Variance tree

Usual procedures for the selection of the common best basis are based on maximum variance criteria (Walczak and Massart, 2000). For instance, the variance spectrum procedure computes at first the variance of all the variables and arranges them into a vector, which has the significance of a spectrum of the variance. The wavelet decomposition is applied onto this vector and the best basis obtained is used to transform and to compress all the objects. Instead, the variance tree procedure applies the wavelet decomposition to all of the objects, obtaining a wavelet tree for each of them. Then, the variance of each coefficient, approximation or detail, is computed, and the variance values are structured into a tree of variances. The best basis derived from this tree is used to transform and to compress all the objects. 

In the second case, the uniform representation of signals in the wavelet domain is not possible and instead, the joint best-basis, in which a small number of wavelets coefficients can describe the majority of data variance, must be selected. This can be done, by applying Coifman and Wickerhausers algorithm of best-basis selection to the so-called variance tree", the elements of which represent the variance of wavelet coefficients with the same addresses (indices) (see Fig. 8) [5]. 

Fig. 8 Schematic repre. teniaiion of variance tree, the elements of which represent the variance of wavelet coefficients of m signals with the. same addresses.

Based on these two accumulator trees, TM and TS, the variance tree , VT, can be constructed, with its elements calculated as ... 

Once the variance tree is constructed, it can be searched for the joint best-basis, using the Coifman-Wickerhauser best-basis selection algorithm with, e.g., the entropy criterion. The entropy cost function (see Chapter 6) for the variance tree coefficients, which occur at the jth level in the i band of the signal decomposition is defined as ... 

Forming the variance tree and computing the new information costs... 

Searching the variance tree for the joint best basis. 

All signals from the NIR data set were decomposed by WPT. The joint best-basis selected for the variance tree, is presented in Fig. 10(a), whereas the variance vector in that basis is visualized in Fig. 10(b). 

Fig. 10 (a) Joint best-hasLs. selected for the variance tree of NIR data set (Egain denotes entropy drop between parents and child hands) (h) Elements of vector v in the joint best-basis (c) Cumulative percentage of the explained variance versus the number of the... 

It can be noticed, there is no big difference in the results for DWT and WPT. The histograms of filters frequencies, and the RMS errors of spectra reconstruction are presented in Fig. 10. Profiles of histograms are very similar for both transforms. This similarity is associated with the fact, that the selected best-basis is very similar to the DWT basis (see Fig. 11), and coefficients of the variance tree are similar to the squared coefficients of the DWT (see Fig. 12(a)). Cumulative percentage of variance for DWT and WPT, presented in Fig. 12(b), can be compared with the analogous figure for PCA compression (Fig. 5). 

Frequency domain splits Fig. II Best-basis selected for the variance tree. 

Fig. 12 (a) The top 200 coefficients (.squared) in the joint basis, and the top 200 elements of variance tree in the joint best basis sorted according to their amplitude and (h) cumulative percentage of the explained variance in the join basis, and in the best joint-... 

Figure 9.5 emphasizes the relationships among three other sums of squares in the ANOVA tree - the sum of squares due to lack of fit, SS f , the sum of squares due to purely experimental uncertainty, SS and the sum of squares of residuals, 55,. Two of the resulting variances, and were used in Section 6.5 where a statistical test was developed for estimating the significance of the lack of fit of a model to a set of data. The null hypothesis... 

Female adult budworm dry weights and the number of survivors of budworm were analyzed by multivariate analysis of variance to test for the effects of site and sex. Stepwise discriminant analysis was used to determine If tree chemical and physical parameters differed between sites (17). 

Figure 12.18 shows a sums of squares and degrees of freedom tree for the data of Table 12.4 and the model of Equation 12.32. The significance of the parameter estimates may be obtained from Equation 10.66 using sr2 and (X X) l to obtain the variance-covariance matrix. The (X X) x matrix for the present example is... 

U[S ] also depends on the distribution of members lifetime. If members lifetime is deterministic (e g., members are First-In-First-Out), all currently active members will occupy consecutive positions in the key tree. In this case, E[S] is identical to E[M], Generally, the higher variance members lifetime has, the larger... 

Examples of mathematical methods include nominal range sensitivity analysis (Cullen Frey, 1999) and differential sensitivity analysis (Hwang et al., 1997 Isukapalli et al., 2000). Examples of statistical sensitivity analysis methods include sample (Pearson) and rank (Spearman) correlation analysis (Edwards, 1976), sample and rank regression analysis (Iman Conover, 1979), analysis of variance (Neter et al., 1996), classification and regression tree (Breiman et al., 1984), response surface method (Khuri Cornell, 1987), Fourier amplitude sensitivity test (FAST) (Saltelli et al., 2000), mutual information index (Jelinek, 1970) and Sobol s indices (Sobol, 1993). Examples of graphical sensitivity analysis methods include scatter plots (Kleijnen Helton, 1999) and conditional sensitivity analysis (Frey et al., 2003). Further discussion of these methods is provided in Frey Patil (2002) and Frey et al. (2003, 2004). 

We consider a fractal arterial tree that consists of several branching levels where each level consists of parallel vessels, Figure 8.5 A. Each vessel is connected to m vessels of the consequent branching level [322]. We make the assumption that the vessel radii and lengths at each level k follow a distribution around the mean values f)k and pk, respectively. The variance of the vessel radii and lengths at each level produces heterogeneity in the velocities. 

Ratio of likelihood scores for selected tree and star phylogeny 2, 8, 9 is treated as a chi-square statistic with one degree of freedom. Alternatively, standard normal test of the mean and variance of the difference of their likelihood scores can be used to compare one tree to another... 

But despite these examples of biochemical evolution and modification in progress, to which more are being added as the science of comparative biochemistry matures, it remains true that the biochemical composition and organization of all the forms of life now present on earth demonstrate unity a unity quite at variance with their more obvious differences in gross structure and behaviour. Most of us would hesitate to compare ourselves with fish, typhoid bacteria, cancer cells, or even oak trees, yet the fact is that we have very much more in common with them than we might have guessed. 

---