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For high dimensional data, where P features for N objects (P >> N) are represented in an NxP matrix X, we describe a clustering algorithm based on the normalized left Gram matrix, G = XX'/P. Under certain regularity conditions, the rows in G that correspond to objects in the same cluster converge to the same mean vector. By clustering on the row means, the algorithm does not require preprocessing by dimension reduction or feature selection techniques and does not require specification of tuning or hyperparameter values. Because it is based on the NxN matrix G, it has a lower computational cost than many methods based on clustering the feature matrix X. When compared to 14 other clustering algorithms applied to 32 benchmarked microarray datasets, the proposed algorithm provided the most accurate estimate of the underlying cluster configuration more than twice as often as its closest competitors.

For multivariate stationary time series many important properties, such as partial correlation, graphical models and autoregressive representations are encoded in the inverse of its spectral density matrix. This is not true for nonstationary time series, where the pertinent information lies in the inverse infinite dimensional covariance matrix operator associated with the multivariate time series. This necessitates the study of the covariance of a multivariate nonstationary time series and its relationship to its inverse. We show that if the rows/columns of the infinite dimensional covariance matrix decay at a certain rate then the rate (up to a factor) transfers to the rows/columns of the inverse covariance matrix. This is used to obtain a nonstationary autoregressive representation of the time series and a Baxter-type bound between the parameters of the autoregressive infinite representation and the corresponding finite autoregressive projection. The aforementioned results lay the foundation for the subsequent analysis of locally stationary time series. In particular, we show that smoothness properties on the covariance matrix transfer to (i) the inverse covariance (ii) the parameters of the vector autoregressive representation and (iii) the partial covariances. All results are set up in such a way that the constants involved depend only on the eigenvalue of the covariance matrix and can be applied in the high-dimensional settings with non-diverging eigenvalues.

We propose NonStGM, a general nonparametric graphical modeling framework for studying dynamic associations among the components of a nonstationary multivariate time series. It builds on the framework of Gaussian Graphical Models (GGM) and stationary time series Gaussian Graphical model (StGM), and complements existing works on parametric graphical models based on change point vector autoregressions (VAR). Analogous to StGM, the proposed framework captures conditional noncorrelations (both intertemporal and contemporaneous) in the form of an undirected graph. In addition, to describe the more nuanced nonstationary relationships among the components of the time series, we introduce the new notion of conditional nonstationarity/stationarity and incorporate it within the graph architecture. This allows one to distinguish between direct and indirect nonstationary relationships among system components, and can be used to search for small subnetworks that serve as the "source" of nonstationarity in a large system. Together, the two concepts of conditional noncorrelation and nonstationarity/stationarity provide a parsimonious description of the dependence structure of the time series.

The Wiener-Hopf equations are a Toeplitz system of linear equations that naturally arise in several applications in time series. These include the update and prediction step of the stationary Kalman filter equations and the prediction of bivariate time series. The celebrated Wiener-Hopf technique is usually used for solving these equations and is based on a comparison of coefficients in a Fourier series expansion. However, a statistical interpretation of both the method and solution is opaque. The purpose of this note is to revisit the (discrete) Wiener-Hopf equations and obtain an alternative solution that is more aligned with classical techniques in time series analysis. Specifically, we propose a solution to the Wiener-Hopf equations that combines linear prediction with deconvolution. The Wiener-Hopf solution requires the spectral factorization of the underlying spectral density function. For ease of evaluation it is often assumed that the spectral density is rational. This allows one to obtain a computationally tractable solution. However, this leads to an approximation error when the underlying spectral density is not a rational function. We use the proposed solution with Baxter's inequality to derive an error bound for the rational spectral density approximation.

The periodogram is a widely used tool to analyze second order stationary time series. An attractive feature of the periodogram is that the expectation of the periodogram is approximately equal to the underlying spectral density of the time series. However, this is only an approximation, and it is well known that the periodogram has a finite sample bias, which can be severe in small samples. In this paper, we show that the bias arises because of the finite boundary of observation in one of the discrete Fourier transforms which is used in the construction of the periodogram. Moreover, we show that by using the best linear predictors of the time series over the boundary of observation we can obtain a "complete periodogram" that is an unbiased estimator of the spectral density. In practice, the "complete periodogram" cannot be evaluated as the best linear predictors are unknown. We propose a method for estimating the best linear predictors and prove that the resulting "estimated complete periodogram" has a smaller bias than the regular periodogram. The estimated complete periodogram and a tapered version of it are used to estimate parameters, which can be represented in terms of the integrated spectral density. We prove that the resulting estimators have a smaller bias than their regular periodogram counterparts. The proposed method is illustrated with simulations and real data.

In time series analysis there is an apparent dichotomy between time and frequency domain methods. The aim of this paper is to draw connections between frequency and time domain methods. Our focus will be on reconciling the Gaussian likelihood and the Whittle likelihood. We derive an exact, interpretable, bound between the Gaussian and Whittle likelihood of a second order stationary time series. The derivation is based on obtaining the transformation which is biorthogonal to the discrete Fourier transform of the time series. Such a transformation yields a new decomposition for the inverse of a Toeplitz matrix and enables the representation of the Gaussian likelihood within the frequency domain. We show that the difference between the Gaussian and Whittle likelihood is due to the omission of the best linear predictions outside the domain of observation in the periodogram associated with the Whittle likelihood. Based on this result, we obtain an approximation for the difference between the Gaussian and Whittle likelihoods in terms of the best fitting, finite order autoregressive parameters. These approximations are used to define two new frequency domain quasi-likelihoods criteria. We show that these new criteria can yield a better approximation of the spectral divergence criterion, as compared to both the Gaussian and Whittle likelihoods. In simulations, we show that the proposed estimators have satisfactory finite sample properties.