![R Code for PCA (Assignment)
OPTIONAL R Code
install.packages("gdata") ## for reading xls files
install.packages("xlsx") ## ” for reading xlsx files
mydata<-read.xlsx("University Ranking.xlsx",1) ## use read.csv for csv files
mydata ## make sure the data is loaded correctly
help(princomp) ## to understand the api for princomp
pcaObj<-princomp(mydata[1:25,2:7], cor = TRUE, scores = TRUE, covmat = NULL)
## the first column in mydata has university names
## princomp(mydata, cor = TRUE) not_same_as prcomp(mydata, scale=TRUE); similar, but different
summary(pcaObj)
loadings(pcaObj)
plot(pcaObj)
biplot(pcaObj)
pcaObj$loadings
pcaObj$scores](https://image.slidesharecdn.com/analyticscertificationtrainingbangalore-190301120804/85/Analytics-certification-training-bangalore-13-320.jpg)
![Data Compression
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kept; c ≤ p](https://image.slidesharecdn.com/analyticscertificationtrainingbangalore-190301120804/85/Analytics-certification-training-bangalore-18-320.jpg)
Analytics certification training bangalore
- 1. Dimension Reduction using Principal Components Analysis (PCA)
- 2. Application of dimension reduction • Computational advantage for other algorithms • Face recognition— image data (pixels) along new axes works better for recognizing faces • Image compression
- 3. Data for 25 undergraduate programs at business schools in US universities in 1995. Use PCA to: 1) Reduce # columns Additional benefits: 2) Identify relation between columns 3) Visualize universities in 2D Univ SAT Top10 Accept SFRatio Expenses GradRate Brown 1310 89 22 13 22,704 94 CalTech 1415 100 25 6 63,575 81 CMU 1260 62 59 9 25,026 72 Columbia 1310 76 24 12 31,510 88 Cornell 1280 83 33 13 21,864 90 Dartmouth 1340 89 23 10 32,162 95 Duke 1315 90 30 12 31,585 95 Georgetown 1255 74 24 12 20,126 92 Harvard 1400 91 14 11 39,525 97 JohnsHopkins 1305 75 44 7 58,691 87 MIT 1380 94 30 10 34,870 91 Northwestern 1260 85 39 11 28,052 89 NotreDame 1255 81 42 13 15,122 94 PennState 1081 38 54 18 10,185 80 Princeton 1375 91 14 8 30,220 95 Purdue 1005 28 90 19 9,066 69 Stanford 1360 90 20 12 36,450 93 TexasA&M 1075 49 67 25 8,704 67 UCBerkeley 1240 95 40 17 15,140 78 UChicago 1290 75 50 13 38,380 87 UMichigan 1180 65 68 16 15,470 85 UPenn 1285 80 36 11 27,553 90 UVA 1225 77 44 14 13,349 92 UWisconsin 1085 40 69 15 11,857 71 Yale 1375 95 19 11 43,514 96 Source: US News & World Report, Sept 18 1995
- 4. Input Output Univ SAT Top10 Accept SFRatio Expenses GradRate PC1 PC2 PC3 PC4 PC5 PC6 Brown 1310 89 22 13 22,704 94 CalTech 1415 100 25 6 63,575 81 CMU 1260 62 59 9 25,026 72 Columbia 1310 76 24 12 31,510 88 Cornell 1280 83 33 13 21,864 90 Dartmouth 1340 89 23 10 32,162 95 Duke 1315 90 30 12 31,585 95 Georgetown 1255 74 24 12 20,126 92 Harvard 1400 91 14 11 39,525 97 JohnsHopkins 1305 75 44 7 58,691 87 Hope is that a fewer columns may capture most of the information from the original dataset PCA
- 5. The Primitive Idea – Intuition First How to compress the data loosing the least amount of information?
- 6. Input • p measurements/ original columns • Correlated Output • p principal components (= p weighted averages of original measurements) • Uncorrelated • Ordered by variance • Keep top principal components; drop rest PCA
- 7. Mechanism The ith principal component is a weighted average of original measurements/columns: Weights (aij) are chosen such that: 1. PCs are ordered by their variance (PC1 has largest variance, followed by PC2, PC3, and so on) 2. Pairs of PCs have correlation = 0 3. For each PC, sum of squared weights =1 pip2i21i1i XaXaXaPC +…++= Univ SAT Top10 Accept SFRatio Expenses GradRate PC1 PC2 PC3 PC4 PC5 PC6 Brown 1310 89 22 13 22,704 94 CalTech 1415 100 25 6 63,575 81 CMU 1260 62 59 9 25,026 72 Columbia 1310 76 24 12 31,510 88 Cornell 1280 83 33 13 21,864 90 Dartmouth 1340 89 23 10 32,162 95 Duke 1315 90 30 12 31,585 95 Georgetown 1255 74 24 12 20,126 92 Harvard 1400 91 14 11 39,525 97 JohnsHopkins 1305 75 44 7 58,691 87
- 8. Demystifying weight computation • Main idea: high variance = lots of information • Goal: Find weights aij that maximize variance of PCi, while keeping PCi uncorrelated to other PCs. • The covariance matrix of the X’s is needed. jiPCPC ),XCov(Xaa),XCov(Xaa )Var(X)Var(X)Var(X)Var(PC ji pp-ipi p-ii pipiii aaa ≠= +…++ ++…++= when,0),(Covarwant,Also 22 112121 2 2 2 21 2 1 pip2i21i1i XaXaXaPC +…++=
- 9. Standardize the inputs Why? • variables with large variances will have bigger influence on result Solution • Standardize before applying PCA Excel: =standardize(cell, average(column), stdev(column)) Univ Z_SAT Z_Top10 Z_Accept Z_SFRatio Z_Expenses Z_GradRate Brown 0.4020 0.6442 -0.8719 0.0688 -0.3247 0.8037 CalTech 1.3710 1.2103 -0.7198 -1.6522 2.5087 -0.6315 CMU -0.0594 -0.7451 1.0037 -0.9146 -0.1637 -1.6251 Columbia 0.4020 -0.0247 -0.7705 -0.1770 0.2858 0.1413 Cornell 0.1251 0.3355 -0.3143 0.0688 -0.3829 0.3621 Dartmouth 0.6788 0.6442 -0.8212 -0.6687 0.3310 0.9141 Duke 0.4481 0.6957 -0.4664 -0.1770 0.2910 0.9141 Georgetown -0.1056 -0.1276 -0.7705 -0.1770 -0.5034 0.5829 Harvard 1.2326 0.7471 -1.2774 -0.4229 0.8414 1.1349 JohnsHopkins 0.3559 -0.0762 0.2433 -1.4063 2.1701 0.0309 MIT 1.0480 0.9015 -0.4664 -0.6687 0.5187 0.4725 Northwestern -0.0594 0.4384 -0.0101 -0.4229 0.0460 0.2517 NotreDame -0.1056 0.2326 0.1419 0.0688 -0.8503 0.8037 PennState -1.7113 -1.9800 0.7502 1.2981 -1.1926 -0.7419 Princeton 1.0018 0.7471 -1.2774 -1.1605 0.1963 0.9141 Purdue -2.4127 -2.4946 2.5751 1.5440 -1.2702 -1.9563 Stanford 0.8634 0.6957 -0.9733 -0.1770 0.6282 0.6933 TexasA&M -1.7667 -1.4140 1.4092 3.0192 -1.2953 -2.1771 UCBerkeley -0.2440 0.9530 0.0406 1.0523 -0.8491 -0.9627 UChicago 0.2174 -0.0762 0.5475 0.0688 0.7620 0.0309 UMichigan -0.7977 -0.5907 1.4599 0.8064 -0.8262 -0.1899 UPenn 0.1713 0.1811 -0.1622 -0.4229 0.0114 0.3621 UVA -0.3824 0.0268 0.2433 0.3147 -0.9732 0.5829 UWisconsin -1.6744 -1.8771 1.5106 0.5606 -1.0767 -1.7355 Yale 1.0018 0.9530 -1.0240 -0.4229 1.1179 1.0245
- 10. Standardization shortcut for PCA • Rather than standardize the data manually, you can use correlation matrix instead of covariance matrix as input • PCA with and without standardization gives different results!
- 11. PCA Transform > Principal Components (correlation matrix has been used here) • PCs are uncorrelated • Var(PC1) > Var (PC2) > ... pip2i21i1i XaXaXaPC +…++= Scaled Data PC Scores Principal Components
- 12. Computing principal scores • For each record, we can compute their score on each PC. • Multiply each weight (aij) by the appropriate Xij • Example for Brown University (using standardized numbers): • PC Score1 for Brown University = (– 0.458)(0.40) +(–0.427)(.64) +(0.424)(–0.87) +(0.391)(.07) + (–0.363)(–0.32) + (–0.379)(.80) = –0.989
- 13. R Code for PCA (Assignment) OPTIONAL R Code install.packages("gdata") ## for reading xls files install.packages("xlsx") ## ” for reading xlsx files mydata<-read.xlsx("University Ranking.xlsx",1) ## use read.csv for csv files mydata ## make sure the data is loaded correctly help(princomp) ## to understand the api for princomp pcaObj<-princomp(mydata[1:25,2:7], cor = TRUE, scores = TRUE, covmat = NULL) ## the first column in mydata has university names ## princomp(mydata, cor = TRUE) not_same_as prcomp(mydata, scale=TRUE); similar, but different summary(pcaObj) loadings(pcaObj) plot(pcaObj) biplot(pcaObj) pcaObj$loadings pcaObj$scores
- 14. Goal #1: Reduce data dimension • PCs are ordered by their variance (=information) • Choose top few PCs and drop the rest! Example: • PC1 captures most ??% of the information. • The first 2 PCs capture ??% • Data reduction: use only two variables instead of 6.
- 15. Matrix Transpose OPTIONAL: R code help(matrix) A<-matrix(c(1,2),nrow=1,ncol=2,byrow=TRUE) A t(A) B<-matrix(c(1,2,3,4),nrow=2,ncol=2,byrow=TRUE) B t(B) C<matrix(c(1,2,3,4,5,6),nrow=3,ncol=2,byrow=TRUE) C t(C)
- 16. Matrix Multiplication OPTIONAL R Code A<- matrix(c(1,2,3,4,5,6),nrow=3,ncol=2,byrow= TRUE) A B<- matrix(c(1,2,3,4,5,6,7,8),nrow=2,ncol=4,byro w=TRUE) B C<-A%*%B D<-t(B)%*%t(A) ## note, B%*%A is not possible; how does D look like?
- 17. Matrix Inverse = =× 10...00 0...100 .......... 00...10 00...01 :matrixIdentity A 1-BThen, matrixidentity,If, IBA OPTIONAL R Code ## How to create nˣn Identity matrix? help(diag) A<-diag(5) ## find inverse of a matrix solve(A)
- 18. Data Compression [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]omponentsPrincipalCPCScorestaedScaledDaApproximat :ionApproximat omponentsPrincipalCPCScores omponentsPrincipalCPCScores ScaledData omponentsPrincipalCScaledDataPCScores 1 T pc cNpN T pppN pp pN pN pppNpN × ×× ×× − × × × × ×× ×= ×= ×= ×= c = Number of components kept; c ≤ p
- 19. Goal #2: Learn relationships with PCA by interpreting the weights • ai1,…, aip are the coefficients for PCi. • They describe the role of original X variables in computing PCi. • Useful in providing context-specific interpretation of each PC.
- 20. PC1 Scores (choose one or more) 1. are approximately a simple average of the 6 variables 2. measure the degree of high Accept & SFRatio, but low Expenses, GradRate, SAT, and Top10
- 21. Goal #3: Use PCA for visualization • The first 2 (or 3) PCs provide a way to project the data from a p-dimensional space onto a 2D (or 3D) space
- 22. Scatter Plot: PC2 vs. PC1 scores
- 23. Monitoring batch processes using PCA • Multivariate data at different time points • Historical database of successful batches are used • Multivariate trajectory data is projected to low-dimensional space >>> Simple monitoring charts to spot outlier
- 24. Your Turn! 1. If we use a subset of the principal components, is this useful for prediction? for explanation? 2. What are advantages and weaknesses of PCA compared to choosing a subset of the variables? 3. PCA vs. Clustering