**2.4 Discharge prediction model analysis**

The results from the model analysis were examined based on the theory of runs which involves the analysis of two processes of sequential and opposite events at a boundary which is within a certain period [16].

The two kinds of "runs" available are run-length and run-sum with the run-length usually used for hydrological analysis related to the prediction of time which is also known as the frequency analysis while the run-sum is for those related to the prediction of rainfall duration and intensity within the analysis season. The overview of these runs is presented in the following **Figure 2**.

The run test was conducted on the simulated data, and since m is a positive runlength while n is a negative run-length as indicated in **Figure 2**, the total run-length (r) is m + n. Moreover, the estimated value of r, E(r), was expressed using the probability (q) for the estimated positive run-length, m, and (p) as the estimated probability value for the negative run-length, n. This relationship was expressed as follows [16]:

$$E(r\_q) = \frac{1}{q(1-p)} = \frac{1}{pq} \tag{6}$$

with boundary conditions 0 < q < 1

$$p = \frac{m}{m+n}; \ = \frac{n}{m+n}; \bar{r}\_q = \frac{1}{k\_r} \sum\_{j=1}^{kr} r\_{q,j} \tag{7}$$

where *rq* is the total run-length by q (probability), j = 1, 2, 3, … , kr; and kr is the total number of run-length.

**Figure 2.**

*Overview of positive run-length, m, positive run-sum, S, negative run-length, n, and negative run-sum, D, on a discrete series [16].*

It is also important to note that the estimated output in the model analysis is expected to be within the tolerance limit formulated as follows:

$$\frac{1}{pq} - \frac{t\_{a/2}}{pq} \left(\frac{p^3 + q^3}{k\_r}\right)^{1/2} \le \bar{r}\_q \le \frac{1}{pq} + \frac{t\_{a/2}}{pq} \left(\frac{p^3 + q^3}{k\_r}\right)^{1/2} \tag{8}$$

where α is the tolerance limit (5%) and t is the normal distribution value from the "t-table."
