## 2 Materials and Methods
### Approach for Drought Assessment Using Satellite and Meteorological Data
#### 2.5.1 Panel Data and Fixed Effect Model  
The cross-sectional data are a set of observations derived from various entities in one fixed point of time. The analysis takes into consideration the differences between the groups of observations from these entities. The time series describe a feature in a temporal aspect; observations are collected at equally spaced time intervals as 'time points,' and the 'lag operator' is defined, which indicates the preceding observation by a number of 'time steps.' In 'panel data,' one cross-sectional entity is considered over time, so the structure is pooled over space as well as over time. It enables us to apply more complex two-dimensional models, taking into account the dynamics of changes in combination with the diversity of the objects. We found it to be appropriate to develop the model  
Figure 8: Temperature condition index (TCI) image, day of year (DOY) 241, 2015.  
Figure 9: Land surface temperature (LST) maximum image, DOY 241.  
using a combination of temporal satellite and spatially diversified meteorological data, described as the Fixed Effects Model [40]. This approach affords the possibility to estimate the linear regression parameters (i.e., slope and intercept) for joint data from \(n\) parallel sources under some assumptions. The first assumption is that the diversified relations within the groups of observations are represented by intercept parameters, which are independent of time, i.e., fixed in time. The second assumption is that the expected values of the variables are also fixed in time.  
The system of equations for the fixed effects model is defined as  
\[Y_{i,t}\ =\ \beta_{0}X_{i,t}+\beta_{1}X_{i,t-1}+\ldots+\beta_{k}X_{i,t-k}+ \alpha_{i}+\varepsilon_{i,t} \tag{3}\]  
where  
\(i\)--entity (one source of data) \(1\ldots\ n\);  
\(n\)--number of entities;  
\(t\)--time point;  
\(\alpha_{i}\)--intercept fixed in time for entity \(i\);  
\(Y_{i,t}\)--dependent variable;  
\(X_{i,t-k}\)--independent variable for entity \(i\) and time point \(t\)-\(k\), \(k=0,\ldots,m\);  
\(m\)--number of lags;  
\(\beta_{k}\)--slope parameter for independent variable \(X_{i,t-k}\), \(k=0,\ldots,m\);  
\(\varepsilon_{i,t}\)--error for entity \(i\) and time point \(t\).  
The intercept \(\alpha_{i}\) is specific for entity \(i\) and is unknown. It does not allow all entity observations to be combined into one system of equations to estimate the parameters \(\beta_{k}\).  
The intragroup transformation of equations system (3) was done. It consists of the subtraction of the expected value in the group (entity) from each variable on the left and right sides of the equations:  
\[Y_{i,t}-\hat{Y}_{i}\ =\ \beta_{0}\Big{(}X_{i,t}-\hat{X}_{i}\Big{)}+\ldots+\ \beta_{k}\Big{(}X_{i,t-k}-\hat{X}_{i}\Big{)}+ \varepsilon_{i,t}-\ \varepsilon_{i} \tag{4}\]  
where  
\(\hat{Y}_{i}\), \(\hat{X}_{i}\)--the fixed-in-time expected values of variables \(X\) and \(Y\);  
\(\varepsilon_{i}=0\), because the mean of the residuum is zero by assumption of the least mean squares method (LMSQ) method.  
The transformation eliminates the element \(\alpha_{i}\) that differentiates the groups of observations because it was constant on the level of one group. Expression (4) is the homogeneous system of linear equations, which can be estimated by the classical least mean squares method (LMSQ). It describes the model based on the anomalies of dependent and independent variables defined in Equation (3). The anomalies of variables erase the spatial diversification and allow for their comparison on a uniform scale.