Journal of Environment and Bio Research
Research Article
Monthly Rainfall Forecast over the Sahel Region Using Sarima Models
Bello E^{1}, Aganbi B^{1}, Gbode E^{2} and Aremu O^{2*}
^{1} Nigeria Meteorological Agency, Nigerian Meteorological Agency, NiMet, Nigeria^{2} Federal University Science and Technology, FUTA, Akure, Nigeria
Received date: April 26, 2018; Accepted date: June 05, 2018; Published date: June18, 2018
*Corresponding author: Aremu Oluwasegun, Federal University Science and Technology, FUTA, Akure, Nigeria, Email: segunaremu007@hotmail.com, s.aremu@nimet.gov.ng
Abstract
Monthly rainfall data spanning a period of thirty six (19812017) years for some stations in the Sahel region of the country (they include, Katsina
Zamfara, Maiduguri, Sokoto, Yobe states) were obtained from the archive of the Nigerian meteorological Agency, NiMet. The data were then
subdivided into segments (19812014, 19812015 and 19812016) in order to train ARIMA models for the states in the region. The models were
subsequently validated and used to forecast monthly rainfall for the aforementioned states. Statistical ttest was then employed to ascertain
differences between the forecast and actual rainfall data for 2015, 2016 and 2017. The results of the ttest carried out at the 1 and 5% level of
significance indicated that there was no significant difference between the forecast and actual rainfall. Accordingly it is recommended that the
developed arima models, which of course differ from one state to the other, could be adopted and used to generate forecast of monthly rainfall for
year 2018. Although statistical ttest indicated no difference between forecast generated from the model and the corresponding actual data, the
models appeared to have performed better over Gusau (in 2016 and 2017), Katsina (2016 and 2017), Potiskum ( in 2016) and Sokoto.
Keywords: Sarima model; NiMet; Climate Data; ACF; PACF; Ttest; pvalue and Null hypothesis;
Introduction
Nigeria’s climate has witnessed significant climate variability leading
to extreme events such as the 2012 flood that dislocated socioeconomic
activities (Afiesimama et al, 2013). Conversely, we have also witnessed
droughtsituation particularly over the Sahel region where rainfall deficit
has been a major source of concern and the bulk of agricultural production
that drives socioeconomic activities is centred. Trends have shown that
climate variability (Leonard K.A mekudzi et al., 2015) can result in late
onset and early cessation of the rainy season in this region. However,
rigorous analysis has shown that natural climate variability alone cannot
explain the longterm trend of changing extremes in temperature and
precipitation (Meehl et al., 2007; Gutowoski et al, 2008; Stott et al, 2010;
and Christidis et al,2011).
While many societies are taking measures to cope with historical
weather extremes, new and more extreme events have the potential to
overwhelm existing structures and programs put in place to mitigate
their impacts (Solomon et al, 2008). The potential for these events to
bring irrevocable and damaging effect to infrastructure, in addition to
precipitating natural resource conflicts, necessitates the need to evolve
tools to quantify and analyse their impact; especially tools or mechanisms
that can account for the trend in climate and its variability. This would in
turn enable effective adaptation strategies. An effective mitigation strategy
is the ability to take proactive approach which is only possible through
evaluation and prediction (Adefolalu DO, 2010).
The primary aim of seasonal rainfall prediction is to:
• Forewarn on imminent and repetitive extreme climate events that could
lead to disasters.
• Protect lives and livelihoods in areas prone to disaster especially to that
are caused by climate episodic events
• Develop and establish indices and indicators of extreme events.
A sizable amount of study has been dedicated to the science of rainfall
prediction ranging from thermodynamic models of Omotosho (1999);
artificial neural network by Cristian et al (2014); arima models by Etuk et al
(2013); Edwin and Martins (2014) examined the stochastic characteristics
of monthly rainfall in Ilorin; to the SARIMA (0, 0, 0) (1, 1, 1)12 models of
Akpakta et al (2015) for monthly rainfall over Umuahia.
However, very little amount of research has been dedicated to forecasting
monthly rainfall and this creates a gap especially in the face of increasing
extreme weather events particularly over the Sahel. It is imperative
therefore, to explore effective approaches to forecasting monthly rainfall,
and Sarima models are a set of veritable tools in this regard capable of
capturing or modelling trend/changes in climate; changes that could be
driving increases in extreme weather or rainfall events.
Arima/Sarima Models
Arima is an acronym for autoregressive integrated moving averages.
The “AR” part of the model is termed the autoregressive component, the
“I” integrated and the “MA” the moving average component.
These models are used to fit time series data for the purpose of
understanding the data and predicting future values in the time series
(people.duke.edu). A major requirement in applying the model to time
series is to ensure that the time series is stationary.
The AR part of ARIMA represents the term in the model obtained
from regressing the variable of interest on itself; the MA part implies a
linear combination of the lagged error terms, while the I part indicates
the ordinary differencing required to make the time series stationary,
although this may have to be coupled with other forms of transformation.
Stationary of a time series implies that the variance of the time series
does not change significantly with time. Various transformations must be
performed on a nonstationary time series to render it stationary before
fitting the data set to arima model. If DickeyFuller test suggest that a time
series is nonstationary, it can be transformed by differencing and taking
logarithm of the data (Wikipedia on DickeyFuller test, 2018).
The models are mostly expressed in the form arima (p, d, q) where
the p, d, q components are positive integers. For the seasonal data the
corresponding seasonal models are given as arima(p,d,q)(P,D,Q)m, where
m are the number of periods in each of the seasons, and like the p,d,q
components, P,D,Q are positive integers.
Study Area
The Sahel Savannah covers the extreme northern part of Nigeria with
the Sudan savannah bordering it to the south. It occupies about 18 130
km2 of the extreme northeast corner of Nigeria. The region is a semiarid
area covering Maiduguri, Yobe, Parts of Kebbi state, Zamfara, and Sokoto
states. It is a home to a large population of animals and is bedevilled by
issues of extreme poverty, climate change, armed conflict and insecurity.
It is characterized by short rainy season with between 380930mm
of rainfall per annum and last for about three to four months (June
September) and a peak in August. There is high inter and intra seasonal
variability in rainfall around the region. It is hot, sunny, dry and windy all
year round with intense heat with temperature ranging between 36 and
42oC in the hot season. The main characteristics of this area are its desert
nature and short grasses. The region is a major player in agricultural
production and is also the worse hit by weather fluctuations. This zone is
characterized by plants such as Cenchrusbiflorus, and Acacia raddiana.
The shrubs that are predominantly scattered in the zone are African myrrh
(Commiphoraafricana) and Leptadeniaspartum. Irrigation farming
is a common practice owing to short duration of rainfall, with the area
dominated by herdsmen. Common crops grown include: maize, sorghum,
cowpea, rice, amongst others.
It is important to state here that the region is a largely agrarian one
and is the centre of the bulk of rain fed agrorelated activities in Nigeria.
Ironically, the area has suffered from the vagaries of extreme weather events
that has had untold impact on the agricultural sector. Mainly because of
the short length of the growing season, intense heat, over grazing, conflict,
population explosion that have resulted in desertification. Hence forecast
of rainfall, particularly on the monthly scale, is of invaluable significance.
Data and Methodology
Data
Data for the research consisting of monthly rainfall for all the stations in the Sahel region, spanning a time interval of at least thirty years (1981 2017), were obtained from the archive of the Nigerian Meteorological Agency, NiMet. These were then segmented into parts viz; from 19812014 for training the model and forecasting 2015 rainfall and then subsequently validating it with the actual rainfall data of 2015; from 19812015 for training the model and forecasting 2016 rainfall and then subsequently validating it with the actual rainfall data of 2016; from 19812016, for training the model and forecasting 2017 rainfall and validating with it the actual rainfall data of 2017; and finally from 19812017, for training the model and forecasting the monthly rainfall for 2018. This was done for all the stations in the Sahel.
Methodology
As have already been stated, the model used in the research is a stochastic model called ARIMA, an acronym for Autoregressive Integrated Moving Averages. Although a detailed literature on the subject matter of ARIMA is not given in this study, a brief definition of its associated basic and essential components outlined in this study, would suffice for the purpose of this research. Robert Nau (people.duke.edu, 2018), Wikipedia (on the rules on identifying the orders of AR and MA terms of arima model, 2018), Cross Validated (on sarima Models, 2018), Dickey Fuller Test (2018) and Statistica (on identifying patterns in a time series data, 2018) amongst others are studies that provides detailed insight into the subject matter of sarima models
General Form/Equation of Arima
The sarima model is generally expressed as SARIMA(p,d,q)(P,D,Q)_{m}
This indeed serves as a solution to the general form of the multiplicative
arima model given by
$$\Phi \left({B}^{m}\right)\varphi \left(B\right){\nabla}^{D}{}_{m}{\nabla}_{d}{Y}_{t}=\Theta \left({B}^{m}\right)\theta \left(B\right){\epsilon}_{t},$$
With ԑt as the white noise process, the terms of this function/model
can be further expressed as follows:
$${\nabla}_{m}{Y}_{t}={Y}_{t}{Y}_{t}m,\nabla {Y}_{t}={Y}_{t}{Y}_{t}{}_{1}\text{(1)}$$
$$\text{\Phi}\left({B}^{m}\right)=1{\text{\Phi}}_{1}{B}^{m}\dots .{\text{\Phi}}_{P}{B}^{Pm}\text{(2)}$$
$$\varphi (B)=1{\varphi}_{1}B{\varphi}_{2}{B}^{2}\text{n}{\varphi}_{P}{B}^{p}\text{(3)}$$
$$\Theta \left({B}^{m}\right)=1+{\Theta}_{1}{B}^{m}+\dots +{\Theta}_{Q}{B}^{Qm}\text{(4)}$$
$$\theta \left(B\right)=1+{\theta}_{1}B+\dots +\dots \dots \dots \dots {\theta}_{q}{B}^{q}\text{(4)}$$
Autocorrelation and Partial Autocorrelation
Most time series patterns can be described in terms of two basic classes of components: trend and seasonality. These two general classes of time series components may coexist in reallife data. The former represents a general systematic linear or (most often) nonlinear component that changes over time and does not repeat within the time range captured by the data set. The latter, however, repeats itself in systematic intervals over time indicating seasonal dependencies that can be measured by Autocorrelation function. Autocorrelation Function (ACF) is the correlation between a time series and lagged version of itself or a display of serial dependencies or coefficients of the time series variable at various lags, while Partial Autocorrelation Function (PACF) is the amount of correlation between the time series and a lagged version of itself that is not explained by lower order lags. For instance if we are regressing a time series variable Y against x1, x2 and x3, the partial autocorrelation between Y and x3 is the amount of correlation between Y and x3 that is not explained by their common correlations with x1 and x2.
Identifying the Orders of Arima Models from Acf and Pacf
A brief approach that was employed to identify the suitable terms of
the arima model in this research would be given in this section. Detail
rules for identifying the terms of Sarima models can be found in (people.
duke.edu).
There is a systematic method of identifying the orders of arima models.
By looking at the ACF and PACF one can tentatively identify the number
of AR and/ or MA terms that are needed. For instance if a time series has
positive autocorrelations out to a high number of order lags (say 12 or
more), then the time series would require a higher order of differencing
to detrend it i.e. seasonal differencing. By mere inspection, one can
determine the number of AR terms needed to explain the autocorrelation
in a time series. Generally, the orders of AR terms are derived from the
PACF plot while the orders of the MA terms are obtained from the ACF.
If the PACF displays a sharp “cutoff ”, while the ACF “decays” more
slowly, the stationarize time series displays an “AR” signature. On the
other hand if the ACF displays a sharp cutoff while the PACF decays
more slowly, the time series displays an MA signature.
In the former case consider adding an AR term to the model which is
equivalent to taking a first order differencing. If the cutoff in the PACF
plot occurs at lag p, this indicates that exactly p AR terms should be added
to the model. On the other hand if the ACF cutoff occurs at lag q, then
consider adding q MA terms to the model. MA terms are commonly
associated with time series that are slightly overdifferenced.
The seasonal part of an Arima model has the same structure as the
nonseasonal part. In identifying seasonal model, the first step is to
determine whether or not a seasonal differencing is required in addition
to or nonseasonal differencing. It is however instructive that using more
than one or two order of differencing should be avoided for both types of
differencing combined.
If the time series has a strong consistent seasonal pattern (this can be
detected in the time series plot or ACF and PACF) it would be reasonable
to take an order of differencing.
The signature SAR/MAR behaviour is similar to those of pure AR/MA
except that the pattern occurs across multiples of lags in the PACF and
ACF. A pure SAR (1) has spikes in the ACF at lags m, 2m, 3m and so on,
while the PACF cuts off at lag m. Conversely a pure SMA(1) process has
spikes in the PACF at lags m, 2m, 3m and so on while ACF cuts off at lag
m.
These are some of the rules considered while designing the arima
model for our time series data.
The observation of nonstationarity quality in most of the time series
necessitated the need to transform the time series using natural logarithm
(“ln”). Although this was done only after adding one to the data in order
to avoid errors that may result from taking the natural logarithms of zeros
in the data set; and our data set contain a lot of zeroes.
The tentative observation of seasonal patterns in the time series and
the ACF and PACF plots necessitated an order of seasonal differencing.
This was able to help detrend the data whose ACF and PACF plots clearly
revealed the orders of AR, SAR, MA and SMA terms required to determine
a parsimonious model. This approach was effective in identifying the
seasonal order in the time series as well as determining a tentative model.
The resulting residuals from the tentative model informed its further
adjustment until the parsimonious model was determined. The residuals
here are ACF, pvalue and QQ plots of residuals from the tentative model
which can be adjusted or accepted as the suitable model for the data
depending on the plots.
A null hypothesis was considered in the case of the pvalue of the
residuals which is that “the ACF of the residuals at various lags is not
significant or zero”; in other words no autocorrelations are present in the
residuals from the model. Expectedly a high pvalue would indicate that
the arima model was able to extract all the autocorrelations in the data and
minimized the error between the observed and simulated data and thus
resulting in residuals with insignificant autocorrelations.
The pvalue in simple terms, is the probability, given the null hypothesis
as in above, of obtaining autocorrelations in residuals equal to the one
observed or more extreme than what was observed. Accordingly, a high
pvalue of the residuals would imply that the null hypothesis should be
accepted.
The associated QuantileQuantile plots indicate the degree to which
the residuals are normally distributed when the data tends to align to a
straight line.
Therefore if a parsimonious arima model is to be determined, the
ensuing residuals should be patternless or random with insignificant
autocorrelations at the various lags; a behaviour which can only be
detected by the QQ plots and autocorrelation functions respectively.
Thus in the course of developing the suitable arima model for our
data, the time series was subjected to this statistical tests whose result
indicated that the residuals are patternless whitenoise with insignificant
autocorrelations at the various lags.
As earlier stated the research would include forecast of monthly rainfall
and validation of the model. This was done by using the monthly rainfall
data from 19812014 as training data to forecast monthly rainfall for 2015
and subsequently validating with actual data of 2015; using the monthly
rainfall data from 19812015 as training data to forecast the rainfall in 2016
and subsequently validating with actual data of 2016; and using monthly
data from 19812016 to forecast rainfall for 2017 and subsequently
validating with actual data of 2017. Thereafter the forecast rainfall was
compared with the observed rainfall using the ttest distribution.
Determining Significant Difference Between Simulated and the Observed Data
The student’s ttest (Wikipedia, 2018) is generally used to test the
differences between the population means of two distributions. This is
often used in cases where the populations are believed to have nearly equal
standard deviations, nearnormally distributed and the drawn samples for
the test is small i.e. n< 30. For instance given two samples with n1, x1
and s1 and n2, x2 and s2 as sample size, mean and standard deviation
respectively, the common standard deviation
${S}_{p}=\sqrt{\frac{\left(n11\right)s{1}^{2}+\left(n21\right){S}^{2}}{\left(n1+n22\right)}}$
And so the standard errors of each of the samples is
${S}_{x1}=\frac{{S}_{p}}{\sqrt{n1}},{S}_{x2}=\frac{{S}_{p}}{\sqrt{n2}},$
and so the sampling error of the
Distribution is
${S}_{\left(x1x2\right)}=\sqrt{{S}_{x1}{}^{2}+{S}_{x2}{}^{2}}$
Thus the tscore is given by
$$tscore=\frac{\stackrel{}{x1}\stackrel{}{x2}}{{S}_{\left(x1x2\right)}}$$
with n1+n22 degrees of freedoms
The calculated tscore is then compared to the tscore with n1+n2
2 degrees of freedoms obtained from ttest tables at the 5% level of
significance.
Thus with as the mean monthly observed rainfall of a given year and
as corresponding mean of monthly rainfall of the forecast, the test was
used to ascertain whether there were significant differences between the
actual and forecast rainfall.
Discussion of Results
The results (figures 15 and tables 1 and 2) of the model validation
based on the monthly rainfall forecast for Potiskum, Sokoto, Maiduguri,
Katsina and Gusau in 2015, 2016 are presented in this section.
Generally, the results of the model validation i.e. comparison between
the forecast and actual rainfall, are reasonable since in most of the cases
there was no significant differences between them, based on the statistical
ttest carried out on the result at a 1 percent level of significance.
Over Potiskum the forecast (Figure 1a) compared favourably well with
the observed in 2016. (Figures 15) (Tables 13)
The reliability of the sarima model (See Table1) designed and found
to be the most parsimonious based on the AIC of the residual (of logged
residual) plot (Table 1), is backed by 99 percent confidence level otherwise
known as the one percent level of significance. At this level of significance,
the difference between the forecast and observed was subjected to ttest
which showed that there was no significant difference (no sig. dif.)
between the forecast and actual rainfall in 2016 (Table 3a) as the calculated
tvalue was less than the critical tvalue (2.09; this is the critical value of t
at the 1% level of significance and at 20 degrees of freedom, used in all the
associated ttest of all the forecast). The standard error (Table 3a) between
the forecast and actual rainfall is relatively low.
The plots of the standardized residuals (Figure 6a) in this case shows
that most of the autocorrelations in the data used in the 2016 forecast
have been removed from the data by the sarima model, sarima (0,0,7)
(1,1,3)12 as buttressed by the corresponding pvalues (Recall that the

2016 
2017 
Potiskum 
Sarima(0,0,5)(1,1,3)12 
Sarima(0,0,5)(1,1,3)12 
Maiduguri 
Sarima(0,0,6)(3,1,1)12 
Sarima(0,0,1)(3,1,1)12 
Sokoto 
Sarima(0,0,1)(3,1,1)12 
Sarima(0,0,1)(3,1,1)12 
Katsina 
Sarima(0,0,7)(3,1,1)12 
Sarima(0,0,7)(3,1,1)12 
Gusau 
Sarima(0,0,7)(3,1,1)12 
Sarima(0,0,7)(3,1,1)12 
null hypothesis of the lags of the lags associate with the residuals is zero).
This reduces the standardized residuals to white noise as reinforced by the
patterns of near randomness in the QQ plots (section 3.5).
The forecast (Figure 1b) over Potiskum for 2017 is fairly or slightly
deviated from the observed. However the difference between them is not
significant (Table 3b).
Similarly the difference between the forecast and observed seems
relatively higher over Potiskum and Sokoto in 2017 (Table 3b, Figure 1b
and Figure 3b). However, the respective indicators (Figure 6b and Figure
8b) quantifying the overall differences indicated that the differences are
not significant at the one percent level of significance.
The forecast over Maiduguri in 2017, matched relatively well with the
actual rainfall with no significant difference at the 1% level (Table 3b). The
sarima bestfit model here, (0, 0, 1) (3, 1, 1)12, which produced the forecast
showed most of the autocorrelations in the data have been extracted as
confirmed by the associated standardized whitenoised residuals (Figure
7b). This is also consistent with the patterns observed in the residual ACF
plots, QQ and pvalue plots.

potiskum 
maiduguri 
sokoto 
katsina 
gusau 


observed 
forecast 
observed 
forecast 
observed 
forecast 
observed 
forecast 
observed 
forecast 
Jan 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 
0.3 
0.0 
Feb 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 
Mar 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 
13.9 
0.0 
17.1 
0.0 
Apr 
1.4 
1.4 
11.9 
1.9 
1.0 
4.1 
20.1 
6.8 
64.5 
6.8 
May 
30.8 
18.0 
78.4 
18.7 
151.8 
53.3 
131.2 
77.9 
89.3 
79.6 
Jun 
71.8 
84.2 
79.1 
62.0 
171.2 
91.5 
149.7 
110.5 
200.7 
110.5 
Jul 
188.6 
152.9 
68.3 
182.6 
280.1 
212.1 
215.3 
180.1 
283.1 
184.6 
Aug 
251.4 
225.8 
306.8 
215.4 
180.7 
235.7 
317.2 
284.3 
310.6 
282.0 
Sep 
123.4 
90.7 
196.5 
117.9 
175.1 
128.5 
76.3 
152.8 
265.0 
155.7 
Oct 
28.9 
15.7 
0.0 
10.1 
0.0 
12.3 
0.0 
23.8 
4.6 
23.3 
Nov 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 
Dec 
696.26 
0 
740.975 
0 
959.875 
0 
0.0 
0 
0 
0 

potiskum 
maiduguri 
sokoto 
katsina 
gusau 


observed 
forecast 
observed 
forecast 
observed 
forecast 
observed 
forecast 
observed 
forecast 
Jan 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 
Feb 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 
Mar 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 
Apr 
0.0 
4.0 
0.0 
1.3 
0.0 
4.3 
0.0 
1.1 
53.0 
5.1 
May 
57.8 
27.1 
52.4 
22.4 
25.4 
61.5 
9.1 
13.1 
83.2 
84.9 
Jun 
99.7 
73.6 
135.9 
110.5 
67.6 
95.4 
110.5 
73.1 
200.7 
118.7 
Jul 
129.3 
184.2 
227.8 
184.6 
125.5 
209.8 
143.2 
166.3 
168.5 
203.5 
Aug 
161.7 
251.8 
236.9 
282.0 
125.4 
244.2 
186.1 
208.0 
223.3 
298.6 
Sep 
48.2 
119.8 
59.2 
155.7 
94.0 
125.0 
195.3 
81.4 
220.3 
176.3 
Oct 
0.0 
18.6 
0.0 
23.3 
2.5 
6.6 
0.0 
7.1 
14.0 
19.8 
Nov 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 
Dec 
0 
0 
0 
0 
0 
0 
0 
0 
0.0 
0.0 

AIC 
Calculated tvalue 
DIFFERENCE OBS &FCST 
STANDARD ERROR 
POTISKUM 
0.3123424 
0.292 
No sig. dif 
13.2 
MAIDUGURI 
0.4325639 
0.326 
No sig. dif 
39.7 
SOKOTO 
0.542171 
0.511 
No sig. dif 
36.1 
KATSINA 
0.5189678 
0.192 
No sig. dif 
26.0 
GUSAU 
0.5185077 
0.780 
No sig. dif 
41.6 

AIC 
Calculated tvalue 
DIFFERENCE OBS &FCST 
STANDARD ERROR 
POTISKUM 
0.2866299 
0.54 
No sig. dif 
30.2 
MAIDUGURI 
0.4475341 
0.16 
No sig. dif 
27.7 
SOKOTO 
0.5383566 
0.93 
No sig. dif 
34.8 
KATSINA 
0.5383566 
0.27 
No sig. dif 
27.8 
GUSAU 
0.2669741 
0.12 
No sig. dif 
29.9 
High pvalues associated with autocorrelation plots, usually indicates
that most of the autocorrelations in the data that produced the forecast has
been extracted, and the approximately straight line pattern in the QQ plot
is indicative of nearrandomly or normaldistributed residuals. Overall,
there was no significant difference at the 1% level, between the actual and
forecast rainfall.
Similarly, over Katsina in 2016 and 2017, the model, arima(0,0,7)
(3,1,1)12 forecast rainfall (Figure 4b, Table 2a and b) which approximately
matched with the actual rainfall, generating randomlydistributed whitenoised
residuals (Figure 9a and b) including high p value plots indicating
that most of the autocorrelations in the data that produced the forecast
have been extracted. This is also consistent with the near whitenoised
residuals in the QQ plot.
Over Gusau, in 2016 and 2017, the forecast (Figure 5a and b) matched
with the actual with no significant difference at the 1% level (table 3a
and b). Most of the autocorrelations have been extracted with the bestfit
sarima model (0,0,7)(3,1,1)12. The results obtained are consistent with the
observed patterns in the ACF, QQ and pvalue plots (9a and b) showing
that the standardized residuals have been reduced to white noise.
Finally taking into account the standard errors of the predictions
(Table 3ab), we recommend that the Sarima models as outlined in table
1, used in producing the forecast for 2017, could be employed to generate
monthly rainfall forecast for 2018 for the aforementioned locations within
the Sahel.