Trend Analysis
The trend analysis depicts the series elevation or dips over time. This study estimated three trends: linear, log-linear, and quadratic trends. To determine which of the three trends best described the series, three distinct models were evaluated.
The time trend in a time series is a linear function of the time t. The model is given by;
$${V}_{t}= {{\beta }_{0}}_{ }+ {\beta }_{1}t+ {e}_{t} \left(1\right)$$
where \({V}_{t}\) is the actual value at time \(t, t=1,\dots ,{ T, e}_{t}\) is the error term, and \({\beta }_{0}\) \({\beta }_{1}\), \({\beta }_{2}\)are the regression coefficients of the actual values over time. A linear trend denotes a steady increase or decrease in the series.
The empirical specification is as follows: if there is a linear trend in all-cause diarrhea disease.
$${Diarrhea Disease}_{t}= {{\beta }_{0}}_{ }+ {\beta }_{1}t+ {e}_{t} \left(2\right)$$
The empirical specification is as follows if climate variables (temperature, rainfall, and relative humidity) empirically show a linear trend.
$${Relative Humidity}_{t} = {{\beta }_{0}}_{ }+ {\beta }_{1}t+ {e}_{t} \left(3\right)$$
$${Temperature}_{t}= {{\beta }_{0}}_{ }+ {\beta }_{1}t+ {e}_{t} \left(4\right)$$
$${Rainfall}_{t}= {{\beta }_{0}}_{ }+ {\beta }_{1}t+ {e}_{t} \left(5\right)$$
If the data displays a quadratic trend, the model is as follows:
$${V}_{t}={\beta }_{0}+ {\beta }_{1}t+ {\beta }_{2}{t}^{2}+{e}_{t} \left(6\right)$$
In this equation, \({V}_{t}\) is the actual value at time \(t, t=1,\dots ,{ T, e}_{t}\) is the error term, and \({\beta }_{0}\) \({\beta }_{1}\), \({\beta }_{2}\)are the regression coefficients of the actual values on time. A series that exhibits a quadratic trend implies that its growth or decline rate is not constant.
If all-cause diarrhea exhibits a quadratic trend, the empirical specification will be as follows:
$${Diarrhea Disease}_{t}={\beta }_{0}+ {\beta }_{1}t+ {\beta }_{2}{t}^{2}+{e}_{t} \left(7\right)$$
Similarly, if climatic variables (Relative Humidity, Temperature, Rainfall) exhibit a quadratic trend, the empirical specification is as follows:
$${Relative Humidity}_{t}={\beta }_{0}+ {\beta }_{1}t+ {\beta }_{2}{t}^{2}+{e}_{t} \left(8\right)$$
$${Tempearture }_{t}={\beta }_{0}+ {\beta }_{1}t+ {\beta }_{2}{t}^{2}+{e}_{t} \left(9\right)$$
$${Rainfall}_{t}={\beta }_{0}+ {\beta }_{1}t+ {\beta }_{2}{t}^{2}+{e}_{t} \left(10\right)$$
The log-linear representation of the model is as follows:
$$\text{l}\text{n}{V}_{t}={\beta }_{0}+ {\beta }_{1}t+ {\beta }_{2}{t}^{2}+{e}_{t} \left(11\right)$$
From the equation above, \({V}_{t}\) is the actual value at time \(t, t=1,\dots ,{ T, e}_{t}\) is the error term and \({\beta }_{0}\) \({\beta }_{1}\), \({\beta }_{2}\)are the regression coefficients of the actual values at time.
If all-cause diarrhea exhibits a log-linear trend, the empirical specification will be as follows:
$$\text{l}\text{n}{Diarrhea Disease}_{t}={\beta }_{0}+ {\beta }_{1}t+ {\beta }_{2}{t}^{2}+{e}_{t} \left(12\right)$$
Empirically, if climatic variables (relative humidity, Temperature, Rainfall) exhibit a log-linear trend, the empirical specification is as follows:
$$\text{l}\text{n}{Relative humidity}_{t}={\beta }_{0}+ {\beta }_{1}t+ {\beta }_{2}{t}^{2}+{e}_{t} \left(13\right)$$
$$\text{l}\text{n}{Temperature}_{t}={\beta }_{0}+ {\beta }_{1}t+ {\beta }_{2}{t}^{2}+{e}_{t} \left(14\right)$$
$$\text{l}\text{n}{Rainfall}_{t}={\beta }_{0}+ {\beta }_{1}t+ {\beta }_{2}{t}^{2}+{e}_{t} \left(15\right)$$