The simple multiscale approach

Wavelet transforms inspect signals at different scales or resolutions. At the coarse level, only the most prominent large features can be seen, whereas at the higher levels finer details are captured. Multiresolution develops representations of a function f(t) at various levels of resolution where each level is expanded in terms of translated scaling functions (t)(2jt — k). As mentioned earlier in this book a sequence of embedded subspaces Vj is created 

It is here assumed that Pjf = f, i.e. the maximum signal resolution is contained in the original function. The wavelet coefficients correspond to the contribution from the projections onto the space of details between two 

An algorithm for the construction of more parsimonious regression and classification models can be found based on these formulas  

Perform multivariate analysis on Record prediction errors etc. end for 

Parsimony is achieved through optimisation of the resolution level of all the spectra in the data set X with respect to prediction ability of the dependent variable y. The process of changing the resolution level of a spectrum is shown in Fig. 4. 

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The simple multiscale approach can be seen as a subset of the optimal scale combination (OSC) method. In OSC the total number of possible scale combinations is generated. In the simple multiscale approach, scales were increased in a systematic fashion 0, 0 1, 0 1 2, 0 1 2 3, ... In OSC all possible combinations of the J + 1 scales are generated and tested. The combination that gives rise to a regression or classification model with the lowest number of coefficients and the lowest prediction error will be selected. Assume i is the number of scales to be selected from a total of K scales. There... 

A systematic exploration of different V parameters would be similar to the OSC approach (see above). However, here restrictions are placed upon the selection of v weights to produce distance matrices with increased resolution of the spectra (i.e. the simple multiscale approach) ... 

Cluster analysis. In this section, the application of the simple multiscale approach to cluster analysis is demonstrated. The masking method will also be used to localise important features. There are several possible cluster analysis algorithms, however only discriminant function analysis (DFA) will be used here. Before discussing the results from the simple multiscale analysis, this section will first present DFA and how it can applied to both unsupervised and supervised classification, followed by how the cluster properties S are measured at each resolution level. 

Measures of cluster structure. To test the simple multiscale approach to cluster analysis, the independent taxonomic information available for the different objects were used either directly or indirectly in the analyses. This is similar to the situation where a taxonomic expert is faced with a data set without the true classification information. In the process of determining interesting clusters the expert is expected to make use of his external knowledge in the assessment of the observed patterns. Thus, here the external class information is used to define a cluster. Having identified taxonomically relevant clusters, the next step is to measure how they relate to each other. The three properties measured for the two data set analysed were ... 

The OSC approach was also tested out here to complement the simple multiscale approach results. Since the number of scales tested for are 9 ( 1,2,..., 9 ), there are 511 different combinations of scales. 474 of these combinations resulted in perfect prediction in the calibration. What scales seem to dominate In order to answer this question, the relative distribution for the different scale combinations was performed. Scale combinations were grouped according to their error produced by DPLS and the distribution of which of the nine scales selected was recorded. Fig. 27 displays the results. 

Multiscale DPLS. Before using the VS-DPLS method on the Euhact data set, it is instructive to first use the simple multiresolution approach. The data set was analysed with DPLS for different scale reconstructions, (j = 0...9 ). At each reconstruction level, a full cross-validated DPLS run estimates the optimal number of factors and calculates a regression model. The regression model is subsequently applied to the unseen validation set. Fig. 25 shows the results from the calibration using cross-validation. We see that the calibration error goes to zero after 5 PLS factors. 

As a matter of fact, one may think of a multiscale approach coupling a macroscale simulation (preferably, a LES) of the whole vessel to meso or microscale simulations (DNS) of local processes. A rather simple, off-line way of doing this is to incorporate the effect of microscale phenomena into the full-scale simulation of the vessel by means of phenomenological coefficients derived from microscale simulations. Kandhai et al. (2003) demonstrated the power of this approach by deriving the functional dependence of the singleparticle drag force in a swarm of particles on volume fraction by means of DNS of the fluid flow through disordered arrays of spheres in a periodic box this functional dependence now can be used in full-scale simulations of any flow device. 

Multiscale regression or wavelet regression [60] is based on the simple idea that the mapping between the independent and dependent variables may involve different resolution levels. Most approaches to multivariate regression and classification only make use of the original data resolution in forming models. The multiscale approach enables the investigator to zoom in and out of the detail structures in the data. 

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