In recent times, functional data analysis (FDA) has been successfully applied in the field of high dimensional data classification. In this paper, we present a novel classification framework using functional data and classwise Principal Component Analysis (PCA). Our proposed method can be used in high dimensional time series data which typically suffers from small sample size problem. Our method extracts a piece wise linear functional feature space and is particularly suitable for hard classification problems.The proposed framework converts time series data into functional data and uses classwise functional PCA for feature extraction followed by classification using a Bayesian linear classifier. We demonstrate the efficacy of our proposed method by applying it to both synthetic data sets and real time series data from diverse fields including but not limited to neuroscience, food science, medical sciences and chemometrics.
The growing popularity of wearable sensors has generated large quantities of temporal physiological and activity data. Ability to analyze this data offers new opportunities for real-time health monitoring and forecasting. However, temporal physiological data presents many analytic challenges: the data is noisy, contains many missing values, and each series has a different length. Most methods proposed for time series analysis and classification do not handle datasets with these characteristics nor do they offer interpretability and explainability, a critical requirement in the health domain. We propose an unsupervised method for learning representations of time series based on common patterns identified within them. The patterns are, interpretable, variable in length, and extracted using Byte Pair Encoding compression technique. In this way the method can capture both long-term and short-term dependencies present in the data. We show that this method applies to both univariate and multivariate time series and beats state-of-the-art approaches on a real world dataset collected from wearable sensors.
Deep neural networks have shown promising results for various clinical prediction tasks. However, training deep networks such as those based on Recurrent Neural Networks (RNNs) requires large labeled data, significant hyper-parameter tuning effort and expertise, and high computational resources. In this work, we investigate as to what extent can transfer learning address these issues when using deep RNNs to model multivariate clinical time series. We consider two scenarios for transfer learning using RNNs: i) domain-adaptation, i.e., leveraging a deep RNN - namely, TimeNet - pre-trained for feature extraction on time series from diverse domains, and adapting it for feature extraction and subsequent target tasks in healthcare domain, ii) task-adaptation, i.e., pre-training a deep RNN - namely, HealthNet - on diverse tasks in healthcare domain, and adapting it to new target tasks in the same domain. We evaluate the above approaches on publicly available MIMIC-III benchmark dataset, and demonstrate that (a) computationally-efficient linear models trained using features extracted via pre-trained RNNs outperform or, in the worst case, perform as well as deep RNNs and statistical hand-crafted features based models trained specifically for target task; (b) models obtained by adapting pre-trained models for target tasks are significantly more robust to the size of labeled data compared to task-specific RNNs, while also being computationally efficient. We, therefore, conclude that pre-trained deep models like TimeNet and HealthNet allow leveraging the advantages of deep learning for clinical time series analysis tasks, while also minimize dependence on hand-crafted features, deal robustly with scarce labeled training data scenarios without overfitting, as well as reduce dependence on expertise and resources required to train deep networks from scratch.
The in-depth analysis of time series has gained a lot of research interest in recent years, with the identification of periodic patterns being one important aspect. Many of the methods for identifying periodic patterns require time series' season length as input parameter. There exist only a few algorithms for automatic season length approximation. Many of these rely on simplifications such as data discretization and user defined parameters. This paper presents an algorithm for season length detection that is designed to be sufficiently reliable to be used in practical applications and does not require any input other than the time series to be analyzed. The algorithm estimates a time series' season length by interpolating, filtering and detrending the data. This is followed by analyzing the distances between zeros in the directly corresponding autocorrelation function. Our algorithm was tested against a comparable algorithm and outperformed it by passing 122 out of 165 tests, while the existing algorithm passed 83 tests. The robustness of our method can be jointly attributed to both the algorithmic approach and also to design decisions taken at the implementational level.
Time series forecasting methods play critical role in estimating the spread of an epidemic. The coronavirus outbreak of December 2019 has already infected millions all over the world and continues to spread on. Just when the curve of the outbreak had started to flatten, many countries have again started to witness a rise in cases which is now being referred as the 2nd wave of the pandemic. A thorough analysis of time-series forecasting models is therefore required to equip state authorities and health officials with immediate strategies for future times. This aims of the study are three-fold: (a) To model the overall trend of the spread; (b) To generate a short-term forecast of 10 days in countries with the highest incidence of confirmed cases (USA, India and Brazil); (c) To quantitatively determine the algorithm that is best suited for precise modelling of the linear and non-linear features of the time series. The comparison of forecasting models for the total cumulative cases of each country is carried out by comparing the reported data and the predicted value, and then ranking the algorithms (Prophet, Holt-Winters, LSTM, ARIMA, and ARIMA-NARNN) based on their RMSE, MAE and MAPE values. The hybrid combination of ARIMA and NARNN (Nonlinear Auto-Regression Neural Network) gave the best result among the selected models with a reduced RMSE, which proved to be almost 35.3% better than one of the most prevalent method of time-series prediction (ARIMA). The results demonstrated the efficacy of the hybrid implementation of the ARIMA-NARNN model over other forecasting methods such as Prophet, Holt Winters, LSTM, and the ARIMA model in encapsulating the linear as well as non-linear patterns of the epidemical datasets.
Hyper-parameters of time series models play an important role in time series analysis. Slight differences in hyper-parameters might lead to very different forecast results for a given model, and therefore, selecting good hyper-parameter values is indispensable. Most of the existing generic hyper-parameter tuning methods, such as Grid Search, Random Search, Bayesian Optimal Search, are based on one key component - search, and thus they are computationally expensive and cannot be applied to fast and scalable time-series hyper-parameter tuning (HPT). We propose a self-supervised learning framework for HPT (SSL-HPT), which uses time series features as inputs and produces optimal hyper-parameters. SSL-HPT algorithm is 6-20x faster at getting hyper-parameters compared to other search based algorithms while producing comparable accurate forecasting results in various applications.
Analyzing time series data is important to predict future events and changes in finance, manufacturing, and administrative decisions. In time series analysis, Gaussian Process (GP) regression methods recently demonstrate competitive performance by decomposing temporal covariance structures. The covariance structure decomposition allows exploiting shared parameters over a set of multiple, selected time series. In this paper, we present two novel GP models which naturally handle multiple time series by placing an Indian Buffet Process (IBP) prior on the presence of shared kernels. We also investigate the well-definedness of the models when infinite latent components are introduced. We present a pragmatic search algorithm which explores a larger structure space efficiently than the existing search algorithm. Experiments are conducted on both synthetic data sets and real-world data sets, showing improved results in term of structure discoveries and predictive performances. We further provide a promising application generating comparison reports from our model results.
Training deep neural networks often requires careful hyper-parameter tuning and significant computational resources. In this paper, we propose ConvTimeNet (CTN): an off-the-shelf deep convolutional neural network (CNN) trained on diverse univariate time series classification (TSC) source tasks. Once trained, CTN can be easily adapted to new TSC target tasks via a small amount of fine-tuning using labeled instances from the target tasks. We note that the length of convolutional filters is a key aspect when building a pre-trained model that can generalize to time series of different lengths across datasets. To achieve this, we incorporate filters of multiple lengths in all convolutional layers of CTN to capture temporal features at multiple time scales. We consider all 65 datasets with time series of lengths up to 512 points from the UCR TSC Benchmark for training and testing transferability of CTN: We train CTN on a randomly chosen subset of 24 datasets using a multi-head approach with a different softmax layer for each training dataset, and study generalizability and transferability of the learned filters on the remaining 41 TSC datasets. We observe significant gains in classification accuracy as well as computational efficiency when using pre-trained CTN as a starting point for subsequent task-specific fine-tuning compared to existing state-of-the-art TSC approaches. We also provide qualitative insights into the working of CTN by: i) analyzing the activations and filters of first convolution layer suggesting the filters in CTN are generically useful, ii) analyzing the impact of the design decision to incorporate multiple length decisions, and iii) finding regions of time series that affect the final classification decision via occlusion sensitivity analysis.
Time series with long-term structure arise in a variety of contexts and capturing this temporal structure is a critical challenge in time series analysis for both inference and forecasting settings. Traditionally, state space models have been successful in providing uncertainty estimates of trajectories in the latent space. More recently, deep learning, attention-based approaches have achieved state of the art performance for sequence modeling, though often require large amounts of data and parameters to do so. We propose Stanza, a nonlinear, non-stationary state space model as an intermediate approach to fill the gap between traditional models and modern deep learning approaches for complex time series. Stanza strikes a balance between competitive forecasting accuracy and probabilistic, interpretable inference for highly structured time series. In particular, Stanza achieves forecasting accuracy competitive with deep LSTMs on real-world datasets, especially for multi-step ahead forecasting.
Spectral density matrix estimation of multivariate time series is a classical problem in time series and signal processing. In modern neuroscience, spectral density based metrics are commonly used for analyzing functional connectivity among brain regions. In this paper, we develop a non-asymptotic theory for regularized estimation of high-dimensional spectral density matrices of Gaussian and linear processes using thresholded versions of averaged periodograms. Our theoretical analysis ensures that consistent estimation of spectral density matrix of a $p$-dimensional time series using $n$ samples is possible under high-dimensional regime $\log p / n \rightarrow 0$ as long as the true spectral density is approximately sparse. A key technical component of our analysis is a new concentration inequality of average periodogram around its expectation, which is of independent interest. Our estimation consistency results complement existing results for shrinkage based estimators of multivariate spectral density, which require no assumption on sparsity but only ensure consistent estimation in a regime $p^2/n \rightarrow 0$. In addition, our proposed thresholding based estimators perform consistent and automatic edge selection when learning coherence networks among the components of a multivariate time series. We demonstrate the advantage of our estimators using simulation studies and a real data application on functional connectivity analysis with fMRI data.
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