Autocovariance functions

Since central second moments are called covariances, the function appearing on the left-hand side of Eq. (3-146) is called the autocovariance function of X(t). Autocorrelation functions will be studied in more detail in Sections 3.14 and 3.16. 

The last equation, which expresses the autocorrelation function of Y(t) in terms of h(t), is often referred to as Campbell s58 theorem. It is useful to note that the autocovariance function of Y(t) is given by the simpler expression... 

The function h(t ) i(t + r)dt is often referred to as the autocorrelation function of the Amotion h(t) however, the reader should be careful to note the difference between the autocorrelation function of h(t)—an integrable function—and the autocorrelation function of Y(t)—a function that is not integrable because it does not die out in time. With this distinction in mind, Campbell s theorem can be expressed by saying that the autocovariance function of a shot noise process is n times the autocorrelation function of the function h(t). 

Arcsine distribution, 105, 111 Assumption of molecular chaos, 17 Asymptotic theory, 384 of relaxation oscillations, 388 Asynchronous excitation, 373 Asynchronous quenching, 373 Autocorrelation function, 146,174 Autocovariance function, 174 Autonomous problems, 340 nonresonance oscillations, 350 resonance oscillations, 350 Autonomous systems, 356 problems of, 323 Autoperiodic oscillation, 372 Averages, 100... 

Marchetti, N., Felinger, A., Pasti, L., Pitrogrande, M.C., Dondi, F. (2004). Decoding two-dimensional complex multicomponent separations by autocovariance function. Anal. Chem. 76, 3055-3068. 

Pietrogrande, M.C., Marchetti, N., Tosi, A., Dondi, E, Righetti, P.G. (2005). Decoding two-dimensional polyacrylamide gel electrophoresis complex maps by autocovariance function a simplified approach for proteomics. Electrophoresis 26, 2739-2748. 

An adaptation of Fourier analysis to 2D separations can be established by calculating the autocovariance function (Marchetti et al., 2004). The theoretical background of that approach is that the power spectrum and the autocovariance function of a signal constitute a Fourier pair, that is, the power spectmm is obtained as the Fourier transform of the autocovariance function. 

The 2D autocovariance function can also be calculated from the 2D chromatogram acquired in digitized form,... 

A nonlinear curve fitting procedure of the experimental (Eq. 4.28) to the theoretical (Eq. 4.27) 2D autocovariance function can serve to perform some fundamental characterization of the 2D separation. The total volume (Vy) and the peak height dispersion (/a() can be readily measured in the chromatogram, thus the number of components (m) and the peak widths (a, and ay) can be estimated (Marchetti et al., 2004). 

The 2D autocovariance function can be utilized in a rather simple manner to get a quick estimation of the number of detectable components and the average peak widths. For that purpose only the maximum amplitude of the autocovariance function and the width at half-height should be measured (Pietrogrande et al., 2005, 2006a). The... 

FIGURE 4.8 Autocovariance function method computed on computer-generated 2D map. (a) Simulated disordered 2D maps containing 100 components, (b) Autocovariance function of the 2D map. Reproduced from Marchetti et al., (2004) with permission from the American Chemical Society. 

Only the central section of the autocovariance function has to be calculated (Fig. 4.9) and simple computations are required to estimate the sample complexity, that is, the... 

When the positions of the spots reveal an ordered pattern on the separation map, the long-term correlations in the autocovariance function can be used to decode the ordered structure of the retention pattern. We can use a simple linear relationship to estimate the position of the th spot (see Eq. 4.1)... 

The above autocovariance function shows a regular repetitive pattern of peaks. Since the distances between adjacent spots in the chromatogram are bx and by in thex and y directions, respectively, these distances and their harmonics will appear in the autocovariance function, yielding an ordered 2D spot series. 

Most often, real 2D chromatograms exhibit a composite ordered and disordered characteristic, that is, a series of disordered spots are superimposed over ordered spot sequences. When the chromatogram is derived from a mixture of several chemical families, a superficial look at the 2D separation map may give the impression of randomness. In that case, the autocovariance function, however, can resolve and help identify the hidden structured nature of the map. 

In Fig. 4.10, the superposition of 10 different ordered spot sequences is shown. All of them have different ax, ay phase values but the same ax, by frequencies. The symmetry of the spot series in the autocovariance function is obvious and the common frequency of the spot trains can be easily identified through the main spot train of the autocovariance function, which runs through the origin. The most intense, major spot train in the autocovariance function follows the direction of the spot trains in the separation map. The spot trains running along thex and y axes give the horizontal and vertical characteristic spot interdistances in the separation map. 

As an X-ray diffraction image map helps identify the lattice structure of a crystal, the autocovariance function of a 2D separation map may help recognize the chemical structure of complex mixtures. 

The applicability of the 2D autocovariance function method and the most relevant results obtained will be discussed in the next section. 

The statistical degree of overlapping (SDO) and 2D autocovariance function (ACVF) methods have been applied to 2D-PAGE maps (Marchetti et al., 2004 Pietrogrande et al., 2002, 2003, 2005, 2006a Campostrini et al., 2005) the means for extracting information from the experimental data and their relevance to proteomics are discussed in the following. The procedures were validated on computer-simulated maps. Their applicability to real samples was tested on reference maps obtained from literature sources. Application to experimental maps is also discussed. 

The 2D autocovariance function was computed on the digitized map signal using Equation 4.28 and the separation parameters were estimated according to Equation 4.30. The results obtained are reported in the following table (Pietrogrande et al., 2005). 

It must be emphasized that the availability of the SMO and 2D autocovariance function methods as two independent statistical procedures to estimate the same parameter, in, the number of proteins, is a helpful tool to verify the reliability of the results obtained. In the case of the 2D PAGE map of colorectal adenocarcinoma cell line (DL-1) an excellent agreement was found between the values obtained from the SMO method—m = 101 10 and m = 105 10—and the 2D autocovariance function procedure—m = 104 10 (Pietrogrande et al., 2006a). 

The 2D autocovariance function was computed on the selected map region that contains the spot train (eight proteins) in addition to 53 proteins randomly located (4.5-7 p7 and 0.65-0.68 log Mr values, enlarged inset in Fig. 4.13a). From the 2D autocovariance function, the number of proteins present in this map region, m, can be correctly estimated m = 53 7 for the original computer-generated map containing 53 SCs (blue line in inset in Fig. 4.13b) and m = 62 8 for the map where the spot train was added (red line in inset in Fig. 4.13b). 

In the 2D autocovariance function plot (Fig. 4.13b) well defined deterministic cones are evident along the Ap7 axis at values ApH 0.2, 0.4, 0.6 pH they are related to the constant interdistances repeated in the spot trains. This behavior is more clearly shown by the intersection of the 2D autocovariance function with the Ap7 separation axis. The inset in Fig. 4.13b reports the 2D autocovariance function plots computed on the same map with (red line) and without (blue line) the spot train. A comparison between the two lines shows that the 2D autocovariance function peaks at 0.2, 0.4, 0.6 ApH (red line) clearly identifying the presence of the spot train singling out this ordered pattern from the random complexity of the map (blue line, from map without the spot train). The difference between the two lines identifies the contribution of the two components to the complex separation the blue line corresponds to the random separation pattern present in the map the red line describes the order in the 2D map due to the superimposed spot train. The high sensitivity of the 2D autocovariance function method in detecting order is noted in fact it is able to detect the presence of only sevenfold repetitiveness hidden in a random pattern of 200 proteins (Pietrogrande et al., 2005). 

Moreover, the two procedures display different and complementary properties so that each of them is the method of choice to obtain specific information on the 2D separations. The SMO procedure is an unique tool to quantitatively estimate the degree of peak overlapping present in a map as well as to predict the influence of different experimental conditions on peak overlapping. The strength of the 2D autocovariance function method lies in its ability to simply single out ordered retention pattern hidden in the complex separation, which can be related to information on the chemical composition of the complex mixture. 

FIGURE 4.13 Identification of a train of spots by the 2D autocovariance function method. (See text for full caption.)... 

Figure 5 shows the PSD with a constant white region, a 1/co (= 1/f) region and a l/eo (= first-order noise) region. Laeven et al. derived the matching autocovariance function ... 

---