomposition Graph

From the plot, it is clear that it has no upward movement indicating that there is no trend. It is

also observed that August has the highest rainfall since it has the highest peaks and December

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the least rainfall since it has the most troughs in the plot. Having obtained all the objectives of

plotting the graph we look forward to analyzing the data.

CHAPTER THREE

DATA ANALYSIS

3.1 MODEL IDENTIFICATION

3.1.1 AutoCorrelation Function (ACF)

The autocorrelation and partial autocorrelation functions are always essential in time series

model identification. Identification and Estimation are necessarily overlapping G. E. P Box et al

(June 1969).

The ACF is mathematical representation of the degree of similarity between a given time series

and a lagged version of itself over successive time intervals. It is the same as calculating the

correlation between two different time series, except that the same time series is used twice -

once in its original form and once lagged one or more time periods.

d = 0 and D = 1

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For the autocorrelation function shows the lowest troughs at lags 6, 18 and 30 and highest

peaks at lags 12, 24 and 36. In that observation, it suggested that the series ACF contains

seasonal component. (Appendix 1).

3.1.2 Partial AutoCorrelation Function (PACF)

The PACF plays an important role in data analyses aimed at identifying the extent of the lag in

an autoregressive model. Where by plotting the partial autocorrelative functions one could

determine the appropriate lags p in an AR (p) model or in an extended ARIMA (p,d,q) model.

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The Partial Autocorrelation Function (PACF) of shows significant partial autocorrelation at lags

1, 3, 4, 5, 6, 7, 11 and 12. This is not a pure white noise and does not fit into any known pattern

except that the significant partial autocorrelation at lag 12 shows that the series has seasonal

variation in it. However one might suspect the presence of non-seasonal auto-regressive

parameter as a result of the high sample autocorrelation coefficients at lag 1 of PACF. (Appendix

2).

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3.2 MODEL FITTING

3.2.1 Model Identification

The objective here is to identify an appropriate subclass of models from the general Seasonal

Auto regressive Integrated Moving Average (SARIMA) family. SARIMA model types are listed

using the standard notation of SARIMA (p,d,q) (P,D,Q)s, where p is the order of

AutoRegression (AR), d is the order of differencing or integration , and q is the order of

moving-average (MA), (P,D,Q) are their seasonal counterparts while s is the length of seasonal

period. When a series is influenced by seasonal factors and recurs on a regular periodic basis.

Following the approach of criteria to judge the best model for a time series, we look at the;

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i Relatively small BIC

ii Relatively high R2

adjusted of the models. The model that the highest properties among the

several mode is the chosen to be tentatively entertained.

Model Type Normalized BIC R2 adjusted Appedices

SARIMA(1,0,0)(1,1,1)12 9.161 .605 Appendix 1

SARIMA(1,0,0)(1,0,0)12 9.386 .482 Appendix3a

SARIMA(1,0,0)(1,0,1)12 9.204 .580 Appendix 3b

SARIMA(1,0,1)(1,1,1)12 9.188 .605 Appendix 3c

SARIMA(1,0,1)(1,0,1)12 9.228 .581 Appendix 3d

SARIMA(1,1,0)(1,1,1)12 9.561 .413 Appendix3e

.

Thus the model to be tentative entertained for the rainfall data is SARIMA (1,0,0)(1,1,1). Since it

has the highest properties of criteria among the all other models identified.

Where

AR (p), the order of the process = 1

MA (q), the order of the process = 0

(d), the order of difference = 0

AR (P), the order of seasonal process =1

MA (Q), the order of the seasonal process = 1

(D), the order of seasonal difference = 1 and

(s), the length of the seasonal period = 12

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3.2.1a Model Fit

Fit Statistic Mean SE Minimum Maximum

Percentile

5 10 25 50 75 90 95

Stationary R-

squared

.334 . .334 .334 .334 .334 .334 .334 .334 .334 .334

R-squared .605 . .605 .605 .605 .605 .605 .605 .605 .605 .605

RMSE 91.907 . 91.907 91.907 91.907 91.907 91.907 91.907 91.907 91.907 91.907

MAPE 358.460 . 358.460 358.460 358.460 358.460 358.460 358.460 358.460 358.460 358.460

MaxAPE 34335.947 . 34335.947 34335.947 34335.947 34335.947 34335.947 34335.947 34335.947 34335.947 34335.947

MAE 60.412 . 60.412 60.412 60.412 60.412 60.412 60.412 60.412 60.412 60.412

MaxAE 546.130 . 546.130 546.130 546.130 546.130 546.130 546.130 546.130 546.130 546.130

Normalized

BIC

9.161 . 9.161 9.161 9.161 9.161 9.161 9.161 9.161 9.161 9.161

This section provides definitions of the goodness-of-fit measures used in picking the best

SARIMA model.

Stationary R-squared can be negative with a range of negative infinity to 1.Negative values mean

that the model under consideration is worse than the baseline model. Positive values mean that

the model under consideration is better than the baseline model. A value of .334 → 1 shows that

the created SARIMA model is good.

An R-squared-value of .605→ 1 supports the claim that the model is good.

While a number of statistics are reported, we will focus on two: MAPE (mean absolute

percentage error) and MaxAPE (maximum absolute percentage error). Absolute percentage error

is a measure of how much a dependent series varies from its model-predicted level and provides

an indication of the uncertainty in your predictions. The mean absolute percentage error varies

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from a minimum of 0.358% to a maximum of 0.358% across all models.

3.2.1b Model Statistics

Model

Number of

Predictors

Model Fit statistics Ljung-Box Q(18)

Number of

Outliers

Stationary

R-squared R-squared RMSE MAPE MAE MaxAPE MaxAE

Norm.

BIC Statistics DF Sig.

Rainfall_

mm-

Model_1

1 .334 .605 91.907 358.46 60.412 34335.947 546.13 9.161 13.465 15 .566 0

The model statistics table provides summary information and goodness-of-fit statistics for each

estimated model. Results for each model are labeled with the model identifier provided in the

model description table.

Although the SARIMA Model offers a number of different goodness-of-fit statistics, we

opt only for the stationary R-squared value. This statistic provides an estimate of the proportion

of the total variation in the series that is explained by the model and is preferable to ordinary R-

squared when there is a trend or seasonal pattern, as is the case here. Larger values of stationary

R-squared (up to a maximum value of 1) indicate better fit. A value of 0.605 → 1 means that the

model does an excellent job of explaining the observed variation in the series.

The Ljung-Box statistic, also known as the modified Box-Pierce statistic, provides an indication

of whether the model is correctly specified. A significance value less than 0.05 implies that there

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is a structure in the observed series which is not accounted for by the model. The value of 0.566

> 0.05 shows that most of the structure of the observed variable (rainfall) if not all are accounted

for by the model.

3.2.1c Chart of Actual and Predicted Rainfall

Model Type: SARIMA(1,0,0)(1,1,1)12

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3.2.2 SARIMA Model Parameters Estimation

The model type is SARIMA(1,0,0)(1,1,1)12. Hence The identified model is represented by

φ (B )Φ (B12

)∇12 xt = θ (B )a t ---- (3.0)

(I-φB )(I-ΦB12)∇12 xt = (I-θB12

)a t ---- (3.1)

Let ∇12 xt = w t

Therefore; (I-φB -ΦB12+φΦB13

)w t = (I-θB12)a t

w t - φw t−1 - Φw t−12 + φΦw t−13 = a t -θat−12 ---- (3.2)

w t = a t + φw t−1 + Φw t−12 - φΦw t−13 - θat−12 ---- (3.3)

Estimate SE t Sig.

Rainfall-Model_1

Rainfall No Transformation Constant 321.065 858.797 .374 .709

AR Lag 1 -.011 .067 -.168 .867

AR, Seasonal Lag 1 .188 .083 2.279 .024

Seasonal Difference 1

MA, Seasonal Lag 1 .991 .678 1.462 .145

MONTH, period 12

No Transformation Numerator Lag 0 -.160 .431 -.370 .712

(1+0.007B )(1-0.202B12)w t = (1-0.991B12

)a t ---- (3.5)

Or

w t = a t -0.007w t−1 +0.202w t−12 -0.003w t−13 +0.991a t−12

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3.3 MODEL DIAGNOSTICS

3.3.1 Descriptive Statistics of Residual

Descriptive Statistics

N Range Min Max Mean

Std.

Deviation Variance Skewness Kurtosis

Stat Stat Stat Stat Stat

Std.

Error Stat Stat Stat

Std.

Error Stat

Std.

Error

ERR 240 860.00 -391.00 469.00 .0094 8.73527 135.32627 18313.199 -.126 .157 1.008 .313

Valid N (listwise) 240

All the results from the descriptive statistics of the residuals are normally distributed. The mean

and skewedness are tending toward 0

3.3.2 Residual ACF Analysis

The most important diagnostic check consists of examining the autocorrelation function of the

residuals which is shown below

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It can be seen that most of the coefficients are fairly small, except the coefficients at lags 5,8 and

11 that have absolute value greater than the absolute error (see Appendix 4). The absolute

standard error of the differenced series is

1√238 = 0.065

It is not surprising to find 2 or 3 “significant” values in 32 autocorrelation coefficients suggesting

that the residuals are reasonable having the property of white noise. So we believe that the model

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is adequate. We are going to test the overall adequacy of the model by the method described in

Box and Jenkins (1970, section 8.2.2), we calculate

Q = n∑t=1

rk2

-------------(3.7)

Where n is the number of the differenced series which is 238, rk is the kth residual

autocorrelation coefficients.

∴ Q = 238 ∑t=1

rk2

= 18.22152

If the model is adequate, Q should be approximately χ2

with (k-p-q) degrees of freedom. The

test would be rejected if the observed value of Q exceeds χk−p−q2

. But χk− p−q2

= χ32−1−12(0 .05 )

= 43.8.

Thus, the observed value of Q is not significant since the value of chi-square is greater than the

value of Q . Then, at this point we conclude that the model is adequate.

3.3.3 Residual Q-Q and P-P Plot

The Q–Q and P-P plots are used to compare the shapes of distributions, providing a graphical

view of how properties such as location, scale, and skewness are similar or different in the two

distributions. If the two distributions being compared are similar, the points in the Q–Q plot will

approximately lie on the line y = x. If the distributions are linearly related, the points in the Q–Q

plot will approximately lie on a line, but not necessarily on the line y = x. Q–Q plots can also be

used as a graphical means of estimating parameters in a location-scale family of distributions.

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In the normal Q-Q- plot, the RESIDUAL variables cluster around the line showing that the

distribution is normal.

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3.3.4 Histogram Chart of RESIDUAL

The histogram for the RESIDUAL looks normal as shown by the Q-Q- and P-P plot.

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3.4 PREDICTION and FORECASTING

3.4.1 Prediction

See Appendix 5

3.4.2 Variable Forecast

Forecast

Model

Jan

2001 Feb 2001

Mar

2001

Apr

2001

May

2001

Jun

2001

Jul

2001

Aug

2001

Sep

2001

Oct

2001

Nov

2001

Dec

2001

Rainfall_

mm-

Model_1

Forecast 48.71 81.14 124.37 192.19 236.23 304.58 349.73 322.22 362.37 306.12 77.74 60.71

UCL 221.69 254.12 297.35 365.17 409.21 477.56 522.71 495.20 535.35 479.10 250.72 233.69

LCL -124.27 -91.85 -48.61 19.21 63.25 131.59 176.75 149.24 189.38 133.14 -95.24 -112.27

For each model, forecasts start after the last non-missing in the range of the requested estimation period, and end at the last period for which

non-missing values of all the predictors are available or at the end date of the requested forecast period, whichever is earlier.

Jan-01 Feb-01 Mar-01 Apr-01 May-01 Jun-01 Jul-01 Aug-01 Sep-01 Oct-01 Nov-01 Dec-010

50

100

150

200

250

300

350

400

f(x) = 0.28739052073966 x − 10444.752446164R² = 0.066638503682077

Rainfall_forecastRainfall_forecast Linear (Rainfall_forecast)

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CHAPTER FOUR

SUMMARY, CONCLUSION AND RECOMMENDATION

4.1 SUMMARY

This work is designed with the aid of building models from the historical rainfall data of Edo

State and forecasting using the model fitted. The first step we took in this research-work was to

obtain raw data of rainfall in Edo state form 1981 – 2000. Analysis was carried-out using SPSS

v19.

Normality test was conducted to know if the data is normally distributed and no log

transformation was carried out on the data. The series plot of the observation (rainfall) was used

to visualize the important features such as the trend and seasonality. The plot shows on long term

movement, but exhibits a periodic pattern.

As the variance of the data did not appear to change with mean level, the series autocorrelation

and partial autocorrelation functions for different series were checked to determine the series that

will best interpret the data. With significant spikes at lag 1 and lag12, then, there is seasonality in

the data. Hence, a SARIMA model (1,0,0)(1,1,1) was developed using a seasonal period of 12.

A SARIMA model (1,0,0)(1,1,1)12 with criteria statistics such as stationary R-squared, R-

squared, MAE, MAPE, MaxAE and MaxAPE showing a well developed model. Prediction and

forecast for 12 lead times were made with the developed model w t = a t -0.007w t−1 +0.202w t−12 -

0.003w t−13 +0.991a t−12 which was compared with the original data to verify that the model that

was fitted was a good one.

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4.2 CONCLUSION

The Edo State monthly rainfall was dominated with a seasonal positive trend. The first and last

quarter of every year has the highest amount of rainfall. The model could effectively used to

predict the future monthly rainfall values of Edo state.

4.3 RECOMMENDATION

The developed model in this work is limited to only monthly rainfall in Edo State. Climatic

factors are many and each changes one another, we cannot claim to have achieved the best result

that will last forever. Then, we recommend that in future, one should carry out the same analysis

to see if other climatic factors like urbanization, green house effect have influence the weather

condition of Edo State.

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