Subspace sampling

Simulate Systems with Barriers Subspace Sampling. 

Figure 9.1 Inverse partial melting problem in the three-dimensional space of elements 1, 2, 3 when the source is known. Projection of the source onto the sample subspace provides the mass-fraction of each phase of the molten source. If one phase is at the origin (sterile phase), every representative point can be shifted by a constant vector.

Therefore, we want to decide which direction, among all possible choices, is common to all sample subspaces, or, at least, which direction represents the best zone of the sample subspaces in a least-square sense. Since a direction can be completely described by its unit vector, we can restrict the solution set to the surface of the unit sphere centered at the origin. Let us call y the solution of unitary modulus and its projection onto the fcth sample subspace (k = 1,..., s) represented by the matrix Ak. It is a simple matter to show that... 

Finding the least-square solution reduces to minimizing the sum S of squared deviations yk — j) between the estimated source solution and its projection onto each sample subspace. Thus, finding the minimum of... 

Proportional focusing of homogeneous subareas or subspaces is necessary if the whole sampling area has high variability as a result of the environment within the features of interest. 

Prior to the actual classification, the FLDC performs a linear mapping to a lower dimensional subspace optimised for class separability, based on the between-class scatter and the within-class scatter of the training set. In classification, each sample is assigned to the class giving the highest log-likelihood using a linear classifier. 

Let LkJk denote the diagonal matrix that contains the k eigenvalues I of the MCD scatter matrix, sorted from largest to smallest. Thus /, > /2 >. .. > Ik. The score distance of the ith sample measures the robust distance of its projection to the center of all the projected observations. Hence, it is measured within the PCA subspace, where due to the knowledge of the eigenvalues, we have information about the covariance structure of the scores. Consequently, the score distance is defined as in Equation 6.6 ... 

Orthogonal outliers have a large orthogonal distance, but a small score distance, as, for example, case 5. They cannot be distinguished from the regular observations once they are projected onto the PCA subspace, but they lie far from this subspace. Consequently, it would be dangerous to replace that sample with its projected value, as its outlyingness would not be visible anymore. 

Figure 9. Decay of the number of surviving trajectories of the M3 against isomerization (see the text for the precise definition of isomerization ). The lower curve represents the dynamics constrained to the Eckart subspace under no gauge field, Eq. (35), while the upper one indicates the true dynamics under the full gauge field, Eq. (37). All of the initial conditions are randomly sampled in configuration space and are taken to be exactly the same for the two sets of dynamics. Number of the sample trajectories is 5000 and their internal energy is set to E = -1.68.

This procedure was used mainly to aehieve a reduction of dimensionality, i.e., to fit a A -dimensional subspace to the original / -variate observations p k). The statistics used to summarize the most important results was the percent of the total variation explained by the first k (usually two or three) components. In the case at hand the interpretation of PCA results was based on the diagrams of coefficients of variables (total concentrations of metals in the samples) and the scatter plot of samples (the stations separated in coastal, intermediate and offshore stations). 

As a result, a trajectory generated by the dynamics of (58) will not sample the entire phase space, but instead will sample a subspace of the entire phase space surface determined by the intersection of the hypersurfaces ylfc(x) = Cfc, where Ck is a set of constants. The microcanonical distribution function, that is generated by these systems, can be constructed from a product of 5-functions that represent these conservation laws ... 

The data used in subspace state-space model development consists of the time series data of output and input variables. For illustration, assume a case with only output data and the objective is to build a model of the form Eq. 4.62. Since the whole data set is already known, it can be partitioned as past and future with respect to any sampling time. Defining a past data window of length K and a future data window of length J that are shifted from the beginning to the end of the data set, stacked vectors of data are formed. The Hankel matrix (Eq. 4.64) is used to develop subspace... 

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