Bayesian computational strategies for inference can be inefficient in approximating the posterior distribution in models that exhibit some form of periodicity. This is because the probability mass of the marginal posterior distribution of the parameter representing the period is usually highly concentrated in a very small region of the parameter space. Therefore, it is necessary to provide as much information as possible to the inference method through the parameter prior distribution. We intend to show that it is possible to construct a prior distribution from the data by fitting a Gaussian process (GP) with a periodic kernel. More specifically, we want to show that it is possible to approximate the marginal posterior distribution of the hyperparameter corresponding to the period in the kernel. Subsequently, this distribution can be used as a prior distribution for the inference method. We use an adaptive importance sampling method to approximate the posterior distribution of the hyperparameters of the GP. Then, we use the marginal posterior distribution of the hyperparameter related to the periodicity in order to construct a prior distribution for the period of the parametric model. This workflow is empirical Bayes, implemented as a modular (cut) transfer of a GP posterior for the period to the parametric model. We applied the proposed methodology to both synthetic and real data. We approximated the posterior distribution of the period of the GP kernel and then passed it forward as a posterior-as-prior with no feedback. Finally, we analyzed its impact on the marginal posterior distribution.

Spectral networks derived from multivariate time series data arise in many domains, from brain science to Earth science. Often, it is of interest to study how these networks change under different conditions. For instance, to better understand epilepsy, it would be interesting to capture the changes in the brain connectivity network as a patient experiences a seizure, using electroencephalography data. A common approach relies on estimating the networks in each condition and calculating their difference. Such estimates may behave poorly in high dimensions as the networks themselves may not be sparse in structure while their difference may be. We build upon this observation to develop an estimator of the difference in inverse spectral densities across two conditions. Using an L1 penalty on the difference, consistency is established by only requiring the difference to be sparse. We illustrate the method on synthetic data experiments, on experiments with electroencephalography data, and on experiments with optogentic stimulation and micro-electrocorticography data.

Time series forecasting is critical in numerous real-world applications, requiring accurate predictions of future values based on observed patterns. While traditional forecasting techniques work well in in-domain scenarios with ample data, they struggle when data is scarce or not available at all, motivating the emergence of zero-shot and few-shot learning settings. Recent advancements often leverage large-scale foundation models for such tasks, but these methods require extensive data and compute resources, and their performance may be hindered by ineffective learning from the available training set. This raises a fundamental question: What factors influence effective learning from data in time series forecasting? Toward addressing this, we propose using Fourier analysis to investigate how models learn from synthetic and real-world time series data. Our findings reveal that forecasters commonly suffer from poor learning from data with multiple frequencies and poor generalization to unseen frequencies, which impedes their predictive performance. To alleviate these issues, we present a novel synthetic data generation framework, designed to enhance real data or replace it completely by creating task-specific frequency information, requiring only the sampling rate of the target data. Our approach, Freq-Synth, improves the robustness of both foundation as well as nonfoundation forecast models in zero-shot and few-shot settings, facilitating more reliable time series forecasting under limited data scenarios. Time series forecasting (TSF) plays a critical role in various areas, such as finance, healthcare, and energy, where accurate predictions of future values are essential for decision-making and planning. Traditionally, in-domain learning has been the common setting for developing forecasting models, where a model is trained using data from the same domain it will later be deployed in (Salinas et al., 2020; Zhou et al., 2021). This ensures that the model captures the patterns, seasonality, and trends specific to the target domain, improving its predictive performance. However, a significant challenge arises when there is scarce or no historical information available for training, limiting the ability to apply traditional in-domain learning approaches (Sarmas et al., 2022; Fong et al., 2020). In such cases, the emergence of zero-shot (ZS) and few-shot (FS) learning settings offer potential solutions. Zero-shot learning enables models to generalize to new, unseen domains without requiring domainspecific data by leveraging knowledge transfer from other domains or tasks.

This study investigated the use of forearm EMG data for distinguishing eight hand gestures, employing the Neural Network and Random Forest algorithms on data from ten participants. The Neural Network achieved 97 percent accuracy with 1000-millisecond windows, while the Random Forest achieved 85 percent accuracy with 200-millisecond windows. Larger window sizes improved gesture classification due to increased temporal resolution. The Random Forest exhibited faster processing at 92 milliseconds, compared to the Neural Network's 124 milliseconds. In conclusion, the study identified a Neural Network with a 1000-millisecond stream as the most accurate (97 percent), and a Random Forest with a 200-millisecond stream as the most efficient (85 percent). Future research should focus on increasing sample size, incorporating more hand gestures, and exploring different feature extraction methods and modeling algorithms to enhance system accuracy and efficiency.

Despite the large research effort devoted to learning dependencies between time series, the state of the art still faces a major limitation: existing methods learn partial correlations but fail to discriminate across distinct frequency bands. Motivated by many applications in which this differentiation is pivotal, we overcome this limitation by learning a block-sparse, frequency-dependent, partial correlation graph, in which layers correspond to different frequency bands, and partial correlations can occur over just a few layers. To this aim, we formulate and solve two nonconvex learning problems: the first has a closed-form solution and is suitable when there is prior knowledge about the number of partial correlations; the second hinges on an iterative solution based on successive convex approximation, and is effective for the general case where no prior knowledge is available. Numerical results on synthetic data show that the proposed methods outperform the current state of the art. Finally, the analysis of financial time series confirms that partial correlations exist only within a few frequency bands, underscoring how our methods enable the gaining of valuable insights that would be undetected without discriminating along the frequency domain.

Time series analysis is a fundamental task in various application domains, and deep learning approaches have demonstrated remarkable performance in this area. However, many real-world time series data exhibit significant periodic or quasi-periodic dynamics that are often not adequately captured by existing deep learning-based solutions. This results in an incomplete representation of the underlying dynamic behaviors of interest. To address this gap, we propose an unsupervised method called Floss that automatically regularizes learned representations in the frequency domain. The Floss method first automatically detects major periodicities from the time series. It then employs periodic shift and spectral density similarity measures to learn meaningful representations with periodic consistency. In addition, Floss can be easily incorporated into both supervised, semi-supervised, and unsupervised learning frameworks. We conduct extensive experiments on common time series classification, forecasting, and anomaly detection tasks to demonstrate the effectiveness of Floss. We incorporate Floss into several representative deep learning solutions to justify our design choices and demonstrate that it is capable of automatically discovering periodic dynamics and improving state-of-the-art deep learning models.

Spectrum estimation is a fundamental methodology in the analysis of time-series data, with applications including medicine, speech analysis, and control design. The asymptotic theory of spectrum estimation is well-understood, but the theory is limited when the number of samples is fixed and finite. This paper gives non-asymptotic error bounds for a broad class of spectral estimators, both pointwise (at specific frequencies) and in the worst case over all frequencies. The general method is used to derive error bounds for the classical Blackman-Tukey, Bartlett, and Welch estimators. In particular, these are first non-asymptotic error bounds for Bartlett and Welch estimators.

Anomaly detection is widely used in a broad range of domains from cybersecurity to manufacturing, finance, and so on. Deep learning based anomaly detection has recently drawn much attention because of its superior capability of recognizing complex data patterns and identifying outliers accurately. However, deep learning models are typically iteratively optimized in a central server with input data gathered from edge devices, and such data transfer between edge devices and the central server impose substantial overhead on the network and incur additional latency and energy consumption. To overcome this problem, we propose a fully-automated, lightweight, statistical learning based anomaly detection framework called LightESD. It is an on-device learning method without the need for data transfer between edge and server, and is extremely lightweight that most low-end edge devices can easily afford with negligible delay, CPU/memory utilization, and power consumption. Yet, it achieves highly competitive detection accuracy. Another salient feature is that it can auto-adapt to probably any dataset without manually setting or configuring model parameters or hyperparameters, which is a drawback of most existing methods. We focus on time series data due to its pervasiveness in edge applications such as IoT. Our evaluation demonstrates that LightESD outperforms other SOTA methods on detection accuracy, efficiency, and resource consumption. Additionally, its fully automated feature gives it another competitive advantage in terms of practical usability and generalizability.

Target detection and recognition is a very challenging task in a wireless environment where a multitude of objects are located, whether to effectively determine their positions or to identify them and predict their moves. In this work, we propose a new method based on a convolutional neural network (CNN) to estimate the range and velocity of moving targets directly from the range-Doppler map of the detected signals. We compare the obtained results to the two dimensional (2D) periodogram, and to the similar state of the art methods, 2DResFreq and VGG-19 network and show that the estimation process performed with our model provides better estimation accuracy of range and velocity index in different signal to noise ratio (SNR) regimes along with a reduced prediction time. Afterwards, we assess the performance of our proposed algorithm using the peak signal to noise ratio (PSNR) which is a relevant metric to analyse the quality of an output image obtained from compression or noise reduction. Compared to the 2D-periodogram, 2DResFreq and VGG-19, we gain 33 dB, 21 dB and 10 dB, respectively, in terms of PSNR when SNR = 30 dB.

We present a computationally-efficient strategy to initialise the hyperparameters of a Gaussian process (GP) avoiding the computation of the likelihood function. Our strategy can be used as a pretraining stage to find initial conditions for maximum-likelihood (ML) training, or as a standalone method to compute hyperparameters values to be plugged in directly into the GP model. Motivated by the fact that training a GP via ML is equivalent (on average) to minimising the KL-divergence between the true and learnt model, we set to explore different metrics/divergences among GPs that are computationally inexpensive and provide hyperparameter values that are close to those found via ML. In practice, we identify the GP hyperparameters by projecting the empirical covariance or (Fourier) power spectrum onto a parametric family, thus proposing and studying various measures of discrepancy operating on the temporal and frequency domains. Our contribution extends the variogram method developed by the geostatistics literature and, accordingly, it is referred to as the generalised variogram method (GVM). In addition to the theoretical presentation of GVM, we provide experimental validation in terms of accuracy, consistency with ML and computational complexity for different kernels using synthetic and real-world data.