Factor and Component Analysis esp. Principal Component Analysis (PCA)

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2 Factor and Component Analysis esp. Principal Component Analysis (PCA)

3 We study phenomena that can not be directly observed –ego, personality, intelligence in psychology –Underlying factors that govern the observed data We want to identify and operate with underlying latent factors rather than the observed data –E.g. topics in news articles –Transcription factors in genomics We want to discover and exploit hidden relationships –“beautiful car” and “gorgeous automobile” are closely related –So are “driver” and “automobile” –But does your search engine know this? –Reduces noise and error in results Why Factor or Component Analysis?

4 We have too many observations and dimensions –To reason about or obtain insights from –To visualize –Too much noise in the data –Need to “reduce” them to a smaller set of factors –Better representation of data without losing much information –Can build more effective data analyses on the reduced- dimensional space: classification, clustering, pattern recognition Combinations of observed variables may be more effective bases for insights, even if physical meaning is obscure Why Factor or Component Analysis?

5 Discover a new set of factors/dimensions/axes against which to represent, describe or evaluate the data –For more effective reasoning, insights, or better visualization –Reduce noise in the data –Typically a smaller set of factors: dimension reduction –Better representation of data without losing much information –Can build more effective data analyses on the reduced- dimensional space: classification, clustering, pattern recognition Factors are combinations of observed variables –May be more effective bases for insights, even if physical meaning is obscure –Observed data are described in terms of these factors rather than in terms of original variables/dimensions Factor or Component Analysis

6 Basic Concept Areas of variance in data are where items can be best discriminated and key underlying phenomena observed –Areas of greatest “signal” in the data If two items or dimensions are highly correlated or dependent –They are likely to represent highly related phenomena –If they tell us about the same underlying variance in the data, combining them to form a single measure is reasonable Parsimony Reduction in Error So we want to combine related variables, and focus on uncorrelated or independent ones, especially those along which the observations have high variance We want a smaller set of variables that explain most of the variance in the original data, in more compact and insightful form

7 Basic Concept What if the dependences and correlations are not so strong or direct? And suppose you have 3 variables, or 4, or 5, or 10000? Look for the phenomena underlying the observed covariance/co-dependence in a set of variables –Once again, phenomena that are uncorrelated or independent, and especially those along which the data show high variance These phenomena are called “factors” or “principal components” or “independent components,” depending on the methods used –Factor analysis: based on variance/covariance/correlation –Independent Component Analysis: based on independence

8 Principal Component Analysis Most common form of factor analysis The new variables/dimensions –Are linear combinations of the original ones –Are uncorrelated with one another Orthogonal in original dimension space –Capture as much of the original variance in the data as possible –Are called Principal Components

9 Some Simple Demos http://www.cs.mcgill.ca/~sqrt/dimr/dimre duction.htmlhttp://www.cs.mcgill.ca/~sqrt/dimr/dimre duction.html

10 What are the new axes? Original Variable A Original Variable B PC 1 PC 2 Orthogonal directions of greatest variance in data Projections along PC1 discriminate the data most along any one axis

11 Principal Components First principal component is the direction of greatest variability (covariance) in the data Second is the next orthogonal (uncorrelated) direction of greatest variability –So first remove all the variability along the first component, and then find the next direction of greatest variability And so on …

12 Principal Components Analysis (PCA) Principle –Linear projection method to reduce the number of parameters –Transfer a set of correlated variables into a new set of uncorrelated variables –Map the data into a space of lower dimensionality –Form of unsupervised learning Properties –It can be viewed as a rotation of the existing axes to new positions in the space defined by original variables –New axes are orthogonal and represent the directions with maximum variability

13 Computing the Components Data points are vectors in a multidimensional space Projection of vector x onto an axis (dimension) u is u.x Direction of greatest variability is that in which the average square of the projection is greatest –I.e. u such that E((u.x) 2 ) over all x is maximized –(we subtract the mean along each dimension, and center the original axis system at the centroid of all data points, for simplicity) –This direction of u is the direction of the first Principal Component

14 Computing the Components E((u.x) 2 ) = E ((u.x) (u.x) T ) = E (u.x.x T.u T ) The matrix C = x.x T contains the correlations (similarities) of the original axes based on how the data values project onto them So we are looking for w that maximizes uCu T, subject to u being unit-length It is maximized when w is the principal eigenvector of the matrix C, in which case –uCu T = u u T = if u is unit-length, where is the principal eigenvalue of the correlation matrix C –The eigenvalue denotes the amount of variability captured along that dimension

15 Why the Eigenvectors? Maximise u T xx T u s.t u T u = 1 Construct Langrangian u T xx T u – λ u T u Vector of partial derivatives set to zero xx T u – λ u = (xx T – λI ) u = 0 As u ≠ 0 then u must be an eigenvector of xx T with eigenvalue λ

16 Singular Value Decomposition The first root is called the prinicipal eigenvalue which has an associated orthonormal (u T u = 1) eigenvector u Subsequent roots are ordered such that λ 1 > λ 2 >… > λ M with rank(D) non-zero values. Eigenvectors form an orthonormal basis i.e. u i T u j = δ ij The eigenvalue decomposition of xx T = UΣU T where U = [u 1, u 2, …, u M ] and Σ = diag[λ 1, λ 2, …, λ M ] Similarly the eigenvalue decomposition of x T x = VΣV T The SVD is closely related to the above x=U Σ 1/2 V T The left eigenvectors U, right eigenvectors V, singular values = square root of eigenvalues.

17 Computing the Components Similarly for the next axis, etc. So, the new axes are the eigenvectors of the matrix of correlations of the original variables, which captures the similarities of the original variables based on how data samples project to them Geometrically: centering followed by rotation –Linear transformation

18 PCs, Variance and Least-Squares The first PC retains the greatest amount of variation in the sample The k th PC retains the k th greatest fraction of the variation in the sample The k th largest eigenvalue of the correlation matrix C is the variance in the sample along the k th PC The least-squares view: PCs are a series of linear least squares fits to a sample, each orthogonal to all previous ones

19 How Many PCs? For n original dimensions, correlation matrix is nxn, and has up to n eigenvectors. So n PCs. Where does dimensionality reduction come from?

20 Dimensionality Reduction Can ignore the components of lesser significance. You do lose some information, but if the eigenvalues are small, you don’t lose much –n dimensions in original data –calculate n eigenvectors and eigenvalues –choose only the first p eigenvectors, based on their eigenvalues –final data set has only p dimensions

21 Eigenvectors of a Correlation Matrix

22 Computing and Using PCA: Text Search Example

23 PCA/FA Principal Components Analysis –Extracts all the factor underlying a set of variables –The number of factors = the number of variables –Completely explains the variance in each variable Parsimony?!?!?! Factor Analysis –Also known as Principal Axis Analysis –Analyses only the shared variance NO ERROR!

24 Vector Space Model for Documents Documents are vectors in multi-dim Euclidean space –Each term is a dimension in the space: LOTS of them –Coordinate of doc d in dimension t is prop. to TF(d,t) TF(d,t) is term frequency of t in document d Term-document matrix, with entries being normalized TF values –t*d matrix, for t terms and d documents Queries are also vectors in the same space –Treated just like documents Rank documents based on similarity of their vectors with the query vector –Using cosine distance of vectors

25 Problems Looks for literal term matches Terms in queries (esp short ones) don’t always capture user’s information need well Problems: –Synonymy: other words with the same meaning Car and automobile –Polysemy: the same word having other meanings Apple (fruit and company) What if we could match against ‘concepts’, that represent related words, rather than words themselves

26 Example of Problems -- Relevant docs may not have the query terms  but may have many “related” terms -- Irrelevant docs may have the query terms  but may not have any “related” terms

27 Latent Semantic Indexing (LSI) Uses statistically derived conceptual indices instead of individual words for retrieval Assumes that there is some underlying or latent structure in word usage that is obscured by variability in word choice Key idea: instead of representing documents and queries as vectors in a t-dim space of terms –Represent them (and terms themselves) as vectors in a lower-dimensional space whose axes are concepts that effectively group together similar words –These axes are the Principal Components from PCA

28 Example Suppose we have keywords –Car, automobile, driver, elephant We want queries on car to also get docs about drivers and automobiles, but not about elephants –What if we could discover that the cars, automobiles and drivers axes are strongly correlated, but elephants is not –How? Via correlations observed through documents –If docs A & B don’t share any words with each other, but both share lots of words with doc C, then A & B will be considered similar –E.g A has cars and drivers, B has automobiles and drivers When you scrunch down dimensions, small differences (noise) gets glossed over, and you get desired behavior

29 LSI and Our Earlier Example -- Relevant docs may not have the query terms  but may have many “related” terms -- Irrelevant docs may have the query terms  but may not have any “related” terms

30 LSI: Satisfying a query Take the vector representation of the query in the original term space and transform it to new space –Turns out d q = x q t T  -1 –places the query pseudo-doc at the centroid of its corresponding terms’ locations in the new space Similarity with existing docs computed by taking dot products with the rows of the matrix D  2 D t

31 Computing and Using LSI (PCA) Definitions –Let t be the total number of index terms –Let N be the number of documents –Let (M ij ) be a term-document matrix with t rows and N columns Entries are TF-based weights

32 Latent Semantic Indexing The matrix (M ij ) can be decomposed into 3 matrices (SVD) as follows: (M ij ) = (U) (S) (V) t (U) is the matrix of eigenvectors derived from (M)(M) t (V) t is the matrix of eigenvectors derived from (M) t (M) (S) is an r x r diagonal matrix of singular values –r = min(t,N) that is, the rank of (Mij) –Singular values are the positive square roots of the eigen values of (M)(M) t (also (M) t (M))

33 Example term ch2ch3 ch4 ch5 ch6 ch7 ch8 ch9 controllability 1 10 0 1001 observability 1 0 0 01 101 realization 1 01 0 1 010 feedback 0 10 0 0 100 controller 0 1 0 0 1 100 observer 0 1 1 0 1 100 transfer function 0 0 0 0 1 10 0 polynomial 0 0 0 0 1 010 matrices 0 0 0 0 1 011 U (9x7) = 0.3996 -0.1037 0.5606 -0.3717 -0.3919 -0.3482 0.1029 0.4180 -0.0641 0.4878 0.1566 0.5771 0.1981 -0.1094 0.3464 -0.4422 -0.3997 -0.5142 0.2787 0.0102 -0.2857 0.1888 0.4615 0.0049 -0.0279 -0.2087 0.4193 -0.6629 0.3602 0.3776 -0.0914 0.1596 -0.2045 -0.3701 -0.1023 0.4075 0.3622 -0.3657 -0.2684 -0.0174 0.2711 0.5676 0.2750 0.1667 -0.1303 0.4376 0.3844 -0.3066 0.1230 0.2259 -0.3096 -0.3579 0.3127 -0.2406 -0.3122 -0.2611 0.2958 -0.4232 0.0277 0.4305 -0.3800 0.5114 0.2010 S (7x7) = 3.9901 0 0 0 0 0 0 0 2.2813 0 0 0 0 0 0 0 1.6705 0 0 0 0 0 0 0 1.3522 0 0 0 0 0 0 0 1.1818 0 0 0 0 0 0 0 0.6623 0 0 0 0 0 0 0 0.6487 V (7x8) = 0.2917 -0.2674 0.3883 -0.5393 0.3926 -0.2112 -0.4505 0.3399 0.4811 0.0649 -0.3760 -0.6959 -0.0421 -0.1462 0.1889 -0.0351 -0.4582 -0.5788 0.2211 0.4247 0.4346 -0.0000 -0.0000 -0.0000 -0.0000 0.0000 -0.0000 0.0000 0.6838 -0.1913 -0.1609 0.2535 0.0050 -0.5229 0.3636 0.4134 0.5716 -0.0566 0.3383 0.4493 0.3198 -0.2839 0.2176 -0.5151 -0.4369 0.1694 -0.2893 0.3161 -0.5330 0.2791 -0.2591 0.6442 0.1593 -0.1648 0.5455 0.2998 This happens to be a rank-7 matrix -so only 7 dimensions required Singular values = Sqrt of Eigen values of AA T T

34 Dimension Reduction in LSI The key idea is to map documents and queries into a lower dimensional space (i.e., composed of higher level concepts which are in fewer number than the index terms) Retrieval in this reduced concept space might be superior to retrieval in the space of index terms

35 Dimension Reduction in LSI In matrix (S), select only k largest values Keep corresponding columns in (U) and (V) t The resultant matrix (M) k is given by –(M) k = (U) k (S) k (V) t k –where k, k < r, is the dimensionality of the concept space The parameter k should be –large enough to allow fitting the characteristics of the data –small enough to filter out the non-relevant representational detail

36 PCs can be viewed as Topics In the sense of having to find quantities that are not observable directly Similarly, transcription factors in biology, as unobservable causal bridges between experimental conditions and gene expression

37 Computing and Using LSI Singular Value Decomposition (SVD): Convert term-document matrix into 3matrices U, S and V  Reduce Dimensionality: Throw out low-order rows and columns Recreate Matrix: Multiply to produce approximate term- document matrix. Use new matrix to process queries OR, better, map query to reduced space MU S VtVt UkUk SkSk VktVkt

38 Following the Example term ch2ch3 ch4 ch5 ch6 ch7 ch8 ch9 controllability 1 10 0 1001 observability 1 0 0 01 101 realization 1 01 0 1 010 feedback 0 10 0 0 100 controller 0 1 0 0 1 100 observer 0 1 1 0 1 100 transfer function 0 0 0 0 1 10 0 polynomial 0 0 0 0 1 010 matrices 0 0 0 0 1 011 U (9x7) = 0.3996 -0.1037 0.5606 -0.3717 -0.3919 -0.3482 0.1029 0.4180 -0.0641 0.4878 0.1566 0.5771 0.1981 -0.1094 0.3464 -0.4422 -0.3997 -0.5142 0.2787 0.0102 -0.2857 0.1888 0.4615 0.0049 -0.0279 -0.2087 0.4193 -0.6629 0.3602 0.3776 -0.0914 0.1596 -0.2045 -0.3701 -0.1023 0.4075 0.3622 -0.3657 -0.2684 -0.0174 0.2711 0.5676 0.2750 0.1667 -0.1303 0.4376 0.3844 -0.3066 0.1230 0.2259 -0.3096 -0.3579 0.3127 -0.2406 -0.3122 -0.2611 0.2958 -0.4232 0.0277 0.4305 -0.3800 0.5114 0.2010 S (7x7) = 3.9901 0 0 0 0 0 0 0 2.2813 0 0 0 0 0 0 0 1.6705 0 0 0 0 0 0 0 1.3522 0 0 0 0 0 0 0 1.1818 0 0 0 0 0 0 0 0.6623 0 0 0 0 0 0 0 0.6487 V (7x8) = 0.2917 -0.2674 0.3883 -0.5393 0.3926 -0.2112 -0.4505 0.3399 0.4811 0.0649 -0.3760 -0.6959 -0.0421 -0.1462 0.1889 -0.0351 -0.4582 -0.5788 0.2211 0.4247 0.4346 -0.0000 -0.0000 -0.0000 -0.0000 0.0000 -0.0000 0.0000 0.6838 -0.1913 -0.1609 0.2535 0.0050 -0.5229 0.3636 0.4134 0.5716 -0.0566 0.3383 0.4493 0.3198 -0.2839 0.2176 -0.5151 -0.4369 0.1694 -0.2893 0.3161 -0.5330 0.2791 -0.2591 0.6442 0.1593 -0.1648 0.5455 0.2998 This happens to be a rank-7 matrix -so only 7 dimensions required Singular values = Sqrt of Eigen values of AA T T

39 U2 (9x2) = 0.3996 -0.1037 0.4180 -0.0641 0.3464 -0.4422 0.1888 0.4615 0.3602 0.3776 0.4075 0.3622 0.2750 0.1667 0.2259 -0.3096 0.2958 -0.4232 S2 (2x2) = 3.9901 0 0 2.2813 V2 (8x2) = 0.2917 -0.2674 0.3399 0.4811 0.1889 -0.0351 -0.0000 -0.0000 0.6838 -0.1913 0.4134 0.5716 0.2176 -0.5151 0.2791 -0.2591 U (9x7) = 0.3996 -0.1037 0.5606 -0.3717 -0.3919 -0.3482 0.1029 0.4180 -0.0641 0.4878 0.1566 0.5771 0.1981 -0.1094 0.3464 -0.4422 -0.3997 -0.5142 0.2787 0.0102 -0.2857 0.1888 0.4615 0.0049 -0.0279 -0.2087 0.4193 -0.6629 0.3602 0.3776 -0.0914 0.1596 -0.2045 -0.3701 -0.1023 0.4075 0.3622 -0.3657 -0.2684 -0.0174 0.2711 0.5676 0.2750 0.1667 -0.1303 0.4376 0.3844 -0.3066 0.1230 0.2259 -0.3096 -0.3579 0.3127 -0.2406 -0.3122 -0.2611 0.2958 -0.4232 0.0277 0.4305 -0.3800 0.5114 0.2010 S (7x7) = 3.9901 0 0 0 0 0 0 0 2.2813 0 0 0 0 0 0 0 1.6705 0 0 0 0 0 0 0 1.3522 0 0 0 0 0 0 0 1.1818 0 0 0 0 0 0 0 0.6623 0 0 0 0 0 0 0 0.6487 V (7x8) = 0.2917 -0.2674 0.3883 -0.5393 0.3926 -0.2112 -0.4505 0.3399 0.4811 0.0649 -0.3760 -0.6959 -0.0421 -0.1462 0.1889 -0.0351 -0.4582 -0.5788 0.2211 0.4247 0.4346 -0.0000 -0.0000 -0.0000 -0.0000 0.0000 -0.0000 0.0000 0.6838 -0.1913 -0.1609 0.2535 0.0050 -0.5229 0.3636 0.4134 0.5716 -0.0566 0.3383 0.4493 0.3198 -0.2839 0.2176 -0.5151 -0.4369 0.1694 -0.2893 0.3161 -0.5330 0.2791 -0.2591 0.6442 0.1593 -0.1648 0.5455 0.2998 U2*S2*V2 will be a 9x8 matrix That approximates original matrix T Formally, this will be the rank-k (2) matrix that is closest to M in the matrix norm sense

40 K=2 K=6 One component ignored 5 components ignored U6S6V6TU6S6V6T U2S2V2TU2S2V2T USV T U4S4V4TU4S4V4T K=4 =U 7 S 7 V 7 T 3 components ignored What should be the value of k?

41 Querying To query for feedback controller, the query vector would be q = [0 0 0 1 1 0 0 0 0]' (' indicates transpose), Let q be the query vector. Then the document-space vector corresponding to q is given by: q'*U2*inv(S2) = Dq Point at the centroid of the query terms’ poisitions in the new space. For the feedback controller query vector, the result is: Dq = 0.1376 0.3678 To find the best document match, we compare the Dq vector against all the document vectors in the 2- dimensional V2 space. The document vector that is nearest in direction to Dq is the best match. The cosine values for the eight document vectors and the query vector are: -0.3747 0.9671 0.1735 -0.9413 0.0851 0.9642 -0.7265 -0.3805 -0.37 0.967 0.173 -0.94 0.08 0.96 -0.72 -0.38

42 Medline data

43 Within.40 threshold K is the number of singular values used

44 What LSI can do LSI analysis effectively does –Dimensionality reduction –Noise reduction –Exploitation of redundant data –Correlation analysis and Query expansion (with related words) Some of the individual effects can be achieved with simpler techniques (e.g. thesaurus construction). LSI does them together. LSI handles synonymy well, not so much polysemy Challenge: SVD is complex to compute (O(n 3 )) –Needs to be updated as new documents are found/updated

45 SVD Properties There is an implicit assumption that the observed data distribution is multivariate Gaussian Can consider as a probabilistic generative model – latent variables are Gaussian – sub-optimal in likelihood terms for non-Gaussian distribution Employed in signal processing for noise filtering – dominant subspace contains majority of information bearing part of signal Similar rationale when applying SVD to LSI

46 LSI Conclusions – SVD defined basis provide P/R improvements over term matching Interpretation difficult Optimal dimension – open question Variable performance on LARGE collections Supercomputing muscle required – Probabilistic approaches provide improvements over SVD Clear interpretation of decomposition Optimal dimension – open question High variability of results due to nonlinear optimisation over HUGE parameter space – Improvements marginal in relation to cost

47 Factor Analysis (e.g. PCA) is not the most sophisticated dimensionality reduction technique Dimensionality reduction is a useful technique for any classification/regression problem –Text retrieval can be seen as a classification problem There are many other dimensionality reduction techniques –Neural nets, support vector machines etc. Compared to them, LSI is limited in the sense that it is a “linear” technique –It cannot capture non-linear dependencies between original dimensions (e.g. data that are circularly distributed).

48 Limitations of PCA Relevant Component Analysis (RCA) Fisher Discriminant analysis (FDA) Are the maximal variance dimensions the relevant dimensions for preservation?

49 Limitations of PCA Should the goal be finding independent rather than pair-wise uncorrelated dimensions Independent Component Analysis (ICA) ICA PCA

50 PCA vs ICA PCA (orthogonal coordinate) ICA (non-orthogonal coordinate)

51 Limitations of PCA The reduction of dimensions for complex distributions may need non linear processing Curvilenear Component Analysis (CCA) –Non linear extension of PCA –Preserves the proximity between the points in the input space i.e. local topology of the distribution –Enables to unfold some varieties in the input data –Keep the local topology

52 Example of data representation using CCA Non linear projection of a horseshoe Non linear projection of a spiral

53 PCA applications -Eigenfaces Eigenfaces are the eigenvectors of the covariance matrix of the probability distribution of the vector space of human faces Eigenfaces are the ‘standardized face ingredients’ derived from the statistical analysis of many pictures of human faces A human face may be considered to be a combination of these standard faces

54 PCA applications -Eigenfaces To generate a set of eigenfaces: 1.Large set of digitized images of human faces is taken under the same lighting conditions. 2.The images are normalized to line up the eyes and mouths. 3.The eigenvectors of the covariance matrix of the statistical distribution of face image vectors are then extracted. 4.These eigenvectors are called eigenfaces.

55 PCA applications -Eigenfaces the principal eigenface looks like a bland androgynous average human face http://en.wikipedia.org/wiki/Image:Eigenfaces.png

56 Eigenfaces – Face Recognition When properly weighted, eigenfaces can be summed together to create an approximate gray-scale rendering of a human face. Remarkably few eigenvector terms are needed to give a fair likeness of most people's faces Hence eigenfaces provide a means of applying data compression to faces for identification purposes. Similarly, Expert Object Recognition in Video

57 Eigenfaces Experiment and Results Data used here are from the ORL database of faces. Facial images of 16 persons each with 10 views are used. - Training set contains 16×7 images. - Test set contains 16×3 images. First three eigenfaces :

58 Classification Using Nearest Neighbor Save average coefficients for each person. Classify new face as the person with the closest average. Recognition accuracy increases with number of eigenfaces till 15. Later eigenfaces do not help much with recognition. Best recognition rates Training set 99% Test set 89%

59 PCA of Genes (Leukemia data, precursor B and T cells) 34 patients, dimension of 8973 genes reduced to 2

60 PCA of genes (Leukemia data) Plot of 8973 genes, dimension of 34 patients reduced to 2

61 Leukemia data - clustering of patients on original gene dimensions

62 Leukemia data - clustering of patients on top 100 eigen-genes

63 Leukemia data - clustering of genes

64 Sporulation Data Example Data: A subset of sporulation data (477 genes) were classified into seven temporal patterns (Chu et al., 1998) The first 2 PCs contains 85.9% of the variation in the data. (Figure 1a) The first 3 PCs contains 93.2% of the variation in the data. (Figure 1b)

65 Sporulation Data The patterns overlap around the origin in (1a). The patterns are much more separated in (1b).

66 PCA for Process Monitoring Variation in processes: Chance Natural variation inherent in a process. Cumulative effect of many small, unavoidable causes. Assignable Variations in raw material, machine tools, mechanical failure and human error. These are accountable circumstances and are normally larger.

67 PCA for process monitoring Latent variables can sometimes be interpreted as measures of physical characteristics of a process i.e., temp, pressure. Variable reduction can increase the sensitivity of a control scheme to assignable causes Application of PCA to monitoring is increasing –Start with a reference set defining normal operation conditions, look for assignable causes –Generate a set of indicator variables that best describe the dynamics of the process –PCA is sensitive to data types

68 Independent Component Analysis (ICA) PCA (orthogonal coordinate) ICA (non-orthogonal coordinate)

69 Cocktail-party Problem Multiple sound sources in room (independent) Multiple sensors receiving signals which are mixture of original signals Estimate original source signals from mixture of received signals Can be viewed as Blind-Source Separation as mixing parameters are not known

70 DEMO: BLIND SOURCE SEPARATION http://www.cis.hut.fi/projects/ica/cocktail/cocktail_en.cgi

71 Cocktail party or Blind Source Separation (BSS) problem –Ill posed problem, unless assumptions are made! Most common assumption is that source signals are statistically independent. This means knowing value of one of them gives no information about the other. Methods based on this assumption are called Independent Component Analysis methods –statistical techniques for decomposing a complex data set into independent parts. It can be shown that under some reasonable conditions, if the ICA assumption holds, then the source signals can be recovered up to permutation and scaling. BSS and ICA

72 Source Separation Using ICA W 11 W 21 W 12 W 22 + + Microphone 1 Microphone 2 Separation 1 Separation 2

73 Original signals (hidden sources) s 1 (t), s 2 (t), s 3 (t), s 4 (t), t=1:T

74 The ICA model s1s1 s2s2 s3s3 s4s4 x1x1 x2x2 x3x3 x4x4 a 11 a 12 a 13 a 14 x i (t) = a i1 *s 1 (t) + a i2 *s 2 (t) + a i3 *s 3 (t) + a i4 *s 4 (t) Here, i=1:4. In vector-matrix notation, and dropping index t, this is x = A * s

75 This is recorded by the microphones: a linear mixture of the sources x i (t) = a i1 *s 1 (t) + a i2 *s 2 (t) + a i3 *s 3 (t) + a i4 *s 4 (t)

76 Recovered signals

77 BSS If we knew the mixing parameters a ij then we would just need to solve a linear system of equations. We know neither a ij nor s i. ICA was initially developed to deal with problems closely related to the cocktail party problem Later it became evident that ICA has many other applications –e.g. from electrical recordings of brain activity from different locations of the scalp (EEG signals) recover underlying components of brain activity Problem: Determine the source signals s, given only the mixtures x.

78 ICA Solution and Applicability ICA is a statistical method, the goal of which is to decompose given multivariate data into a linear sum of statistically independent components. For example, given two-dimensional vector, x = [ x 1 x 2 ] T, ICA aims at finding the following decomposition where a 1, a 2 are basis vectors and s 1, s 2 are basis coefficients. Constraint: Basis coefficients s 1 and s 2 are statistically independent.

79 Application domains of ICA Blind source separation Image denoising Medical signal processing – fMRI, ECG, EEG Modelling of the hippocampus and visual cortex Feature extraction, face recognition Compression, redundancy reduction Watermarking Clustering Time series analysis (stock market, microarray data) Topic extraction Econometrics: Finding hidden factors in financial data

80 Image denoising Wiener filtering ICA filtering Noisy image Original image

81 Blind Separation of Information from Galaxy Spectra

82 Approaches to ICA Unsupervised Learning –Factorial coding (Minimum entropy coding, Redundancy reduction)Factorial coding –Maximum likelihood learningMaximum likelihood learning –Nonlinear information maximization (entropy maximization)Nonlinear information maximization –Negentropy maximizationNegentropy maximization –Bayesian learning Statistical Signal Processing –Higher-order moments or cumulants –Joint approximate diagonalization –Maximum likelihood estimation

83 Feature Extraction in ECG data (Raw Data)

84 Feature Extraction in ECG data (PCA)

85 Feature Extraction in ECG data (Extended ICA)

86 Feature Extraction in ECG data (flexible ICA)

87 In the heart, PET can: –quantify the extent of heart disease –can calculate myocardial blood flow or metabolism quantitatively PET image acquisition

88 Static image –accumulating the data during the acquisition Dynamic image –for the quantitative analysis (ex. blood flow, metabolism) –acquiring the data sequentially with a time interval

89 Dynamic Image Acquisition Using PET temporal spatial frame 1 frame 2 frame 3 frame n

90 yNyN Left Ventricle Right Ventricle Myocardium Noise Dynamic Frames Elementary Activities Independent Components Unmixing Mixing u1u1 u2u2 u3u3 uNuN y1y1 y2y2 y3y3 g(u)

91 Independent Components Right Ventricle Left Ventricle Tissue

92 Other PET applications In cancer, PET can: distinguish benign from malignant tumors stage cancer by showing metastases anywhere in body In the brain, PET can: calculate cerebral blood flow or metabolism quantitatively. positively diagnose Alzheimer's disease for early intervention. locate tumors in the brain and distinguish tumor from scar tissue. locate the focus of seizures for some patients with epilepsy. find regions related with specific task like memory, behavior etc.

93 Factorial Faces for Recognition Eigenfaces Factorial Faces PCA/FA Factorial Coding/ICA It finds a linear data representation that best model the covariance structure of face image data (second-order relations) It finds a linear data representation that Best model the probability distribution of Face image data (high-order relations) (IEEE BMCV 2000)

94 Eigenfaces PCA Low Dimensional Representation =a1×a1×+a2×a2×+a3×a3×+a4×a4×+++an×an×

95 Factorial Code Representation: Factorial Faces Efficient Low Dimensional Representation PCA (Dimensionality Reduction) Factorial Code Representation =b1×b1×+b2×b2×+b3×b3×+b4×b4×+++bn×bn×

96 Experimental Results Figure 1. Sample images in the training set. (neutral expression, anger, and right-light-on from first session; smile and left-light-on from second session) Figure 2. Sample images in the test set. (smile and left-light-on from first session; neutral expression, anger, and right-light-on from second session)

97 Eigenfaces vs Factorial Faces Figure 3. First 20 basis images: (a) in eigenface method; (b) factorial code. They are ordered by column, then, by row. (a)(b)

98 Figure 1. The comparison of recognition performance using Nearest Neighbor: the eigenface method and Factorial Code Representation using 20 or 30 principal components in the snap-shot method; Experimental Results

99 Figure 2. The comparison of recognition performance using MLP Classifier: the eigenface method and Factorial Code Representation using 20 or 50 principal components in the snap-shot method; Experimental Results

100 PCA vs ICA Linear Transform –Compression –Classification PCA –Focus on uncorrelated and Gaussian components –Second-order statistics –Orthogonal transformation ICA –Focus on independent and non-Gaussian components –Higher-order statistics –Non-orthogonal transformation

101 Gaussians and ICA If some components are gaussian and some are non-gaussian. –Can estimate all non-gaussian components –Linear combination of gaussian components can be estimated. –If only one gaussian component, model can be estimated ICA sometimes viewed as non-Gaussian factor analysis