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Projection Pursuit

Projection Pursuit. PCA and FDA are linear, PP may be linear or non-linear. Find interesting “criterion of fit”, or “figure of merit” function, that allows for low-dim (usually 2D or 3D) projection. Projection Pursuit (PP). General transformation with parameters W.

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Projection Pursuit

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  1. Projection Pursuit

  2. PCA and FDA are linear, PP may be linear or non-linear. Find interesting “criterion of fit”, or “figure of merit” function, that allows for low-dim (usually 2D or 3D) projection. Projection Pursuit (PP) General transformation with parameters W. Index of “interestingness” Interesting indices may use a priori knowledge about the problem: 1. mean nearest neighbor distance – increase clustering of Y(j)2. maximize mutual information between classes and features 3. find projection that have non-Gaussian distributions. The last index does not use a priori knowledge; it leads to the Independent Component Analysis (ICA).ICA features are not only uncorrelated, but also independent.

  3. ICA is a special version of PP, recently very popular. Gaussian distributions of variable Y are characterized by 2 parameters: mean value: variance: These are the first 2 moments of distribution; all higher are 0 for G(Y). Kurtosis One simple measure of non-Gaussianity of projections is the 4-th moment (cumulant) of the distribution, called kurtosis, measures “skewedness” of the distribution. For E{Y}=0 kurtosis is: Super-Gaussian distribution: long tail, peak at zero, k4(y)>0, like binary image data. sub-Gaussian distribution is more flat and has k4(y)<0, like speech signal data.

  4. Features Yi, Yjare uncorrelated if covariance is diagonal, or: Correlation and independence Variables are statistically independent if their joint probability distribution is a product of probabilities for all variables: Uncorrelated features are orthogonal. Statistically independent features Yi, Yj for any functions give: This is much stronger condition than correlation; in particular the functions may be powers of variables; any non-Gaussian distribution after PCA transformation will still have correlated features.

  5. Example: PCA and PP based on maximal kurtosis: note nice separation of the blue class. PP/ICA example

  6. Some remarks • Many formulations of PP and ICA methods exist. • PP is used for data visualization and dimensionality reduction. • Nonlinear projections are frequently considered, but solutions are more numerically intensive. • PCA may also be viewed as PP, max (for standardized data): Index I(Y;W) is based here on maximum variance. Other components are found in the space orthogonal to W1TX Same index is used, with projection on space orthogonal to k-1PCs.

  7. How do we find multiple Projections • Statistical approach is complicated: • Perform a transformation on the data to eliminate structure in the already found direction • Then perform PP again • Neural Comp approach: Lateral Inhibition

  8. High Dimensional Data Dimension Reduction Visualisation Classification Analysis Feature Extraction

  9. Projection Pursuit what: An automated procedure that seeks interesting low dimensional projections of a high dimensional cloud by numerically maximizing an objective function or projection index. Huber, 1985

  10. Projection Pursuit why: Curse of dimensionality • Less Robustness • worse mean squared error • greater computational cost • slower convergence to limiting distributions • … • Required number of labelled samples increases with dimensionality.

  11. What is an interesting projection In general: the projection that reveals more information about the structure. In pattern recognition: a projection that maximises class separability in a low dimensional subspace.

  12. Projection Pursuit Dimensional Reduction Find lower-dimensional projections of a high-dimensional point cloud to facilitate classification. Exploratory Projection Pursuit Reduce the dimension of the problem to facilitate visualization.

  13. Projection Pursuit How many dimensions to use • for visualization • for classification/analysis Which Projection Index to use • measure of variation (Principal Components) • departure from normality (negative entropy) • class separability(distance, Bhattacharyya, Mahalanobis, ...) • …

  14. Projection Pursuit Which optimization method to choose We are trying to find the global optimum among local ones • hill climbing methods (simulated annealing) • regular optimization routines with random starting points.

  15. Timetable for Dimensionality reduction • Begin 16 April 1998 • Report on the state-of-the-art. 1 June 1998 • Begin software implementation 15 June 1998 • Prototype software presentation 1 November 1998

  16. ICA demos • ICA has many applications in signal and image analysis. • Finding independent signal sources allows for separation of signals from different sources, removal of noise or artifacts. Observations X are a linear mixture W of unknown sources Y Both W and Y are unknown! This is a blind separation problem. How can they be found? If Y are Independent Components and W linear mixing the problem is similar to FDA or PCA, only the criterion function is different. Play with ICALab PCA/ICA Matlab software for signal/image analysis:http://www.bsp.brain.riken.go.jp/page7.html

  17. ICA demo: images & audio Example from Cichocki’s lab, http://www.bsp.brain.riken.go.jp/page7.html X space for images: take intensity of all pixels  one vector per image, or take smaller patches (ex: 64x64), increasing # vectors • 5images:originals,mixed, convergence ofICA iterations X space for signals: sample the signal for some time Dt • 10 songs:mixed samplesandseparated samples

  18. Self-organization PCA, FDA, ICA, PP are all inspired by statistics, although some neural-inspired methods have been proposed to find interesting solutions, especially for their non-linear versions. • Brains learn to discover the structure of signals: visual, tactile, olfactory, auditory (speech and sounds). • This is a good example of unsupervised learning: spontaneous development of feature detectors, compressing internal information that is needed to model environmental states (inputs). • Some simple stimuli lead to complex behavioral patterns in animals; brains use specialized microcircuits to derive vital information from signals – for example, amygdala nuclei in rats sensitive to ultrasound signals signifying “cat around”.

  19. Models of self-organizaiton SOM or SOFM (Self-Organized Feature Mapping) – self-organizing feature map, one of the simplest models. How can such maps develop spontaneously? Local neural connections: neuronsinteract strongly with those nearby, but weakly with those that are far (in addition inhibiting some intermediate neurons). History: von der Malsburg and Willshaw (1976), competitive learning, Hebb mechanisms, „Mexican hat” interactions, models of visual systems. Amari (1980) – models of continuous neural tissue. Kohonen (1981) - simplification, no inhibition; leaving two essential factors: competition and cooperation.

  20. Computational Intelligence: Methods and Applications Lecture 8 Projection Pursuit &Independent Component Analysis Włodzisław Duch SCE, NTU, Singapore Google: Duch

  21. Computational Intelligence: Methods and Applications Lecture 6Principal Component Analysis. Włodzisław Duch SCE, NTU, Singapore http://www.ntu.edu.sg/home/aswduch

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