Statistical inference of spectral estimation for continuous-time MA processes with finite second moments

If we follow an asexually reproducing population through time, then the amount of time that has passed since the most recent common ancestor (MRCA) of all current individuals lived will change as time progresses. The resulting “MRCA age” process has been studied previously when the population has a constant large size and evolves via the diffusion limit of standard Wright–Fisher dynamics. For any population model, the sample paths of the MRCA age process are made up of periods of linear upward drift with slope +1 punctuated by downward jumps. We build other Markov processes that have such paths from Poisson point processes on ℝ++×ℝ++ with intensity measures of the form λ⊗μ where λ is Lebesgue measure, and μ (the “family lifetime measure”) is an arbitrary, absolutely continuous measure satisfying μ((0, ∞))=∞ and μ((x, ∞))<∞ for all x>0. Special cases of this construction describe the time evolution of the MRCA age in (1+β)-stable continuous state branching processes conditioned on nonextinction—a particular case of which, β=1, is Feller’s continuous state branching process conditioned on nonextinction. As well as the continuous time process, we also consider the discrete time Markov chain that records the value of the continuous process just before and after its successive jumps. We find transition probabilities for both the continuous and discrete time processes, determine when these processes are transient and recurrent and compute stationary distributions when they exist. Moreover, we introduce a new family of Markov processes that stands in a relation with respect to the general (1+β)-stable continuous state branching process and its conditioned version that is similar to the one between the family of Bessel-squared diffusions and the unconditioned and conditioned Feller continuous state branching process.

ABSTRACTThis paper presents a new method for the causal quarter-plane region of support two-dimensional (2-D) moving average (MA) model parameter estimation. The new approach is based on approximation of 2-D MA by the 2-D AR model. To achieve this aim, the corresponding relations are extended to a 2-D case and the related algorithm is presented. In this method, a 2-D series with the MA model has been approximated by a 2-D AR model with higher order and then the parameters of the AR model are estimated by the new method that is presented. Then the relation between the parameters of the 2-D AR and 2-D MA model is obtained and finally by using this relation, the parameters of the 2-D MA model are obtained. Since the proposed method does not involve complex and time consuming matrix computations, it is computationally efficient. The presented method also has good accuracy in standard deviation and mean value; a fact that has been shown by applying this method to a numerical example and presenting the results of simulation.

ABSTRACTIn this paper, we call attention of two observed features in practical applications of the Generalized Autoregressive Moving Average (GARMA) model due to the structure of its linear predictor. One is the multicollinearity which may lead to a non-convergence of the maximum likelihood, using iteratively reweighted least squares, and the inflation of the estimator's variance. The second is that the inclusion of the same lagged observations into the autoregressive and moving average components confounds the interpretation of the parameters. A modified model, GAR-M, is presented to reduce the multicollinearity and to improve the interpretation of the parameters. The expectation and variance under stationarity conditions are presented for the identity and logarithm link function. In a general sense, simulation studies show that the maximum likelihood estimators based on the GARMA and GAR-M models are equivalent but the GAR-M estimators presented a little lower standard errors and some restrictions in the parametric space are imposed to guarantee the stationarity of the process. Also, a real data analysis illustrates the GAR-M fit for daily hospitalization rates of elderly people due to respiratory diseases from October 2012 to April 2015 in São Paulo city, Brazil.

In this paper we study the spectral properties of an auto-regressive (AR) spectral estimator proposed by Cadzow, which uses a large number of correlations to set up the normal equations. This Set of overdetermined equations is then solved in a least squares sense. The main contribution of this paper is the derivation of asymptotic relationships, i.e., as the number of samples goes to infinity, between the number of correlations used, the model order and the signal-to-noise ratio of the signal, and the characteristics of the resulting spectral estimate when the signal under study is composed of sinusoids in noise. The characteristics studied are the height, bandwidth, and area of the peaks in the estimated spectrum. The method is shown to be a spectral density estimator like the ME method, where spectral areas rather than spectral values should be interpreted as estimates of power. The role of the number of correlations as a signal-to-noise ratio enhancer is discussed. Computer simulations are presented which verify the theoretical results.

The aim of this study is to extend the scope and applicability of the level-crossing method to discrete-time stochastic processes and generalize it to enable us to study multiple discrete-time stochastic processes. In previous versions of the level-crossing method, problems with it correspond to the fact that this method had been developed for analyzing a continuous-time process or at most a multiple continuous-time process in an individual manner. However, since all empirical processes are discrete in time, the already-established level-crossing method may not prove adequate for studying empirical processes. Beyond this, due to the fact that most empirical processes are coupled; their individual study could lead to vague results. To achieve the objectives of this study, we first find an analytical expression for the average frequency of crossing a level in a discrete-time process, giving the measure of the time experienced for two consecutive crossings named as the "waiting time." We then introduce the generalized level-crossing method by which the consideration of coupling between the components of a multiple process becomes possible. Finally, we provide an analytic solution when the components of a multiple stochastic process are independent Gaussian white noises. The comparison of the results obtained for coupled and uncoupled processes measures the strength and efficiency of the coupling, justifying our model and analysis. The advantage of the proposed method is its sensitivity to the slightest coupling and shortest correlation length.

In this chapter we consider problems of statistical inference for time series based on frequency-domain properties of the series. The fundamental tool used is the periodogram, which is defined in Section 10.1 for any time series {x 1,..., x n }. Section 10.2 deals with statistical tests for the presence of “hidden periodicities” in the data. Several tests are discussed, corresponding to various different models and hypotheses which we may wish to test. Spectral analysis for stationary time series, and in particular the estimation of the spectral density, depends very heavily on the asymptotic distribution as n → ∞ of the periodogram ordinates of the series {X 1,..., X n }. The essential results are contained in Theorem 10.3.2. Under rather general conditions, the periodogram ordinates I n (λ i ) at any set of frequencies λ 1,..., λ m , 0 < λ 1 < ⋯ < λ m < π, are asymptotically independent exponential random variables with means 2πf(λ i ), were f is the spectral density of {X t }. Consequently the periodogram I n is not a consistent estimator of 2πf. Consistent estimators can however be constructed by applying linear smoothing filters to the periodogram. The asymptotic behaviour of the resulting discrete spectral average estimators can be derived from the asymptotic behaviour of the periodogram as shown in Section 10.4. Lag-window estimators of the form \({\left( {2\pi } \right)^{ - 1}}\sum\nolimits_{\left| h \right| \leqslant r} {w\left( {h/r} \right)\hat \gamma \left( h \right)} {e^{ - ih\omega }}\), where w(x), −1 ≤ x ≤ 1, is a suitably chosen weight function, are also discussed in Section 10.4 and compared with discrete spectral average estimators. Approximate confidence intervals for the spectral density are given in Section 10.5. An alternative approach to spectral density estimation, based on fitting an AR MA model to the data and computing the spectral density of the fitted process, is discussed in Section 10.6. An important role in the development of spectral analysis has been played by the fast Fourier transform algorithm, which makes possible the rapid calculation of the periodogram for very large data sets. An introduction to the algorithm and its application to the computation of autocovariances is given in Section 10.7. The chapter concludes with a discussion of the asymptotic behaviour of the maximum likelihood estimators of the coefficients of an ARMA(p, q) process.

SummaryThis paper introduces continuous‐time random processes whose spectral density is unbounded at some non‐zero frequencies. The discretized versions of these processes have asymptotic properties similar to those of discrete‐time Gegenbauer processes. The paper presents some properties of the covariance function and spectral density as well as a theory of statistical estimation of the mean and covariance function of such processes. Some directions for further generalizations of the results are indicated.

This paper presents a novel method of estimating the optimal steady state Kalman filter gain of a linear discrete time-invariant system from a non-optimal Kalman filter residual sequence. The relation between the optimal residual sequence and a signal derived from the non-optimal residual sequence is described by a Moving Average (MA) model whose coefficients are expressed in terms of the state space parameters and the optimal steady state Kalman filter gain. In order to identify the MA model, a whitening filter of the derived signal, which corresponds to an AutoRegressive (AR) model of the signal, is first identified using the least-squares method. Then the inverse filter of the whitening filter, which corresponds to the MA model, is calculated. From the coefficients of the identified MA model, the optimal steady state Kalman filter gain can be obtained. Numerical example is provided to illustrate the feasibility of this approach.

A stochastic process or random process consists of chronologically ordered random variables {X t ; t ≥ 0}. For simplicity we assume that the process starts at time t = 0 in X 0 = 0. In this chapter, we consider exclusively processes in discrete time, i.e. processes which are observed at equally spaced points of time t = 0, 1, 2,... . Typical examples are daily, monthly or yearly observed economic data as stock prices, rates of unemployment or sales amount.

Using the control problem of a two-state Markov process in discrete time as an example, we consider the basic stages concerning the application of theory of conditional Markov processes to synthesize optimal algorithms of the control of stochastic systems. It is assumed that the control changes the statistical properties of the states of a controlled plant. The numerical method for solving the problem and the results of solving particular example are presented. The special features of the solution of this problem compared to the well-known problem in continuous time are discussed.

We develop a general framework for stationary marked point processes in discrete time. We start with a careful analysis of the sample paths. Our initial representation is a sequence $$\{(t_j,k_j): j\in {\mathbb {Z}}\}$$ of times $$t_j\in {\mathbb {Z}}$$ and marks $$k_j\in {\mathbb {K}}$$ , with batch arrivals (i.e., $$t_j=t_{j+1}$$ ) allowed. We also define alternative interarrival time and sequence representations and show that the three different representations are topologically equivalent. Then, we develop discrete analogs of the familiar stationary stochastic constructs in continuous time: time-stationary and point-stationary random marked point processes, Palm distributions, inversion formulas and Campbell’s theorem with an application to the derivation of a periodic-stationary Little’s law. Along the way, we provide examples to illustrate interesting features of the discrete-time theory.

Three important parametric models for describing the correlation functions and spectra of stationary stochastic processes are the autoregressive (AR), moving average (MA), and autoregressive-moving average (ARMA) models. Quite recently, the MATLAB toolbox “ARMASA” has been made publicly available. This toolbox provides state-of-the-art algorithms to perform automatic identification and selection between the models based on the estimated prediction error. ARMASA works on a single segment of data, whereas in some applications, the data are available as multiple segments. We could process each segment independently and average the estimated autocorrelation functions or spectra afterward. Better performance, however, can be expected when all segments are processed simultaneously, for two reasons. Initially, the bias in the estimated model parameters depends on the number of observations in a segment. Averaging models from segments does not reduce this bias, in contrast to identification from all the data at once. In either case, the variance of the estimated model parameters will decrease. By considering all the data at once this variance reduction can be considered in the order selection stage, decreasing the selection bias. In this paper, we extend the parameter estimation algorithms of the ARMASA toolbox to the case of multiple segments. AR parameter estimation from segments has been considered before, but not for the MA and ARMA models. In addition, in the AR case, model order selection can be done easily when the segments are independent. But, in the application we are primarily interested in, the segments are formed by lines of scanning transmission electron microscopy (STEM) images, which can be correlated with each other. Because the correlation among data segments influences the effective number of independent observations available it becomes necessary to estimate that number. This is not an easy problem in general. For observations from a stochastic process with an isotropic 2-D autocorrelation function we can, however, formulate an accurate estimator in terms of the line autocorrelation function. This situation arises, for example, in STEM images of amorphous specimens. From the AR model identification for segments, the extension to the MA and ARMA models is rather straightforward because MA and ARMA algorithms are based on Durbin's methods, whose first step is the estimation of a long AR model. We compare the segment-based algorithms to conventional ARMASA with simulations of stationary stochastic processes. An illustrative example involving simulated STEM images shows large potential benefits.

A comparative study and assessment of doppler ultrasound spectral estimation techniques part II: Methods and results

Signal processing of flutter flight test data enables verification of aircraft flutter design, and the signal from a flutter flight test excited by atmospheric turbulence is a particularly important form of the flutter test. Owing to the randomness of atmospheric turbulence excitation, multi-channel analysis of turbulence responses at various positions in the same component can improve the analytical accuracy of flutter signal processing. The relationship between the maximum singular value of the multi-channel turbulence response power spectral density matrix and the system self-power spectral density function is elucidated herein using a frequency domain decomposition method. However, there is a contradiction in the power spectral density function between the spectral line density and the spectral line smoothing calculated based on the periodogram of the frequency domain decomposition. By applying an autoregressive spectral model, the power spectral density function of the turbulence response is calculated to achieve spectral line smoothing and sufficient spectral line density. Additionally, the power spectral density function is then used to construct the power spectral density function matrix of the multi-channel turbulence response, and the maximum singular value curve is calculated based on the singular value decomposition of each spectral pin. Finally, the modal parameters of the turbulence response signal are estimated via multi-modal frequency domain fitting. The developed approach is validated based on simulations and flutter flight test turbulence response signals.

A coprime sensor array (CSA) is a sparse array geometry that interleaves two spatially undersampled uniform linear arrays (ULAs) with coprime undersampling factors. The CSA spans the same aperture and has equivalent resolution as a fully populated ULA, but uses fewer sensors. Previous works have introduced three CSA processors as spatial power spectral density (PSD) estimators: the Product, Min, and Blend processors. However, two of these processors display increased PSD esti-mate variance over the ULA PSD estimate. This dissertation focuses on reducing the variance of the CSA PSD estimators. PSD estimation traditionally relies on averag-ing uncorrelated coherent measurements (snapshots) to reduce variance. However,non-stationary underwater sonar environments often preclude increasing the num-ber of snapshots required to achieve a desirable PSD variance. In the traditional signal processing literature, there are two alternative methods to reduce variance without additional snapshot cost. The multitaper (MT) method and Welch's over-lapping segment averaging (WOSA) method improve PSD variance by O(K) at the expense of an O(K) resolution decrease by averaging K uncorrelated PSD estimates. Multitaper obtains these uncorrelated estimates by windowing the array with K or-thogonal tapers that span the entire array aperture. In contrast, WOSA obtains the uncorrelated estimates by subdividing the array into K possibly overlapping segments.This dissertation extends these two existing ULA variance reduction techniques to the CSA processors, making four main contributions. The first proposes the multitapered Product processor (MT-Product) that estimates the spatial PSD with reduced variance with respect to the traditional CSA Product processor but still uses fewer sensors than a fully populated ULA. The second proposes the multita-pered Min processor (MT-Min) that reduces the PSD estimate variance further than either MT-Product or Product. The MT-Min estimator has variance comparable to a multitapered ULA. The third proposes the multitapered Blend processor (MT-Blend) that blends MT-Product's attenuation of multiple source cross-terms with MT-Min's low variance while still achieving an O(K) variance reduction over the traditional Blend processor. The final contribution proposes the Welch overlapping segment averaging Product processor (WOSA-Product) that estimates the spatial PSD with reduced variance with respect to the traditional CSA Product processor but still uses fewer sensors than a fully populated ULA. WOSA-Product is gener-ally able to form a larger number of uncorrelated PSD estimates than MT-Product. Closed-form statistics for spatially white Gaussian processes are derived for MT-Product, MT-Min, and WOSA-Product. Simulations verify the variance reduction predicted by the analytical derivations each processor, and the effects of WOSA-Product segment length on resolution, variance reduction, and peak sidelobe levels are discussed.