The mean daily discharges are as shown in Figure 2; the discernible features are the flood peaks and the seasonal periodicity, with both large and small discharges during the year. Besides these general characteristics, the study of the autocorrelation and the spectral analysis provide the first important indications on the aperiodicity of the signal under examination. Figure 3 shows the autocorrelation for values of the delay time (lag) between 1 day and about 5 years; after a rapid decrease, the autocorrelation function displays a regular behaviour, which represents the effect of the seasonal characteristic of the discharges, due, besides the rainfall regime, to other hydrologic forcing which breathes with the season. Contrary to expectation, there is no brisk fall of the autocorrelation as shown in Figure 4, which might indicate a complex behaviour, having characteristic timescales of a few days.

Similarly, the autocorrelation of the first difference signal (Figure 4) does not show much of a rapid monotonic decrease in autocorrelation. Despite this though, apart from the periodicity, the behaviour of the series appears essentially aperiodic and erratic; for example, once the annual periodic component has been filtered out from the series through least squares methods, the spectral density (Figure 5) exhibits a typical broadband behaviour. The spectrum does not show any privileged frequencies, but rather a linear decay that links the

whole range of frequency components. The fact that the spectrum is continuous with a pronounced and wide base underscores the aperiodicity of the series; but the problem however is, how much does the character of this complex aperiodicity and irregularity may translate to complex nonlinearity. This therefore portends the need to investigate if the dynamics of the discharges of a river could have a dominant chaotic signature on which high-dimension linear and nonlinear dynamics may be grafted.

The first step in the search for a deterministic behaviour is that of attempting to reconstruct the dynamics in phase space. Having available the time series of only one of the variables present in the phenomenon, that is, the discharge x ( t i ) , the delay time method proposed by Takens [10] and Packard et al. [11] can be used to reconstruct the attractor; this is based on the fact that the interaction between the variables is such that every component contains information on the complex dynamics of the system. Choosing a delay time τ, usually a multiple of sampling period Δ t , the method entails the construction of a series of M = ( m − 1 ) τ / Δ t vectors, of the dimension m, of the form

x ( t i ) = { x ( t i ) , x ( t i + τ ) , ⋯ , x [ t i + ( m − 1 ) τ ] } (1)

where, m ( m = 2 , 3 , ⋯ ) is called the embedding dimension.

To construct a well-behaved phase-space by delay time, a careful choice of τ is critical. The delay time τ is commonly selected by using the autocorrelation function (ACF) method where ACF first attains zeros or drops below a small value, say 1/e⁴, or the mutual information (MI) method according to Fraser and Swinney [12] where the MI attains a minimum. Here, the delay time τ is taken as the lag that first generates a zero autocorrelation, which is when the autocorrelation function crosses the zero line (Mpitsos et al., [13] ). In practice, the estimate of τ is usually application and author dependent; for instance, some authors take the delay time as 1 day (Porporato and Ridolfi, [14] ), 2 days (Jayawardena and Lai, [15], 7 days (Islam and Sivakumar, [16] ), 10 days (Elshorbagy et al., [17] ), 20 days (Wilcox et al., [18] ), 91 days (Wang, [19] ), and 146 days (Pasternack, [20] ). These differences may arise from the nature of the autocorrelation function; to compare the influence of the delay time τ on the construction of state-phase, the state-phase maps can be plotted for differing τ values. The best τ value should make the state-phase plot best unfolded. Towards this end, the state-phase map is constructed for different τ values (τ = 1, 7, 10, 30, and 78), these are as displayed in Figure 6 along with the 3-dimensional state-phase map based on τ equals to 78, i.e., when the autocorrelation function first crossed the zero line (Figure 6). Figure 6 shows that the best unfolding can be obtained when τ = 78; therefore, τ = 78 is adopted for estimating the correlation dimension of the daily streamflow process in this study.

The time delay coordinate method (Packard, et al. [11]; Takens, [10] ) was used to reconstruct the phase-space from the scalar time series; in this case, because of the nature of the data; i.e., a univariate series. This is informed by the fact that to describe the temporal evolution of a dynamical system in a multi-dimensional phase-space with a scalar time series, there is need to employ some techniques to unfold the multi-dimensional structure using the available data. Thus, the approach in the proceeding section is further complimented by applying the method for the determination of minimal sufficient dimension (m) as proposed by Kennel et al. [21], called the “False Nearest Neighbour(FNN)” method. That is, supposing the point X i = [ Y i − p + 1 , ⋯ , Y i ] has a neighbour X j = [ Y j − p + 1 , ⋯ , Y j ] in a p-dimensional space then the distance ‖ X i − X j ‖ is calculated in order to compute:

R i = | Y i + 1 − Y j + 1 | ‖ X i − X j ‖ (2)

If R i exceeds a given threshold R T (a suitable value is 10 ≤ R T ≤ 50 ), the point X i is marked as having a False Nearest Neighbour. As a consequence, the embedding dimension p is high enough if the fraction of points that have False Nearest Neighbours is actually zero, or sufficiently small, say, smaller than a criterion R f . In this case, the False Nearest Neighbour threshold R T was set to 10 (as reported in Wang [19]. Based on this, the fraction of False Nearest Neighbours as a function of the embedding dimension was calculated based on phase-space reconstruction using embedding dimension. Here, the minimal embedding dimension was taken as 8; this implies that the state of the streamflow process can be determined by eight lagged observed values as shown in Figure 7.

Following from the analysis, eight lagged values of input variables were used when fitting the ANN model to the series; specifically, this implied that based on the phase-space reconstruction, the discharges Q t − 7 , Q t − 6 , ⋯ , Q t of day t-7 to day t. The eight lagged input values were used to forecast the discharge from time t + 1, i.e., the next day, to t + 5; i.e., 5-ahead values, using a multiple-output approach rather than a single-output. The idea here is just to explore the ANN model forecast behaviour over a high lead time (Figure 8).

To address the thrust of the study, two model configurations were considered corresponding to two model architectural variants with different nodal configurations. Precisely, single and double hidden layers were thus considered to examine the implications of model structural complexity. The ANN models adopted were 1) 8 7 5 single-hidden layer with 7 nodes; i.e., 8 input nodes in the input layer, 7 nodes in the hidden layer, and 5 output nodes in the output layer 2) 8 5 2 5 double-hidden layers with 5 and 2 nodes, respectively and 3) 8 4 3 5 double-hidden layers with 4 and 3 nodes, respectively; though after several trials in an attempt to choose comparable network structures.

For the purposes of the stated aim of the study, the multi-layer feedforward back propagation network was used. Specifically, network training was implemented using the trainbr (Bayesian regularisation: Br) function, traingdm (Gradient descent with momentum: gdm) function, trainlm (Levenberg-Marquardt: lm function in MATLAB Neural Network Toolbox. Since in neural network training, the transfer function is of critical relevance and predictability of future behaviour is a direct consequence of the correct identification of it, for the identified network structure, the tansigmoid and purelin transfer functions were used in the hidden and output layers, respectively. The purelin transfer function was considered for the output layer because it allows the network outputs to take on any value, whereas the last layer of a multi-layer network with sigmoid neurons constrains the network outputs to a small range.

Before applying the ANN, both input and output data were pre-processed and normalised in the range [−1 1]. The scaling strategy was adopted based on the findings of Wang [19] and Otache [22]; rescaling was done to scale the data series to fall within this bound. Scaling of the original data, say x ′ t to the network range was done by

x t = ( U x − L x ) x ′ t + ( M x L x − m x U x ) M x − m x (3)

where, x ′ t = the original input data, x t = the input data scaled to the network range, M x and m x are respectively the maximum and the minimum of the original input data, while U x and L x are the upper and the lower network ranges for the network input, respectively. Similarly, the original output, say y ′ t is scaled to the network range by

y t = ( U y − L y ) y ′ t + ( M y L y − m y U y ) M y − m y (4)

where, y t the systems’ output is scaled to the network range, M y and m y are respectively the maximum and minimum values of the original output data y ′ t , whereas U y and L y are respectively the upper and the lower network ranges for the network output. After scaling the inputs and outputs, the resulting output, say y ^ t is in the scaled domain. Hence, there is need to rescale the output y ^ t back to its original domain; this is by inverting Equation (3) and using y ^ ′ t as

y ^ ′ t = ( M y − m y ) y ^ t − ( M y L y − m y U y ) U y − L y (5)