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We obtained historical data of rainfall in Warri Town for the period 2003-2012 for the purpose of model identification and those of 2013 for forecast validation of the identified model. Model identification was by visual inspection of both the sample ACF and sample PACF to postulate many possible models and then use the model selection criterion of Residual Sum of Square (RSS), Akaike’s Information Criterion (AIC) complemented by the Schwartz’s Bayesian Criterion (SBC), to choose the best model. The chosen model was the Seasonal ARIMA (1, 1, 1) (0, 1, 1) process which met the criterion of model parsimony with RSS value of 81.098,773, AIC value of 281.312,35 and SBC value of 289.330,84. Model adequacy checks showed that the model was appropriate. We used the model to forecast rainfall for 2013 and the result compared very well with the observed empirical data for 2013.

The influence of rainfall on flooding with downstream implication of erosion, water quality, agriculture, sewage system, and tourism among others cannot be over emphasized. For these reasons, early warning of rainfall is very important in the use and management of water resources. [

Efforts are being intensified to use time series autoregressive moving average (ARMA) models to forecast hydrological data. The use of ARMA models is justified as a result of its theoretical base in hydrological studies. For example, the parameter value of AR (1) model is the same as the constant due to [

Warri city is located in latitude 5˚31'N and longitude 5˚45'E with two distinct seasons: the rainy season (May- October) and dry season (November-April). It has mean annual temperature of 32.8˚C and an annual rain amount of 2673.8 mm. Warri is a major oil city located in the Niger Delta region of Nigeria.

The main focus of this work is on determining appropriate Seasonal ARIMA model that can adequately predict rainfall for Warri city.

The seasonal multiplicative ARIMA (Autoregressive, Integrated Moving Average) model is of the form

where

y_{t} is the observed temperature data at time t,

Several researchers and scientist have used these models for several technical and scientific studies. [

We obtained historical data of average monthly rainfall for Warri town for the period 2003-2012 for the purpose of model identification and those of 2013 for forecast validation of the chosen model from the National Metrological Center, Oshodi-Nigeria

To detect possible presence of seasonality, trend, time varying variance and other nonlinear phenomena, we inspect the time plot of the observed data side by side with the plots of sample autocorrelation functions (ACF) and sample partial autocorrelation functions (PACF). This will help us determine possible order of differencing and the necessity of logarithmic transform to stabilize variance. Non stationary behavior is indicated by the refusal of both the ACF values,

Model identification is by comparing the theoretical patterns of the ACF and PACF of the various ARIMA models with that of the sample ACF and PACF computed using empirical data. A suitable model is inferred by matching these patterns. Generally [

However, the mixed Seasonal ARIMA model is difficult to identify by visual methods of ACF and PACF plots only. In this work, we use the model identification discussed above to give a rough guess of possible values p, q, P, and Q from which several models shall be postulated and then use the model selection criterion of Residual Sum of Square RSS (Box and Jenkins, 1976), Akaike’s Information Criterion AIC [

The SBC computation is based on the mathematical formula

Having identified a suitable Seasonal ARIMA model, the next stage is the parameters estimation of the identified model and this is done through an exact maximum likelihood estimate due to [

The test for model adequacy stage requires residual analysis and this is done by inspecting the ACF of the residual obtained by fitting the identified model. If the model is adequate then residuals should be a white noise process. Under the assumption that the residual is a white noise process, the standard error of the autocorrelation

functions should be approximately

tion functions should fall within the range

process is not white noise.

To decide on the presence of trend and time varying variances, we inspect the time plot of Warri rainfall data in

Examination of

On inspecting

We complete the data preparation process by additionally performing a first order seasonal difference and the time plot is shown in

Visual examination of

The estimated order of the model parameters p, q, P and Q are identified by visual inspection of ACF and PACF of the stationary process of rainfall shown in

We expect the ACF in

order one i.e. q = 1 and a seasonal moving average parameter of order two i.e. Q = 2. Similarly from the PACF in

Since our strategy is not to have mixed seasonal factors, we postulate two initial models. The two models are: Seasonal ARIMA (3, 1, 1) (1, 1, 0) and Seasonal ARIMA (3, 1, 1) (0, 1, 2). We extend the search to models around these models and based on the model selection criterion of RSES, AIC and SBC, the best model is selected.

The result is shown in

From

From

We estimated the parameter values of the chosen model as shown below.

Model | RSES | AIC | SBC |
---|---|---|---|

Seasonal ARIMA (3, 1, 1)(1, 1, 1) | 82.022,153 | 289.643,06 | 305.680,03 |

Seasonal ARIMA (3, 1, 1)(1, 1, 0) | 97.429,173 | 303.626,69 | 316.990,83 |

Seasonal ARIMA (3, 1, 1)(0, 1, 1) | 79.556,756 | 283.111,35 | 296.475,49 |

Seasonal ARIMA (2, 1, 1)(1, 1, 1) | 83.110,148 | 289.322,14 | 302.686,28 |

Seasonal ARIMA (2, 1, 1)(1, 1, 0) | 96.695,759 | 300.852,39 | 311.543,71 |

Seasonal ARIMA (2, 1, 1)(0, 1, 1) | 80.785,406 | 282.605,43 | 293.296,74 |

Seasonal ARIMA (1, 1, 1)(1, 1, 1) | 82.035,822 | 286.248,57 | 296.939,88 |

Seasonal ARIMA (1, 1, 1)(1, 1, 0) | 97.249,013 | 299.467,15 | 307.485,63 |

Seasonal ARIMA (1, 1, 1)(0, 1, 1) | 81.098,773 | 281.312,35 | 289.330,84 |

Seasonal ARIMA (3, 1, 1)(0, 1, 2) | 79.745,649 | 287.008,11 | 303.045,08 |

Seasonal ARIMA (3, 1, 1)(1, 1, 2) | 79.320,155 | 288.354,37 | 307.064,17 |

Seasonal ARIMA (2, 1, 1)(0, 1, 2) | 79.874,139 | 285.607,37 | 298.971,52 |

Seasonal ARIMA (2, 1, 1)(1, 1, 2) | 79.578,425 | 286.994,24 | 303.031,21 |

Seasonal ARIMA (1, 1, 1)(0, 1, 2) | 79.937,258 | 283.884 | 294.575,32 |

Seasonal ARIMA (1, 1, 1)(1, 1, 2) | 79.683,933 | 284.905,83 | 298.269,97 |

From

To verify the suitability of the model, we plot the autocorrelation values of the residual against lag as shown in

We note that on inspection of

A t-distribution test of equality of mean based on

Variables in the Model | ||||
---|---|---|---|---|

B | SEB | T-Ratio | Approx. | Prob. |

AR1 | 0.315,427,75 | 0.096,022,14 | 3.284,948,0 | 0.001,390,64 |

MA1 | 0.997,165,70 | 0.834,721,79 | 1.194,608,4 | 0.234,957,39 |

SMA1 | 0.826,972,02 | 0.132,568,69 | 6.238,064,2 | 0.000,000,00 |

Month | Jan. | Feb. | March | April | May | June | July | Aug. | Sept. | Oct. | Nov. |
---|---|---|---|---|---|---|---|---|---|---|---|

Forecast | 19.26 | 43.27 | 75.77 | 193.07 | 256.19 | 402.69 | 402.039 | 322.85 | 384.63 | 220.01 | 50.64 |

Observed | 15.7 | 21.2 | 98.4 | 189.5 | 218.9 | 379.8 | 351.7 | 292.4 | 362.4 | 236.6 | 108.4 |

Difference | −0.96 | −22.07 | 22.63 | −3.57 | −37.29 | −22.89 | −50.34 | −30.45 | −22.23 | 16.59 | 57.76 |

We have shown that time series ARIMA models can be used to model and forecast the rain in Warri Town. The identified Seasonal ARIMA (1, 1, 1) (0, 1, 1) has proved to be adequate in forecasting rain for at least one year.

Daniel Eni,Fola J. Adeyeye, (2015) Seasonal ARIMA Modeling and Forecasting of Rainfall in Warri Town, Nigeria. Journal of Geoscience and Environment Protection,03,91-98. doi: 10.4236/gep.2015.36015