latex: Match with Overleaf current work

latex/chapters/id/03_methodology/steps/data_acquisition.tex

@@ -1,9 +1,10 @@
-Dataset yang digunakan dalam penelitian ini bersumber dari basis data getaran yang dipublikasi oleh \textcite{abdeljaber2017}.
+Dataset yang digunakan dalam penelitian ini bersumber dari basis data getaran yang dipublikasi oleh \textcite{abdeljaber2017}. Dataset tersebut dapat diakses dan diunduh melalui tautan DOI berikut:  
+\url{https://doi.org/10.17632/52rmx5bjcr.1}
 
 Dataset terdiri dari dua folder:
 \begin{itemize}
-  \item \texttt{Dataset A/} – biasanya digunakan untuk pelatihan (training)
-  \item \texttt{Dataset B/} – biasanya digunakan untuk pengujian (testing)
+  \item \texttt{Dataset A/} – digunakan untuk pelatihan (training)
+  \item \texttt{Dataset B/} – digunakan untuk pengujian (testing)
 \end{itemize}
 
 Setiap folder berisi 31 berkas dalam format \texttt{.TXT}, yang dinamai sesuai dengan kondisi kerusakan struktur. Pola penamaan berkas adalah sebagai berikut:
@@ -20,14 +21,34 @@ Sepuluh baris pertama dari setiap berkas berisi metadata yang menjelaskan konfig
   \item \textbf{Kolom 2–31:} Magnitudo percepatan dari \textit{joint} 1 hingga 30
 \end{itemize}
 
-Setiap sinyal di-\textit{sampling} pada frekuensi $f_s = 1024$ Hz dan direkam selama durasi total $T = 256$ detik, sehingga menghasilkan:
+Setiap sinyal di-\textit{sampling} pada frekuensi $f_s = 1024$ Hz dan direkam selama $t = 256$ detik, sehingga menghasilkan:
+\begin{align}
+\gls{not:signal} &= \gls{not:sampling_freq} \cdot \gls{not:time_length} \nonumber \\
+  &= 1024 \cdot 256 \nonumber \\
+  &= 262144 \quad \text{sampel per kanal} \label{eq:sample}
+\end{align}
 
-\begin{equation*}
-N = f_s \cdot T = 1024 \times 256 = 262{,}144 \quad \text{sampel per kanal}
-\end{equation*}
+Dengan demikian, setiap berkas \verb|zzzAD|$n$\verb|.TXT| dapat direpresentasikan sebagai matriks:
+\begin{equation}
+    \mathbf{D}^{(n)} \in \mathbb{R}^{262144 \times 30}, \quad n = 1, \dots, 30
+\end{equation}
+di mana $n$ mengacu pada indeks kasus (1–30 = kerusakan pada \textit{joint} ke-$n$), dan berkas tanpa kerusakan pada seluruh \textit{joint}, \verb|zzzAU|\verb|.TXT|, direpresentasikan dengan matriks:
+\begin{equation}
+    \mathbf{U} \in \mathbb{R}^{262144 \times 30}
+\end{equation}
 
-Dengan demikian, setiap berkas dapat direpresentasikan sebagai matriks:
-\begin{equation*}
-\mathbf{X}^{(c)} \in \mathbb{R}^{262{,}144 \times 31}, \quad c = 0, 1, \dots, 30
-\end{equation*}
-di mana $c$ mengacu pada indeks kasus (0 = sehat, 1–30 = kerusakan pada \textit{joint}n ke-$c$), dan setiap baris merepresentasikan pengukuran berdasarkan waktu di seluruh 30 kanal sensor.
+
+Kemudian \textit{dataset} A dapat direpresentasikan sebagai matriks:
+\begin{equation}
+  \gls{not:dataset_A}
+  =
+  \Bigl\{
+      \mathbf{U} \in \mathbb{R}^{262144 \times 30}
+  \Bigr\}
+  \;\cup\;
+  \Bigl\{
+      \mathbf{D}^{(n)} \in \mathbb{R}^{262144 \times 30}
+      \;\bigm|\;
+      n = 1, \dots, 30
+  \Bigr\}.
+\end{equation}

latex/chapters/id/03_methodology/steps/preprocessing/data_augmentation.tex

@@ -0,0 +1,24 @@
+We now introduce a simple “data‐augmentation” logic across repeated tests as:
+\[
+\mathbf{c}_{j}^{(i)}
+\;=\;
+\Bigl[S_{0+j}^{(i)},\,S_{5+j}^{(i)},\,S_{10+j}^{(i)},\,S_{15+j}^{(i)},\,S_{20+j}^{(i)},\,S_{25+j}^{(i)}\Bigr]^{T}
+\;\in\mathbb{R}^{6}\!,
+\]
+where \(S_{k}^{(i)}\) is the \(k\)th sensor’s time‐frequency feature vector (after STFT+log‐scaling) from the \(i\)-th replicate of scenario \(j\).
+
+For each fixed scenario \(j\), collect the five replicates into the set
+\[
+\mathcal{D}^{(j)}
+=\bigl\{\mathbf{c}_{j}^{(1)},\,\mathbf{c}_{j}^{(2)},\,\mathbf{c}_{j}^{(3)},\,\mathbf{c}_{j}^{(4)},\,\mathbf{c}_{j}^{(5)}\bigr\},
+\]
+so \(|\mathcal{D}^{(j)}|=5\).  Across all six scenarios, the total augmented dataset is
+\[
+\mathcal{D}
+=\bigcup_{j=0}^{5}\mathcal{D}^{(j)}
+=\bigl\{\mathbf{c}_{j}^{(i)}: j=0,\dots,5,\;i=1,\dots,5\bigr\},
+\]
+with \(\lvert\mathcal{D}\rvert = 6 \times 5 = 30\) samples.
+
+Each \(\mathbf{c}_{j}^{(i)}\) hence represents one ``column‐based’’ damage sample,
+and the collection \(\mathcal{D}\) serves as the input set for subsequent classification.  

latex/chapters/id/03_methodology/steps/preprocessing/data_representation.tex

@@ -0,0 +1,9 @@
+Let $\mathcal{G}$ represent the $6 \times 5$ structural grid, where each node is denoted with row and column as $N_{r,c}$ with $r \in \{1,2,...,6\}$ and $c \in \{1,2,...,5\}$.\\
+
+\begin{figure}[ht]
+    \centering
+    % \includegraphics[width=\textwidth]{}
+    \input{chapters/img/specimen}
+    \caption{Diagram joint and sensors placement}
+    \label{fig:specimen}
+\end{figure}

latex/chapters/id/03_methodology/steps/preprocessing/index.tex

@@ -0,0 +1,66 @@
+\subsection{Signal Normalization}
+Each raw acceleration time series 
+\(\mathbf{a}_{k}(n)\), \(n = 0,1,\dots,N-1\) with \(N=262144\) samples (collected at \(f_s=1024\) Hz over 256 s) :contentReference[oaicite:0]{index=0} is first standardized to zero mean and unit variance:
+\[
+\tilde a_{k}(n)
+=\frac{a_{k}(n)-\mu_{k}}{\sigma_{k}},
+\quad
+\mu_{k}=\frac{1}{N}\sum_{n=0}^{N-1}a_{k}(n),
+\quad
+\sigma_{k}=\sqrt{\frac{1}{N}\sum_{n=0}^{N-1}\bigl(a_{k}(n)-\mu_{k}\bigr)^{2}}.
+\]
+
+\subsection{Framing and Windowing}
+The normalized signal \(\tilde a_{k}(n)\) is chopped into overlapping frames of length \(W\) samples with hop size \(H\). The \(p\)-th frame is
+\[
+x_{k,p}[m]
+=\tilde a_{k}(pH + m)\,w[m],
+\quad
+m=0,1,\dots,W-1,
+\]
+where \(w[m]\) is a chosen window function (e.g., Hamming).
+
+\subsection{Short-Time Fourier Transform (STFT)}
+For each frame \(x_{k,p}[m]\), compute its STFT:
+\[
+S_{k}(f,p)
+=\sum_{m=0}^{W-1}x_{k,p}[m]\;e^{-j2\pi\,f\,m/W},
+\]
+where \(f=0,1,\dots,W-1\) indexes frequency bins :contentReference[oaicite:1]{index=1}.
+
+\subsection{Spectrogram and Log-Magnitude}
+Form the magnitude spectrogram
+\[
+M_{k}(f,p)
+=\bigl|S_{k}(f,p)\bigr|,
+\]
+and apply log scaling for numerical stability:
+\[
+L_{k}(f,p)
+=\log\bigl(1 + M_{k}(f,p)^{2}\bigr).
+\]
+This yields a time–frequency representation
+\(\mathbf{L}_{k}\in\mathbb{R}^{F\times P}\), with \(F\) frequency bins and \(P\) frames.
+
+\subsection{Feature Matrix Assembly}
+For each column \(j\in\{1,\dots,5\}\), select only the two endpoint sensors:
+\[
+\mathbf{L}_{\text{bot},j} = \mathbf{L}_{(j)},\quad
+\mathbf{L}_{\text{top},j} = \mathbf{L}_{(25+j)},
+\]
+and stack them:
+\[
+\mathbf{F}_{j}
+=
+\begin{bmatrix}
+\mathbf{L}_{\text{bot},j} \\[6pt]
+\mathbf{L}_{\text{top},j}
+\end{bmatrix}
+\;\in\mathbb{R}^{2F\times P}.
+\]
+Finally, flatten into a feature vector:
+\[
+\mathbf{f}_{j}
+=\operatorname{vec}\bigl(\mathbf{F}_{j}\bigr)
+\;\in\mathbb{R}^{2FP}.
+\]

latex/chapters/id/03_methodology/steps/preprocessing/limited_sensor_approach.tex

@@ -0,0 +1,20 @@
+For the vertical column approach with limited sensors 
+% we
+are defined as column vector $\mathbf{c}_j$:
+
+\begin{equation}
+\mathbf{c}_j^{(i,d)} = [S_{0+j}^{(i+d)}, S_{5+j}^{(i+d)}, S_{10+j}^{(i+d)}, S_{15+j}^{(i+d)}, S_{20+j}^{(i+d)}, S_{25+j}^{(i+d)}]^T
+\end{equation}
+\begin{equation}
+\mathbf{D}^{(i)} = [\mathbf{c}_0^{(i,i+1)}, \mathbf{c}_1^{(i,i+6)}, \mathbf{c}_2^{(i,i+11)}, \mathbf{c}_3^{(i,i+16)}, \mathbf{c}_4^{(i,i+21)}]^T
+\end{equation}
+
+where $j \in \{0, 1,2,3,4\}$ represents the column index.
+
+For the limited sensor case focusing on endpoints only, we use:
+
+\begin{equation}
+\mathbf{c}^{\text{limited}}_j = [S_{0+(j-1)}, S_{25+(j-1)}]^T
+\end{equation}
+
+representing only the lower sensor (sensor A) and upper sensor (sensor B) of column $j$.

latex/chapters/id/03_methodology/steps/preprocessing/time_series_data.tex

@@ -0,0 +1,23 @@
+Untuk setiap sensor $S_k$ dengan $k \in \{0,1,2,...,29\}$ diletakkan pada \textit{node} $N_{k}$, deret deret akselerasi waktu didefinisikan sebagai:
+
+\begin{equation}
+\mathbf{a}_{k}(t) = [a_{k}(t_1), a_{k}(t_2), \ldots, a_{k}(t_{262144})]
+\end{equation}
+
+% where $N = 262144$ samples at a sampling frequency of 1024 Hz over 256 seconds.
+% $k \in \{i,(i+1),...,(i+(r\times j))\}$
+
+Satu dataset utuh untuk setiap skenario ($A|B$) dapat direpresentasikan sebagai matrix $\mathbf{X}_d \in \mathbb{R}^{30 \times 262144}$:
+
+\begin{equation}
+\mathbf{{X}_d}^\intercal = 
+\begin{bmatrix}
+\mathbf{a}_{0}(t) \\
+\mathbf{a}_{1}(t) \\
+\mathbf{a}_{2}(t) \\
+\vdots \\
+\mathbf{a}_{29}(t)
+\end{bmatrix}
+\end{equation}
+
+di mana $d \in \{0, 1, 2, \ldots, 30\}$ merepresentasikan skenario kerusakan, dengan $d=0$ mengindikasikan tanpa kasus kerusakan.