Open Journal of Mo dern Hydrology, 2011, 1, 1-9
doi:10.4236/ojmh.2011.11001 Published Online July 2011 (
Copyright © 2011 SciRes. OJMH
ARMA Modelling of Benue River Flow Dynamics:
Comparative Study of PAR Model
Otache Y. Martins, M. A. Sadeeq, I. E. Ahaneku
Department of Agricultural & Bioresources Engineering, Federal University of Technology, Minna, Nigeria
E-mail: k
Received May 2, 2011; revised June 5, 2011; accepted July 6, 2011
The seemingly complex nature of river flow and the significant variability it exhibits in both time and space,
have largely led to the development and application of the stochastic process concept for its modelling, fore-
casting, and other ancillary purposes. Towards this end, in this study, attempt was made at stochastic model-
ling of the daily streamflow process of the Benue River. In this regard, Autoregressive Moving Average
(ARMA) models and its derivative, the Periodic Autoregressive (PAR) model were developed and used for
forecasting. Comparative forecast performances of the different models indicate that despite the shortcom-
ings associated with univariate time series, reliable forecasts can be obtained for lead times, 1 to 5 day-ahead.
The forecast results also showed that the traditional ARMA model could not robustly simulate high flow re-
gimes unlike the periodic AR (PAR). Thus, for proper understanding of the dynamics of the river flow and
its management, especially, flood defense, in the light of this study, the traditional ARMA models may not
be suitable since they do not allow for real-time appraisal. To account for seasonal variations, PAR models
should be used in forecasting the streamflow processes of the Benue River. However, since almost all
mechanisms involved in the river flow processes present some degree of nonlinearity thus, how appropriate
the stochastic process might be for every flow series may be called to question.
Keywords: Time Scale, Streamflow, Autoregressive Model, Fuzzy Cluster, Forecasting, Dynamics
1. Introduction
Time series modelling for either data generation or fore-
casting of hydrologic variables is an important step in
planning and operational analysis of water resource sys-
tems. But, providing good forecast functions for time
dependent data has become a common problem. It is
particularly acute in environmental and ecologic studies
in which the ability to predict is closely allied to the suc-
cessful allocation of the resources needed to control the
environment. Operational hydrological forecasting and
water resource management require efficient tools to
provide accurate estimates of future river level condi-
tions and meet real world demand. The use of stochastic
time-series models for hydrologic forecasting has
evolved greatly. Although many studies have given flow
simulation considerable attention, it is important to rec-
ognise that simulation is not an end in itself but rather a
means to an end, the end being an optimal water resource
design. Despite some notable applications and case stud-
ies [1], relatively few studies have reported on the use of
flow simulation or stochastic modelling in general in
solving engineering problems [2]. More attention needs
to be given to the uses of synthetic data, such as using
the data with optimizing techniques to obtain optimal
operating policies for storage or a set of storages.
In the context of the above, stochastic linear models
are fitted to hydrologic data for two main reasons: to
enable forecasts of the data one or more time periods
ahead to allow for the generation of sequences of syn-
thetic data. In the same context, deterministic models are
of importance in forecasting flows over very short time
intervals such as hours or even days. But even in these
situations if the parameters of a deterministic model have
no physical interpretation and cannot be measured in the
field, a deterministic model offers few advantages over a
stochastic model [3]; since probability limits for fore-
casts may readily be obtained, there may be advantages
in using a stochastic model. Stochastic streamflow mod-
els are often used in simulation studies to evaluate the
likely future performance of water resources systems.
Several stochastic models have been proposed for mod-
elling hydrological time series and generating synthetic
streamflows. These include Autoregressive Moving Av-
erage (ARMA) models [4], disaggregation models [5],
models based on the concept of pattern recognition [6].
Most of the time-series modelling procedures fall within
the framework of multivariate autoregressive moving
average (ARMA) models.
Generally, Autoregressive (AR) and Autoregressive
Integrated Moving Average (ARIMA) models have an
important place in the stochastic modelling of hydrologic
data. Such models are of value in handling what might be
described as the short-run problem; that of modelling the
seasonal variability in a stochastic flow series. In recent
years, their importance to practical water resource prob-
lems has been over-shadowed by more sophisticated
types of models that are designed to preserve long-run
dependencies, perhaps of the order of decades, in hydro-
logic series. In most of these models, the Hurst h has
been used to characterize the long-run dependencies.
Although the long-run problem is important, the short-
run problem, perhaps of the order of months to a few
years, is important, too. Thus, as river flow dynamics at
some time-space scales are not as irregular and complex
as those at other time-space scales, the need and appro-
priateness of the stochastic process concept for ‘every’
river flow, and hydrologic and geomorphic series calls
for a second look at the totality of the whole assertion. In
view of these conflicting paradigms, the question of
whether a given river (or any hydrologic and geomorphic)
series can be modelled appropriately by stochastic me-
thods underscores the premise of this study. Considering
all of this, the focus of the study is to model the daily
flow sequence of the Benue River using Autoregressive
Integrated Moving Average (ARIMA) and its two de-
rivatives, the ARMA and the Periodic Autoregressive
(PAR) models. Here, the feature of interest is to investi-
gate the suitability of either model on the basis of some
selected forecast performance criteria.
2. Materials and Methods
2.1. Hydrology of the Benue River
The Benue River is the major tributary of the Niger
River. It is approximately 1,400 km long and almost
navigable during the rainy season (between July and Oc-
tober). Hence, it is an important transportation route in
the regions it flows through. Its headwaters rises in the
Adamawa Plateau of the Northern Cameroon, flows into
Nigeria south of the Mandara Mountains through the
east-central part of Nigeria before entering the Niger
River at Lokoja (Figure 1(a)). The wide flood plain is
used for agriculture, with main crops being sugar cane
Figure 1. (a) Map of Nigeria showing Benue River and its
traverse; (b) General hydrologi c al y e a r flow regime.
and rice. There is only one high-water season because of
its southerly location; this normally occurs from May to
October, while on the other hand, the low-water period is
from December to June. Figure 1(b) explains the hydro-
logical flow regime of the Benue River in line with the
general climatic pattern. There are definite wet and dry
seasons which give rise to changes in river flow and sa-
linity regimes. The flood of the Benue River (upper,
middle, and downstream) lasts from July to October, and
sometimes up to early November.
2.2. Data Base Management
In this study, historical time series for gauging stations at
the base of the Benue River (i.e., Lower Benue River
Basin) at Makurdi (7˚44N, 8˚32E) was used. A total of
26 years (1974-2000) water stage and discharge data
were collected. A shorter time scale was considered for
Copyright © 2011 SciRes. OJMH
the modelling and forecasting. To this end, daily average
discharges were used only.
2.3. ARMA Modelling of the Daily Flows
The procedure of fitting deseasonalised ARMA models
to daily streamflow as used in this study involves two
basic steps; i.e., deseasonalisation and ARMA model
construction. To do this, the flow series was logarithmi-
cally transformed and deseasonalised by subtracting the
seasonal mean values and dividing by the seasonal stan-
dard deviations of the logarithmic transformed series. To
alleviate the stochastic fluctuations of both the daily and
monthly means and standard deviations, they were
smoothened by the first 8 Fourier harmonics respectively
before being used for standardisation. To broaden the
choices of the models in the modelling exercise, the pos-
sibility of the traditional autoregressive integrated mov-
ing average (ARIMA) model was examined. Unlike the
deseasonalisation pre-processing method, the logarithmic
transformed flow series was differenced before fitting the
appropriate ARIMA model. The objective is to appraise
the impact the pre-processing may have on the overall
forecasting results for the respective models adopted.
Figure 2. ACF of residuals from: (a) ARMA (20,1), and (b)
ARIMA (8,2,3) models.
Based on the Autocorrelation function (ACF) and Par-
tial Autocorrelation function (PACF) structures of the
flow series as well as the model selection criterion AIC,
an ARMA (20,1), ARIMA (8,2,3) were fitted to the flow
series. The parameters were estimated using the arima.
mle function in S-Plus version 6 (Insightful Cooperation,
2001) software. In order to examine the goodness of fit
of the ARMA (20,1), ARIMA (8,2,3) models respec-
tively, the ACF of the residuals from the models were
inspected. The ACF plots in Figure 2 show that there is
no significant autocorrelation left in the residuals from
both ARMA-type models. The adequacy of the models
was further examined by using Ljung-Box test on the
residual series. The Ljung-Box test results for ARMA
(20,1), ARIMA (8,2,3), are shown in Figure 3. The p-
values’ exceedance of 0.05 indicates the acceptance of
the null hypothesis of model adequacy at the 5% signifi-
cance level.
2.4. PAR Model Building
A lot of contrasting difficulties are usually encountered
in the development of different types of PAR models; for
instance, model order, lose of generality, and the over-
whelmingly burdensome and practically infeasible com-
putation of compatibility between neighbouring days.
Because of this, the method for fitting PAR model based
on cluster analysis as espoused by Wang [7] and Otache
[8] was adopted. The fuzzy clustering method was ap-
plied to partition the days over an annual cycle in order
to build the PAR model. The Fuzzy Clustering Method
(FCM) approach partitions a set of n vectors ,
, into c fuzzy clusters; this implies that each data
point belongs to a cluster to a degree specified by a
membership grade ij between 0 and 1. Thus, a matrix
U consisting of the elements ij can be defined based
on the assumption that the summation of degrees of be-
longing for a data point is equal to 1, i.e., 1i
1,, n
j1, ,n
. The objective of the FCM algorithm is to
find c cluster centers such that the cost functions of dis-
similarity (or distance measure) are minimized. The cost
function is defined by
,, ,cn
Uv vud
where, vi is the cluster center of the fuzzy group i;
iji j
dvx is the Euclidean distance between the ith
cluster and the jth data point, and m 1 is a weighting
exponent, taken as 2 here. The necessary conditions for
Equation (1) to reach minimum are:
ij j
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Figure 3. Ljung-Box lack-of-fit tests for: (a) ARMA (20,1), and (b) ARIMA (8,2,3) models.
Figure 4. Membership grades of the days over the year for
the daily streamflow base d on fuzzy c lustering.
ij kkj
In partitioning the days over the year with the cluster-
ing approach, the raw average daily discharge data and
the autocorrelation values at different lag times, say 1 -
10 days were used. The discharge data and the autocor-
relation coefficients were organized as a matrix X of size
where N is the number of years and 10
the autocorrelation values at 10 lags. To eliminate the
influence of large differences among data values on the
cluster result, the daily discharges were first logarithmi-
cally transformed before carrying out the cluster analysis.
Figure 4 shows the FCM clustering result. The entire
daily discharge over the annual cycle was partitioned
into three, basically conforming to the flow dynamics
which is made up of low, medium, and high flows. The
medium flow regime is a watershed or rather, the transi-
tion between the low and high flows. Based on the parti-
tioning results, one AR model was fitted to a partition.
Before fitting the respective AR models, the daily
streamflow series was deseasonalised. The orders of the
AR models were determined according to AIC criterion
[9], with the PACF acting as a basis for the model
choices. The partitioning of the daily streamflow in terms
of days over an annual cycle is shown in Table 1. Based
on the minimum AIC, Table 2 shows the orders of the
AR models for each partition while Table 3 indicate the
concise definition of each partition in terms of the intrin-
sic flow pattern. Collectively, the respective AR models
constitute the PAR model. During the forecasting proc-
ess, a specific AR model is applied depending on what
season’s partition the date to be forecasted is in.
10 365N
2.5. Forecast Performance Measures
Since forecast accuracy is best assessed by retrospective
comparison of forecasts actually made or that which
have been made, and the values observed during the
forecast period, the following measures were used to
evaluate model performances in the respective cases.
Mean Absolute Error: 1
Mean Absolute Percentage Error: 1
Root Mean Squared Error:
Mean Squared Relative Error:
ii i
Coefficient of Efficiency:
Coefficient of Determination:
 
Seasonally Adjusted Coefficient of Efficiency:
Table 1. Partitioning of days based on FCM method.
Partition 1 2 3
Day span 1-55, 268 - 365 56 - 105, 225 - 267 106 - 224
Table 2. Selected AR orders for the PAR model according
to minimum AIC value.
Partition 1 2 3
AIC 15 9 4
Table 3. Flow partitions and respective definition of flow
Partition 1 2 3
Flow pattern Low flows Moderately high flows High flows
where, Q
and i are the n modelled and observed
flows respectively;
QQ and are the mean of the ob-
served and modelled flows respectively and
(mod is the modulus, an operator used for calculating the
remainder) is the season, ranging from 0 to S - 1; and S is
the total number of season. The forecast exercise was
done by using the models developed. In all the cases, the
forecast horizon covers a two-year period; i.e., the last
two years. It suffices to note that model building was on
a rolling-forward basis.
3. Results and Discussion
The forecast results were evaluated based on the stated
measures of performance for each model under differing
flow regimes as appropriate; namely, Wet (April-Octo-
ber), and Dry (November-March).The evaluation results
for 1 to 10-day ahead forecasts with the ARMA(20,1) are
listed in Tables 4-6; ARIMA (8,2,3) in Tables 7-9, and
PAR in Tables 10-12. Based on the performance statis-
tics, the following observations can be made.
In terms of the values of CE, it is obvious that both the
ARMA (20,1), ARIMA (8,2,3), and PAR models indi-
cate that satisfying forecasts can be achieved for lead
times up to 10 days considering the whole year; that is,
there is a possibility of making long-term forecasts of the
streamflow process with the respective models. But con-
cisely, this shows how much the CE statistic can fla-
grantly exaggerate the forecast accuracy of the model;
SACE statistic in contrast, indicates the contrary. Using a
threshold of 0.9 [10], the SACE statistic shows that
with the ARMA (20,1), satisfactory forecast can be ob-
tained for up to a 10-day ahead lead time, and for
ARIMA (8,2,3), it is 5 days; whereas with the PAR
model, it is around 7 days. Realistically, baring model
uncertainty resulting from problems of externalities (say,
data quality problems, data size, non-stationarity and
seasonality issues), based on the SACE statistic, reliable
forecasts can plausibly be made up to a lead time of 7
It is important to note that there is obvious presence of
significant seasonal variation in forecast accuracy. The
forecast accuracy for dry season is relatively much
higher than that of the wet season. Using the MAE statis-
tic (threshold value, say, 150), with the ARMA (20,1),
satisfactory forecasts on the average, can be made for up
to 3 - 5 days, that is for both wet and dry season. On the
same basis, the performance of the ARIMA (8,2,3) is
abysmal for the wet season period; in the dry season pe-
riod, reliable forecasts are possible for at most 4 days
while with the PAR model, around 3 - 6 days for both
wet and dry season periods. When assessing the per-
formance of a streamflow forecasting model, it is not
only important to evaluate the average prediction error
but also the distribution of prediction errors as shown by
the results here. It is important to know whether the
model is predicting higher flows badly or the lower
agnitude flows badly, which may help in further refin- m
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Table 4. Forecast performance of ARMA (20,1) model for Whole year (Daily flows).
1 66.01 0.019 136.72 0.001 0.999 0.914 0.999
2 90.85 0.025 193.32 0.002 0.998 0.913 0.998
3 115.25 0.030 247.59 0.002 0.996 0.911 0.998
4 140.22 0.036 303.02 0.003 0.995 0.910 0.997
5 160.77 0.041 347.91 0.003 0.993 0.908 0.996
6 186.62 0.047 402.48 0.004 0.991 0.906 0.995
7 206.37 0.051 443.88 0.005 0.989 0.905 0.994
8 232.23 0.058 497.25 0.006 0.987 0.903 0.993
9 252.53 0.063 539.58 0.007 0.984 0.900 0.992
10 271.06 0.068 577.97 0.008 0.982 0.898 0.991
Table 5. Forecast performance of ARMA (20,1) model for Wet season (Daily flows).
1 84.05 0.023 146.24 0.002 0.999 0.999
2 113.90 0.029 202.53 0.002 0.998 0.999
3 143.61 0.034 256.17 0.003 0.997 0.998
4 173.25 0.040 312.30 0.003 0.995 0.998
5 196.60 0.044 355.70 0.004 0.994 0.997
6 228.18 0.051 412.46 0.005 0.992 0.996
7 251.23 0.055 453.29 0.005 0.991 0.996
8 283.70 0.062 509.38 0.007 0.989 0.994
9 308.34 0.067 553.07 0.008 0.987 0.994
10 330.57 0.072 592.29 0.009 0.985 0.993
Table 6. Forecast performance of ARMA (20,1) model for Dry season (Daily flows).
1 40.43 0.013 121.97 0.001 0.997 0.998
2 58.19 0.018 179.46 0.001 0.995 0.998
3 75.05 0.024 234.91 0.001 0.991 0.997
4 98.40 0.030 289.35 0.002 0.987 0.996
5 109.99 0.035 336.55 0.002 0.982 0.996
6 127.73 0.041 387.91 0.003 0.976 0.994
7 142.81 0.046 430.21 0.004 0.971 0.993
8 159.29 0.051 479.54 0.004 0.964 0.991
9 173.44 0.056 519.87 0.005 0.958 0.990
10 186.72 0.061 557.05 0.006 0.951 0.989
Table 7. Forecast performance of ARIMA (8,2,3) model for Whole year (Daily flows).
1 72.53 0.023 114.73 0.001 0.999 0.914 0.999
2 138.67 0.043 215.35 0.003 0.997 0.912 0.999
3 206.90 0.064 321.16 0.005 0.994 0.909 0.997
4 276.83 0.085 430.53 0.010 0.990 0.906 0.996
5 347.88 0.106 541.99 0.015 0.984 0.900 0.994
6 420.33 0.127 656.66 0.022 0.977 0.893 0.991
7 493.59 0.148 772.39 0.030 0.969 0.886 0.989
8 568.38 0.170 892.39 0.039 0.958 0.877 0.985
9 644.18 0.192 1014.21 0.051 0.946 0.865 0.982
10 721.30 0.213 1139.19 0.063 0.932 0.852 0.978
ing the model. While both the CE and r2 values in all the
instances considered are high, indicating the quality and
explanatory power of the ARMA (20,1), ARIMA (8,2,3),
and PAR models in the respective cases, the high values
of MAE, especially for the wet season (high flow period)
portray a different picture to the contrary. The ARMA
(20,1), ARIMA (8,2,3) and PAR failed to adequately
capture the high flow dynamics of the streamflow, thus
stressing the need for incorporating exogenous inputs in
the streamflow forecasting exercise.
Generally, the statistical performance criteria, RMSE,
r2, and CE are global statistics and do not provide any
robust information on the distribution of errors; precisely,
CE, MSRE, RMSE, MAE, and r2 are all measures that
incorporate both systematic and random errors. For in-
stance, it is noted that a CE value of zero indicates that
the observed mean is as good a predictor as the model,
hile a negative value implies that the observed mean is w
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Table 8. Forecast performance of ARIMA (8,2,3) model for Wet season (Daily flows).
1 100.51 0.022 142.56 0.001 0.999 0.999
2 192.42 0.040 269.32 0.002 0.996 0.999
3 287.95 0.059 401.93 0.005 0.993 0.998
4 386.62 0.079 539.36 0.008 0.987 0.997
5 487.30 0.101 679.58 0.014 0.980 0.996
6 590.87 0.122 824.49 0.021 0.971 0.994
7 695.62 0.144 970.80 0.029 0.960 0.992
8 804.48 0.167 1123.59 0.040 0.946 0.990
9 915.02 0.191 1278.75 0.053 0.930 0.987
10 1028.61 0.215 1438.56 0.067 0.912 0.984
Table 9. Forecast performance of ARIMA (8,2,3) model for Dry season (Daily flows).
1 32.87 0.025 54.89 0.001 0.999 0.999
2 62.50 0.048 96.46 0.003 0.998 0.998
3 92.03 0.071 142.73 0.006 0.996 0.997
4 121.24 0.093 189.12 0.011 0.994 0.996
5 150.29 0.114 235.68 0.017 0.991 0.994
6 178.62 0.135 280.91 0.023 0.987 0.992
7 207.29 0.155 326.22 0.031 0.983 0.989
8 233.78 0.174 368.53 0.039 0.978 0.986
9 260.34 0.193 411.05 0.048 0.973 0.982
10 285.77 0.211 451.72 0.057 0.968 0.979
Table 10. Forecast performance of PAR model for Whole year (Daily flows).
1 67.69 0.019 142.05 0.001 0.998 0.914 0.999
2 96.09 0.026 203.73 0.002 0.997 0.912 0.998
3 122.97 0.032 263.42 0.002 0.996 0.911 0.998
4 150.62 0.038 324.05 0.003 0.994 0.910 0.997
5 176.60 0.043 382.51 0.003 0.992 0.907 0.996
6 202.85 0.049 438.79 0.004 0.990 0.906 0.995
7 226.87 0.054 492.02 0.005 0.987 0.903 0.993
8 252.07 0.061 543.92 0.006 0.984 0.900 0.992
9 275.05 0.067 593.04 0.008 0.981 0.898 0.990
10 297.34 0.072 640.10 0.009 0.978 0.893 0.989
Table 11. Forecast performance of PAR model for Wet season (Daily flows).
1 84.49 0.023 150.60 0.002 0.999 0.999
2 117.42 0.028 210.92 0.002 0.998 0.999
3 148.28 0.034 268.75 0.003 0.996 0.998
4 179.69 0.039 328.64 0.003 0.995 0.997
5 209.27 0.043 386.83 0.004 0.993 0.997
6 239.33 0.049 443.29 0.004 0.991 0.996
7 267.08 0.053 497.01 0.005 0.989 0.995
8 296.31 0.060 549.47 0.007 0.987 0.994
9 323.05 0.065 599.42 0.010 0.984 0.992
10 349.06 0.070 647.43 0.010 0.982 0.991
a better predictor than the model [11]. But for hydro-
logical time series that often exhibit strong seasonality,
the general concern is whether the model is better than
seasonal mean values of the series rather than the overall
observed mean. This phenomenon cannot adequately be
addressed by CE as in Equation (8); similarly, it is noted
that the value of CE calculated for a whole year is higher
than the average of CE values calculated for separate
seasons, which illogically implies that the model per-
formance for the whole year is better than for most sepa-
rate seasons [7]. These problems arise from the inade-
quacy of the definition of CE in dealing with seasonal
processes. Basically, CE’s definition is premised on the
assumption that the process of interest is stationary [12];
but, hydrological time series usually exhibit strong sea-
sonality. It is interesting to note that when strong season-
ality exists, especially, when the mean value changes
ith season, for most of the seasons (such as days or w
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Table 12. Forecast performance of PAR model for Dry season (Daily flows).
1 43.90 0.014 128.97 0.001 0.997 0.998
2 65.87 0.022 193.08 0.001 0.994 0.998
3 87.12 0.029 255.67 0.001 0.989 0.997
4 109.43 0.037 317.44 0.002 0.984 0.997
5 130.30 0.043 376.32 0.003 0.978 0.996
6 151.16 0.050 432.34 0.004 0.971 0.995
7 169.88 0.056 484.88 0.005 0.963 0.994
8 189.36 0.063 535.96 0.006 0.955 0.993
9 207.03 0.069 583.88 0.007 0.947 0.992
10 224.05 0.075 629.55 0.009 0.938 0991
Figure 5. Comparison between the overall standard devia-
tion and seasonal standard deviation over an annual cycle.
months) in a year, the value of the overall standard de-
viation is larger than the values of seasonal standard de-
viation [7,8]. Figure 5 illustrates this disparity resulting
from the existence of strong seasonality. As shown by
Figure 5, the computed overall standard deviation (using
the overall mean) is about 3721.97 m3·s–1 whereas the
average of daily standard deviations (calculated for the
average discharges in each day over the year) is 977.38
m3·s–1. Thus Equation (10) (i.e. , SACE), the seasonally
adjusted coefficient of efficiency, espoused by Wang [7]
which requires the use of seasonal mean values can
overcome the shortcomings of the traditional CE.
To a large extent, the performance of a hydrologic
model is seriously dependent on several factors, among
which is the quality and information content of the data
used vis-à-vis the form of pre-processing or transforma-
tion adopted. Most univariate time series models are de-
veloped under the assumption of second-order stationar-
ity; thus if a strong seasonal component causes a series to
be non-stationary, the traditional approach is to either
pass it through a linear time-invariant filter, where the
output is assumed to be stationary. But there are many
instances of hydrologic time series that cannot be filtered
or standardised to achieve second-order stationarity be-
cause the entire correlation structure of the series may be
dependent on season. Considering this therefore, it is
important to look at the forecast accuracy of the ARMA
(20,1), ARIMA (8,2,3) and PAR models against the
backdrop of the pre-processing strategy adopted here
preparatory to the forecasting process since there is evi-
dence of strong seasonality in the flow series. As re-
ported in Kavvas and Delleur [13], McKerchar and
Delleur [3] and Delleur et al., [14], both from analytical
and empirical results, seasonal and/or non-seasonal dif-
ferencing, although very effective in the removal of hy-
drologic periodicities, distorts the original spectrum of
the time series, thus making it impractical or impossible
to fit an ARMA model for hydrologic simulation or syn-
thetic generation. Resulting from this, the forecasting
capabilities of either seasonally differenced or non-sea-
sonally differenced models may be impaired since they
do not take into account the seasonal variation in the
standard deviations as well as the seasonal structure in-
herent in the time series. A similar argument may be
made for the deseasonalisation pre-processing approach;
the deseasonalised modelling has some associated theo-
retical difficulties. The principal setback is the stationar-
ity assumption usually made for deseasonalised series,
which is not likely to be satisfied; this agrees with the
findings of Moss and Bryson [15]. These difficulties can
be overcome by employing periodic models, which allow
the model parameters, as well as model orders, to vary
depending on the season of the year.
Thus considering all the issues highlighted, the inabil-
ity of the ARMA (20,1), ARIMA (8,2,3), and PAR mod-
els to adequately capture the dynamics of the flow proc-
ess here can be understood. Despite this though, consid-
ering the defects of both the ARMA (20,1) and ARIMA
(8,2,3) resulting from the pre-processing style respec-
tively, in the context of realistic forecasting, the PAR
model as used here performs comprehensively better, as
it has a higher potential to account for the variability in
both seasonal deviations and seasonal correlation struc-
4. Conclusions
Data-driven models based on univariate time series were
used for forecasting in this study, namely: traditional
ARMA-type and the periodic AR (PAR) models. Com-
parative forecast performances of the various models
show that despite the limitation associated with univari-
ate time series, reliable forecasts can be obtained for lead
times, one to 5-day-ahead on the average for all the
models used. The forecast results also brought to the fore
the inadequacy of the traditional ARMA model. It was
unable to robustly simulate high flow regimes unlike the
periodic AR (PAR). Because of this, it is imperative that
in order to account for seasonal variations, PAR models
should be used in forecasting the daily streamflow proc-
ess of the Benue River. However, the stochastic model-
ling does show that the ARMA type models could be
used as preliminary models for the basis of understand-
ing the dynamics of the streamflow process.
In the light of the results obtained in this study, suffice
it to note that one limitation of this study is smallness of
the data size used for modelling the streamflow process.
Thus, to enhance the performance of the models and es-
tablish the generality of the conclusions drawn, it is
strongly recommended that larger data size be used, and
too, explanatory exogenous variables (e.g. precipitation)
be included during the modelling exercise, i.e., multi-
variate modelling. To this end, in order to improve the
accuracy of long-range forecasts, investigation of the
linkage between streamflow processes and ancillary hy-
droclimatic factors would be inevitable. In addition,
since predictability is an important aspect of the dynam-
ics of hydrological processes, though not considered in
this study, a definition of the predictability of the stream-
flow processes is a necessity; at least to put forward a
predictable horizon for the entire respective flow dy-
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