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[DOC] Initial Discussion for Theoritical Foundation #61

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opened 2025-05-14 05:13:45 +00:00 by nuluh · 3 comments
nuluh commented 2025-05-14 05:13:45 +00:00 (Migrated from github.com)
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Documentation Type

Thesis Chapter/Section

Description

Need to document the log for theoritical foundation along the writing process.

Current State

No response

Proposed Changes

\section{Theoretical Foundation}

This chapter discusses the theoretical basis used in this research, including signal processing techniques and machine learning algorithms. These foundations form the core of the method used for vibration-based damage localization in a beam structure.

\subsection{Short-Time Fourier Transform (STFT)}

The Short-Time Fourier Transform (STFT) is a fundamental technique used to analyze non-stationary signals, such as those generated by structures under dynamic load or white noise excitation. While the traditional Fourier Transform provides frequency-domain information, it lacks time resolution. STFT overcomes this limitation by applying the Fourier Transform over short overlapping segments of the signal, thereby producing a time-frequency representation.

Mathematically, the STFT of a signal $x(t)$ is given by:
\begin{equation}
X(t, \omega) = \int_{-\infty}^{\infty} x(\tau) w(\tau - t) e^{-j \omega \tau} d\tau
\end{equation}
where $w(\tau - t)$ is a window function centered at time $t$, and $\omega$ is the angular frequency.

In this study, the STFT is employed to extract the time-frequency features of the vibration signals collected from the structure. These features are then used as inputs to machine learning classifiers. This process captures localized frequency content over time, which is crucial in identifying structural changes due to damage.

\subsection{Classification Algorithms}

This research evaluates five classical machine learning algorithms to perform the classification task of damage localization. Each algorithm has different strengths and limitations, and their performance is benchmarked to identify the most suitable one for the given dataset.

\subsubsection{Support Vector Machine (SVM)}

SVM is a supervised learning algorithm that seeks an optimal hyperplane that separates data into classes with maximum margin. SVM performs well in high-dimensional spaces and is robust to overfitting, especially in cases with a clear margin of separation.

SVM is appropriate for vibration signal classification due to its capability to handle nonlinear decision boundaries when equipped with kernel functions.

\subsubsection{K-Nearest Neighbors (KNN)}

KNN is a non-parametric, instance-based learning algorithm. It classifies a new data point based on the majority vote of its $k$ nearest neighbors in the feature space. Although simple, KNN can be effective when the data is well-distributed and class boundaries are smooth.

Its performance is sensitive to the choice of $k$ and distance metric. For high-dimensional data like STFT features, dimensionality reduction or careful scaling may be required.

\subsubsection{Decision Tree (DT)}

Decision Tree is a rule-based classifier that splits data into classes using feature thresholds. It builds a tree where each internal node represents a feature, each branch a decision rule, and each leaf a class label. DTs are easy to interpret and can capture non-linear relationships.

However, they are prone to overfitting, especially with noisy or small datasets.

\subsubsection{Random Forest (RF)}

Random Forest is an ensemble learning method based on constructing multiple decision trees during training and outputting the mode of the classes for classification. It improves the generalization capability of individual trees and reduces overfitting.

RF is suitable for damage detection as it provides robustness to noise and variance, making it ideal for real-world sensor data.

\subsubsection{Naïve Bayes (NB)}

Naïve Bayes is a probabilistic classifier based on Bayes' theorem, assuming feature independence. Despite its simplicity, it often performs well in high-dimensional problems and with small datasets.

NB is particularly effective when class-conditional independence holds approximately, which may occur when STFT features are well-separated in distribution.

\bigskip
These theoretical foundations provide the methodological framework for implementing and evaluating the proposed damage localization system in this research. The combination of time-frequency analysis using STFT and classical machine learning classifiers enables an efficient and interpretable approach to structural health monitoring.

Documentation Location

latex/chapters/02_literature_review

Priority

Critical (required for thesis)

Target Audience

Thesis Committee/Reviewers

References

No response

Additional Notes

No response

### Documentation Type Thesis Chapter/Section ### Description Need to document the log for theoritical foundation along the writing process. ### Current State _No response_ ### Proposed Changes ```tex \section{Theoretical Foundation} This chapter discusses the theoretical basis used in this research, including signal processing techniques and machine learning algorithms. These foundations form the core of the method used for vibration-based damage localization in a beam structure. \subsection{Short-Time Fourier Transform (STFT)} The Short-Time Fourier Transform (STFT) is a fundamental technique used to analyze non-stationary signals, such as those generated by structures under dynamic load or white noise excitation. While the traditional Fourier Transform provides frequency-domain information, it lacks time resolution. STFT overcomes this limitation by applying the Fourier Transform over short overlapping segments of the signal, thereby producing a time-frequency representation. Mathematically, the STFT of a signal $x(t)$ is given by: \begin{equation} X(t, \omega) = \int_{-\infty}^{\infty} x(\tau) w(\tau - t) e^{-j \omega \tau} d\tau \end{equation} where $w(\tau - t)$ is a window function centered at time $t$, and $\omega$ is the angular frequency. In this study, the STFT is employed to extract the time-frequency features of the vibration signals collected from the structure. These features are then used as inputs to machine learning classifiers. This process captures localized frequency content over time, which is crucial in identifying structural changes due to damage. \subsection{Classification Algorithms} This research evaluates five classical machine learning algorithms to perform the classification task of damage localization. Each algorithm has different strengths and limitations, and their performance is benchmarked to identify the most suitable one for the given dataset. \subsubsection{Support Vector Machine (SVM)} SVM is a supervised learning algorithm that seeks an optimal hyperplane that separates data into classes with maximum margin. SVM performs well in high-dimensional spaces and is robust to overfitting, especially in cases with a clear margin of separation. SVM is appropriate for vibration signal classification due to its capability to handle nonlinear decision boundaries when equipped with kernel functions. \subsubsection{K-Nearest Neighbors (KNN)} KNN is a non-parametric, instance-based learning algorithm. It classifies a new data point based on the majority vote of its $k$ nearest neighbors in the feature space. Although simple, KNN can be effective when the data is well-distributed and class boundaries are smooth. Its performance is sensitive to the choice of $k$ and distance metric. For high-dimensional data like STFT features, dimensionality reduction or careful scaling may be required. \subsubsection{Decision Tree (DT)} Decision Tree is a rule-based classifier that splits data into classes using feature thresholds. It builds a tree where each internal node represents a feature, each branch a decision rule, and each leaf a class label. DTs are easy to interpret and can capture non-linear relationships. However, they are prone to overfitting, especially with noisy or small datasets. \subsubsection{Random Forest (RF)} Random Forest is an ensemble learning method based on constructing multiple decision trees during training and outputting the mode of the classes for classification. It improves the generalization capability of individual trees and reduces overfitting. RF is suitable for damage detection as it provides robustness to noise and variance, making it ideal for real-world sensor data. \subsubsection{Naïve Bayes (NB)} Naïve Bayes is a probabilistic classifier based on Bayes' theorem, assuming feature independence. Despite its simplicity, it often performs well in high-dimensional problems and with small datasets. NB is particularly effective when class-conditional independence holds approximately, which may occur when STFT features are well-separated in distribution. \bigskip These theoretical foundations provide the methodological framework for implementing and evaluating the proposed damage localization system in this research. The combination of time-frequency analysis using STFT and classical machine learning classifiers enables an efficient and interpretable approach to structural health monitoring. ``` ### Documentation Location latex/chapters/02_literature_review ### Priority Critical (required for thesis) ### Target Audience Thesis Committee/Reviewers ### References _No response_ ### Additional Notes _No response_
meowndor commented 2025-05-16 05:17:31 +00:00 (Migrated from github.com)
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The STFT function better to display in this notation


\text{STFT}\{x(t)\}(t, \omega) = \int_{-\infty}^{\infty} x(\tau) w(\tau - t) e^{-j\omega\tau} d\tau
The STFT function better to display in this notation $$ \text{STFT}\{x(t)\}(t, \omega) = \int_{-\infty}^{\infty} x(\tau) w(\tau - t) e^{-j\omega\tau} d\tau $$
nuluh commented 2025-05-25 05:03:48 +00:00 (Migrated from github.com)
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  • Fungsi windowing dan Hann Window dijadikan satu paragraf. Paragraf kedua dari Hann Window dimulai dari "Pemilihan jenis jendela ... waktu-frekuensi" karena hanya ada satu fungsi windowing yang digunakan.
  • Notasi jangan lupa dimasukkan pada daftar simbol dan harus sinkron/konsisten
  • Memungkinan adana kutipan untuk kelebihan dari dasar teori yang diangkat
  • Fungsi windowing (incl Hann window) dijadikan satu section pada STFT.
  • Notasi/Fungsi matematika (general) untuk setiap algoritma klasifikasi.
- Fungsi windowing dan Hann Window dijadikan satu paragraf. Paragraf kedua dari Hann Window dimulai dari "Pemilihan jenis jendela ... waktu-frekuensi" karena hanya ada satu fungsi windowing yang digunakan. - Notasi jangan lupa dimasukkan pada daftar simbol dan harus sinkron/konsisten - Memungkinan adana kutipan untuk kelebihan dari dasar teori yang diangkat - Fungsi windowing (incl Hann window) dijadikan satu section pada STFT. - Notasi/Fungsi matematika (general) untuk setiap algoritma klasifikasi.
nuluh commented 2025-05-25 05:06:11 +00:00 (Migrated from github.com)
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The STFT function better to display in this notation


\text{STFT}\{x(t)\}(t, \omega) = \int_{-\infty}^{\infty} x(\tau) w(\tau - t) e^{-j\omega\tau} d\tau

Advisors said bettter to preserve the integrals version

> The STFT function better to display in this notation > > $$ > \text{STFT}\{x(t)\}(t, \omega) = \int_{-\infty}^{\infty} x(\tau) w(\tau - t) e^{-j\omega\tau} d\tau > $$ Advisors said bettter to preserve the integrals version
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