Elizabeth Ann Maharaj

Cluster of Time Series

p-value of the test that is applied to every pair of given time series.

Autocorrelation-based fuzzy clustering of time series

The traditional approaches to clustering a set of time series are generally applicable if there is a fixed underlying structure to the time series so that each will belong to one cluster or the other. However, time series often display dynamic behaviour in their evolution over time. This dynamic behaviour should be taken into account when attempting to cluster time series. For instance, during a certain period, a time series might belong to a certain cluster; afterwards its dynamics might be closer to that of another cluster. In this case, the traditional clustering approaches are unlikely to find and represent the underlying structure in the given time series. This switch from one time state to another, which is typically vague, can be naturally treated following a fuzzy approach. This paper proposes a fuzzy clustering approach based on the autocorrelation functions of time series, in which each time series is not assigned exclusively to only one cluster, but it is allowed to belong to different clusters with various membership degrees.

A Significance Test for Classifying ARMA Models

Given that the Euclidean distance between the parameter estimates of autoregressive expansions of autoregressive moving average models can be used to classify stationary time series into groups, a test of hypothesis is proposed to determine whether two stationary series in a particular group have significantly different generating processes. Based on this test a new clustering algorithm is also proposed. The results of Monte Carlo simulations are given.

Fuzzy clustering of time series in the frequency domain

Traditional and fuzzy cluster analyses are applicable to variables whose values are uncorrelated. Hence, in order to cluster time series data which are usually serially correlated, one needs to extract features from the time series, the values of which are uncorrelated. The periodogram which is an estimator of the spectral density function of a time series is a feature that can be used in the cluster analysis of time series because its ordinates are uncorrelated. Additionally, the normalized periodogram and the logarithm of the normalized periodogram are also features that can be used. In this paper, we consider a fuzzy clustering approach for time series based on the estimated cepstrum. The cepstrum is the spectrum of the logarithm of the spectral density function. We show in our simulation studies for the typical generating processes that have been considered, fuzzy clustering based on the cepstral coefficients performs very well compared to when it is based on other features.

Comparison of non-stationary time series in the frequency domain

In this paper we compare two non-stationary time series using non-parametric procedures. Evolutionary spectra are estimated for the two series. Randomization tests are performed on groups of spectral estimates for both related and independent time series. Simulation studies show that in certain cases the tests perform reasonably well. The tests are applied to observed geological and financial time series.

Comparison and Classification of Stationary Multivariate Time Series

Time series often have patterns that form a basis for comparing them or classifying them into groups. In this paper we present procedures to compare and classify stationary multivariate time series. Simulations studies show that the procedures perform fairly well for reasonably long series.

Wavelets-based clustering of multivariate time series

Crisp and fuzzy clustering methods based on a combination of univariate and multivariate wavelet features are considered for the clustering of multivariate time series. The performance of each of these methods is evaluated for stationary and variance nonstationary multivariate time series with different error correlation structures. The main outcomes of the simulation studies are are as follows: the superior performance of this approach for both the crisp and fuzzy cluster methods compared to some of the other approaches for clustering multivariate time series; the very good performance of the fuzzy relational method, overall, to cluster longer time series when all of them do not appear to group exclusively into well separated clusters. We consider an application to multivariate greenhouse gases time series and show that the crisp and fuzzy clustering methods considered are well validated.

Discrimination of locally stationary time series using wavelets

Time series are sometimes generated by processes that change suddenly from one stationary regime to another, with no intervening periods of transition of any significant duration. A good example of this is provided by seismic data, namely, waveforms of earthquakes and explosions. In order to classify an unknown event as either an earthquake or an explosion, statistical analysts might be helped by having at their disposal an automatic means of identifying, at any time, which pattern prevails. Several authors have proposed methods to tackle this problem by combining the techniques of spectral analysis with those of discriminant analysis. The goal is to develop a discriminant scheme for locally stationary time series such as earthquake and explosion waveforms, by combining the techniques of wavelet analysis with those of discriminant analysis.

Wavelet-based Fuzzy Clustering of Time Series

Traditional procedures for clustering time series are based mostly on crisp hierarchical or partitioning methods. Given that the dynamics of a time series may change over time, a time series might display patterns that may enable it to belong to one cluster over one period while over another period, its pattern may be more consistent with those in another cluster. The traditional clustering procedures are unable to identify the changing patterns over time. However, clustering based on fuzzy logic will be able to detect the switching patterns from one time period to another thus enabling some time series to simultaneously belong to more than one cluster. In particular, this paper proposes a fuzzy approach to the clustering of time series based on their variances through wavelet decomposition. We will show that this approach will distinguish between time series with different patterns in variability as well identifying time series with switching patterns in variability.

Discriminant analysis of multivariate time series: Application to diagnosis based on ECG signals

In analysing ECG data, the main aim is to differentiate between the signal patterns of healthy subjects and those of individuals with specific heart conditions. We propose an approach for classifying multivariate ECG signals based on discriminant and wavelet analyses. For this purpose we use multiple-scale wavelet variances and wavelet correlations to distinguish between the patterns of multivariate ECG signals based on the variability of the individual components of each ECG signal and on the relationships between every pair of these components. Using the results of other ECG classification studies in the literature as references, we demonstrate that our approach applied to 12-lead ECG signals from a particular database compares favourably. We also demonstrate with real and synthetic ECG data that our approach to classifying multivariate time series out-performs other well-known approaches for classifying multivariate time series.