The accuracy of the model can be evaluated by calculating the residuals E by subtracting the rainfall data from the calculated values. As seen in Figure 7, the model fits the data with some variation representing the remaining stochastic component of time series. Ponatinib chemical structure The model accuracy can be evaluated quantitatively by estimating the mean absolute errorFigure 7Model error E calculated by subtracting the monthly total rainfall data from the calculated values. The data represent that obtained from the weather station of Kuwait Airport for the time duration from January 1965 (corresponding to month number 0) to …MAE=1n��i=1n|Ei|.
(5)It is worth mentioning that this statistical criterion for evaluating model accuracy is more convenient for rainfall data with zero values, while other well-known criteria such as the Nash-Sutcliffe equation and the Mean Absolute Percentage Error are inappropriate, as the actual value in their expressions is found in the denominator by which the equations become undefined for zero rainfalls. It is found that the overall model error estimated by (5) is about MAE = 12.6mm.The remaining stochastic component can be tested for any possible persistence and/or random structure. To do this, the residuals are first standardized as follows:Zi=Ei?E����,(6)where E�� is mean residual of series Ei, which is for this data equal to E�� = ?5.36mm; and �� is standard deviation of residual of series Ei, equal to �� = 21.3mm. The series Zi then has zero mean and unit standard deviation.
The correlogram r, also known as the autocorrelation function ACF, can then be obtained for the series Zi at lag L fromrL=(1/(n?L))��i=1n?L(Zi?Z��)(Zi+L+Z��)(1/n)��i=1n(Zi?Z��)2,(7)where Z�� is mean of the sample n values of series Zi. The lag L is usually taken for values up to n/4. The resulting values of rL will range between ?1 and +1. Figure 8 shows the estimated correlogram. Here, two lines of upper and lower confidence bands were drawn with significance level �� = 0.05 to examine whether the data are randomFigure 8Correlogram up to lag 135 for series Zi. The dashed lines represent upper and lower bounds with significance level �� = 0.05.C.B.=��z1?��/2n,(8)where z is quantile function of standard normal distribution. If the correlogram is higher (lower) than this upper (lower) band, the null hypothesis that there is no autocorrelation at and beyond a given lag is rejected at the given significance level. As can be seen, the correlogram exhibits an alternating sequence of positive and negative spikes. Such a pattern is the correlogram signature of a repetitive oscillating movement, suggesting that the remaining stochastic component Dacomitinib in this data is due to the presence of other periods that have not been taken into account in the model.