Analysis of rainfall occurrence in consecutive days using Markov models with covariate dependence in selected regions of Bangladesh

Abstract

The objectives of this study are to reveal the regional variation and to identify the responsible factors for making the transition of rainfall status in two consecutive days in Bangladesh at three selected regions. The covariate-dependent first-order Markov models have been used to find out how the change of rainfall status (rain and no rain) is affected by the selected factors. Among the considered factors, we found that the relative humidity, sunshine hour, maximum temperature, and cloud cover show positive association with the transition type no rain to rain for all three stations. However, the minimum temperature, maximum temperature, and sea level pressure have a negative effect on this transition type in all selected stations with the exception that the maximum temperature in Rajshahi and the minimum temperature in Rajshahi and Sylhet do not have any significant effect. On the other hand, with the exception, the maximum temperature in Rajshahi and sunshine hour in Dhaka and Sylhet, the relative humidity, maximum temperature, sunshine hour, and cloud cover show positive association with the transition type rain to rain, and the minimum temperature and sea level pressure show negative association with this transition type. The regional variations and responsible factors for changing the rainfall status also disclosed among the three selected regions over four seasons such as pre-monsoon, monsoon, post-monsoon, and winter. In all four seasons, the transition of the rainfall status experienced statistically significant positive impact for the exploratory variables relative humidity, sunshine hour, and cloud cover and negative impact for maximum temperature, minimum temperature, and sea level pressure with some exceptions. However, the humidity and cloud cover play the most important role in making the transition of the rainfall status between two consecutive days.

Appendices

Appendix

Table 6 The intra-seasonal variation among fitted first-order Markov models for three different rainfall regions for the rainfall data during 1970 to 2014

Annex A

The rth-order Markov model satisfy the following property.

$$ P\left({Y}_{ij}={y}_{ij}|{Y}_{ij-k}={y}_{ij-k},...,{Y}_{ij-r}={y}_{ij-r},...,{Y}_{ij-1}={y}_{ij-1}\right)=P\left({Y}_{ij}={y}_{ij}|{Y}_{ij-r}={y}_{ij-r},...,{Y}_{ij-1}={y}_{ij-1}\right)\ \mathrm{for}\ j>k>r. $$

The Markov model (Islam et al. 2009) can be defined for the time points t_j, t_{j − 1}, t_{j − 2}, ..., t_{j − r} with the outcomes Y_{i, j} = s_0,Y_{i, j − 1} = s_1,Y_{i, j − 2} = s_2,..., Y_{i, j − r} = s_r as

$$ {P}_{s_1{s}_2,\dots, {s}_r,{s}_0}=P\left({Y}_{ij}={s}_0|{Y}_{i,j-1}={s}_{1,}\dots, {Y}_{i,j-r}={s}_{r,}X\right)=\frac{e^{X{\beta}_{s_1{s}_2\dots {s}_r}}}{1+{e}^{X{\beta}_{s_1{s}_2\dots {s}_r}}},{s}_1,{s}_2,\dots, {s}_r=0,1. $$

(1)

The first (r = 1)-order Markov models (Islam et al. 2009) also can be derived from this generalized Markov model (Eq. 1) by considering two time points as

$$ {P}_{s_1,{s}_0}=P\left({Y}_{ij}={s}_0|{Y}_{i,j-1}={s}_1,X\right)=\frac{e^{X{\beta}_{s_1}}}{1+{e}^{X{\beta}_{s_1}}},{s}_0,{s}_1=0,1. $$

(2)

The likelihood function for the first-order Markov model can be written as:

$$ {\displaystyle \begin{array}{l}L\left(\beta \right)=\prod \limits_{i=1}^n\prod \limits_{j=1}^J\prod \limits_{s_{1=0}}^1\prod \limits_{s_{0=0}}^1\left[{P_{s_1{s}_0}}^{\partial_{ij{s}_1{s}_0}}\right]\\ {}=\prod \limits_{i=1}^n\prod \limits_{j=1}^J\left[{p_{ij01}}^{\partial_{ij01}}{\left(1-{p}_{ij01}\right)}^{\left(1-{\partial}_{ij01}\right)}{p_{ij11}}^{\partial_{ij11}}{\left(1-{p}_{ij11}\right)}^{\left(1-{\partial}_{ij11}\right)}\right]\\ {}\ln \left(L\left(\beta \right)\right)=\sum \limits_{i=1}^n\sum \limits_{j=1}^J\left[\left\{{\partial}_{ij01}\left\{X{\beta}_0\right\}-\ln \left(1+{e}^{x{\beta}_0}\right)\right\}+\left\{{\partial}_{ij11}\left\{X{\beta}_1\right\}-\ln \left(1+{e}^{x{\beta}_1}\right)\right\}\right]\\ {}\ln \left(L\left(\beta \right)\right)=\ln \left(L\left({\beta}_0\right)\right)+\ln \left(L\left({\beta}_1\right)\right),\end{array}} $$

(3)

where ${\partial}_{ij}=\left\{\begin{array}{c}1,\mathrm{if}\ \mathrm{the}\ \mathrm{transition}\ \mathrm{occurs}\ \mathrm{for}\ {\mathrm{i}}^{th}\ \mathrm{unit}\ \mathrm{at}\ {\mathrm{j}}^{\mathrm{th}}\ \mathrm{follow}-\mathrm{up}\ \mathrm{time},\kern3em \\ {}0,\mathrm{otherwise}.\kern20.75em \end{array}\right.$

After differentiating the log likelihood function with respect to β_0l and β_1l we will get the following differential equations.

$$ {\displaystyle \begin{array}{l}\frac{\partial \ln \left(L\left({\beta}_0\right)\right)}{\partial {\beta}_{0l}}=0,\\ {}\frac{\partial \ln \left(L\left({\beta}_1\right)\right)}{\partial {\beta}_{1l}}=0,\end{array}} $$

where l = 0, 1, ..., p. Now, we can derive the MLE of βdenoted as ${\hat{\beta}}_0=\left({\hat{\beta}}_{00},{\hat{\beta}}_{01},...,{\hat{\beta}}_{0p}\right)$ and ${\hat{\beta}}_1=\left({\hat{\beta}}_{10},{\hat{\beta}}_{11},...,{\hat{\beta}}_{1p}\right)$by solving these differential questions. The second (r = 2) 2nd order, (r = 3) 3^rdorder and so on can be defined in similar ways. The parameter estimation and testing approaches in detail are available in Islam and Chowdhury (2006, 2008, 2017); Chowdhury et al. (2005) and Islam et al. (2013, 2014).