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Sensor Data Analysis Using Principal Component Analysis

Learn how to analyze sensor data using Principal Component Analysis (PCA) to identify patterns and reduce dimensionality. We cover covariance matrix calculation, eigenvectors, and more.

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Sensor Data Analysis Using Principal Component Analysis

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  1. EE-240/2009 PCA

  2. PCA Principal Component Analysis

  3. Sensor 3 Sensor 4 x = 1.8196 5.6843 6.8238 4.7767 1.0397 4.1195 6.0911 4.2638 1.1126 4.6507 7.8633 5.5043 1.3735 6.0801 8.0462 5.6324 2.4124 3.6608 7.1652 5.0156 3.2898 4.7301 6.0169 4.2119 2.9436 4.2940 7.4612 5.2228 1.6561 3.1509 5.2104 3.6473 0.4757 1.4661 6.6255 4.6379 2.1659 1.6211 9.2418 6.4692 1.4852 3.6537 7.3074 5.1152 0.7544 3.3891 6.1951 4.3365 2.3136 3.5918 7.6207 5.3345 2.4068 2.5844 6.8016 4.7611 2.5996 4.2271 7.7247 5.4073 1.8136 4.2080 6.7446 4.7212 N Medida 9 Medida 14 m

  4. x= 0.5632 0.0525 0.1992 0.9442 -1.4904 0.0169 0.5112 0.2334 2.3856 0.4861 0.5951 0.0604 0.6180 1.5972 -0.5145 -0.6150 1.7430 1.4551 -1.0344 -0.9598 1.5369 1.3969 0.1178 -0.5428 0.8500 0.9795 -1.4907 -0.7346 -0.0970 -0.7409 -0.9971 -0.8030 -0.1868 -0.2996 -0.5746 -0.8211 1.2367 2.3720 0.3793 0.1620 0.0675 -0.3321 1.3183 0.7651 -0.8986 -0.5059 -0.1140 1.3910 -0.6703 0.1073 2.3948 1.4875 1.0090 1.2579 0.0914 -0.3765 -0.1388 -0.7678 -2.3396 -1.2983 0.9069 1.4772 -0.3849 -0.0941 -0.3874 -0.4512 0.0444 -0.1314 -0.2293 -0.6040 0.4161 0.5944 -3.0050 -1.9646 -1.1580 -0.1741 -0.6379 -1.7942 1.0003 -0.4333 0.2530 -0.1742 1.8173 1.5607 -1.7688 -2.0968 0.1575 -0.1781 -0.5782 0.0264 -2.5254 -1.8191 0.5193 0.9366 1.7935 1.6216 -1.7429 -1.9603 0.4392 -0.3092]

  5. clear all; N=50; sigma2=[1 0.8 ; 0.8 1]; mi=repmat([0 0],N,1); xx=mvnrnd(mi,sigma2); xmean=mean(xx,1); [lin,col]=size(xx); x=xx-repmat(xmean,lin,1); [p,lat,exp]=pcacov(x); plot(x(:,1),x(:,2),'+'); hold on plot([p(1,1) 0 p(1,2)],[p(2,1) 0 p(2,2)])

  6. Como determinar?

  7. cov = [1 0 ; 0 1] cov=[1 0.9 ; 0.9 1]

  8. cov = [1 0 ; 0 1] cov=[1 0.9 ; 0.9 1]

  9. cov = [1 0 ; 0 1] cov=[1 0.9 ; 0.9 1]

  10. Para Gaussianas: Não Correlacionados  Independentes Matriz de Covariança  Diagonal cov = [1 0 ; 0 1]

  11. Dado XNm Obter P, tal que Y = XP é diagonal. No caso de e-valores distintos: P-1 A P = L Dada uma matriz Amm , os auto-valores  e os auto-vetores v são caracterizados por ou seja, Como (s) é um polinômio de grau m, (s) =0 possui m raízes, 1, 2, ... , m associados a v1, v2, ... , vm

  12. Dado XNm Obter P, tal que Y = XP vj T PTP = vi i = j A = XTX P-1 A P = L é diagonal. vi e-vetores de XTX ( normalizados  vi  = 1) P é simétrica P-1 = PT

  13. Dado XNm Obter P, tal que Y = XP vj T PTP = vi é diagonal. PT = P i  j A = XTX P-1 A P = L vi e-vetores de XTX ( normalizados  vi  = 1) P é simétrica P-1 = PT

  14. Dado XNm Obter P, tal que Y = XP vj T PTP = vi é diagonal. A = XTX P-1 A P = L vi e-vetores de XTX ( normalizados  vi  = 1) PTP = I P é simétrica P-1 = PT

  15. Dado XNm Obter P, tal que Y = XP é diagonal. A = XTX P-1 A P = L vi e-vetores de XTX ( normalizados  vi  = 1) P é simétrica P-1 = PT P é ortogonal

  16. Singular Value Decomposition Dado XNm Obter P, tal que Y = XP é diagonal.

  17. Dado XNm Obter P, tal que Y = XP é diagonal. P = Matriz de e-vec de (XTX) V = Matriz à direita no SVD

  18. x= 0.5632 0.0525 0.1992 0.9442 -1.4904 0.0169 0.5112 0.2334 2.3856 0.4861 0.5951 0.0604 0.6180 1.5972 -0.5145 -0.6150 1.7430 1.4551 -1.0344 -0.9598 1.5369 1.3969 0.1178 -0.5428 0.8500 0.9795 -1.4907 -0.7346 -0.0970 -0.7409 -0.9971 -0.8030 -0.1868 -0.2996 -0.5746 -0.8211 1.2367 2.3720 0.3793 0.1620 0.0675 -0.3321 1.3183 0.7651 -0.8986 -0.5059 -0.1140 1.3910 -0.6703 0.1073 2.3948 1.4875 1.0090 1.2579 0.0914 -0.3765 -0.1388 -0.7678 -2.3396 -1.2983 0.9069 1.4772 -0.3849 -0.0941 -0.3874 -0.4512 0.0444 -0.1314 -0.2293 -0.6040 0.4161 0.5944 -3.0050 -1.9646 -1.1580 -0.1741 -0.6379 -1.7942 1.0003 -0.4333 0.2530 -0.1742 1.8173 1.5607 -1.7688 -2.0968 0.1575 -0.1781 -0.5782 0.0264 -2.5254 -1.8191 0.5193 0.9366 1.7935 1.6216 -1.7429 -1.9603 0.4392 -0.3092]

  19. >> [P, Lambda]=eig(xx) P = -0.7563 0.6543 -0.6543 -0.7563 Lambda = 113.9223 0 0 12.1205 >> Lambda = inv(P)*xx*P OK Lambda = 113.9223 0 0.0000 12.1205 Lambda (1) >> Lambda (2) x= 0.5632 0.0525 0.1992 0.9442 -1.4904 0.0169 0.5112 0.2334 2.3856 0.4861 0.5951 0.0604 0.6180 1.5972 -0.5145 -0.6150 1.7430 1.4551 -1.0344 -0.9598 1.5369 1.3969 0.1178 -0.5428 0.8500 0.9795 -1.4907 -0.7346 -0.0970 -0.7409 -0.9971 -0.8030 -0.1868 -0.2996 -0.5746 -0.8211 1.2367 2.3720 0.3793 0.1620 0.0675 -0.3321 1.3183 0.7651 -0.8986 -0.5059 -0.1140 1.3910 -0.6703 0.1073 2.3948 1.4875 1.0090 1.2579 0.0914 -0.3765 -0.1388 -0.7678 -2.3396 -1.2983 0.9069 1.4772 -0.3849 -0.0941 -0.3874 -0.4512 0.0444 -0.1314 -0.2293 -0.6040 0.4161 0.5944 -3.0050 -1.9646 -1.1580 -0.1741 -0.6379 -1.7942 1.0003 -0.4333 0.2530 -0.1742 1.8173 1.5607 -1.7688 -2.0968 0.1575 -0.1781 -0.5782 0.0264 -2.5254 -1.8191 0.5193 0.9366 1.7935 1.6216 -1.7429 -1.9603 0.4392 -0.3092] xx = 70.3445 50.3713 50.3713 55.6982

  20. P = -0.7563 0.6543 -0.6543 -0.7563 >> xnew = x * P Pelim = -0.7563 0.0 -0.6543 0.0 >> xelim = x * Pelim

  21. Médodo da Diagonalização: P = -0.7563 0.6543 -0.6543 -0.7563 >> xn=x/sqrt(N-1) >> [u,sigma,v]=svd(xn) v = -0.7563 0.6543 -0.6543 -0.7563 Lambda = 113.9223 0 0.0000 12.1205 OK sigma = 1.5248 0 0 0.4973 >> sigmadiag =sqrt(Lambda/(N-1)) sigmadiag = 1.5248 0 0.0000 0.4973 OK x= 0.5632 0.0525 0.1992 0.9442 -1.4904 0.0169 0.5112 0.2334 2.3856 0.4861 0.5951 0.0604 0.6180 1.5972 -0.5145 -0.6150 1.7430 1.4551 -1.0344 -0.9598 1.5369 1.3969 0.1178 -0.5428 0.8500 0.9795 -1.4907 -0.7346 -0.0970 -0.7409 -0.9971 -0.8030 -0.1868 -0.2996 -0.5746 -0.8211 1.2367 2.3720 0.3793 0.1620 0.0675 -0.3321 1.3183 0.7651 -0.8986 -0.5059 -0.1140 1.3910 -0.6703 0.1073 Médodo da SVD 2.3948 1.4875 1.0090 1.2579 0.0914 -0.3765 -0.1388 -0.7678 -2.3396 -1.2983 0.9069 1.4772 -0.3849 -0.0941 -0.3874 -0.4512 0.0444 -0.1314 -0.2293 -0.6040 0.4161 0.5944 -3.0050 -1.9646 -1.1580 -0.1741 -0.6379 -1.7942 1.0003 -0.4333 0.2530 -0.1742 1.8173 1.5607 -1.7688 -2.0968 0.1575 -0.1781 -0.5782 0.0264 -2.5254 -1.8191 0.5193 0.9366 1.7935 1.6216 -1.7429 -1.9603 0.4392 -0.3092] x= 0.5632 0.0525 0.1992 0.9442 .... ....

  22. >> [pc,latent,explained]=pcacov(x) latent = 10.6734 3.4814 pc = -0.7563 0.6543 -0.6543 -0.7563 explained = 75.4046 24.5954 >> help pcacov PCACOV Principal Component Analysis using the covariance matrix. [PC, LATENT, EXPLAINED] = PCACOV(X) takes a the covariance matrix, X, and returns the principal components in PC, the eigenvalues of the covariance matrix of X in LATENT, and the percentage of the total variance in the observations explained by each eigenvector in EXPLAINED.

  23. >> [pc,latent,explained]=pcacov(x) latent = 10.6734 3.4814 pc = -0.7563 0.6543 -0.6543 -0.7563 explained = 75.4046 24.5954 >> help pcacov PCACOV Principal Component Analysis using the covariance matrix. [PC, LATENT, EXPLAINED] = PCACOV(X) takes a the covariance matrix, X, and returns the principal components in PC, the eigenvalues of the covariance matrix of X in LATENT, and the percentage of the total variance in the observations explained by each eigenvector in EXPLAINED. P = -0.7563 0.6543 -0.6543 -0.7563 v = -0.7563 0.6543 -0.6543 -0.7563 OK

  24. >> help pcacov PCACOV Principal Component Analysis using the covariance matrix. [PC, LATENT, EXPLAINED] = PCACOV(X) takes a the covariance matrix, X, and returns the principal components in PC, the eigenvalues of the covariance matrix of X in LATENT, and the percentage of the total variance in the observations explained by each eigenvector in EXPLAINED. >> [pc,latent,explained]=pcacov(x) latent = 10.6734 3.4814 pc = -0.7563 0.6543 -0.6543 -0.7563 explained = 75.4046 24.5954 Lambda = 113.9223 0 0 12.1205 sqrlamb = 10.6734 0 0 3.4814 >> sqrlamb=sqrt(evalor)

  25. >> help pcacov PCACOV Principal Component Analysis using the covariance matrix. [PC, LATENT, EXPLAINED] = PCACOV(X) takes a the covariance matrix, X, and returns the principal components in PC, the eigenvalues of the covariance matrix of X in LATENT, and the percentage of the total variance in the observations explained by each eigenvector in EXPLAINED. >> [pc,latent,explained]=pcacov(x) latent = 10.6734 3.4814 pc = -0.7563 0.6543 -0.6543 -0.7563 explained = 75.4046 24.5954 sqrlamb = 10.6734 0 0 3.4814 percent = 75.4046 24.5954 >> e=[Lambda(1,1) ; Lambda(2,2)] >> soma=sum(e) >> percent=e*100/soma

  26. >> [pc,score,latent1,tsquare]=princomp(x) score = -0.4603 0.3288 -0.7684 -0.5837 1.1161 -0.9879 -0.5393 0.1579 -2.1222 1.1932 -0.4896 0.3437 ..... ..... pc = -0.7563 0.6543 -0.6543 -0.7563 latent1 = 2.3249 0.2474 tsquare = 0.5281 1.6315 4.4813 0.2260 7.6929 0.5806 ..... >> help princomp PRINCOMP Principal Component Analysis (centered and scaled data). [PC, SCORE, LATENT, TSQUARE] = PRINCOMP(X) takes a data matrix X and returns the principal components in PC, the so-called Z-scores in SCORES, the eigenvalues of the covariance matrix of X in LATENT, and Hotelling's T-squared statistic for each data point in TSQUARE.

  27. >> help princomp PRINCOMP Principal Component Analysis (centered and scaled data). [PC, SCORE, LATENT, TSQUARE] = PRINCOMP(X) takes a data matrix X and returns the principal components in PC, the so-called Z-scores in SCORES, the eigenvalues of the covariance matrix of X in LATENT, and Hotelling's T-squared statistic for each data point in TSQUARE. >> [pc,score,latent1,tsquare]=princomp(x) score = -0.4603 0.3288 -0.7684 -0.5837 1.1161 -0.9879 -0.5393 0.1579 -2.1222 1.1932 -0.4896 0.3437 ..... ..... pc = -0.7563 0.6543 -0.6543 -0.7563 latent1 = 2.3249 0.2474 tsquare = 0.5281 1.6315 4.4813 0.2260 7.6929 0.5806 ..... P = -0.7563 0.6543 -0.6543 -0.7563

  28. >> help princomp PRINCOMP Principal Component Analysis (centered and scaled data). [PC, SCORE, LATENT, TSQUARE] = PRINCOMP(X) takes a data matrix X and returns the principal components in PC, the so-called Z-scores in SCORES, the eigenvalues of the covariance matrix of X in LATENT, and Hotelling's T-squared statistic for each data point in TSQUARE. >> [pc,score,latent1,tsquare]=princomp(x) score = -0.4603 0.3288 -0.7684 -0.5837 1.1161 -0.9879 -0.5393 0.1579 -2.1222 1.1932 -0.4896 0.3437 ..... ..... sco = -0.4603 0.3288 -0.7684 -0.5838 1.1161 -0.9880 -0.5393 0.1580 -2.1223 1.1933 -0.4896 0.3437 >> sco=x*P

  29. >> help princomp PRINCOMP Principal Component Analysis (centered and scaled data). [PC, SCORE, LATENT, TSQUARE] = PRINCOMP(X) takes a data matrix X and returns the principal components in PC, the so-called Z-scores in SCORES, the eigenvalues of the covariance matrix of X in LATENT, and Hotelling's T-squared statistic for each data point in TSQUARE. >> [pc,score,latent1,tsquare]=princomp(x) score = -0.4603 0.3288 -0.7684 -0.5837 1.1161 -0.9879 -0.5393 0.1579 -2.1222 1.1932 -0.4896 0.3437 ..... ..... pc = -0.7563 0.6543 -0.6543 -0.7563 latent1 = 2.3249 0.2474 tsquare = 0.5281 1.6315 4.4813 0.2260 7.6929 0.5806 ..... >> eig(x'*x/(N-1)) ans = 0.2474 2.3249

  30. caracteriza a região de confiança 100 a% >> help princomp PRINCOMP Principal Component Analysis (centered and scaled data). [PC, SCORE, LATENT, TSQUARE] = PRINCOMP(X) takes a data matrix X and returns the principal components in PC, the so-called Z-scores in SCORES, the eigenvalues of the covariance matrix of X in LATENT, and Hotelling's T-squared statistic for each data point in TSQUARE. >> [pc,score,latent1,tsquare]=princomp(x) score = -0.4603 0.3288 -0.7684 -0.5837 1.1161 -0.9879 -0.5393 0.1579 -2.1222 1.1932 -0.4896 0.3437 ..... ..... pc = -0.7563 0.6543 -0.6543 -0.7563 latent1 = 2.3249 0.2474 tsquare = 0.5281 1.6315 4.4813 0.2260 7.6929 0.5806 .....

  31. Não gaussianidade = OK?

  32. Não gaussianidade = OK?

  33. Não gaussianidade = OK?

  34. >> xm=mean(xx); >> [lin,col]=size(xx); >> xm=repmat(xm,lin,1); >> xx=xx-xm;

  35. x = -0.4326 -1.6656 0.1253 0.2877 -1.1465 1.1909 1.1892 -0.0376 0.3273 0.1746 -0.1867 0.7258 -0.5883 2.1832 -0.1364 0.1139 1.0668 0.0593 -0.0956 -0.8323 N >> [u,sigma,v]=svd(x) sigma = 3.2980 0 0 2.0045 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 u = 0.4460 0.4456 -0.2868 0.4908 0.1650 0.0392 0.1066 -0.0386 0.4553 -0.1743 -0.0726 -0.1012 -0.4848 0.1055 -0.0323 -0.2554 -0.7703 -0.0485 0.0638 0.2687 -0.4458 0.3769 0.6421 0.2784 0.0623 -0.0995 -0.3056 -0.0404 0.2425 0.0424 0.1146 -0.5628 0.1922 0.7577 -0.0733 0.0123 0.0424 0.0239 -0.2207 0.0494 -0.0222 -0.1814 0.0253 -0.0595 0.9785 -0.0107 -0.0308 0.0038 -0.0560 0.0280 -0.2271 -0.0149 -0.1378 0.0660 0.0063 0.9434 -0.1717 -0.0146 0.0532 0.0484 -0.6853 -0.0321 -0.4187 0.2038 0.0205 -0.1705 0.4827 -0.0445 0.1649 0.1443 -0.0450 0.0488 -0.0384 0.0320 0.0076 -0.0097 -0.0300 0.9956 0.0281 0.0029 0.0758 -0.5182 0.1585 -0.2137 -0.0664 0.0039 0.0167 0.0200 0.8044 0.0516 0.2334 0.1651 0.1094 -0.0129 0.0120 0.0625 0.1882 0.0107 -0.0038 0.9309 v = 0.2876 -0.9578 -0.9578 -0.2876 N  N

  36. Muito Obrigado!

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