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Temporal Data Analysis A.A. 2011/2012 MAY BE STATIONARY AND RANDOM OBVIOUSLY NONSTATIONARY OR NONRANDOM QUALIFICATION A. MEAN AND MEAN SQUARE VALUE ANALYSIS TEST FOR STATIONARITY NONSTATIONARY E. NONSTATIONARY AND TRANSIENT DATA ANALYSIS B. AUTOCORRELATION ANALYSIS STATIONARY TEST FOR PERIODICITIES PERIODIC OR ALMOST PERIODIC F. PERIODIC AND ALMOST- PERIODIC DATA ANALYSIS C. POWER SPECTRAL DENSITY ANALYSIS STATIONARY AND RANDOM (WITH PERIOCITIES IDENTIFIED OR REMOVED) D. PROBABILITY DENSITY ANALYSIS TEST FOR NORMALITY G. SPECIALIZED DATA ANALYSIS NON-GAUSSIAN Figure 5.1: General procedure for analyzing individual sample records 72 M.Orlandini
A.A. 2011/2012 Temporal Data Analysis 5.1.2 Autocorrelation Analysis The next suggested analysis is autocorrelation, as indicated by Block B. The autocorrelation function of stationary data is the inverse Fourier transform of the power spectral density function (see (3.25)). Thus the determination of the autocorrelation function will technically not yield any new information over the power spectrum. There might be applications, however, when an autocorrelogram would present the desired information in a more convenient format. Autocorrelation functions can be a useful tool for detecting periodicities in otherwise random data. Furthermore, autocorrelation functions might be computed as an intermediate step in the calculation of power spectral estimates. 5.1.3 Power Spectral Density Analysis Perhaps the most important single descriptive characteristic of stationary random data is the power spectral density function, which defines the frequency composition of the data. For constant parameter linear physical systems, the output power spectrum is equal to the input power spectrum multiplied by the square of the gain factor of the system. Thus power spectra measurements can yield information concerning the dynamic characteristics of the system. The total area under a power spectrum (that is ∫ |F (ω)| 2 dω) is equal to the mean square value. To be more general, the mean square value of the data in any frequency range of concern is determined by the area under the power spectrum bounded by the limits of that frequency range. Obviously, the measurement of power spectra data, as indicated in Block C, will be valuable for many analysis objectives, like detection of periodicities and as an intermediate step in the calculation of autocorrelation functions. 5.1.4 Probability Density Analysis The last fundamental analysis included in the procedure is probability density analysis, as indicated by Block D. Probability density analysis is often omitted from a data analysis procedure because of the tendency to assume that all random phenomena are normally distributed (this analysis is performed in the so-called Exploratory Data Analysis – EDA). In some cases, however, random data may deviate substantially from the Gaussian form. If such deviations are detected by a test for normality, then the probability density function of the data must be measured to establish the actual probabilistic characteristics of the data. Furthermore, a probability density function estimate is sometimes used as a basis for a normality test. 5.1.5 Nonstationary and Transient Data Analysis All of the analysis techniques discussed so far apply only to sample records of stationary data. If the data are determined to be nonstationary during the qualification phase of the processing, then special analysis techniques will be required as indicated by Block E. Note that certain classes of nonstationary data can be sometimes be analyzed using the same equipment or computer M.Orlandini 73
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Temporal Data Analysis: Theory and
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Course Outline Table of Contents Li
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A.A. 2011/2012 Temporal Data Analys
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List of Figures 1.1 Classification
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List of Definitions 2.1 Even and od
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List of Theorems 2.1 Convolution th
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Part I Theory 1
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