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Dimension reduction (2) Projection pursuit ICA NCA Partial Least Squares Blais. “The role of the environment in synaptic plasticity…..” (1998) Liao et al. PNAS. (2003) Barker & Raynes. J. Chemometrics 2013.

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Projection pursuit A very broad term: finding the most “interesting” direction of projection. How the projection is done depends on the definition of “interesting”. If it is maximal variation, then PP leads to PCA. In a narrower sense: Finding non-Gaussian projections. For most high-dimensional clouds, most low-dimensional projections are close to Gaussian important information in the data is in the directions for which the projected data is far from Gaussian.

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Projection pursuit It boils down to objective functions – each kind of “interesting” has its own objective function(s) to maximize.

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PCA Projection pursuit with multi-modality as objective. Projection pursuit

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One objective function to measure multi-modality: It uses the first three moments of the distribution. It can help finding clusters through visualization. To find w, the function is maximized over w by gradient ascent:

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Projection pursuit Can think of PCA as a case of PP, with the objective function: For other PC directions, find projection onto space orthogonal to the previously found PCs.

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Projection pursuit Some other objective functions (y is the RV generated by projection w’x) The Kurtosis as defined here has value 0 for normal distribution. Higher Kertusis: peaked and fat-tailed. m/link/kurtosis.htm

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ICA finds a unique solution by requiring the factors to be statistically independent, rather than just uncorrelated. Lack of correlation only determines the second-degree cross- moment, while statistical independence means for any functions g1() and g2(), For multivariate Gaussian, uncorrelatedness = independence Independent component analysis Again, another view of dimension reduction is factorization into latent variables.

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ICA Multivariate Gaussian is determined by second moments alone. Thus if the true hidden factors are Gaussian, then still they can be determined only up to a rotation. In ICA, the latent variables are assumed to be independent and non-Gaussian. The matrix A must have full column rank.

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Independent component analysis ICA is a special case of PP. The key is again for y being non-Gaussian. Several ways to measure non-Gaussianity: (1)Kurtotis (zero for Gaussian RV, sensitive to outliers) (2)Entropy (Gaussian RV has the largest entropy given the first and second moments) (3) Negentropy: y gauss is a Gaussian RV with the same covariance matrix as y.

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ICA To measure statistical independence, use mutual information, Sum of marginal entropies minus the overall entropy Non-negative ； Zero if and only if independent.

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ICA The computation: There is no closed form solution, hence gradient descent is used. Approximation to negentropy (for less intensive computation and better resistance to outliers) Two commonly used G(): v is standard gaussian. G() is some nonquadratic function. When G(x)=x 4 this is Kurtosis.

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ICA FastICA: Center the X vectors to mean zero. Whiten the X vectors such that E(xx’)=I. This is done through eigen value decomposition. Initialize the weight vector w Iterate: w + =E{xg(w T x)}-E{g’(w T x)}w w=w + /||w + || until convergence g() is the derivative of the non-quadratic function

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Figure 14.28: Mixtures of independent uniform random variables. The upper left panel shows 500 realizations from the two independent uniform sources, the upper-right panel their mixed versions. The lower two panels show the PCA and ICA solutions, respectively. ICA

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Other than dimension reduction, hidden factor model, there is another way to understand a model like this: It can be understood as explaining the data by a bipartite network --- a control layer and an output layer. Unlike PCA and ICA, NCA doesn’t assume a fully linked loading matrix. Rather, the matrix is sparse. The non-zero locations are pre-determined by biological knowledge about regulatory networks. For example, Network component analysis

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Motivation: Instead of blindly search for lower dimensional space, a priori information is incorporated into the loading matrix.

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NCA X NxP =A NxK P KxP +E NxP Conditions for the solution to be unique: (1)A is full column rank; (2)When a column of A is removed, together with all rows corresponding to non-zero values in the column, the remaining matrix is still full column rank; (3)P must have full row rank

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NCA Fig. 2. A completely identifiable network (a) and an unidentifiable network (b). Although the two initial [A] matrices describing the network matrices have an identical number of constraints (zero entries), the network in b does not satisfy the identifiability conditions because of the connectivity pattern of R 3. The edges in red are the differences between the two networks.

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NCA Notice that both A and P are to be estimated. Then the criteria of identifiability is in fact untestable. The compute NCA, minimize the square loss function: Z 0 is the topology constraint matrix – i.e. which position of A is non-zero. It is based on prior knowledge. It is the network connectivity matrix.

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NCA Solving NCA: This is a linear decomposition system which has the bi- convex property. It is solved by iteratively solving for A and P while fixing the other one. Both steps use least squares. Convergence is judged by the total least-square error. The total error is non-increasing in each step. Optimality is guaranteed if the three conditions for identifiability are satisfied. Otherwise a local optimum may be found.

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PLS Finding latent factors in X that can predict Y. X is multi-dimensional, Y can be either a random variable or a random vector. The model will look like: where T j is a linear combination of X PLS is suitable in handling p>>N situation.

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PLS Data: Goal:

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PLS Solution: a k+1 is the (k+1) th eigen vector of Alternatively, The PLS components minimize Can be solved by iterative regression.

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PLS Example: PLS v.s. PCA in regression: Y is related to X 1

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