Calculating the required cash in bank branches: a Bayesian-data mining approach

Abstract

The issue of sufficiency of cash in bank branches is considered as an important issue especially for branch managers; because, not only the insufficiency of daily cash results in lack of response to needs of customers, but also may its excess result in increase in costs for banks. Hence, banks are always attempting to determine their required cash based on their daily operation. For this purpose, in this paper, 18 branches of a certain bank in a period of five months, due to diversity of the branches, have been classified by two methods of hierarchical clustering and Bayesian hierarchical clustering in similar clusters, and then by considering the results obtained from clustering, amounts of entered and consumed branch cash have been estimated by neural network (via classic and Bayesian approach), so that the cash required by branches can be calculated. The error criteria of the estimates show that calculations by applying Bayesian neural network method with considering Bayesian clustering have the least error compared to other methods.

Appendix

Expanding (6) and considering hyperparameters, the density function for the weight according to Bayes’ theorem is as follow:

$$ P(w|D,\alpha ,\beta ,F) = \frac{P(D|w,\beta ,F)P(w|\alpha ,F)}{P(D|\alpha ,\beta ,F)} $$

Also, the probability densities are:

$$ \begin{aligned} P(D|w,\beta ,F) & = \frac{1}{{Z_{D} (\beta )}}\exp ( - \beta E_{D} ) \\ P(w|\alpha ,F) & = \frac{1}{{Z_{W} (\alpha )}}\exp ( - \alpha E_{W} ) \\ \end{aligned} $$

$$ Z_{D} (\beta ) = (\pi /\beta )^{N/2} ,\quad Z_{W} (\alpha ) = (\pi /\alpha )^{W/2} $$

Replacing the above probabilities into the first one and knowing that $P(D|\alpha ,\beta ,F)$ is normalization factor, will have:

$$ P(w|D,\alpha ,\beta ,F) = \frac{1}{{Z_{M} (\alpha ,\beta )}}\exp ( - M(w)) $$

So, it will be obtained:

$$ \begin{aligned} P(D|\alpha ,\beta ,F) & = \frac{P(D|w,\beta ,F)P(w|\alpha ,F)}{P(w|D,\alpha ,\beta ,F)} \\ & = \frac{{\left[ {\frac{1}{{Z_{D} (\beta )}}\exp ( - \beta E_{D} )} \right]\left[ {\frac{1}{{Z_{W} (\alpha )}}\exp ( - \alpha E_{W} )} \right]}}{{\frac{1}{{Z_{M} (\alpha ,\beta )}}\exp ( - M(w))}} \\ & \quad \frac{{Z_{M} (\alpha ,\beta )}}{{Z_{D} (\beta )Z_{W} (\alpha )}} \cdot \frac{{\exp ( - \beta E_{D} - \alpha E_{W} )}}{\exp ( - M(w))} = \frac{{Z_{M} (\alpha ,\beta )}}{{Z_{D} (\beta )Z_{W} (\alpha )}} \\ \end{aligned} $$

Calculating $Z_{M} (\alpha ,\beta )$ through a Taylor series expansion, we can expand $M(w)$ around the minimum point of the posterior density $w_{\text{MAP}}$. So:

$$ Z_{M} \simeq (2\pi )^{W/2} (\det ((H_{\text{MAP}} )^{ - 1} )^{1/2} )\exp ( - M(w_{\text{MAP}} )) $$

$H = \beta \nabla^{\text{2}} E_{D} + \alpha \nabla^{\text{2}} E_{W}$ is the Hessian matrix of the function $M$.

With placement and taking the derivative, the optimum of $\alpha$ and $\beta$ will be obtained as below:

$$ \begin{aligned} \alpha_{\text{MAP}} & = \frac{\gamma }{{2E_{W} (w_{\text{MAP}} )}} \\ \beta_{\text{MAP}} & = \frac{N - \gamma }{{2E_{D} (w_{\text{MAP}} )}} \\ \end{aligned} $$

where $\gamma = W - \alpha_{\text{MAP}} {\text{tr}}(H_{\text{MAP}} )^{ - 1}$.