A Generic Approach to Covariance Function Estimation Using ARMA-Models

Abstract

Covariance function modeling is an essential part of stochastic methodology. Many processes in geodetic applications have rather complex, often oscillating covariance functions, where it is difficult to find corresponding analytical functions for modeling. This paper aims to give the methodological foundations for an advanced covariance modeling and elaborates a set of generic base functions which can be used for flexible covariance modeling. In particular, we provide a straightforward procedure and guidelines for a generic approach to the fitting of oscillating covariance functions to an empirical sequence of covariances. The underlying methodology is developed based on the well known properties of autoregressive processes in time series. The surprising simplicity of the proposed covariance model is that it corresponds to a finite sum of covariance functions of second-order Gauss-Markov (SOGM) processes. Furthermore, the great benefit is that the method is automated to a great extent and directly results in the appropriate model. A manual decision for a set of components is not required. Notably, the numerical method can be easily extended to ARMA-processes, which results in the same linear system of equations. Although the underlying mathematical methodology is extensively complex, the results can be obtained from a simple and straightforward numerical method.

Estimation of Interference Correlation in mmWave Cellular Systems

“…From (11) we can write the interference vector as the sum of LN g correlated Gaussian sources, arriving at the BS with receive phase shifts β i,ℓ , for i = 0, . .…”

“…Point 4) is obtained by observing that the LS estimate of x(t) from N (t), given the estimate Â, is obtained as (see (11))…”

“…These models neither exemplify any particular propagation scenario nor are suited to capture the underlying composite interference signals. In fact, they are typically proposed as the simplest analytical model that can be easily adapted to empirically derived covariances [11]. Nevertheless, exponential correlation modeling has some drawbacks: on one hand, the fitting may not be straightforward given that they are not linear functions, on the other, they may miss relevant components, e.g., at high frequencies.…”

Local and Global Measures of Information Storage for the Assessment of Heartbeat-Evoked Cortical Responses

“…Here, we stick to the AR representation because of the difficulty to identify ARMA models and to provide the local formulation of information measures for this class of models. Nevertheless, the parametric representation and the exploitation of the extended Yule-Walker equations allow to take into account long lags in the representation of the past history of the process, covering long memories until the covariance decays, while this is not possible using the model-free approach [69]. In fact, the estimation of information measures is more difficult using the nearest neighbor estimator and should thus be limited to a small number of past lags; the obtained results in this case could be strongly affected by the choice of the dimension of the embedding vector as well as by the number of neighbors.…”

Future-Aware Trend Alignment for Sales Predictions

“…The AR and ARMA models are suitable for stationary time series, but most time series data are non-stationary, so various linear and non-linear time series models [4], namely autoregressive integrated moving average (ARIMA) [5], seasonal-ARIMA, the seasonally decomposed autoregressive (STL-ARIMA) algorithm [6], the autoregressive conditional heteroscedasticity model (ARCH), and generalized autoregressive conditional heteroskedasticity (GARCH), have come into being. Moreover, autoregressionmethods that can model cointegration (autoregressive distributed lag (ARDL)) [7] or estimate covariance functions (stochastic autoregressive moving average (ARMA) processes) [8] have been proposed. The above regressive algorithms are not domain-specific, and they have been applied in various fields, such as sales forecasting [9] and wind speed prediction [10].…”