booklet format - inaf iasf bologna

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
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