# Time Series Anomaly Detection Using Signal Processing and Deep Learning
## 2 Literature Review  
Time series anomaly detection has been an area of interest recently due to its importance in various applications. Traditional approaches, such as statistical methods, rely on building models that assume particular properties of the time series data. For example, the ARIMA and its variants are commonly used for anomaly detection by modeling the temporal behavior of the data and identifying deviations from predicted trends [22; 23]. However, these methods are limited in handling complex and non-stationary time series data. More recently, machine learning approaches have gained prominence for anomaly detection in time series. Unsupervised methods, in particular, reconstruction-based approaches, where Autoencoder models are widely used to learn compressed representations of the input data from which the original data are remapped. Autoencoders [24] consist of an encoder, which maps the input data into a lower-dimensional latent space, and a decoder, which reconstructs the input from the latent representation. In the context of time series anomaly detection, Autoencoders are trained on normal data to learn the regular patterns of the time series. During inference, data points that result in high reconstruction errors are detected as anomalies. The performance of Autoencoders is highly dependent on the quality of the training data, and they might not learn meaningful latent presentations if the data have a lot of noise or is sparse. Variants of Autoencoders, such as Denoising Autoencoders and Variational Autoencoders (VAEs), have been introduced to improve robustness and generalization. VAEs [25] are explicitly trained to model a probabilistic distribution of the input data, but it is challenging to appropriately learn the underlying data distribution or long-term dependencies in time series data without further prior knowledge incorporation [26]. Denoising Autoencoders [27] learn robust latent representations by reconstructing the original input data from a noisy version of the input. Here, prior assumptions about the level and type of noise are necessary to ensure modeling success. Different deep learning network architecture variants, such as temporal convolutional Autoencoders [28] and LSTM-based Autoencoders [29; 30] have been explored to better capture the temporal dependencies in time series data during reconstruction. These models either assume a particular structure in the time series data or have restricting network flexibility that limits the performance. Other traditional approaches for time series analysis leverage signal processing techniques to decompose time series signals into time and frequency components. Fourier Transform (FFT) and Short-Time Fourier Transform (STFT) enable frequency-domain analysis [31] to identify anomalies through irregularities in the spectral domain. Wavelet Transform (WT) [32] extends this by providing a multi-scale time-frequency representation to allow the localized detection of anomalies across different resolutions. Hybrid machine learning approaches can tackle both time and frequency features to address complex patterns in time series data and increase model performance. For example, time-frequency information derived from Wavelets can be combined with Convolutional Neural Networks (CNNs) to detect irregularities in the data [33; 34]. Similarly, the network architectures of the LSTM and Squeeze and Excitation Network (SENet) were combined with an FFT-constructed frequency matrix to improve time series anomaly detection in [35]. Traditional Autoencoders may inadvertently reconstruct anomalies due to their ability to capture complex features in latent spaces. Dr. [PERSON] et al [36] proposed a novel approach to mitigate this by integrating Wavelet Transform into the Autoencoder architecture as a regularization mechanism. Another way to look at time series data is through Functional Data Analysis (FDA) [37]. It provides powerful tools for time series anomaly detection by treating time series data as functions rather than discrete observations.