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Dimension reduction (1). Overview PCA Factor Analysis EDR space SIR. References: Applied Multivariate Analysis. http://www.stat.ucla.edu/~kcli/sir-PHD.pdf. Overview. The purpose of dimension reduction: Data simplification Data visualization

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## Dimension reduction (1)

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**Dimension reduction (1)**Overview PCA Factor Analysis EDR space SIR References: Applied Multivariate Analysis. http://www.stat.ucla.edu/~kcli/sir-PHD.pdf**Overview**• The purpose of dimension reduction: • Data simplification • Data visualization • Reduce noise (if we can assume only the dominating dimensions are signals) • Variable selection for prediction**Overview**An analogy:**PCA**• Explain the variance-covariance structure among a set of random variables by a few linear combinations of the variables; • Does not require normality!**Reminder of some results for random vectors**Proof of the first (and second) point of the previous slide.**PCA**The eigen values are the variance components: Proportion of total variance explained by the kth PC:**PCA**The geometrical interpretation of PCA:**PCA**PCA using the correlation matrix, instead of the covariance matrix? This is equivalent to first standardizing all X vectors.**PCA**Using the correlation matrix avoids the domination from one X variable due to scaling (unit changes), for example using inch instead of foot. Example:**PCA**Selecting the number of components? Based on eigen values (% variation explained). Assumption: the small amount of variation explained by low-rank PCs is noise.**Factor Analysis**If we take the first several PCs that explain most of the variation in the data, we have one form of factor model. L: loading matrix F: unobserved random vector (latent variables). ε: unobserved random vector (noise)**Factor Analysis**Orthogonal factor model assumes no correlation between the factor RVs. is a diagonal matrix**Factor Analysis**Rotations in the m-dimensional subspace defined by the factors make the solution non-unique: PCA is one unique solution, as the vectors are sequentially selected. Maximum likelihood estimator is another solution:**Factor Analysis**As we said, rotations within the m-dimensional subspace doesn’t change the overall amount of variation explained. Do rotation to make the results more interpretable:**Factor Analysis**Varimax criterion: Find T such that is maximized. V is proportional to the summation of the variance of the squared loadings. Maximizing V makes the squared loadings as spread out as possible --- some are real small, and some are real big.**Factor Analysis**Orthogonal simple factor rotation: Rotate the orthogonal factors around the origin until the system is maximally aligned with the separate clusters of variables. Oblique Simple Structure Rotation: Allow the factors to become correlated. Each factor is rotated individually to fit a cluster.**MDS**Multidimensional scaling is a dimension reduction procedure that maps the distances between observations to a lower dimensional space. Minimize this objective function: D: distance in the original space d: distance in the reduced dimension space. Numerical method is used for the minimization.**EDR space**Now we start talking about regression. The data is {xi, yi} Is dimension reduction on X matrix alone helpful here? Possibly, if the dimension reduction preserves the essential structure about Y|X. This is suspicious. Effective Dimension Reduction --- reduce the dimension of X without losing information which is essential to predict Y.**EDR space**The model: Y is predicted by a set of linear combinations of X. If g() is known, this is not very different from a generalized linear model. For dimension reduction purpose, is there a scheme which can work on almost any g(), without knowledge of its actual form?**EDR space**The general model encompasses many models as special cases:**EDR space**Under this general model, The space B generated by β1, β2, ……, βK is called the e.d.r. space. Reducing to this sub-space causes no loss of information regarding predicting Y. Similar to factor analysis, the subspace B is identifiable, but the vectors aren’t. Any non-zero vector in the e.d.r. space is called an e.d.r. direction.**EDR space**This equation assumes almost the weakest form, to reflect the hope that a low-dimensional projection of a high-dimensional regresservariable contains most of the information that can be gathered from a sample of modest size. It doesn’t impose any structure on how the projected regresservariables effect the output variable. Most regression models assume K=1, plus additional structures on g().**EDR space**The philosophical point of Sliced Inverse Regression: the estimation of the projection directions can be a more important statistical issue than the estimation of the structure of g() itself. After finding a good e.d.r. space, we can project data to this smaller space. Then we are in a better position to identify what should be pursued further : model building, response surface estimation, cluster analysis, heteroscedasticity analysis, variable selection, ……**SIR**Sliced Inverse Regression. In regular regression, our interest is the conditional density h(Y|X). Most important is E(Y|x) and var(Y|x). SIR treats Y as independent variable and X as the dependent variable. Given Y=y, what values will X take? This takes us from a p-dimensional problem (subject to curse of dimensionality) back to a 1-dimensional curve-fitting problem: E(xi|y), i=1,…, p**SIR**covariance matrix for the slice means of x, weighted by the slice sizes Find the SIR directions by conducting the eigenvalue decomposition of with respect to : sample covariance for xi ’s**SIR**An example response surface found by SIR.**SIR and LDA**Reminder: Fisher’s linear discriminant analysis seeks a projection direction that maximized class separation. When the underlying distributions are Gaussian, it agrees with the Bayes decision rule. It seeks to maximize: Between-group variance: Within-group variance:**SIR and LDA**The solution is the first eigen vector in this eigen value decomposition: If we let , the LDA agrees with SIR up to a scaling.**Multi-class LDA**Structure-preserving dimension reduction in classification. Within-class scatter: Between-class scatter: Mixture scatter: a: observations, c: class centers Kim et al. Pattern Recognition 2007, 40:2939**Multi-class LDA**Maximize: The solution come from the eigen value/vectors of When we have N<<p, Sw is singular. Let Kim et al. Pattern Recognition 2007, 40:2939**Multi-class LDA**Kim et al. Pattern Recognition 2007, 40:2939

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