The estimation of continuous time models with mixed frequency data

This paper derives exact representations for discrete time mixed frequency data generated by an underlying multivariate continuous time model. Allowance is made for different combinations of stock and flow variables as well as deterministic trends, and the variables themselves may be stationary or nonstationary (and possibly cointegrated). The resulting discrete time representations allow for the information contained in high frequency data to be utilised alongside the low frequency data in the estimation of the parameters of the continuous time model. Monte Carlo simulations explore the finite sample performance of the maximum likelihood estimator of the continuous time system parameters based on mixed frequency data, and a comparison with extant methods of using data only at the lowest frequency is provided. An empirical application demonstrates the methods developed in the paper and it concludes with a discussion of further ways in which the present analysis can be extended and refined

Exact Discrete Representations of Linear Continuous Time Models with Mixed Frequency Data

The time aggregation of vector linear processes containing (i) mixed stock- ow data and (ii) aggregated at mixed frequencies, is explored, focusing on a method to translate the parameters of the underlying continuous time model into those of an equivalent model of the observed data. Based on manipulations of a general state-space form, the results may be used to model multiple frequencies or aggregation schemes. Estimation of the continuous time parameters via the ARMA representation of the observable data vector is discussed and demonstrated in an application to model stock price and dividend data. Simulation evidence suggests that these estimators have superior properties to the traditional approach of concentrating the data to a single low frequency

Novel Fourier Quadrature Transforms and Analytic Signal Representations for Nonlinear and Non-stationary Time Series Analysis

The Hilbert transform (HT) and associated Gabor analytic signal (GAS) representation are well-known and widely used mathematical formulations for modeling and analysis of signals in various applications. In this study, like the HT, to obtain quadrature component of a signal, we propose the novel discrete Fourier cosine quadrature transforms (FCQTs) and discrete Fourier sine quadrature transforms (FSQTs), designated as Fourier quadrature transforms (FQTs). Using these FQTs, we propose sixteen Fourier-Singh analytic signal (FSAS) representations with following properties: (1) real part of eight FSAS representations is the original signal and imaginary part is the FCQT of the real part, (2) imaginary part of eight FSAS representations is the original signal and real part is the FSQT of the real part, (3) like the GAS, Fourier spectrum of the all FSAS representations has only positive frequencies, however unlike the GAS, the real and imaginary parts of the proposed FSAS representations are not orthogonal to each other. The Fourier decomposition method (FDM) is an adaptive data analysis approach to decompose a signal into a set of small number of Fourier intrinsic band functions which are AM-FM components. This study also proposes a new formulation of the FDM using the discrete cosine transform (DCT) with the GAS and FSAS representations, and demonstrate its efficacy for improved time-frequency-energy representation and analysis of nonlinear and non-stationary time series.Comment: 22 pages, 13 figure

A Unified Approach to the STFT, TFDs and Instantaneous Frequency

Spectral analysis of time varying signals is traditionally performed with the short time Fourier transformation (STFT). In the last few years, many authors have advocated the use of time frequency distributions for this task. This paper has 2 main aims. The first is to reformulate Cohen-class of time frequency representations (TFRs) into discrete-time, discrete-frequency, computer-implemented form. The second aim is to show how, in this form, many of the properties of the continuous-time, continuous-frequency formulation are either lost or altered. Intuitions applicable in the continuous-time case examined here. The properties of the discrete variable formulation examined are the presence and form of cross-terms, instantaneous frequency (IF) estimation and relations between Cohen's class TFRs. We define a parameterized class of distributions which is a blending between the STFT and wigner ville distribution (WVD). The two main conclusions to be drawn are that all TFRs of Cohen's class implementable in the form (which includes all commonly used TFRs) posses cross terms and that IF estimation using periodic moments of these TFRs is purposeless, since simpler methods obtain the same results

Induction Machine Diagnosis using Stator Current Advanced Signal Processing

International audienceInduction machines are widely used in industrial applications. Safety, reliability, efficiency and performance are major concerns that direct the research activities in the field of electrical machines. Even though the induction machines are very reliable, many failures can occur such as bearing faults, air-gap eccentricity and broken rotor bars. Therefore, the challenge is to detect them at an early stage in order to prevent breakdowns. In particular, stator current-based condition monitoring is an extensively investigated field for cost and maintenance savings. In fact, several signal processing techniques for stator current-based induction machine faults detection have been studied. These techniques can be classified into: spectral analysis approaches, demodulation techniques and time-frequency representations. In addition, for diagnostic purposes, more sophisticated techniques are required in order to determine the faulty components. This paper intends to review the spectral analysis techniques and time-frequency representations. These techniques are demonstrated on experimental data issued from a test bed equipped with a 0.75 kW induction machine. Nomenclature O&M = Operation and Maintenance; WTG = Wind Turbine Generator; MMF = Magneto-Motive Force; MCSA = Motor Current signal Analysis; PSD = Power Spectral Density; FFT = Fast Fourier Transform; DFT = Discrete Fourier Transform; MUSIC = MUltiple SIgnal Characterization; ESPRIT = Estimation of Signal Parameters via Rotational Invariance Techniques; SNR = Signal to Noise Ratio; MLE = Maximum Likelihood Estimation; STFT = Short-Time Fourier Transform; CWT = Continuous Wavelet Transform; WVD = Wigner-Ville distribution; HHT = Hilbert-Huang Transform; DWT = Discrete Wavelet Transform; EMD = Empirical Mode Decomposition; IMF = Intrinsic Mode Function; AM = Amplitude Modulation; FM = Frequency Modulation; IA = Instantaneous Amplitude; IF = Instantaneous Frequency; í µí± ! = Supply frequency; í µí± ! = Rotational frequency; í µí± ! = Fault frequency introduced by the modified rotor MMF; í µí± ! = Characteristic vibration frequencies; í µí± !"# = Bearing defects characteristic frequency; í µí± !" = Bearing outer raceway defect characteristic frequency; í µí± !" = Bearing inner raceway defect characteristic frequency; í µí± !" = Bearing balls defect characteristic frequency; í µí± !"" = Eccentricity characteristic frequency; í µí± ! = Number of rotor bars or rotor slots; í µí± = Slip; í µí°¹ ! = Sampling frequency; í µí± = Number of samples; í µí±¤[. ] = Time-window (Hanning, Hamming, etc.); í µí¼ = Time-delay; í µí¼ ! = Variance; ℎ[. ] = Time-window

Two Dimensional Signal Representation Using Prolate Spheroidal Functions

The most widely used methods of signal representation are the time function and the frequency function or spectrum representations. This work is concerned with the development of a representation which is a combination of these two. Two previous attempts at defining this type of signal representation, which is referred to as two dimensional representation, have been made and a summary and evaluation of these attempts is presented. The primary objective of the work reported here was to develop a practical two dimensional representation which has the desired two dimensional conceptual properties as well as mathematical convenience. The representations defined are based on the angular prolate spheroidal functions. These functions have a number of desirable properties among which are the followings they are orthogonal over both a finite and the infinite interval, they are bandlimited, and they have certain properties concerning their maximal proximity to being timelimited. The procedure used in defining the first two dimensional representation is to make an orthogonal expansion, using the prolate spheroidal functions, of each timelimited portion of each bandlimited portion of the signal to be represented. The second two dimensional representation is defined from an orthogonal expansion of each bandlimited portion of each timelimited portion of the signal to be represented. For both of these, the summation over all time intervals and all frequency intervals results in the complete representation of the signal. It is seen from this that since it is not possible to timelimit and bandlimit simultaneously, these limiting processes have been carried out serially. Due to the peculiar properties of the prolate spheroidal functions, as the number of orthogonal function terms is increased, the representation of a timelimited function converges first in a certain bandwidth, and the representation of a band- limited function converges first in a certain time interval. It is demonstrated that both series representations will converge to either a timelimited, or a bandlimited portion of the represented signal upon inclusion of the proper terms. Following this, several applications of the representations are presented. First, it is shown that the result of the convolution of 2 two dimensionally represented functions may be determined at discrete values of time from the expansion coefficients alone. The spectrum of the product of two functions may be determined in a similar manner at discrete values of frequency. As a result, it is possible to determine the contribution made to the output of a linear system at any time due to the portion of the input in any time and frequency interval. A technique is also developed for the solution of this same problem for the more general time variable linear system with the output being determined in continuous form rather than only at discrete values. It is somewhat more difficult to calculate the coefficients in this case, however. Another application demonstrated is a method by which the value of Woodward\u27s ambiguity function may be calculated for discrete values of the time and frequency variables. The two dimensional nature of the representation is demonstrated by two numerical examples using very elementary time functions. A further numerical example is provided for the case of the determination of the output of a linear system at discrete values of time. This work is concluded by a brief listing of further problems which seem amenable to solution as a result of this type of analysis. This list includes such problems as biological system signal analysis, signal design, and random process representation

Low rank Green's function representations applied to dynamical mean-field theory

Several recent works have introduced highly compact representations of single-particle Green's functions in the imaginary time and Matsubara frequency domains, as well as efficient interpolation grids used to recover the representations. In particular, the intermediate representation with sparse sampling and the discrete Lehmann representation (DLR) make use of low-rank compression techniques to obtain optimal approximations with controllable accuracy. We consider the use of the DLR in dynamical mean-field theory (DMFT) calculations, and in particular, show that the standard full Matsubara frequency grid can be replaced by the compact grid of DLR Matsubara frequency nodes. We test the performance of the method for a DMFT calculation of Sr$_2$RuO$_4$ at temperature $50$K using a continuous-time quantum Monte Carlo impurity solver, and demonstrate that Matsubara frequency quantities can be represented on a grid of only $36$ nodes with no reduction in accuracy, or increase in the number of self-consistent iterations, despite the presence of significant Monte Carlo noise.Comment: 5 pages, 4 figure