PSI - Issue 78

Akshay Rai et al. / Procedia Structural Integrity 78 (2026) 891–898

893

• Time-series signals x s ( t ), containing L number of time steps for each signal. • Scalar values for both internal ( T 0 ) and external ( T 1 ) temperatures.

2.2. Data preprocessing

In the data preprocessing phase, raw sensor measurements are transformed into a format suitable for anomaly detection. It ensures numerical stability, e ffi ciency, and the extraction of vital features. The process begins with data acquisition from various sensors on the structure, including nine accelerometers that capture vibration responses as time-series signals ( x s ( t ) ∈ R L ), with 72,657 data points each at a sampling frequency of 40 Hz. Additionally, thermocouple sensors record internal ( T 0 ) and external ( T 1 ) temperatures every 30 minutes, resulting in 48 sam ples daily. After data acquisition, the next stage involves processing the signals and spectral representation. This process starts with filtering to eliminate undesirable noise and focus on the most significant dynamic content. Each accelerometer signal is processed using a 4th-order Butterworth band-pass filter, which features a lower cuto ff fre quency of 1.5 Hz and an upper cuto ff frequency of 15 Hz. This particular frequency range is chosen to e ff ectively reduce low-frequency drift and high-frequency noise, thereby capturing the most pertinent structural dynamics. The filtered time-series signals are transformed from the time domain to the frequency domain by computing their cross-spectral density (CSD) using Welch’s method. This process reduces the original 72,657-point time series to 257 spectral bins, maintaining key frequency-domain characteristics while lowering data dimensionality. The CSD captures inter-sensor correlations and is represented as S xx ( f ) ∈ C S × S , encoding correlations between S accelerometer sensors. The CSD between signals x i and x j at frequency f is defined as S x i x j ( f ) = F ( E [ x i ( t ) · x j ( t + τ )]) , (1) where F denotes the Fourier Transform of the cross-correlation function. This results in a tensor S x ∈ C S × S × F , with F being 257 frequency bins. The next critical step is Dominant Singular Value Decomposition (SVD) applied to each frequency slice f k of the CSD matrix, resulting in S x ( f k ) = U ( f k ) Σ ( f k ) V H ( f k ) , (2) Only the most significant singular value, σ k , is retained, yielding a compact signature d = [ σ 1 ,σ 2 , . . . ,σ k ] ∈ R F . This captures the strongest inter-sensor dynamics across frequencies, resulting in a 2D matrix of shape ( N , 257) for the dataset. The input to the Convolutional Autoencoder (CAE) model is formed by concatenating this SVD signature with temperature data, represented as X = [ d , T 1 , T 0 ] ∈ R F + 2 . To ensure numerical stability and e ff ective model training, the combined SVD signature and temperature data undergo normalisation using two strategies. Dominant-SVs are normalised with per-sample normalisation, where each sample d i is scaled independently using its min and max values. This method retains relative variations and enhances features like mode generation. In contrast, temperature channels ( T 1 and T 0 ) use per-dataset normalisation, applying global min and max values. This uniform scaling is essential for models trained and tested on di ff erent distributions, preserving the intrinsic temporal variability of environmental conditions. In autoencoder-based anomaly detection frameworks, traditional metrics such as Mean Squared Error (MSE) and Mean Absolute Error (MAE) are commonly employed. However, these metrics struggle to capture nuanced spectral characteristics, especially in dominant-SVs representations of structural response data and temperature variations. This is particularly critical for large, historical masonry structures, where subtle frequency domain deviations may indicate early signs of damage or anomalies. To improve anomaly detection, a refined metric using spectral compensation factors is introduced. These factors adjust for spectral alignment, frequency spread, and entropy variations between actual and reconstructed datasets. The proposed compensation metrics, which incorporate statistical divergence measures for comprehensive evalu ation of reconstruction fidelity, include: 1. Spectral Centroid Ratio (SCR): This metric measures the ”centre of mass” of a signal’s power spectral distri bution. The spectral centroid (SC) for a dominant-SV signal S ( f ) is calculated as: SC = F i = 1 f i · S ( f i ) F i = 1 S ( f i ) (3) 2.3. Temperature-compensated error metrics for anomaly detection