Tema 1 - PCA

8/7/2019 Tema 1 - PCA

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Teme Curs Data Mining

Profesor indrumator :Victor Neagoe

Nica GeorgianaMaster SIVA

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Data Mining

Tema 1

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Avnd la dispoziie baza de date SPECTF.mat formata din imagini Single ProtonEmission Computed Tomography am realizat o serie de procesri asupra acesteia,procesri ce au la baz algoritmul PCA.

n continuare va fi prezentat pe scurt baza de date. Aceasta este formata din doua

clase : normal si anormal (notat 0 sau 1) si este formata din 267 seturi de imagini fiecareimagine avand 44 de trasaturi.Toti vectorii sunt astfel 44-dimensionali.

Vectorii din baza de date au fost impartiti in 2 loturi:- SPECTF_train.mat, continand 80 de instante din clasa initiala, 40 catalogati cu

sanatos si 40 cu bolnav;- SPECTF_test.mat, continand 187 de instante din clasa initiala, 15 etichetati cu

sanatos si 172 cu bolnav;-

Ambele loturi sunt etichetate, stiindu-se fiecare vector din ce clasa provine.

n cele ce urmeaz vor fi prezentate rezultatele obinute la fiecare pas de procesare.

1. Calculul vectorului medie pentru vectorul de antrenare.

med_antrenare =

63.666767.888963.244464.2000

62.222265.888967.044463.822264.333361.422264.155668.422267.511166.088965.733362.888958.422264.488965.488967.155664.711166.777864.2000

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45.888952.888965.155664.800063.6222

66.733364.444463.444454.977860.466760.822266.422266.844465.111153.333362.0222

63.800065.533368.800067.466768.466768.911164.844468.244467.000067.733368.2667

69.066766.088967.800065.733366.822266.577867.933367.888967.688966.911168.288963.933366.422265.688967.933366.244467.533367.111167.5111

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65.755664.622267.022265.733368.1778

69.377866.644464.577866.977865.177865.6667

2. Determinarea matricei de covarianta pentru vectorul de antrenare i reprezentarea sagrafic.

Elementele matricei de covarian se determin astfel:

n aceast formul, n are semnificaia numrului devectori de antrenare, xkii xkj reprezint elementele

din ectorii de antrenare i i j corespunztoare caracteristicii k.

covarianta_antrenare =

Columns 1 through 5

76.4000 60.8292 44.8708 61.9000 58.479260.8292 73.0806 57.3153 57.3125 59.1889

44.8708 57.3153 100.5289 46.0100 78.656961.9000 57.3125 46.0100 69.9900 60.337558.4792 59.1889 78.6569 60.3375 114.347265.6542 59.8306 51.9028 66.8875 61.438956.9708 60.9903 60.9564 58.5575 67.706953.1917 61.9389 64.5119 48.1075 62.322268.3625 61.9458 59.9792 70.0000 89.670851.4417 52.6764 80.6169 62.9900 79.922246.7792 58.1097 75.1536 53.1450 72.780665.4667 61.7764 54.6669 61.2650 68.722263.9333 71.0819 66.3547 60.1050 71.748663.6167 58.0181 60.2753 62.0275 78.276454.1000 58.3583 61.1492 56.4175 61.583361.3542 57.6681 63.5778 55.5500 61.451458.4792 56.2639 88.2669 57.4900 86.097268.2792 55.1306 55.3578 65.7450 70.063960.7542 61.9931 56.8828 52.5700 59.276461.0292 64.4972 57.4911 64.6950 64.355659.0333 61.2444 46.9022 56.2075 45.3611

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58.2458 59.1986 59.3181 52.2750 68.265348.1500 56.4875 55.4225 45.2025 62.612576.4042 53.8931 15.0153 59.8375 65.476433.2792 54.6306 41.2903 35.6250 51.301457.6917 60.8972 69.6161 61.5950 83.2556

53.7375 64.8250 86.6775 62.7850 86.837546.4292 63.3014 89.0894 59.8925 93.259768.2750 62.4333 49.5242 61.5925 58.233356.1333 52.9778 49.5764 49.5375 52.981954.7083 60.2778 45.8389 52.7125 42.956952.9208 56.0236 88.6031 50.9650 91.877839.0750 65.0167 116.3608 56.8850 112.954237.8667 62.4389 104.2369 50.9325 94.934754.0542 64.3764 73.1419 56.7650 83.647251.7833 60.4528 60.5464 53.9550 62.344449.9833 62.7319 72.1472 56.1000 79.3736

35.5500 58.2583 108.5542 62.3250 104.133354.9167 55.4639 67.4969 62.2100 85.959757.3500 61.4375 76.6150 60.5475 90.612557.0875 57.2958 57.0267 61.4775 69.645861.3500 61.9000 55.3025 60.3850 69.262562.8375 60.3417 57.3483 62.5350 63.916757.1500 65.8417 64.2358 61.2475 61.629260.7333 62.8569 55.0622 65.4350 65.2861

Columns 6 through 10

65.6542 56.9708 53.1917 68.3625 51.441759.8306 60.9903 61.9389 61.9458 52.676451.9028 60.9564 64.5119 59.9792 80.616966.8875 58.5575 48.1075 70.0000 62.990061.4389 67.7069 62.3222 89.6708 79.922277.5806 60.1278 52.9014 75.1083 69.863960.1278 67.9739 59.3419 64.0542 61.676952.9014 59.3419 73.1572 55.7333 51.654775.1083 64.0542 55.7333 97.0000 78.158369.8639 61.6769 51.6547 78.1583 104.512253.0722 64.0711 60.6656 61.7583 67.438161.9764 63.3769 62.7047 67.9833 57.262262.5694 59.3997 70.8886 67.1792 60.028661.6181 67.0978 59.3839 73.4708 63.366457.4458 60.2067 56.6858 58.0875 62.850860.2306 57.7903 55.8639 65.0083 62.263963.5514 61.1019 60.9797 76.5958 92.924768.6806 60.1503 51.5114 74.3333 67.433962.1806 56.2253 59.2239 64.0333 53.2089

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64.8597 61.7211 53.9406 67.3958 67.200659.1819 54.9447 54.0336 55.4917 40.106160.6611 61.1306 54.7653 63.2167 59.215355.4250 55.9575 53.8075 57.7625 52.002565.3931 48.3403 31.0889 68.1083 52.7639

29.0931 45.9528 35.5389 33.0583 20.376458.4722 64.9461 59.4031 73.3083 72.350665.3625 66.8925 64.7550 73.2375 88.735062.6639 65.6469 63.1247 75.4083 91.102263.0083 58.9192 55.5358 62.9250 56.263355.4778 52.2139 51.8819 50.9667 58.894453.0403 55.3264 56.6444 50.1542 42.244461.0486 55.7131 51.7228 69.3667 98.217857.8167 68.8508 75.9467 78.1625 103.526752.9764 66.4794 68.7572 62.0083 94.904761.7389 65.8269 66.1797 73.1708 71.6372

55.8903 54.9539 58.7469 57.9667 60.511958.2569 61.7472 58.4736 69.2417 64.461166.1958 64.6542 51.1958 90.9500 126.558362.1139 62.1994 52.0147 70.4333 82.832264.4750 64.3925 65.3050 78.4500 79.685063.2958 63.8367 52.3533 69.2500 62.585862.2750 64.5800 58.7550 67.5500 56.585069.7792 61.0508 54.9717 71.7375 65.926763.1542 63.4133 60.4092 63.1375 66.064265.5444 65.0647 55.1161 67.1042 63.3711

Columns 11 through 1546.7792 65.4667 63.9333 63.6167 54.100058.1097 61.7764 71.0819 58.0181 58.358375.1536 54.6669 66.3547 60.2753 61.149253.1450 61.2650 60.1050 62.0275 56.417572.7806 68.7222 71.7486 78.2764 61.583353.0722 61.9764 62.5694 61.6181 57.445864.0711 63.3769 59.3997 67.0978 60.206760.6656 62.7047 70.8886 59.3839 56.685861.7583 67.9833 67.1792 73.4708 58.087567.4381 57.2622 60.0286 63.3664 62.850878.8239 56.8256 62.0553 65.7797 60.348356.8256 73.9372 64.4786 67.8039 59.438362.0553 64.4786 101.6406 61.6869 64.676765.7797 67.8039 61.6869 80.0206 59.575860.3483 59.4383 64.6767 59.5758 65.560056.2597 59.1264 63.2069 63.6931 56.333365.0256 64.8372 71.4286 72.1414 56.9633

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53.6572 63.6089 58.7844 66.6006 56.735852.6697 57.7714 64.6719 56.9381 56.598360.3114 59.6131 69.1428 63.6547 64.335853.0253 57.8061 60.5831 53.5519 52.669257.9069 58.1778 60.9639 61.0861 56.9667

55.4575 53.3150 61.2675 51.8775 54.942525.4472 61.2639 57.6069 59.7681 51.120844.9847 44.3389 54.0569 41.6431 40.545870.1864 66.2506 65.7553 71.2047 61.360872.6925 62.8225 73.8950 66.3350 67.282574.9706 57.3022 71.5336 62.5814 65.868351.2858 64.8508 66.3517 62.1383 58.347547.8986 57.8069 62.5722 50.6778 57.129250.0486 56.0944 62.8222 55.7778 53.691764.1894 54.3803 65.1439 56.6886 61.834299.9717 55.3642 81.1908 72.4892 69.1450

89.0406 56.1047 72.5761 62.9589 68.410869.8006 65.5372 72.1536 66.0039 63.113356.5136 61.5244 70.7822 54.4953 60.076765.6653 56.0611 68.8931 59.3569 56.891790.0083 49.1958 64.4542 63.0333 66.425068.2981 59.4572 63.3061 64.5864 64.603370.7550 66.2600 71.8575 70.6600 62.820063.8158 60.9358 58.3592 65.3108 59.517563.8800 65.2850 61.7825 66.6475 59.670058.7217 60.6142 65.4408 65.4017 57.082564.8217 63.1267 68.4283 60.6267 63.3200

60.3828 64.2711 63.5006 69.0794 62.7242Columns 16 through 20

61.3542 58.4792 68.2792 60.7542 61.029257.6681 56.2639 55.1306 61.9931 64.497263.5778 88.2669 55.3578 56.8828 57.491155.5500 57.4900 65.7450 52.5700 64.695061.4514 86.0972 70.0639 59.2764 64.355660.2306 63.5514 68.6806 62.1806 64.859757.7903 61.1019 60.1503 56.2253 61.721155.8639 60.9797 51.5114 59.2239 53.940665.0083 76.5958 74.3333 64.0333 67.395862.2639 92.9247 67.4339 53.2089 67.200656.2597 65.0256 53.6572 52.6697 60.311459.1264 64.8372 63.6089 57.7714 59.613163.2069 71.4286 58.7844 64.6719 69.142863.6931 72.1414 66.6006 56.9381 63.654756.3333 56.9633 56.7358 56.5983 64.3358

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70.5056 75.7389 63.3181 62.7056 58.347275.7389 134.8372 72.1589 58.2839 58.538163.3181 72.1589 77.9406 61.6281 60.982262.7056 58.2839 61.6281 72.5906 57.532258.3472 58.5381 60.9822 57.5322 75.1239

58.4944 44.9186 53.8794 60.1044 56.237857.2986 57.9153 59.7736 61.8486 59.194453.8000 47.1525 48.7450 60.2700 53.757535.9806 45.2639 58.3431 41.5181 74.522234.0181 30.8639 22.0056 28.6431 46.609759.0597 74.9631 62.8947 54.0197 63.961461.2750 90.4225 63.6300 56.1300 72.305058.5639 85.7522 62.1039 57.1164 70.120657.6583 57.7133 61.9483 57.7483 65.923352.4653 50.6569 52.8028 54.9403 58.136153.0903 39.9944 46.4903 52.4653 58.7986

55.1361 96.5303 63.9186 53.1936 69.289460.8042 103.6142 60.1467 55.5842 73.884252.8389 87.5547 49.9864 48.9114 72.065657.9639 67.7122 58.0589 60.4839 65.875653.1778 59.6744 50.9803 55.4928 60.551159.1319 68.6486 60.7069 61.8944 60.052858.1458 103.7708 58.7583 45.0083 77.645846.5514 61.3197 61.7189 47.5314 73.648159.6500 81.4975 64.2300 58.2800 66.305058.0958 55.7233 62.3158 58.8908 62.803356.9750 51.2475 61.2050 60.1300 62.6800

64.7167 68.6642 64.6217 63.2217 62.696757.1792 61.2017 57.1842 56.6842 66.771759.7444 61.4586 62.7119 57.8494 69.6453


59.0333 58.2458 48.1500 76.4042 33.279261.2444 59.1986 56.4875 53.8931 54.630646.9022 59.3181 55.4225 15.0153 41.290356.2075 52.2750 45.2025 59.8375 35.625045.3611 68.2653 62.6125 65.4764 51.301459.1819 60.6611 55.4250 65.3931 29.093154.9447 61.1306 55.9575 48.3403 45.952854.0336 54.7653 53.8075 31.0889 35.538955.4917 63.2167 57.7625 68.1083 33.058340.1061 59.2153 52.0025 52.7639 20.376453.0253 57.9069 55.4575 25.4472 44.984757.8061 58.1778 53.3150 61.2639 44.338960.5831 60.9639 61.2675 57.6069 54.0569

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53.5519 61.0861 51.8775 59.7681 41.643152.6692 56.9667 54.9425 51.1208 40.545858.4944 57.2986 53.8000 35.9806 34.018144.9186 57.9153 47.1525 45.2639 30.863953.8794 59.7736 48.7450 58.3431 22.0056

60.1044 61.8486 60.2700 41.5181 28.643156.2378 59.1944 53.7575 74.5222 46.609781.9906 60.9014 59.9825 32.3694 54.344460.9014 76.4472 71.3875 56.0236 56.061159.9825 71.3875 85.1650 32.2750 52.637532.3694 56.0236 32.2750 234.1306 52.368154.3444 56.0611 52.6375 52.3681 155.755648.3503 57.4569 51.9450 48.4097 41.047250.1550 59.8750 53.0475 53.8000 47.200048.2636 63.7278 62.0550 48.8764 44.438958.8567 58.5792 50.3175 79.7708 43.7708

52.8222 62.4181 64.3125 62.6278 36.065356.5847 50.2181 48.6375 44.1653 46.165334.6589 64.2847 54.6775 94.4736 33.086140.7383 56.9833 49.0225 28.5167 46.816740.3586 56.4153 52.7075 46.4139 49.663954.1686 61.8278 60.6775 53.0264 43.301452.1872 55.8931 57.9425 46.1653 48.015354.7931 63.5889 60.4000 36.0569 58.869434.1792 53.2542 50.4125 42.2083 36.033337.2036 58.3903 49.0850 90.9889 33.176446.9550 58.9000 55.1225 51.7000 37.6375

60.8242 62.1792 60.3025 44.2458 39.495860.0175 63.6000 60.4350 51.9375 47.775061.6758 63.5833 63.5475 41.4417 37.616759.2258 59.1708 56.2475 50.8042 53.704258.3731 61.0264 53.0225 69.7569 53.6944


57.6917 53.7375 46.4292 68.2750 56.133360.8972 64.8250 63.3014 62.4333 52.977869.6161 86.6775 89.0894 49.5242 49.576461.5950 62.7850 59.8925 61.5925 49.537583.2556 86.8375 93.2597 58.2333 52.981958.4722 65.3625 62.6639 63.0083 55.477864.9461 66.8925 65.6469 58.9192 52.213959.4031 64.7550 63.1247 55.5358 51.881973.3083 73.2375 75.4083 62.9250 50.966772.3506 88.7350 91.1022 56.2633 58.894470.1864 72.6925 74.9706 51.2858 47.8986

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66.2506 62.8225 57.3022 64.8508 57.806965.7553 73.8950 71.5336 66.3517 62.572271.2047 66.3350 62.5814 62.1383 50.677861.3608 67.2825 65.8683 58.3475 57.129259.0597 61.2750 58.5639 57.6583 52.4653

74.9631 90.4225 85.7522 57.7133 50.656962.8947 63.6300 62.1039 61.9483 52.802854.0197 56.1300 57.1164 57.7483 54.940363.9614 72.3050 70.1206 65.9233 58.136148.3503 50.1550 48.2636 58.8567 52.822257.4569 59.8750 63.7278 58.5792 62.418151.9450 53.0475 62.0550 50.3175 64.312548.4097 53.8000 48.8764 79.7708 62.627841.0472 47.2000 44.4389 43.7708 36.065382.3739 78.6550 75.5831 57.0483 53.911178.6550 106.7650 102.8575 60.2950 51.7625

75.5831 102.8575 114.7072 54.5683 50.031957.0483 60.2950 54.5683 71.3100 60.916753.9111 51.7625 50.0319 60.9167 85.988947.2611 51.3500 45.2319 59.6667 47.888974.5019 103.9100 107.5453 58.0592 63.105693.2592 126.8650 129.8117 50.3325 39.495882.0656 115.4675 118.1372 54.2983 51.806970.0131 81.2350 83.4272 60.5883 55.306959.9011 66.3700 63.7919 59.5017 63.888962.7653 69.5750 77.7861 53.8792 48.534789.5458 126.6000 130.8958 46.8250 38.1167

74.0981 86.0650 87.3672 62.9908 60.694481.2175 92.0650 89.6950 57.4075 51.225064.8658 62.7975 64.5508 58.0425 55.629265.4175 62.1150 60.5950 61.6200 56.962562.5342 63.6775 62.4367 60.8075 57.858365.5092 73.5150 70.0117 61.3700 56.645861.3578 66.9150 63.9811 65.5867 54.5972


54.7083 52.9208 39.0750 37.8667 54.054260.2778 56.0236 65.0167 62.4389 64.376445.8389 88.6031 116.3608 104.2369 73.141952.7125 50.9650 56.8850 50.9325 56.765042.9569 91.8778 112.9542 94.9347 83.647253.0403 61.0486 57.8167 52.9764 61.738955.3264 55.7131 68.8508 66.4794 65.826956.6444 51.7228 75.9467 68.7572 66.179750.1542 69.3667 78.1625 62.0083 73.1708

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42.2444 98.2178 103.5267 94.9047 71.637250.0486 64.1894 99.9717 89.0406 69.800656.0944 54.3803 55.3642 56.1047 65.537262.8222 65.1439 81.1908 72.5761 72.153655.7778 56.6886 72.4892 62.9589 66.0039

53.6917 61.8342 69.1450 68.4108 63.113353.0903 55.1361 60.8042 52.8389 57.963939.9944 96.5303 103.6142 87.5547 67.712246.4903 63.9186 60.1467 49.9864 58.058952.4653 53.1936 55.5842 48.9114 60.483958.7986 69.2894 73.8842 72.0656 65.875656.5847 34.6589 40.7383 40.3586 54.168650.2181 64.2847 56.9833 56.4153 61.827848.6375 54.6775 49.0225 52.7075 60.677544.1653 94.4736 28.5167 46.4139 53.026446.1653 33.0861 46.8167 49.6639 43.3014

47.2611 74.5019 93.2592 82.0656 70.013151.3500 103.9100 126.8650 115.4675 81.235045.2319 107.5453 129.8117 118.1372 83.427259.6667 58.0592 50.3325 54.2983 60.588347.8889 63.1056 39.4958 51.8069 55.306967.4139 30.5931 45.5208 46.0569 55.856930.5931 145.9122 126.8933 124.2853 76.680345.5208 126.8933 204.2400 165.9217 96.414246.0569 124.2853 165.9217 161.7822 92.579755.8569 76.6803 96.4142 92.5797 84.862255.1139 59.8106 66.4408 68.6719 67.0119

48.8472 62.2264 89.7958 73.6986 68.811138.0042 137.4292 179.4375 159.7458 92.245844.6444 100.0128 105.3692 104.9522 77.057247.0500 91.3350 112.9775 97.9800 78.310050.6667 54.9442 64.0475 62.6908 65.623356.5000 52.4975 64.1275 62.0425 67.272553.9833 56.3683 58.3025 54.1092 60.664258.3958 63.6808 76.8400 75.2592 66.939260.9597 56.6614 64.6858 62.9286 65.6836


51.7833 49.9833 35.5500 54.9167 57.350060.4528 62.7319 58.2583 55.4639 61.437560.5464 72.1472 108.5542 67.4969 76.615053.9550 56.1000 62.3250 62.2100 60.547562.3444 79.3736 104.1333 85.9597 90.612555.8903 58.2569 66.1958 62.1139 64.475054.9539 61.7472 64.6542 62.1994 64.3925

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58.7469 58.4736 51.1958 52.0147 65.305057.9667 69.2417 90.9500 70.4333 78.450060.5119 64.4611 126.5583 82.8322 79.685056.5136 65.6653 90.0083 68.2981 70.755061.5244 56.0611 49.1958 59.4572 66.2600

70.7822 68.8931 64.4542 63.3061 71.857554.4953 59.3569 63.0333 64.5864 70.660060.0767 56.8917 66.4250 64.6033 62.820053.1778 59.1319 58.1458 46.5514 59.650059.6744 68.6486 103.7708 61.3197 81.497550.9803 60.7069 58.7583 61.7189 64.230055.4928 61.8944 45.0083 47.5314 58.280060.5511 60.0528 77.6458 73.6481 66.305052.1872 54.7931 34.1792 37.2036 46.955055.8931 63.5889 53.2542 58.3903 58.900057.9425 60.4000 50.4125 49.0850 55.1225

46.1653 36.0569 42.2083 90.9889 51.700048.0153 58.8694 36.0333 33.1764 37.637559.9011 62.7653 89.5458 74.0981 81.217566.3700 69.5750 126.6000 86.0650 92.065063.7919 77.7861 130.8958 87.3672 89.695059.5017 53.8792 46.8250 62.9908 57.407563.8889 48.5347 38.1167 60.6944 51.225055.1139 48.8472 38.0042 44.6444 47.050059.8106 62.2264 137.4292 100.0128 91.335066.4408 89.7958 179.4375 105.3692 112.977568.6719 73.6986 159.7458 104.9522 97.9800

67.0119 68.8111 92.2458 77.0572 78.310073.5489 58.7722 61.5417 62.6769 62.395058.7722 87.8806 75.9667 62.1361 64.600061.5417 75.9667 228.4750 106.6958 105.825062.6769 62.1361 106.6958 108.9622 82.565062.3950 64.6000 105.8250 82.5650 95.040056.3092 62.6917 66.4625 64.5308 63.597559.3825 61.3125 56.7625 65.1275 64.640057.7158 59.0833 62.6250 55.6442 65.815062.9533 60.2458 77.7750 64.6567 67.315059.8797 61.0556 62.2417 66.7261 62.3525


57.0875 61.3500 62.8375 57.1500 60.733357.2958 61.9000 60.3417 65.8417 62.856957.0267 55.3025 57.3483 64.2358 55.062261.4775 60.3850 62.5350 61.2475 65.435069.6458 69.2625 63.9167 61.6292 65.2861

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63.2958 62.2750 69.7792 63.1542 65.544463.8367 64.5800 61.0508 63.4133 65.064752.3533 58.7550 54.9717 60.4092 55.116169.2500 67.5500 71.7375 63.1375 67.104262.5858 56.5850 65.9267 66.0642 63.3711

63.8158 63.8800 58.7217 64.8217 60.382860.9358 65.2850 60.6142 63.1267 64.271158.3592 61.7825 65.4408 68.4283 63.500665.3108 66.6475 65.4017 60.6267 69.079459.5175 59.6700 57.0825 63.3200 62.724258.0958 56.9750 64.7167 57.1792 59.744455.7233 51.2475 68.6642 61.2017 61.458662.3158 61.2050 64.6217 57.1842 62.711958.8908 60.1300 63.2217 56.6842 57.849462.8033 62.6800 62.6967 66.7717 69.645360.8242 60.0175 61.6758 59.2258 58.3731

62.1792 63.6000 63.5833 59.1708 61.026460.3025 60.4350 63.5475 56.2475 53.022544.2458 51.9375 41.4417 50.8042 69.756939.4958 47.7750 37.6167 53.7042 53.694464.8658 65.4175 62.5342 65.5092 61.357862.7975 62.1150 63.6775 73.5150 66.915064.5508 60.5950 62.4367 70.0117 63.981158.0425 61.6200 60.8075 61.3700 65.586755.6292 56.9625 57.8583 56.6458 54.597250.6667 56.5000 53.9833 58.3958 60.959754.9442 52.4975 56.3683 63.6808 56.6614

64.0475 64.1275 58.3025 76.8400 64.685862.6908 62.0425 54.1092 75.2592 62.928665.6233 67.2725 60.6642 66.9392 65.683656.3092 59.3825 57.7158 62.9533 59.879762.6917 61.3125 59.0833 60.2458 61.055666.4625 56.7625 62.6250 77.7750 62.241764.5308 65.1275 55.6442 64.6567 66.726163.5975 64.6400 65.8150 67.3150 62.352571.5900 66.8975 66.4600 60.9975 63.089266.8975 70.9900 63.8400 63.3900 65.077566.4600 63.8400 76.5650 61.5275 62.835860.9975 63.3900 61.5275 71.6650 65.498363.0892 65.0775 62.8358 65.4983 75.5956

Dup cum se poate observa, matricea obinut este o matrice ptratic de 13 linii i 13coloane. Semnificaia pe care o are semnul unui element oarecare de formaxi,j este dupcum urmeaz: un semn pozitiv dezvluie o variaie n acelai sens (fie ambele cresc, fiedescresc) ale variabilelor de ordin i i j; un semn negativ indic variaia n sensuri

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contrare. Valoarea zero a unui element din matricea de covariaie indic independenastatistic ntre cele dou variabile de ordin i ij.

Reprezentarea grafic a matricei de covariaie calculat pentru vectorul de antrenare

n graficul de mai sus au fost reprezentate valorile de pe liniile din matricea de covariaien funcie de numrul caracteristicii.

3. Calculul valorilor proprii si sortarea acestora n sens descresctor.

val_proprii =

1.0e+003 *

-0.00000.00000.00000.00010.00010.00020.00020.0003

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0.00040.00070.00090.00100.0013

0.00160.00190.00220.00240.00290.00320.00370.00450.00510.00580.0070

0.00720.00940.01150.01190.01410.01430.01780.01950.02310.02900.0317

0.03880.04820.07040.07830.08710.09120.16790.24400.45532.8885

val_proprii_sortate =

1.0e+003 *

2.88850.45530.24400.1679

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0.09120.08710.07830.07040.0482

0.03880.03170.02900.02310.01950.01780.01430.01410.01190.01150.0094

0.00720.00700.00580.00510.00450.00370.00320.00290.00240.00220.0019

0.00160.00130.00100.00090.00070.00040.00030.00020.00020.00010.00010.00000.00000.0000

4. Matricea vectorilor propii corespunzatori valorilor proprii ordonate descrescator Matricea transformatei KarhunenLove.

vect_proprii_sortati =

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Columns 1 through 7

0.1292 -0.1942 0.0651 -0.1272 0.0431 -0.0019 0.10540.1390 -0.1064 -0.0606 0.0872 -0.0651 0.0002 0.07260.1560 0.1464 -0.1599 0.0047 0.0217 -0.2243 0.0542

0.1341 -0.1073 0.0239 -0.0987 0.0250 0.2370 -0.04410.1720 0.0683 0.0386 -0.0038 0.2533 0.0541 0.09530.1413 -0.1115 0.0327 -0.1598 0.0070 0.0832 -0.14940.1414 -0.0718 -0.0612 0.0042 -0.0124 0.1089 0.01760.1335 -0.0547 -0.1448 0.0081 -0.1581 -0.1045 0.29810.1582 -0.0513 0.0416 -0.1852 0.2281 0.2186 -0.08190.1629 0.1235 0.0613 -0.2032 0.0434 -0.0408 -0.24280.1473 0.0494 -0.1399 0.0488 -0.0726 0.1253 0.03130.1397 -0.1334 -0.0140 -0.0384 0.0159 0.0261 0.15560.1536 -0.0812 -0.0610 0.0762 -0.0807 -0.1772 0.24350.1464 -0.0845 -0.0161 -0.0759 0.1295 0.1620 0.1691

0.1385 -0.0563 -0.0279 0.0120 -0.1523 -0.0047 -0.01150.1343 -0.0878 -0.1104 -0.1373 0.1293 -0.1029 0.03050.1627 0.0951 -0.0232 -0.2382 0.4670 -0.3626 0.20950.1392 -0.0981 0.0217 -0.2330 0.1291 0.0371 0.00360.1307 -0.1253 -0.1008 -0.1090 -0.0501 -0.1266 -0.00270.1486 -0.0669 0.0681 0.0315 -0.1150 0.1033 -0.05420.1202 -0.1857 -0.1684 0.0548 -0.0237 0.0144 -0.12420.1381 -0.1153 -0.0395 0.0562 0.0515 -0.1573 -0.22060.1276 -0.1103 -0.1394 0.0802 -0.0638 -0.2204 -0.37080.1289 -0.2176 0.7975 0.1471 0.0446 -0.0242 0.11150.1011 -0.1207 -0.0721 0.7565 0.4419 0.0100 -0.1113

0.1547 0.0211 -0.0237 -0.0453 0.0480 0.1087 0.10240.1753 0.1449 0.0177 0.0268 -0.0092 -0.0185 0.06070.1750 0.1755 0.0011 0.0380 -0.0062 -0.0508 -0.08940.1359 -0.1563 0.0749 -0.0061 -0.0777 -0.0157 0.07470.1251 -0.1438 0.0339 -0.0095 -0.2549 -0.3500 -0.19930.1165 -0.1546 -0.0911 0.0745 -0.1745 0.1089 0.11420.1732 0.2103 0.2741 -0.0166 -0.0092 -0.3969 -0.12960.1995 0.3905 -0.0699 0.1097 -0.0338 0.0989 0.28550.1846 0.3124 0.0205 0.1901 -0.2549 -0.0470 0.09610.1596 0.0216 -0.0313 0.0376 -0.1220 0.0436 0.05720.1374 -0.0649 -0.0554 0.0668 -0.1656 -0.1288 0.01050.1471 -0.0092 -0.1251 0.0830 0.1567 -0.0041 -0.01910.1961 0.4345 0.0739 -0.0238 0.0679 0.2577 -0.37570.1622 0.0811 0.2229 0.0237 -0.2504 0.1303 -0.03000.1662 0.0860 0.0044 -0.0537 0.0127 0.0473 0.13290.1408 -0.0823 -0.0687 -0.0515 -0.0343 0.1547 -0.16340.1414 -0.1129 -0.0570 0.0191 -0.0784 0.1454 -0.02190.1413 -0.1081 -0.0893 -0.1266 0.0616 0.0010 -0.15920.1472 -0.0484 -0.0493 0.0700 -0.1014 0.0563 -0.0315

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0.1447 -0.1126 0.0221 0.0364 0.0041 0.1479 0.0094


-0.0593 -0.0366 -0.0792 -0.2014 0.1106 -0.1784 0.2250

-0.1382 -0.1853 -0.0182 -0.0867 -0.0386 0.0430 0.1975-0.0294 0.0994 -0.1970 0.1472 -0.1323 -0.0690 0.2625-0.1439 0.0223 0.0659 0.0582 0.3124 0.0756 -0.03960.4789 -0.0527 0.1678 0.0310 -0.0501 0.0075 0.0598-0.1250 -0.1437 -0.0155 -0.0271 0.0952 0.1223 -0.09880.0451 0.1564 -0.1459 0.0310 -0.1229 0.1670 -0.04420.0352 -0.0753 0.0131 -0.0841 -0.1836 0.1488 0.01370.1216 -0.2304 0.2279 -0.0375 -0.1126 -0.0194 0.1437-0.1387 0.1795 0.1081 0.2507 -0.0276 0.0698 -0.01780.0826 0.1701 -0.1567 0.0589 -0.1283 -0.2700 -0.04910.0422 0.1832 0.1206 -0.1606 -0.0548 0.2390 0.2667

-0.1234 -0.3381 0.4249 0.0964 0.1010 -0.3237 -0.22260.1191 0.2407 -0.0929 -0.0195 -0.1998 -0.1096 -0.2452-0.0527 0.1050 -0.0169 0.1023 0.0138 0.0299 -0.0450-0.1328 0.0185 -0.1269 0.0320 -0.1107 -0.1679 0.0136-0.2268 0.2121 0.0442 0.0905 -0.0893 0.0731 -0.15140.0377 -0.0006 -0.1692 0.0584 0.3028 0.0295 0.11140.0437 -0.2883 -0.1759 0.0092 -0.0589 -0.0272 0.1151-0.1610 -0.0293 -0.0233 0.0963 0.0710 -0.1285 -0.1477-0.1486 -0.1441 -0.2555 -0.2828 0.1012 -0.0892 0.10190.1899 -0.0179 -0.2020 -0.0196 -0.0748 -0.0184 -0.11220.2595 -0.1249 0.0728 -0.1050 -0.2377 0.0239 -0.2736

-0.0116 -0.1255 -0.1071 0.0614 -0.2558 -0.0512 -0.0421-0.0967 0.1482 0.0754 -0.0580 0.1400 -0.0052 0.03820.1265 0.2229 0.1592 -0.1837 0.0893 -0.1705 0.1114-0.1512 -0.0094 0.0107 -0.1800 0.1722 0.2831 -0.11860.0465 -0.2060 -0.0617 0.0231 0.1165 0.4184 -0.1976-0.1291 0.0444 -0.0408 0.0200 0.0396 -0.0181 0.12070.0851 0.2664 0.2122 0.0956 0.0992 -0.2240 0.0874-0.2334 -0.0245 0.0045 0.0907 -0.2553 0.0498 -0.08690.0338 -0.0176 -0.1716 -0.2505 0.1655 -0.0075 0.10140.0513 -0.1755 -0.1667 0.0506 0.1402 -0.2669 -0.1513-0.0152 0.1165 -0.1461 -0.0577 -0.0505 0.0458 -0.00210.1306 -0.0916 0.1050 0.0158 -0.2446 0.2373 0.1364-0.0458 0.0804 0.3730 0.1473 0.0397 0.1553 0.15610.1483 -0.2717 -0.1043 0.5285 0.1615 0.0390 0.2085-0.2878 -0.0893 0.1729 -0.0732 -0.3085 -0.1796 0.21560.2045 0.1664 0.0276 0.1659 0.2675 -0.0408 -0.02750.1306 -0.0068 0.1741 -0.3573 0.1167 0.0584 -0.14090.1379 0.0809 -0.0808 0.0103 0.0436 -0.0559 -0.01950.1576 0.0800 -0.0527 -0.0672 -0.0114 -0.0267 0.0741

19

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-0.0493 -0.0052 0.0536 -0.1343 0.0335 -0.1134 -0.3308-0.1857 0.0654 0.0345 -0.0706 0.0162 0.0553 0.0759-0.1109 0.1167 -0.1078 0.2010 -0.0315 0.1579 -0.2087


-0.1290 -0.0872 0.1617 0.0649 -0.2456 0.1019 0.0048-0.2018 0.0572 -0.2343 -0.3427 -0.0463 0.3125 0.1790-0.1980 -0.1150 0.1135 0.1450 -0.2104 -0.3168 -0.05080.0568 -0.0163 -0.1322 0.0968 0.0058 0.0854 0.04590.0703 -0.2135 0.1088 0.1489 0.0734 0.0799 -0.05860.0330 0.3991 0.1564 -0.0146 -0.1450 -0.2607 -0.1409-0.1326 0.0384 -0.2078 0.0184 -0.1476 0.0961 -0.2133-0.0497 0.2860 -0.0799 0.2183 -0.0615 0.0674 -0.11730.0492 0.0677 0.0989 0.0656 0.0632 0.0507 0.0316-0.0833 0.2308 0.0461 0.1911 -0.0267 -0.0395 -0.0674

0.0685 0.1167 -0.2501 -0.0128 -0.1179 -0.1457 0.24030.0640 0.0621 -0.0765 0.1073 0.0032 -0.0790 -0.00420.0208 -0.1919 -0.1487 0.1585 -0.2981 -0.1938 -0.0422-0.0288 -0.0931 0.0697 0.0575 0.0813 0.0701 0.0181-0.2106 -0.2159 -0.1787 0.1102 0.2552 -0.3728 -0.0169-0.0565 -0.2451 0.1179 -0.0216 0.1994 0.0091 -0.24540.2724 0.0390 -0.1291 -0.2087 -0.0641 0.0584 0.1985-0.0993 -0.0617 0.0423 0.1733 -0.0235 0.0102 0.0598-0.1437 0.0357 0.1052 -0.3293 0.3135 -0.1984 -0.0760-0.1170 -0.2200 -0.0262 -0.0884 0.1645 0.1001 0.01800.5646 -0.0758 -0.1020 0.2482 0.1261 -0.0757 0.1297

-0.1133 0.1164 0.0614 0.1808 -0.1389 0.0514 0.3184-0.0641 -0.0165 -0.0798 0.0136 -0.0227 0.0819 0.16320.0782 0.1132 -0.1674 0.0046 0.0630 -0.1196 -0.0702-0.0874 0.0543 0.1482 0.0016 -0.0036 -0.0697 -0.1090-0.1745 -0.0073 -0.3606 -0.1358 0.1143 0.0880 0.0234-0.0493 -0.0786 0.0620 0.0730 0.0855 0.0434 -0.1097-0.0163 -0.2176 -0.2523 0.1510 0.0053 0.1470 -0.04380.0977 -0.0915 0.1906 0.0234 -0.1333 0.2362 0.03730.1204 0.1898 -0.0075 0.1622 0.2703 0.2944 -0.3049-0.1408 -0.1000 0.2762 0.1176 -0.0245 0.2911 0.0313-0.1285 -0.1090 0.0526 -0.0646 0.0116 0.0242 0.12270.0519 0.3408 0.1466 0.1174 0.3045 0.0911 0.11460.2798 -0.0456 0.0554 -0.2078 -0.1668 0.0690 -0.27070.2246 -0.1184 0.1507 -0.0565 -0.0486 -0.1104 -0.00490.1266 -0.0041 0.1663 -0.2071 0.2035 -0.1299 0.35800.0210 0.1193 -0.1520 -0.1567 -0.0631 0.1242 -0.1491-0.0255 -0.0682 -0.0397 0.0465 -0.0118 0.0501 0.01200.0018 -0.0795 0.1641 -0.0921 -0.2914 -0.0356 0.2071-0.1759 0.1069 0.1062 -0.0946 0.0735 -0.1811 -0.0895

20

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0.2293 -0.1641 -0.1329 -0.1872 -0.0378 -0.1484 -0.24400.0278 0.0869 0.0730 -0.1285 -0.0559 -0.0039 0.04940.0281 0.0765 0.1459 -0.3220 -0.1925 0.0274 -0.1895-0.1092 0.1934 -0.1861 0.0538 -0.0190 -0.1291 0.06070.0131 -0.0566 0.1384 -0.0538 0.1998 -0.0572 0.1724


-0.0149 -0.1036 -0.0501 -0.0437 0.0401 -0.3021 -0.19400.0555 0.2045 -0.1333 -0.0418 -0.0589 0.1041 -0.20230.4519 -0.0729 0.1214 -0.0140 -0.1354 0.2298 -0.25590.0359 -0.1692 -0.0410 -0.0744 -0.1122 0.1525 0.0671-0.0737 -0.0348 0.0111 -0.1506 0.0102 0.1021 -0.2202-0.0277 -0.0554 0.0153 0.0570 -0.2583 0.0840 -0.1813-0.1573 0.1384 0.1811 0.0923 -0.0388 -0.0706 -0.1054-0.0142 -0.0175 -0.1302 0.0373 0.3227 0.1201 0.0146

0.1605 0.1062 -0.1677 -0.0392 0.1441 -0.1169 -0.25310.1003 0.2081 -0.1660 -0.1274 0.1166 -0.1491 0.27920.0669 -0.0385 -0.2042 -0.1384 -0.0637 -0.2196 0.0909-0.1200 -0.1982 -0.1415 0.1350 0.0244 0.2054 0.1973-0.2061 0.1592 0.0136 -0.0849 -0.1053 0.0689 0.06190.0446 0.0073 0.0283 -0.0860 -0.0367 0.2768 0.0083-0.1594 -0.1117 -0.0002 0.0530 0.3205 -0.1258 -0.04280.0649 -0.0535 -0.2196 0.1536 -0.1679 -0.3206 0.1214-0.1146 0.0109 0.0378 0.1646 0.1057 -0.1143 -0.1034-0.3853 -0.1564 -0.0175 -0.1393 -0.1346 0.0984 0.0724-0.1698 0.1025 -0.0087 -0.1440 0.2267 0.0373 0.0196

0.0449 0.1017 -0.0585 0.1538 0.1266 0.0825 -0.10210.1881 0.1155 0.0224 0.1823 0.0651 0.0936 0.0034-0.1074 0.3139 0.3189 -0.1337 0.2025 0.0679 0.1070-0.0894 -0.4342 -0.0809 0.3038 -0.1356 -0.0647 -0.09130.1295 -0.1273 0.1175 -0.0286 -0.0316 -0.0409 0.0290-0.0399 0.0412 -0.1259 -0.0363 0.0607 -0.0543 0.04470.1752 0.0763 0.1948 0.0055 -0.1654 0.0357 0.1318-0.0160 0.0637 0.4967 0.1186 -0.0258 -0.2400 -0.19460.3035 -0.0424 -0.2137 -0.3123 0.0387 -0.1509 0.02480.0301 -0.1021 0.0218 -0.0407 0.3077 -0.0099 0.1674-0.0022 0.1458 -0.0144 -0.0034 -0.0931 0.0340 -0.18890.0829 -0.1126 0.1472 0.0529 -0.1479 0.0203 0.2505-0.0597 0.0556 -0.1294 -0.0276 -0.2136 0.1550 0.24380.0037 -0.1098 0.0697 -0.0089 -0.0716 -0.1096 0.0172-0.2187 -0.1344 -0.1477 -0.1693 0.0733 0.1023 -0.1317-0.0617 0.3826 -0.0337 0.0654 -0.3220 -0.1766 0.21040.1417 -0.2448 0.2162 -0.2342 0.0116 -0.0139 0.00720.0480 -0.1150 0.1157 0.3005 -0.0012 0.0797 0.1993-0.2123 -0.0724 0.1043 0.0749 0.1026 0.1514 0.0240

21

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0.0777 0.1024 -0.1278 0.3872 0.1889 -0.0783 -0.12350.1572 0.0720 -0.0845 0.2949 0.0948 0.0873 0.21490.0165 0.0287 0.2209 -0.0904 0.0561 0.0299 0.1001-0.1387 -0.0685 0.0642 -0.1434 -0.0999 -0.3339 0.05240.2122 -0.1133 0.0303 -0.2148 0.1069 0.1208 0.0776

-0.0578 -0.0330 -0.0442 -0.0972 -0.1147 -0.1029 -0.1740-0.1159 0.2020 -0.2356 0.0183 -0.1913 0.2524 -0.2127


0.0609 0.0465 -0.0380 0.1652 0.2231 -0.0055 0.1312-0.0268 0.3556 -0.1682 0.0268 0.0283 0.0312 -0.0313-0.0150 -0.0706 0.1518 -0.1143 -0.1261 -0.0555 -0.07320.2377 0.0397 0.1698 -0.1299 -0.0445 0.0074 0.00420.1018 0.3671 0.0507 -0.0381 -0.0508 0.1464 -0.01990.0312 0.2194 -0.1495 0.2293 -0.1001 -0.1449 -0.0142

-0.2459 0.1855 -0.0121 -0.0605 0.1894 0.0698 -0.2084-0.0175 -0.0719 -0.2908 -0.1747 -0.2785 -0.0642 -0.0439-0.1468 -0.2468 0.0714 0.1125 -0.0359 -0.0210 0.31350.1792 0.2433 0.0833 0.1420 -0.0196 -0.0136 0.0219-0.2628 -0.0790 -0.0516 -0.0927 0.2386 -0.0138 0.21580.0375 0.0099 -0.0150 -0.0357 -0.0823 0.0722 0.21490.0622 -0.0176 0.0145 -0.0594 0.0457 -0.0154 0.0442-0.0603 -0.1434 -0.1767 0.2043 0.1544 -0.1090 -0.0247-0.0910 0.3090 -0.0139 0.0637 0.0222 -0.2134 0.18710.2698 0.0291 -0.3140 0.0251 0.0382 0.0597 -0.1805-0.0100 -0.0336 0.0883 -0.0351 -0.1337 0.1172 -0.0164

-0.2468 -0.2217 -0.1325 -0.0013 -0.0113 0.0118 -0.32040.1448 -0.1103 0.2561 -0.2005 0.0657 0.1475 -0.1106-0.1629 -0.2255 -0.0506 0.3300 -0.4770 -0.0893 -0.02190.0804 0.0919 -0.0512 -0.0492 0.0730 -0.0170 -0.08000.2880 -0.1035 -0.1036 0.1842 0.0401 0.0218 0.01580.0214 0.0359 0.2080 0.0790 -0.0268 -0.0689 -0.07620.0399 -0.0336 -0.0195 -0.0648 0.0340 0.0151 -0.0889-0.0639 -0.0049 -0.0277 -0.0216 -0.0707 0.0081 -0.05910.2818 -0.0042 0.1495 0.0743 -0.1094 -0.2552 -0.1084-0.0252 -0.1047 -0.0267 -0.1329 0.0977 -0.1266 0.24740.0021 -0.1337 -0.0633 -0.0683 0.0499 0.0364 -0.1342-0.2126 0.1299 0.4568 0.0567 0.0698 -0.2738 -0.1774-0.1258 -0.1153 -0.0029 -0.1358 0.1135 0.0298 0.01310.0498 0.0312 0.0603 -0.0464 -0.0416 0.3539 0.2325-0.1999 0.0921 -0.0559 -0.0284 -0.1101 0.0274 0.3336-0.1127 0.2126 0.0874 0.0354 -0.1146 0.0681 -0.08040.3132 -0.1239 0.0348 0.3162 0.1085 -0.0904 0.0761-0.2109 -0.0150 0.1190 0.0959 -0.0909 -0.1835 -0.1914-0.0547 0.0006 -0.3264 0.1363 0.1524 0.0238 -0.1377

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0.0159 -0.1442 -0.0506 0.0794 0.1926 -0.1191 0.2080-0.0063 -0.0565 -0.1356 -0.2083 0.0929 -0.0241 -0.13170.1091 0.0101 -0.1216 -0.2089 -0.0494 0.2024 -0.1652-0.0238 -0.1437 0.0645 0.0683 0.3241 0.1205 -0.0954-0.1448 0.1127 -0.0234 0.1046 -0.2016 0.4276 0.1601

0.1878 -0.1479 -0.0436 -0.3453 -0.3289 -0.2236 0.0935-0.1650 0.0723 -0.0447 -0.2513 -0.0118 -0.1078 -0.01710.0092 -0.2232 0.2873 0.1742 -0.0249 0.4072 -0.10830.1091 -0.0107 0.1156 -0.2458 0.1839 -0.1260 0.1738


-0.1979 -0.0723 0.3388 0.1719 -0.0659 -0.0654 -0.18870.2312 0.2638 0.0472 -0.0914 -0.0817 -0.0288 0.1011-0.0415 0.1476 0.0375 0.0455 -0.0208 0.0319 0.0438-0.3455 -0.1804 0.0820 0.0934 -0.0941 -0.3268 0.3715

0.0902 0.0999 -0.1530 0.0779 0.0705 -0.3234 0.03590.0439 -0.1229 -0.0358 -0.1445 0.4701 -0.0614 -0.0828-0.4577 -0.0571 -0.0076 0.1418 -0.0083 0.2310 0.0559-0.1154 -0.1908 -0.0883 -0.0298 -0.1668 -0.0784 -0.0262-0.0455 -0.1303 -0.1755 0.0125 -0.0477 0.3650 0.1648-0.0002 0.3022 0.1532 -0.1279 -0.2478 0.1176 0.1450-0.0753 0.1161 -0.2307 -0.1196 0.2560 -0.2476 0.2086-0.0395 0.1704 -0.0468 -0.1521 0.0556 0.0356 -0.1134-0.0538 0.0698 0.0262 -0.1251 -0.0170 0.1245 -0.05920.1617 -0.1821 0.4486 -0.3733 -0.1130 -0.0652 0.04620.2879 -0.1922 0.0711 0.2713 0.0381 0.0278 0.0527

-0.0505 -0.0746 -0.3653 -0.1285 -0.0956 -0.0768 -0.07550.0437 -0.0369 0.1196 0.1309 0.1664 -0.0133 0.00630.2390 0.2143 -0.1247 0.0820 -0.0314 0.2365 0.1783-0.1225 -0.0846 0.1553 -0.2330 0.1828 0.0127 0.2159-0.2192 0.2491 -0.0141 0.1081 0.0835 -0.2141 0.01890.0945 0.0543 0.0837 0.0809 -0.0468 0.1203 0.0898-0.0385 -0.0315 -0.2085 0.1415 0.0532 -0.1109 -0.0820-0.0690 0.0839 0.0611 -0.1286 -0.1467 0.1057 0.07090.0257 0.0974 -0.0286 0.0563 -0.0584 0.0180 0.0504-0.0268 -0.0581 0.0568 -0.0596 0.0628 0.0654 0.09140.0484 -0.2035 -0.1550 -0.0001 0.1386 0.2881 -0.04060.0766 0.1439 -0.1442 -0.3385 -0.1398 -0.1244 0.17960.0425 -0.0487 0.1490 -0.0248 0.2855 0.0632 -0.23260.0097 -0.0018 -0.2177 -0.2708 0.0993 -0.0773 -0.24620.0775 0.0023 0.0914 0.0657 0.1675 -0.0766 -0.01920.1788 -0.0469 0.0124 0.1835 0.2879 0.1658 0.1676-0.1202 -0.2486 -0.0085 -0.0328 -0.0773 0.0043 -0.0086-0.0764 -0.0176 0.0574 0.0316 -0.1002 0.1311 -0.11960.0180 0.0586 -0.0936 0.1221 -0.0032 0.1443 0.1958

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0.0661 -0.1801 0.0967 0.1411 -0.0971 -0.1716 0.1185-0.2240 -0.0117 -0.0303 0.0236 -0.0665 0.0662 -0.02540.1113 -0.0651 0.0511 0.0707 -0.1031 -0.1125 -0.1079-0.0052 -0.0493 0.0709 0.0068 0.0784 -0.0978 -0.15130.1214 -0.1927 0.0114 -0.1701 0.0354 0.1138 0.0945

-0.0107 0.3156 0.0498 0.2549 0.0825 -0.1217 -0.0751-0.0904 0.0777 0.0309 -0.1630 -0.0803 0.0996 -0.25200.0674 0.2403 0.2597 0.0941 0.0124 -0.0150 -0.17980.1893 -0.1074 -0.1529 0.2462 -0.2470 -0.0331 0.04690.2836 -0.2184 -0.1034 0.0006 -0.2916 -0.2218 -0.1878-0.1071 0.0557 -0.1478 0.0755 -0.1363 0.1197 -0.3836


0.3367 -0.1783 -0.1223-0.2174 -0.1087 -0.0879

-0.1225 -0.0522 -0.0647-0.2940 -0.1760 -0.07810.1850 0.1452 0.2125-0.0053 0.0056 0.0817-0.0780 0.3672 0.09970.1331 -0.2967 0.2492-0.2585 0.0346 0.00850.1959 0.0373 0.12430.1490 -0.0645 0.03910.0684 0.2911 -0.5477-0.0657 0.1080 0.0038

-0.1187 0.1172 0.0498-0.1674 -0.0439 -0.0529-0.2060 0.0475 -0.0539-0.0406 -0.0695 0.01420.0699 -0.1849 0.02590.1782 0.0610 -0.03810.2072 0.1929 -0.01450.1175 0.1737 0.1740-0.1691 -0.0246 -0.24550.0474 -0.1368 -0.00530.0075 -0.0536 -0.03480.0256 -0.0895 -0.00480.2651 -0.1075 0.06090.1167 -0.0515 -0.03690.0189 0.0262 -0.1192-0.1931 -0.0134 0.1281-0.0777 -0.0697 -0.11510.0790 -0.0234 0.1260-0.0541 0.1378 0.2416

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-0.0646 0.0123 -0.2661-0.0493 -0.0061 0.02820.0637 -0.1631 -0.21010.0227 0.0609 0.12040.0696 0.0907 0.0883

0.0207 -0.0570 -0.02350.0348 0.0503 -0.0696-0.1995 -0.1078 0.1725-0.0893 -0.4068 0.0013-0.2731 0.2000 0.22650.1840 0.1788 -0.23440.0216 0.2337 0.13550.1455 -0.1721 0.0506

5. Reprezentarea grafic a valorilor proprii sortate.

6. Variatia energiei conservate si a erorii patratice in functie de numarul variabil decomponente retinute.

Se definete factorul de conservare a energiei ca raportul dintre suma primelor celor maimari m valori proprii reinute i suma tuturor valorilor proprii:

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%100

1

1

=

==n

ii

m

ii

Facorul de prezervare a energiei n funcie de numrul de componente reinute

Dup cum se poate observa, prima component principal are un aport considerabil(75%) n ceea ce privete conservarea energiei, celelalte componente aducnd un aportdin ce n ce n ce mai mic, astfel nct se poate afirma c de la a 8-a componentcontribuiile sunt neglijabile.

Se definete eroarea ptratic medie, er = 1

Eroarea ptratic medie n funcie de numrul de componente reinute7. Aflarea numarului de componente retinute pentru a indeplini urmaroatele cerinte:

a) ultima componenta retinuta sa reprezinte 1% din valoarea primei componenteretinute (criteriul pragului cu prag = 1%*valoarea maxima)

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b) varianta > 95%*energia totala (criteriul energiei totale)c) varianta > 99%

componente_retinute_a = 8Avnd n vedere rezultatele obinute la punctul 3, era de ateptat un asemenea

rezultat.componente_retinute_b = 4componente_retinute_c = 7

8. Clasificarea vectorilor de test pe baza componentelor retinute la punctele 7.b) si7.c) folosind algoritmul Neareast Neighbour si algoritmul celei mai apropiate medii.Evaluati scorul de recunoastere corecta. Comparati rezultatele obtinute cu celecorespunzatoare folosirii tuturor ceor 13 caracteristici ce compun baza e date.

Rezultatele obinute folosind algoritmul Neareast Neighbour:clasa =

scor_recunoastere =

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Se poate observa c, desi ne-am fi asteptat la o crestere liniara a scorului derecunoastere odata cu cresterea numarului de componente retinute, acest lucru sedovedeste a fi adevarat pana la un punct. In cazul in care folosim doar primele 4componente se obtine un rezultat mai slab decat atunci cand se folosesc 7 componente,ns atunci cnd sunt luate n considerare toate cele 13 acest lucru pare s introduc erori

de clasificare.Rezultatele obinute folosind algoritmul Neareast Mean:

Vectorii medie corespunztori fiecrei clase n variantele 4, 7, respectiv 13componente reinute:

avg_1 =

avg_2 =

avg_3 =

Distanele fa de mediile claselor corespunztoare fiecrui vector de test n variantele4, 7, respectiv 13 componente reinute:

dist_1 =

dist_2 =

dist_3 =

clasele =

4, 7, respectiv 13 componente reinute

scor_nearest_mean =

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Se poate observa c, dei scorul obinut 4 componte reinute este acelai (0.9326) nambele varinte, performanele algoritmului Nearest Neighbour sunt mult mai sczuteatunci cnd sunt folosite mai multe componente, semn c, dei acesea sunt mai puinsemnificatie n termeni de energie nmagazinat, intriduc totui erori masive

Anexe

Codul surs folosit

% 1. Calculul vectorului medie pentru vectorul de antrenaremed_antrenare = mean(b,2)mat = b;% 2. Determinarea matricei de covarianta pentru vectorul de antrenarecovarianta_antrenare = zeros(size(b,2));sigma = zeros(size(b,2));for i = 1:size(b,2)

for j = 1:size(b,2)for k = 1:size(b,2)sigma(i,j) = sigma(i,j) + (mat(i,k) - med_antrenare(i)) *

(mat(j,k) - med_antrenare(j));end

endendcovarianta_antrenare = sigma/size(b,2)figure(1),plot(covarianta_antrenare)% 3. Calculul valorilor proprii si sortarea acestora[v,d] = eig(covarianta_antrenare);val_proprii = diag(d)

[val_proprii_sortate, index] = sort(val_proprii,'descend')% 4. Matricea vectorilor propii corespunzatori valorilor propriiordonate% descrescator -- Matricea transformatei K-Lvect_proprii_sortati = zeros(size(val_proprii));for i = 1:size(val_proprii)

vect_proprii_sortati(:,i) = v(:,index(i));endvect_proprii_sortati% 5. Graficul valorilor proprii sortate

figure(2)plot(val_proprii_sortate)% 6. Variatia energiei conservate si a erorii patratice in functie de% numarul variabil de componente retinutefact_cons_en = zeros(45,1);energie_totala = sum(val_proprii_sortate);for i = 1:45

energie_conservata(i) = 0;

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for j = 1:ienergie_conservata(i) = energie_conservata(i) +

val_proprii_sortate(j);endfact_cons_en(i) = energie_conservata(i)/energie_totala;

ender_partatica_medie = zeros(size(b,1),1);er_partatica_medie = ones(size(b,2),1) - fact_cons_en;figure(3)plot(fact_cons_en)figure(4)plot(er_partatica_medie)% 7. Aflarea numarului de componente retinute pentru a indeplini% urmaroatele cerinte:% a) ultima componenta retinuta sa reprezinte 1% din valoarea primei% componente retinute (criteriul pragului cu prag = 1%*valoareamaxima)% b) varianta > 95%*energia totala (criteriul energiei totale)

% c) varianta > 99%prag_a = val_proprii_sortate(1)/100;m = 1;while ((val_proprii_sortate(m)>= prag_a)&& (m < size(b,1)))

m = m+1;endcomponente_retinute_a = m-1val_proprii_sortatem = 1;prag_cons_energie_b = 0.95;while ((fact_cons_en(m)

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dist_22 = zeros(size(mat_test,2),size(mat,2));dist_33 = zeros(size(mat_test,2),size(mat,2));pozitie_minim = zeros(size(mat_test,2),1);componente(1,1) = componente_retinute_b;componente(2,1) = componente_retinute_c;componente(3,1) = 45;for i = 1:size(mat_test,2)

for j = 1:size(mat,2)for k = 1:componente(1)

dist_11(i,j) = dist_11(i,j) + (mat_test(index(k),i) -mat(index(k),j))^2;

end% if (dist_11(i,j)

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% if (dist_33(i,j)

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avg_1(j,m) = avg_1(j,m) + mat(index(j),i + nr_comp(1));else

avg_1(j,m) = avg_1(j,m) + mat(index(j),i +nr_comp(1)+nr_comp(2));

endend

endendavg_1(:,m) = avg_1(:,m)/nr_comp(m);endfor m = 1:3for i = 1:nr_comp(m) % vectori din clasa 1

for j = 1:componente(2) % componente retinuteavg_2(j,m) = avg_2(j,m) + mat(index(j),i);

endendavg_2(:,m) = avg_2(:,m)/nr_comp(m);end

for m = 1:3for i = 1:nr_comp(m) % vectori din clasa 1

for j = 1:componente(3) % componente retinuteavg_3(j,m) = avg_3(j,m) + mat(index(j),i);

endendavg_3(:,m) = avg_3(:,m)/nr_comp(m);endavg_1avg_2avg_3dist_1 = zeros(size(mat_test,2),3);dist_2 = zeros(size(mat_test,2),3);dist_3 = zeros(size(mat_test,2),3);for m = 1:3for i = 1:size(mat_test,2)

for j = 1:componente(1)dist_1(i,m) = dist_1(i,m) + (mat_test(index(j),i) -

avg_1(j,m))^2;end

endendcls_1 = zeros(size(mat_test,2),1);dist_1

scor_nearest_mean = zeros(3,1);for i = 1:size(mat_test,2)

for j = 1:3if (dist_1(i,j) == min(dist_1(i,:)))

cls_1(i) = j;if (cls_1(i) == clasa_reala(i))

scor_nearest_mean(1) = 1 + scor_nearest_mean(1);end

endend

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endfor m = 1:3for i = 1:size(mat_test,2)


avg_2(j,m))^2;end

endendcls_2 = zeros(size(mat_test,2),1);dist_2for i = 1:size(mat_test,2)




end

endendfor m = 1:3for i = 1:size(mat_test,2)


avg_3(j,m))^2;end

endendcls_3 = zeros(size(mat_test,2),1);dist_3for i = 1:size(mat_test,2)




endend

endclasele = [cls_1,cls_2,cls_3]cls_1cls_2cls_3

scor_nearest_mean = scor_nearest_mean/size(mat_test,2)

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Tema 1 - PCA

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