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\markboth{Pratiwi, H., dkk} %Jika lebih dari dua penulis, tuliskan sebagai Nama Penulis Pertama dkk.
{Comparison of Weight Matrix...}

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\title{COMPARISON OF WEIGHT MATRIX IN HOTSPOT MODELING IN WEST KALIMANTAN USING THE GSTAR METHOD}

\author{HESTY PRATIWI$^{a}$, NURFITRI IMRO'AH$^{a}$\footnote{penulis korespondensi},NUR'AINUL MIFTAHUL HUDA$^{b}$, MUHAMMAD YAHYA AYYASH$^{a}$\\ }

\address{
$^{a}$ Statistics Department, Universitas Tanjungpura, Pontianak, Indonesia,\\
$^{b}$ Mathematics Department, Universitas Tanjungpura, Pontianak, Indonesia.\\
email : \email{h1091211020@student.untan.ac.id, nurfitriimroah@math.untan.ac.id, nurainul@fmipa.untan.ac.id, h1091211042@student.untan.ac.id }}

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\begin{abstract}
\begin{center}
Diterima ..... \quad Direvisi ..... \quad Dipublikasikan ..... %tanggal-tanggal tersebut \textbf{dikosongkan} saja
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\bigskip

\textbf{Abstrak}. %Dalam bahasa Indonesia
Penelitian ini membahas perbandingan efektivitas antara matriks bobot seragam dan Queen contiguity dalam pemodelan jumlah titik api di Provinsi Kalimantan Barat menggunakan metode Generalized Space-Time Autoregressive (GSTAR). Kebakaran hutan dan lahan di Kalimantan Barat merupakan masalah serius yang menimbulkan kerusakan ekologis dan dampak negatif lainnya. Data yang digunakan adalah data titik api dari empat kabupaten di Kalimantan Barat selama periode Januari 2018 hingga Maret 2023. Hasil penelitian menunjukkan bahwa matriks bobot Queen contiguity memberikan hasil yang lebih akurat dibandingkan dengan matriks bobot seragam, dengan nilai Root Mean Squared Error (RMSE) dan Mean Absolute Deviation (MAD) yang lebih rendah. Temuan ini diharapkan dapat membantu dalam perencanaan strategi mitigasi kebakaran hutan dan lahan yang lebih efektif.
\keywords{GSTAR, Matriks Bobot Seragam, Queen Contiguity}
\bigskip

\textbf{Abstract}. % Dalam bahasa Inggris
\textit{This research aims to investigate the usefulness of the Generalized Space-Time Autoregressive (GSTAR) approach in predicting the number of fire hotspots in West Kalimantan Province. Specifically, the study compares the performance of the Queen contiguity method and the uniform weight matrix. Fires in the forests and on the land in West Kalimantan are severe problems that cause harm to the environment and other adverse effects. Data on fire hotspots were collected from four different regencies in West Kalimantan between January 2018 and March 2023 to provide the information used in this study. Compared to the uniform weight matrix, the study results reveal that the Queen contiguity weight matrix produces more accurate results. This is evidenced by the fact that the Root Mean Squared Error (RMSE) and Mean Absolute Deviation (MAD) values are lower in the Queen contiguity weight matrix. Based on these findings, more effective techniques for preventing forest and land fires are anticipated to be considered for planning purposes.}

\end{abstract}

\keywords{GSTAR, Uniform Weight Matrix, Queen Contiguity}

\section{Introduction}
Forest and land fires are natural disasters that often occur in Indonesia. This occurrence is common in West Kalimantan, one of the regions we are discussing. These fires have several detrimental effects on life, including the destruction of ecosystems, a reduction in biodiversity, a loss in the economic worth of forests and land productivity, climate change, and the appearance of smoke pollution, which impairs health and transportation by sea and air \cite{ref4}.

Indicators of forest and land fires can be identified through hotspots. According to the data analysis, four districts have the largest average number of hotspots. These districts are the Ketapang District, the Sanggau District, the Sintang District, and the Landak District—in that order. As a result, modelling and estimating the number of hotspots is of utmost significance for planning mitigation and response measures. It is effective.

In spatial analysis, selecting the spatial weight matrix is a crucial step that can influence the modelling results. The Queen Contiguity and the uniform weight matrix, respectively, are two methods that are frequently utilized. All neighbours are given the same weight in a uniform weight matrix, which does not consider distance or other spatial relationships. The Queen Contiguity matrix, on the other hand, takes into account neighbours that share edges or corners with the unit of analysis. As a result, it offers a more comprehensive depiction of spatial interconnections \cite{ref5}.

In modelling spatial-temporal data, the Generalized Space-Time Autoregressive (GSTAR) method is frequently utilized due to its capacity to capture the complexity and dynamism of data dispersed throughout space and time. By utilizing the GSTAR approach, the number of hotspots can be anticipated with greater precision, which in turn contributes to developing more efficient mitigation strategies\cite{ref2}.

This research aims to assess the performance of two different types of weight matrices, uniform weight and Queen Contiguity, in modelling the number of hotspots in West Kalimantan Province using the GSTAR technique. By carrying out this comparison, a more optimal approach can be developed for modelling and predicting the distribution of hotspots. This would allow the information obtained to be more valuable in the efforts being made to prevent and manage forest and land fires in the region.


\section{Theoretical Basis}
This research uses the Generalized Space-Time Autoregressive (GSTAR) method of an order [1;1], which uses a uniform weight matrix and a queen contiguity weight matrix.
\section*{2.1. GSTAR Model [1;1]}
The GSTAR model with first-order spatial lag and first-order autoregressive or GSTAR [1;1] has the following general formula \cite{ref8}.
\begin{eqnarray}
	Y_t &=& \Phi_{10}Y_{t-1}+\Phi_{11}W_{(*)}Y_{t-1}+e_{t} \label{pers1}
\end{eqnarray}
where,
\begin{eqnarray*}
		Y_t &=&\begin{bmatrix}
		Y_{t}^{(1)}\\
		Y_{t}^{(2)}\\
		\vdots\\
		Y_{t}^{(n)}
	\end{bmatrix}
	;
	\Phi_{10}=
	\begin{bmatrix}
		\Phi_{10}^{(1)}	&0	&\dots	&0\\
		0	&\Phi_{10}^{(2)}	&\dots	&0\\
		\vdots	&\vdots	&\ddots	&\vdots\\
		0	&0	&\dots	&\Phi_{10}^{(2)}\\
	\end{bmatrix}
	;
	\Phi_{11}=
	\begin{bmatrix}
		\Phi_{11}^{(1)}	&0	&\dots	&0\\
		0	&\Phi_{11}^{(2)}	&\dots	&0\\
		\vdots	&\vdots	&\ddots	&\vdots\\
		0	&0	&\dots	&\Phi_{11}^{(2)}\\
	\end{bmatrix}
	;
	e_{t}=
	\begin{bmatrix}
		e_{t}^{(1)}\\
		e_{t}^{(2)}\\
		\vdots\\
		e_{t}^{(n)}
	\end{bmatrix}
\end{eqnarray*}

$W_((*))$ is a weight matrix, and $\phi$ is a diagonal matrix where each entry on the diagonal is a parameter. Equation \eqref{pers1} can also be written.
\begin{eqnarray}
	Y_t = (\Phi_{10}+\Phi_{11}W_{(*)})Y_{t-1}+e_{t}
\end{eqnarray}

\section*{2.2. Selection of Weight Matrix in GSTAR Model}
The weight matrix is a characteristic of the GSTAR space-time model that can be used to identify the model. This matrix describes the relationships that exist between different geographical areas \cite{ref6}. In a general sense, the weight matrix can be expressed in the following form \cite{ref1}:
\begin{eqnarray*}
	\textbf{W}&=&
	\begin{bmatrix}
		0	&w_{12}	&\dots	&w_{1j}\\
		w_{12}	&0	&\dots	&w_{2j}\\
		\vdots	&\vdots	&\ddots	&\vdots\\
		w_{i1}	&w_{i2}	&\dots	&w_{ij}
	\end{bmatrix}
\end{eqnarray*}

\section*{2.2.1. Uniform Weight Matrix}
This weight matrix gives the same value for each site, which is why it is frequently utilized for data containing similar places or consistent distances between locations \cite{ref9}.
\begin{equation}
	w_{ij}=\frac{1}{m} \label{pers2}
\end{equation}
\begin{eqnarray*}
	W&=&
	\begin{bmatrix}
		0&\frac{1}{m}&\dots	&\frac{1}{m}\\
		\frac{1}{m}	&0	&\dots&\frac{1}{m}\\
		\vdots	&\vdots	&\ddots	&\vdots\\
		\frac{1}{m}	&\frac{1}{m}	&\dots	&0
	\end{bmatrix}
\end{eqnarray*}
where $m= n-1$, $n$ is the number of locations and $w_{ij}$=$0$ if $i=j$.\\
\section*{2.2.2. Queen Contiguity Weight Matrix}
The weight of this location is derived from regions near one another or whose corner points meet; hence, these areas are assigned the value $w_{ij}=1$, while the value $w_{ij}=0$ is assigned to other areas. First, the matrix needs to be normalized so that the sum of each row in the matrix equals one \cite{ref7}.
\begin{eqnarray}
	w_{ij}=\frac{c_{ij}}{c_{i}}
\end{eqnarray}
where\\
$c_{ij}$: $i$ is the row element, $j$ is the column\\
$c_{i}$: total elements of the $i$-row\\
$i,j:1,2,…,n$ where n: number of locations
\section*{2.3. Parameter Estimation}
The GSTAR model can be written as a linear model, and the Ordinary Least Square (OLS) method can be utilized to estimate the autoregressive parameters of the model \cite{ref10}. Because the GSTAR model is a regression model, the Ordinary Least Squares (OLS) approach is typically utilized to estimate the parameters that are associated with this model. The following linear equation can be constructed using the GSTAR [1;1] formula.
\begin{eqnarray}
	Y=X\beta+e	
\end{eqnarray}
This equation can then be modified with different locations so that for the ith location, it can be formulated in the following equation \cite{ref3}. 
\begin{eqnarray}
	Y_i=X_i \beta_i+e_i	
\end{eqnarray}
where, $Y_i$ is the observation for the $i$-th location $(i=1,2,…,N)$, and $\beta_i=(\phi_{10}^{(1)}$,$\phi_{11}^{(1)})$. If for $V_i (t)=\Sigma_{{j}\neq{i}}W_{ij} Y_j (t)$, then equation (6) can be translated into the following matrix.
\begin{eqnarray*}
	\begin{bmatrix}
		Y_i(1)\\
		Y_i(2)\\
		\vdots\\
		Y_i(T)	
	\end{bmatrix}
=
	\begin{bmatrix}
		Y_{i}(1)& V_{i}(0)\\
		Y_{i}(0)& V_{i}(1)\\
		\vdots& \vdots\\
		Y_i(T-1)&V_i(T-1)	
	\end{bmatrix}
	\begin{bmatrix}
		\phi_{10}^{(1)}\\
		\phi_{11}^{(1)}
	\end{bmatrix}
	+
	\begin{bmatrix}
	e_{i}(1)\\
	e_{i}(2)\\
	\vdots\\
	e_{i}(n)
	\end{bmatrix}
	\end{eqnarray*}
By virtue of the fact that $\beta_i=(\phi_{10}^{(1)}$ is, in fact, an autoregressive parameter for both time and space, the vector of estimation that is obtained through the use of the OLS method can be expressed as follows \cite{ref11}: 
\begin{eqnarray}
	\hat{\beta} =
	\begin{bmatrix}
		X'X\end{bmatrix}^{-1}X'Y
\end{eqnarray}
\section{Discussion}
\section*{3.1. Descriptive Analysis}
\begin{figure}[htbp]
	\centering\includegraphics[width=7cm]{gambar 1.png}
	\caption{Hotspot Distribution Map} \label{gambar1}
\end{figure}
The map that may be found in Figure 1 depicts the distribution of hotspots across the several districts that make up West Kalimantan Province. The degree of fluctuation in the frequency of fires is depicted on this map through the use of colour coding, which provides a clear visualisation of the spatial distribution of fires that occur.

Based on Figure \ref{gambar1},it is evident that there are six regions or cities that are highlighted in yellow, and the average number of hotspots in these regions is less than 5,304. There are four regions or cities that have an average number of fire hotspots measuring between 5,304 and 8,000, and these areas are highlighted in orange. On the red map, there are an average of more than 8,000 hotspots, which are comprised of four districts or cities.

Secondary data were obtained from the website sipongi.menlhk.go.id before being used in this investigation. An average of more than 8,000 hotspots may be found in West Kalimantan, which includes Ketapang Regency, Landak Regency, Sanggau Regency, and Sintang Regency $(n=4)$. This research makes use of data on the distribution of hotspots in West Kalimantan, which constitutes the high category. The hotspot distribution data that was utilised is data from the period beginning in January 2018 and ending in March 2023 $(T=75)$. In the course of this investigation, four different variables were utilised, namely:\\
1. $Y_t^{(1)}$: Number of hotspots in Landak Regency\\
2. $Y_t^{(2)}$: Number of hotspots in Ketapang Regency\\
3. $Y_t^{(3)}$: Number of hotspots in Sanggau Regency\\
4. $Y_t^{(4)}$: Number of hotspots in Sintang Regency\\
The descriptive statistics of the data that were utilised in the GSTAR modelling are presented in Table 1. It is important to note that the displayed data contain the minimum, maximum, average, and standard deviation values for each district that was examined.
\begin{table}[htbp]
	\begin{center}
		\begin{small}
			\caption{Descriptive Statistics on the Number of Hotspot}\label{tabel1}
			\begin{tabular}{|c|c|c|c|c|c|}
				\hline	Regency/City & Total Observations & Minimum & Maximum & Mean& Standard Deviation	\\
				\hline Ketapang & 75 &	0	&6208 &358 &1075	\\
				\hline Landak & 75 &	0	&2402 &109 &356	\\
				\hline Sanggau & 75 &	0	&5305 &225 &808	\\
				\hline Sintang & 75 &	0	&2903 &130 &410	\\
				\hline								
			\end{tabular}
		\end{small}
	\end{center}
\end{table}

Based on Table \ref{tabel1}, the total number of monthly data observations in this study was 75 for each district, and it is well known that the distribution of hotspots that occurred in the four districts of West Kalimantan Province varied substantially. This is something that is currently recognised. A value of zero hotspot cases is the bare minimum for each of the four districts. With 6,208 cases, Ketapang Regency had the highest maximum value, while Landak Regency had the lowest maximum value, with only 2,402 cases available. Ketapang Regency in Indonesia has the highest standard deviation value, while Landak Regency has the lowest number.
\begin{figure}[b]
	\centering
	\begin{subfigure}[b]{0.45\textwidth}
		\centering
		\includegraphics[width=1\textwidth]{ketapang.png}
		\caption{}\label{Gambar3a}
	\end{subfigure}
	\begin{subfigure}[b]{0.45\textwidth}
		\centering
		\includegraphics[width=1\textwidth]{landak.png}
		\caption{}\label{Gambar3b}
	\end{subfigure}
	\begin{subfigure}[b]{0.45\textwidth}
		\centering
		\includegraphics[width=1\textwidth]{sanggau.png}
		\caption{}\label{Gambar3c}
	\end{subfigure}
	\begin{subfigure}[b]{0.45\textwidth}
		\centering
		\includegraphics[width=1\textwidth]{sintang.png}
		\caption{}\label{Gambar3e}
	\end{subfigure}
	\caption{Time Series Data Plot Number of Hotspot}
	\label{Gambar2}
\end{figure}

Based on Figure \ref{Gambar2},
for the purpose of gaining a better understanding of the rise and fall in the total data on the distribution of hotspots over time, a time series graphic was created for each site like this:
\section*{3.2. Data Stationarity}
The stationarity of the data is the most important requirement that must be satisfied in order to conduct our time series analysis. Both the mean and the variance of the data that is used must be stationary. If the data are not stationary in terms of their mean and variance, then it is necessary to do processes including data transformation and differencing.
\section*{3.2.1. Stationarity In Variance}
based on figure \ref{Gambar2},due to the fact that the data does not exhibit a steady variance, it is necessary to transform the data. Considering that the data in question has a value of zero, the square root transformation should be utilised. The formula that is used to perform the square root transformation is as follows:
\begin{eqnarray}
	Z_t=\sqrt{Y_t}
\end{eqnarray}
where, $Z_t$ is transformed data, $Y_t$ is observation data, $t=1,2,…,T$
\section*{3.2.2. Stationarity Within the Mean}
The Augmented Dickey-Fuller Test, also known as the ADF test, is a method that can be utilised to determine whether or not the data is stationary within the average. You can view the results of the ADF tests conducted in the Ketapang, Landak, Sanggau, and Sintang Regencies in the table that is located below.
\begin{table}[htbp]
	\begin{center}
		\begin{small}
			\caption{Test ADF Data}\label{tab2}
			\begin{tabular}{|c|c|c|c|}
				\hline	Regency/City & P-value &Decision	& Information	\\
				\hline Ketapang & $0.04$ & $H_{0}$ rejected & Stationary	\\
				\hline Landak & $0.03$ &$H_{0}$ rejected	&Stationary	\\
				\hline Sanggau & $0.02$ &$H_{0}$ rejected	&Stationary	\\
				\hline Sintang & $0.01$ &$H_{0}$ rejected	&Stationary	\\	
				\hline				
			\end{tabular}
		\end{small}
	\end{center}
\end{table}

Based on Table \ref{tab2}, every district possesses a p-value for the hotspot data that is less than 0.05, which indicates that all of the data is stationary in the mean. In order to proceed to the next level of the analysis of the data.
\section*{3.3. Calculation of Weight Matrix in GSTAR Model}
Calculating the weight matrix that will be used is the next stage. The GSTAR model's weight matrix describes the interaction between the various sites. Both the queen contiguity weight matrix and the uniform weight matrix are utilised in this study. Both of these weight matrices are distinct from one another. Through the utilisation of Equation \eqref{pers2}, the uniform weight matrix is computed. Considering that there are four different places, the value of m in this equation is three. At the same time, a queen contiguity weight matrix is produced using information about the neighbourhood. These districts are assigned a value of 1 in this matrix, while districts that do not have any direct borders are assigned a value of 0. This matrix is used to categorise districts. The value 0 is used to fill the diagonal of the matrix, and the matrix that is generated is structured as follows:
\begin{eqnarray*}
	W_{(U)}=
	\begin{bmatrix}
		0    &0.33	&0.33	&0.33\\
		0.33	&0	&0.33	&0.33\\
		0.33	&0.33	&0	&0.33\\
		0.33	&0.33	&0.33	&0
	\end{bmatrix}\\
	W_{(Q)}=
	\begin{bmatrix}
		0	&0	&0.5	&0.5\\
		0	&0	&1	&0\\
		0.33	&0.33	&0	&0.33\\
		1	&0	&0	&0
	\end{bmatrix}\\
\end{eqnarray*}
where,\\
$W_{(U)}$: uniform weight matrix\\
$W_{(Q)}$: queen contiguity weight matrix
\section*{3.4. Parameter Estimation}
Table 3 presents the parameter estimates that were derived from the GSTAR model that was utilised for the analysis of hotspot data in four districts located within the province of West Kalimantan.
	\begin{table}[ht]
		\centering
		\caption{Parameter Estimation Results and Residual Diagnostics}\label{tabel3}
		\begin{tabular}{|c|c|c|c|c|c|c|c|}
			\hline
			Weight & \multicolumn{3}{c}{Parameter}&&Independence&Normality & Information \\
			\hline
			\multirow{4}{*}{Uniform} 
			& $\phi_{10}^{(1)}$ & 0,197 & $\phi_{11}^{(1)}$&1,103 & \multirow{4}{*}{\includegraphics[width=2cm]{acf seragam.png}} & \multirow{4}{*}{\includegraphics[width=2cm]{qq seragam.png}} & \multirow{4}{*}{White Noise} \\
			& $\phi_{10}^{(2)}$ & 1.219 & $\phi_{11}^{(2)}$&-0.353 &&& \\
			& $\phi_{10}^{(3)}$ & 0.883 &$\phi_{11}^{(3)}$&-0.207 &&& \\
			& $\phi_{10}^{(4)}$ & 0.209 &$\phi_{11}^{(4)}$&0.447 &&& \\
			\hline
			\multirow{4}{*}{Queen Contiguity} 
			& $\phi_{10}^{(1)}$ & 0.195 & $\phi_{11}^{(1)}$&1.016 & \multirow{4}{*}{\includegraphics[width=2cm]{acf queen.png}} & \multirow{4}{*}{\includegraphics[width=2cm]{qq queen.png}} & \multirow{4}{*}{White Noise} \\
			& $\phi_{10}^{(2)}$ & 2.025 & $\phi_{11}^{(2)}$&-0.894 & &&\\
			& $\phi_{10}^{(3)}$ & 0.883 &$\phi_{11}^{(3)}$&-0.207 & &&\\
			& $\phi_{10}^{(4)}$ & 1.113 &$\phi_{11}^{(4)}$&-0.223 & &&\\
			\hline
		\end{tabular}
	\end{table}

Based on Table \ref{tabel3}, as a result of the fact that it is known that the models derived from both matrices satisfy the white noise assumption, it is possible to utilise the two models to make predictions regarding the number of fire hotspots in the regions of Landak Regency, Ketapang Regency, Sanggau Regency, and Sintang Regency.
\section*{3.5. Selection of the Best Model}
One of the most crucial things to do while conducting a modelling analysis of the number of hotspots in West Kalimantan Province using the GSTAR approach is to select which model is the most appropriate to employ. When picking the optimal model, it is necessary to conduct many assessments of statistical criteria. These evaluations are designed to guarantee that the chosen model has the highest level of accuracy in forecasting the number of hotspots. The outcomes of picking the most suitable model on the basis of these criteria are represented in the table below. The statistical values that were obtained are then utilised to determine which model is the most appropriate. This table offers information about the models that were tested, along with the statistical values that were obtained.
\begin{table}[htbp]
	\begin{center}
		\begin{small}
			\caption{Model Accuracy Value}\label{tabel4}
			\begin{tabular}{|c|c|c|c|}
				\hline	Weight & RMSE &MAPE	& MAD	\\
				\hline Uniform & $4.795$ & \textbf{27.88} & 2.418	\\
				\hline Queen Contiguity & \textbf{4.406} &$27.91$	&\textbf{2.275}\\	
				\hline				
			\end{tabular}
		\end{small}
	\end{center}
\end{table}

Based on Table \ref{tabel4}, the GSTAR model, which makes use of the queen contiguity weight matrix and has RMSE and MAD values that are lower than those of the uniform weight matrix, is the most effective. Based on the queen contiguity weight matrix, the GSTAR [1;1] equation model can be expressed as follows:
\begin{eqnarray*}
	\begin{bmatrix}
		Y_t^{(1)}\\
		Y_t^{(2)}\\
		Y_t^{(3)}\\
		Y_t^{(4)}
	\end{bmatrix}
	=
	\begin{pmatrix}
		\begin{bmatrix}
			0.195&0&0&0\\
			0&2.025&0&0\\	
			0&0&0.883&0\\
			0&0&0&1.113
		\end{bmatrix}
		+
		\begin{bmatrix}
			1.016&0&0&0\\
			0&-0,894&0&0\\	
			0&0&-0.207&0\\
			0&0&0&-0.223
		\end{bmatrix}
		\begin{bmatrix}
			0	&0	&0.5	&0.5\\
			0	&0	&1	&0\\
			0.33	&0.33	&0	&0.33\\
			1	&0	&0	&0
		\end{bmatrix}
	\end{pmatrix}
	\begin{bmatrix}
		Y_{t-1}^{(1)}\\
		Y_{t-1}^{(2)}\\
		Y_{t-1}^{(3)}\\
		Y_{t-1}^{(4)}
	\end{bmatrix}\\
\end{eqnarray*}
Landak : $Y_t^{(1)} = 0.195Y_{t-1}^{(1)}+0.508Y_{t-1}^{(3)}+0.508Y_{t-1}^{(4)}$\\
Ketapang : $Y_t^{(2)} = 2.025Y_{t-1}^{(2)}-0.894Y_{t-1}^{(3)}$\\
Sanggau : $Y_t^{(3)} = -0.069Y_{t-1}^{(1)}-0.069Y_{t-1}^{(2)}+0.883Y_{t-1}^{(3)}-0.069Y_{t-1}^{(4)}$\\
Sintang : $Y_t^{(4)} = -0.223Y_{t-1}^{(1)}+1.113Y_{t-1}^{(4)}$
\section*{3.6. Contrasting the Actual Values with the Estimated Values}
As shown in Figure \ref{Gambar3}, a comparison plot is displayed between the actual data and the estimated data that was derived from the results of the GSTAR modelling for the number of hotspots in the several districts that make up West Kalimantan Province. The purpose of this plot is to determine how accurate the model is in forecasting the number of hotspots. By contrasting the actual data with the estimated data, we are able to determine the degree to which the GSTAR model is able to recreate the actual data patterns.
\begin{figure}
	\centering
	\begin{subfigure}[c]{0.45\textwidth}
		\centering
		\includegraphics[width=1\textwidth]{Ketapang1.png}
		\caption{}\label{Gambar4a}
	\end{subfigure}
	\begin{subfigure}[c]{0.45\textwidth}
		\centering
		\includegraphics[width=1\textwidth]{Landak1.png}
		\caption{}\label{Gambar4b}
	\end{subfigure}
	\begin{subfigure}[c]{0.45\textwidth}
		\centering
		\includegraphics[width=1\textwidth]{Sanggau1.png}
		\caption{}\label{Gambar4c}
	\end{subfigure}
	\begin{subfigure}[c]{0.45\textwidth}
		\centering
		\includegraphics[width=1\textwidth]{Sintang1.png}
		\caption{}\label{Gambar4d}
	\end{subfigure}
	\caption{Comparison Plot of Actual and Estimated Data}
	\label{Gambar3}
\end{figure}
Figure \ref{Gambar3} illustrates regardless of whether or not the model is effective. There is a high degree of congruence or overlap between the actual data and the anticipated data lines, which is an indication that the model predictions are extremely accurate. In addition, this graphic assists in determining whether or not the model is properly capturing trend patterns in the historical data regarding the amount of hotspots being used. The fact that the GSTAR model is able to follow the pattern of changes in the number of hotspots over time is demonstrated by the fact that the real data trends and the estimated data trends are consistent with one another. By comparing these two data sets, it is possible to determine the areas in which the model may need to be more accurate. For example, the presence of outliers or substantial variations that were not predicted properly by the model can be seen as examples of inaccuracies in the model. The presence of significant discrepancies between the real data and the estimated data may be an indication that the model requires some modifications or enhancements.
\section{Conclusion}
As a result of the findings of the investigation that has been carried out, it is possible to conclude that the Queen Contiguity weight matrix is superior to the uniform weight matrix when it comes to modeling the distribution of hotspots in the regions of Landak, Ketapang, Sanggau, and Sintang. The lower values of Root Mean Squared Error (RMSE) and Mean Absolute Deviation (MAD) prove this. According to the findings of the study, the distribution of hotspots in the Regencies of Ketapang, Landak, Sanggau, and Sintang is not only influenced by previous events that occurred in their place but it is also influenced by previous events that occurred in a variety of other locations. For instance, according to the developed model, the number of hotspots in Ketapang Regency is affected by previous events within Ketapang Regency itself, with a coefficient of 0.195. Additionally, Sanggau and Sintang Regency, each of which had a value of 0.508, also had an impact on the number of hotspots in Ketapang Regency—applying the Queen Contiguity weight matrix results in more precise forecasts when modeling the number of fire hotspots in West Kalimantan Province. As a result, these findings can serve as a reference for developing more efficient forest and land fire mitigation plans. This research would assist the government, and other relevant parties make more informed decisions about managing forest fire catastrophes. Therefore, selecting the appropriate weight matrix is of utmost significance in spatial-temporal analysis to acquire a more accurate and trustworthy model. 

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\end{document}
