Robert Bos

Autoregressive spectral estimation by application of the Burg algorithm to irregularly sampled data

Many methods have been developed for spectral analysis of irregularly sampled data. Currently, popular methods such as Lomb-Scargle and resampling tend to be biased at higher frequencies. Slotting methods fail to consistently produce a spectrum that is positive for all frequencies. In this paper, a new estimator is introduced that applies the Burg algorithm for autoregressive spectral estimation to unevenly spaced data. The new estimator can be perceived as searching for sequences of data that are almost equidistant, and then analyzing those sequences using the Burg algorithm for segments. The estimated spectrum is guaranteed to be positive. If a sufficiently large data set is available, results can be accurate up to relatively high frequencies.

Autoregressive spectral analysis when observations are missing

A new missing data algorithm ARFIL gives good results in spectral estimation. The log likelihood of a multivariate Gaussian random variable can always be written as a sum of conditional log likelihoods. For a complete set of autoregressive AR(p) data the best predictor in the likelihood requires only p previous observations. If observations are missing, the best AR predictor in the likelihood will in general include all previous observations. Using only those observations that fall within a finite time interval will approximate this likelihood. The resulting non-linear estimation algorithm requires no user provided starting values. In various simulations, the spectral accuracy of robust maximum likelihood methods was much better than the accuracy of other spectral estimates for randomly missing data.

Time-series analysis if data are randomly missing

Maximum-likelihood (ML) theory presents an elegant asymptotic solution for the estimation of the parameters of time-series models. Unfortunately, the performance of ML algorithms in finite samples is often disappointing, especially in missing-data problems. The likelihood function is symmetric with respect to the unit circle for the estimated zeros of time-series models. As a consequence, the unit circle is either a local maximum or a local minimum in the likelihood of moving-average (MA) models. This is a trap for nonlinear optimization algorithms that often converge to poor models, with estimated zeros precisely on the unit circle. With ML estimation, it is much easier to estimate a long autoregressive (AR) model with only poles. The parameters of that long AR model can then be used to estimate MA and autoregressive moving-average (ARMA) models for different model orders. The accuracy of the estimated AR, MA, and ARMA spectra is very good. The robustness is excellent as long as the AR order is less than 10 or 15. For still-higher AR orders until about 60, the possible convergence to a useful model will depend on the missing fraction and on the specific properties of the data at hand.

Estimating Time Series Models from Irregularly Sampled Data

The maximum likelihood estimation of the parameters of the continuous time model for irregularly sampled data is sensitive to initial conditions. Simulations often converge to a good solution if the true parameters are used as the starting values for the nonlinear search of the minimum of the negative log likelihood. From realizable starting values the convergence to a model with an accurate spectrum is rare if more than three parameters have to be estimated. A discrete time spectral estimator is introduced that applies the principles of the algorithm for automatic equidistant missing data analysis to unevenly spaced data. This time series estimator approximates the irregular data by a number of equidistantly resampled missing data sets, with a special nearest neighbor method. Slotted nearest neighbor resampling replaces a true observation time instant by the nearest equidistant resampling time point, if and only if the true time is within half the slot width. A smaller slot will reduce bias. A refined slotted resampling method is introduced, which uses a slot width that is only a fraction of the resampling time, giving multiple data sets with equidistant missing data time sequences which are shifted over the slot width. The highest frequency with accurate spectral estimates can be beyond the mean data rate

Application of autoregressive spectral analysis to missing data problems

Time series solutions for spectral analysis in missing data problems use reconstruction of the missing data, or a maximum likelihood approach that analyzes only the available measured data. Maximum likelihood estimation yields the most accurate spectra. An approximate maximum likelihood algorithm is presented that uses only previous observations falling in a finite interval to compute the likelihood, instead of all previous observations. The resulting nonlinear estimation algorithm requires no user-provided initial solution, is suited for order selection, and can give very accurate spectra even if less than 10% of the data remains.

Likelihood-Based Hypothesis Tests for Brain Activation Detection From MRI Data Disturbed by Colored Noise: A Simulation Study

Functional magnetic resonance imaging (fMRI) data that are corrupted by temporally colored noise are generally preprocessed (i.e., prewhitened or precolored) prior to functional activation detection. In this paper, we propose likelihood-based hypothesis tests that account for colored noise directly within the framework of functional activation detection. Three likelihood-based tests are proposed: the generalized likelihood ratio (GLR) test, the Wald test, and the Rao test. The fMRI time series is modeled as a linear regression model, where one regressor describes the task-related hemodynamic response, one regressor accounts for a constant baseline and one regressor describes potential drift. The temporal correlation structure of the noise is modeled as an autoregressive (AR) model. The order of the AR model is determined from practical null data sets using Akaikes information criterion (with penalty factor 3) as order selection criterion. The tests proposed are based on exact expressions for the likelihood function of the data. Using Monte Carlo simulation experiments, the performance of the proposed tests is evaluated in terms of detection rate and false alarm rate properties and compared to the current general linear model (GLM) test, which estimates the coloring of the noise in a separate step. Results show that theoretical asymptotic distributions of the GLM, GLR, and Wald test statistics cannot be reliably used for computing thresholds for activation detection from finite length time series. Furthermore, it is shown that, for a fixed false alarm rate, the detection rate of the proposed GLR test statistic is slightly, but (statistically) significantly improved compared to that of the common GLM-based tests. Finally, simulations results reveal that all tests considered show seriously inferior performance if the order of the AR model is not chosen sufficiently high to give an adequate description of the correlation structure of the noise, whereas the effects of (slightly) overmodeling are observed to be less harmful.

Estimation of autoregressive spectra with randomly missing data

A Gaussian likelihood is completely determined by the data covariance matrix, which can be characterized for stationary random processes by the parameters of an autoregressive (AR) model. The best AR predictor includes all previous observations if data are incomplete, in contrast with consecutive data where p previous observations determine the best AR(p) prediction. The missing data likelihood will be approximated with only those observations that fall within a finite time interval. The resulting nonlinear estimation algorithm requires no user provided initial solution, is suited for order selection and can give very accurate spectra even if less than 10% of the data remains.

DESIGNING A KALMAN FILTER WHEN NO NOISE COVARIANCE INFORMATION IS AVAILABLE

Abstract A problem when designing Kalman filters using first principles models is often that these models lack a description of the noises that affect the states and the measurements. In these cases, the Kalman filter has to be estimated from data. For this purpose many algorithms have been presented in the literature. All methods in the literature assume that the system under consideration has an observability matrix that has no small singular values. In this paper it will be shown that small singular values can lead to poor performance of estimated Kalman filters. Also a method will be introduced for estimating the Kalman filter in the case that the system has small singular values. This method is able to construct a good filter, even if the first principles model is badly observable.

Robust estimation of the noise variance from background MR data

In the literature, many methods are available for estimation of the variance of the noise in magnetic resonance (MR) images. A commonly used method, based on the maximum of the background mode of the histogram, is revisited and a new, robust, and easy to use method is presented based on maximum likelihood (ML) estimation. Both methods are evaluated in terms of accuracy and precision using simulated MR data. It is shown that the newly proposed method outperforms the commonly used method in terms of mean-squared error (MSE).

AR spectral estimation by application of the Burg algorithm to irregularly sampled data

Many methods have been developed for spectral analysis of irregularly sampled data. Current popular methods such as Lomb-Scargle and resampling tend to be biased at higher frequencies. Slotting methods fail to consistently produce a spectrum that is positive for all frequencies. In this paper, a new estimator is introduced that applies the Burg algorithm for AR spectral estimation to unevenly spaced data. The new estimator can be perceived as searching for sequences of data that are almost equidistant, and then analyzing those sequences using the Burg algorithm for segments. The estimated spectrum is guaranteed to be positive. If a sufficiently large data set is available, results can be accurate even at higher frequencies.