## Importance of components:\n##                            PC1      PC2     PC3     PC4     PC5     PC6    PC7\n## Standard deviation     136.532 38.14735 3.06642 1.27492 0.90474 0.64734 0.3054\n## Proportion of Variance   0.927  0.07237 0.00047 0.00008 0.00004 0.00002 0.0000\n## Cumulative Proportion    0.927  0.99938 0.99985 0.99993 0.99997 0.99999 1.0000\n##                           PC8    PC9\n## Standard deviation     0.2859 0.2159\n## Proportion of Variance 0.0000 0.0000\n## Cumulative Proportion  1.0000 1.0000<\/code><\/pre>\n\n\n\n
O desvio padr\u00e3o (Standard deviation) da j<\/em>-\u00e9sima componente corresponde a \u221a\u03bbj<\/sub><\/em>.<\/p>\n\n\n\n
Como as vari\u00e1veis est\u00e3o em diferentes escalas, o mais adequado \u00e9 utilizar a matriz de correla\u00e7\u00e3o.<\/p>\n\n\n\n
# PCA com a matriz de correla\u00e7\u00e3o (vari\u00e1veis padronizadas)\npca_corr <- prcomp(dados_pca, center = TRUE, scale = TRUE)\nsummary(pca_corr)<\/code><\/pre>\n\n\n\n
## Importance of components:\n##                           PC1    PC2     PC3     PC4     PC5     PC6     PC7\n## Standard deviation     2.3782 1.4429 0.71008 0.51481 0.42797 0.35184 0.32413\n## Proportion of Variance 0.6284 0.2313 0.05602 0.02945 0.02035 0.01375 0.01167\n## Cumulative Proportion  0.6284 0.8598 0.91581 0.94525 0.96560 0.97936 0.99103\n##                           PC8     PC9\n## Standard deviation     0.2419 0.14896\n## Proportion of Variance 0.0065 0.00247\n## Cumulative Proportion  0.9975 1.00000<\/code><\/pre>\n\n\n\n
fviz_eig(pca_corr)<\/code><\/pre>\n\n\n\n
<\/figure><\/div>\n\n\n\n
Interpreta\u00e7\u00e3o:<\/p>\n\n\n\n
A 1\u00aa componente explica 62,84% da vari\u00e2ncia total dos dados. Portanto, quase 2\/3 das informa\u00e7\u00f5es contidas nas 9 vari\u00e1veis do banco de dados podem ser encapsulada apenas nessa componente. A 2\u00aa componente explica 23,13% da vari\u00e2ncia total. Com apenas 2 componentes 85,98% da vari\u00e2ncia dos dados \u00e9 explicada.<\/p>\n\n\n\n
# Coeficientes das componentes principais (autovetores da matriz de correla\u00e7\u00e3o)\nsummary(pca_corr)$rotation #Pode-se trocar os sinais<\/code><\/pre>\n\n\n\n
##                       PC1         PC2         PC3          PC4        PC5\n## milhas_galao   -0.3931477  0.02753861 -0.22119309 -0.006126378 -0.3207620\n## cilindros       0.4025537  0.01570975 -0.25231615  0.040700251  0.1171397\n## deslocamento    0.3973528 -0.08888469 -0.07825139  0.339493732 -0.4867849\n## potencia_bruta  0.3670814  0.26941371 -0.01721159  0.068300993 -0.2947317\n## relacao_et     -0.3118165  0.34165268  0.14995507  0.845658485  0.1619259\n## peso            0.3734771 -0.17194306  0.45373418  0.191260029 -0.1874822\n## tempo_14       -0.2243508 -0.48404435  0.62812782 -0.030329127 -0.1482495\n## marchas        -0.2094749  0.55078264  0.20658376 -0.282381831 -0.5624860\n## carburador      0.2445807  0.48431310  0.46412069 -0.214492216  0.3997820\n##                        PC6         PC7         PC8         PC9\n## milhas_galao    0.72015586 -0.38138068 -0.12465987  0.11492862\n## cilindros       0.22432550 -0.15893251  0.81032177  0.16266295\n## deslocamento   -0.01967516 -0.18233095 -0.06416707 -0.66190812\n## potencia_bruta  0.35394225  0.69620751 -0.16573993  0.25177306\n## relacao_et     -0.01536794  0.04767957  0.13505066  0.03809096\n## peso           -0.08377237 -0.42777608 -0.19839375  0.56918844\n## tempo_14        0.25752940  0.27622581  0.35613350 -0.16873731\n## marchas        -0.32298239 -0.08555707  0.31636479  0.04719694\n## carburador      0.35706914 -0.20604210 -0.10832772 -0.32045892<\/code><\/pre>\n\n\n\n
No caso em que todos os coeficientes apresentarem o mesmo sinal (positivo OU negativo) a componente \u00e9 chamada de \u00edndice global. Quando os sinais s\u00e3o opostos a componente \u00e9 chamada de \u00edndice comparativo.<\/p>\n\n\n\n
^Y1<\/sub> = \u22120.393 Z1<\/sub> + 0.403 Z2<\/sub> + 0.397 Z3<\/sub> + 0.367 Z4<\/sub> \u2212 0.312 Z5<\/sub> + 0.373 Z6<\/sub> \u2212 0.224 Z7<\/sub> \u2212 0.209 Z8<\/sub> + 0.245 Z9<\/sub><\/p>\n\n\n\n
^Y1<\/sub> = 0.393 Z1<\/sub> \u2212 0.403 Z2<\/sub> \u2212 0.397 Z3<\/sub> \u2212 0.367 Z4<\/sub> + 0.312 Z5<\/sub> \u2212 0.373 Z6<\/sub> + 0.224 Z7<\/sub> + 0.209 Z8<\/sub> \u2212 0.245 Z9<\/sub><\/p>\n\n\n\n
OBS: Zi<\/sub><\/em> \u00e9 a vari\u00e1vel i<\/em> padronizada.<\/li><\/ul>\n\n\n\n
Interpreta\u00e7\u00e3o PC1:<\/p>\n\n\n\n
Em geral, os coeficientes est\u00e3o equilibrados (todos est\u00e3o relativamente pr\u00f3ximos), sendo que, a vari\u00e1vel Z2<\/sub><\/em> apresentou o maior peso (-0.403). Al\u00e9m disso, trata-se de um \u00edndice de compara\u00e7\u00e3o entre os seguintes grupos de vari\u00e1veis:<\/p>\n\n\n\n
Grupo 1 – Z1<\/sub>, Z5<\/sub>, Z7<\/sub> e Z8<\/sub> (coeficientes positivos)<\/li>
Grupo 2 – Z2<\/sub>, Z3<\/sub>, Z4<\/sub>, Z6<\/sub> e Z9<\/sub> (coeficientes negativos)<\/li><\/ul>\n\n\n\n
Quanto maior o valor das vari\u00e1veis do Grupo 1, maior o valor do escore de ^Y1<\/sub>.<\/p>\n\n\n\n
Quanto maior o valor das vari\u00e1veis do Grupo 2, menor o valor do escore de ^Y1<\/sub>.<\/p>\n\n\n\n
Escores<\/strong><\/h3>\n\n\n\n
Cada automovel tem 1 escore para cada componente principal.<\/p>\n\n\n\n
summary(pca_corr)$x<\/code><\/pre>\n\n\n\n
##                             PC1        PC2         PC3         PC4         PC5\n## Mazda RX4           -0.66422351  1.1734476 -0.20431724 -0.12601751  0.75200784\n## Mazda RX4 Wag       -0.63719807  0.9769448  0.11077779 -0.08567709  0.65668822\n## Datsun 710          -2.29973601 -0.3265893 -0.21014955 -0.10862524 -0.07622329\n## Hornet 4 Drive      -0.21529670 -1.9768101 -0.32946822 -0.30806225 -0.24391787\n## Hornet Sportabout    1.58697405 -0.8287285 -1.03299254  0.14738418 -0.22270405\n## Valiant              0.04960512 -2.4466838  0.11177774 -0.87154914 -0.12574876\n## Duster 360           2.71439677  0.3610529 -0.65206041  0.09633337  0.29674234\n## Merc 240D           -2.04370658 -0.8006412  0.84898795 -0.27451338 -0.26307848\n## Merc 230            -2.29506729 -1.3056004  1.96848450  0.05055875 -0.45988113\n## Merc 280            -0.38252133  0.5811211  0.88632274  0.07026946  0.45835852\n## Merc 280C           -0.36652708  0.4121971  1.14860950  0.06150898  0.48309076\n## Merc 450SE           1.88466875 -0.7241198 -0.20604588 -0.21856675  0.27996207\n## Merc 450SL           1.67107231 -0.7144354 -0.32644071 -0.28933625  0.28061777\n## Merc 450SLC          1.77692371 -0.8411687 -0.08557921 -0.28421711  0.34961695\n## Cadillac Fleetwood   3.64958983 -0.9480878  0.88315862  0.21645793 -0.34788247\n## Lincoln Continental  3.71033756 -0.8426945  0.93230325  0.34099021 -0.34260485\n## Chrysler Imperial    3.33196300 -0.4805609  0.67061959  0.65189724 -0.43940743\n## Fiat 128            -3.45236266 -0.4327074 -0.22604214  0.10018032 -0.33470301\n## Honda Civic         -3.85477722  0.7084152 -0.22670973  1.19340342  0.53954318\n## Toyota Corolla      -3.85488283 -0.3872111 -0.25488964  0.21962306 -0.30372397\n## Toyota Corona       -1.90375523 -1.5725638  0.06620424  0.07989679  0.50126570\n## Dodge Challenger     1.80402354 -1.1340965 -1.00776416 -0.58796239  0.09903732\n## AMC Javelin          1.46483534 -0.9777629 -0.76680342 -0.03308788  0.26871378\n## Camaro Z28           2.60135738  0.7649595 -0.48915140  0.95247550  0.53065965\n## Pontiac Firebird     1.87424485 -0.9791561 -0.89787633  0.22438738 -0.50770999\n## Fiat X1-9           -3.14830645 -0.2552569 -0.36230545  0.06406082  0.03361267\n## Porsche 914-2       -2.77939557  1.6373369 -0.35969974  0.31886540 -0.43251030\n## Lotus Europa        -2.90895427  1.3962368 -0.91635036 -0.90254314 -0.75861156\n## Ford Pantera L       1.54812696  3.0206982 -0.51945216  0.86560850 -0.86048411\n## Ferrari Dino         0.08049995  2.8346567  0.34481747 -1.14659658  0.29944552\n## Maserati Bora        2.96252801  3.9993896  0.70296512 -0.73000448 -0.22756074\n## Volvo 142E          -1.90443632  0.1084190  0.39906976  0.31285789  0.11738974\n##                             PC6         PC7          PC8          PC9\n## Mazda RX4           -0.12506777 -0.42357334 -0.003259165 -0.167051112\n## Mazda RX4 Wag       -0.06619437 -0.44849307  0.056643244 -0.071592094\n## Datsun 710          -0.56693648  0.38612406 -0.202035744  0.114505030\n## Hornet 4 Drive       0.08382435  0.03299362 -0.023714111 -0.145255757\n## Hornet Sportabout    0.18280435 -0.05793795  0.152342587 -0.154646072\n## Valiant             -0.23043022  0.22451528  0.098663134 -0.004233901\n## Duster 360           0.27763557  0.44227307 -0.306373481 -0.186980810\n## Merc 240D           -0.19042527 -0.39416400 -0.187088365 -0.010461330\n## Merc 230             0.20443847  0.53713423  0.413455512 -0.169005773\n## Merc 280            -0.07984989 -0.26113412  0.204105964  0.110461785\n## Merc 280C           -0.16066456 -0.07979514  0.352641772  0.027108266\n## Merc 450SE           0.17135058 -0.08914480  0.092140434  0.396034809\n## Merc 450SL           0.33682412  0.03346598  0.182323579  0.196526577\n## Merc 450SLC          0.13926264  0.20632469  0.295340402  0.147796262\n## Cadillac Fleetwood  -0.24002207 -0.31053111 -0.171865268 -0.251117818\n## Lincoln Continental -0.22646211 -0.28589695 -0.239313268 -0.028994385\n## Chrysler Imperial    0.31045750 -0.38304409 -0.359765688  0.223097923\n## Fiat 128             0.57303421 -0.24650594 -0.066340528  0.220271421\n## Honda Civic          0.37207104 -0.20055288  0.087333576 -0.241702175\n## Toyota Corolla       0.83750899 -0.10186868  0.104053562  0.042833437\n## Toyota Corona       -0.07212137  0.74680802 -0.408144457 -0.082722856\n## Dodge Challenger    -0.33920894 -0.14045443  0.156086022 -0.050247532\n## AMC Javelin         -0.31479492  0.03753417  0.370979414 -0.043466032\n## Camaro Z28           0.05970074  0.38212238 -0.289612990  0.082069840\n## Pontiac Firebird     0.20785973 -0.32709161  0.027471038 -0.130958896\n## Fiat X1-9           -0.09586730  0.10352270 -0.020876499  0.021084764\n## Porsche 914-2       -0.69006515 -0.26313120 -0.105695694  0.085027267\n## Lotus Europa         0.05473409 -0.03491081 -0.236552376 -0.046341050\n## Ford Pantera L      -0.50704173  0.37940892  0.548070377  0.053196712\n## Ferrari Dino        -0.08124583 -0.26924964 -0.123537656 -0.047915313\n## Maserati Bora        0.65580986  0.49422807 -0.082329298 -0.053112079\n## Volvo 142E          -0.48091826  0.31102454 -0.315146031  0.165790892<\/code><\/pre>\n\n\n\n
Gr\u00e1ficos<\/strong><\/h3>\n\n\n\n
Interpreta\u00e7\u00e3o geral dos gr\u00e1ficos:<\/p>\n\n\n\n
<\/figure><\/div>\n\n\n\n
Os quadrantes com circulos azuis indicam que uma componente explicou mais as vari\u00e1veis presentes nesse quadrante do que a outra componente.<\/p>\n\n\n\n
# Gr\u00e1fico de indiv\u00edduos (autom\u00f3veis)\nfviz_pca_ind(pca_corr,\n             col.ind = \"cos2\", #Cor pela qualidade de representa\u00e7\u00e3o\n             gradient.cols = c(\"#00AFBB\", \"#E7B800\", \"#FC4E07\"),\n             repel = TRUE, # Texto n\u00e3o sobreposto\n             legend.title = \"Representa\u00e7\u00e3o\"\n)<\/code><\/pre>\n\n\n\n
<\/figure><\/div>\n\n\n\n
# Gr\u00e1fico das vari\u00e1veis\nfviz_pca_var(pca_corr,\n             col.var = \"contrib\", # Cor por contribui\u00e7\u00f5es para o PC\n             gradient.cols = c(\"#00AFBB\", \"#E7B800\", \"#FC4E07\"),\n             repel = TRUE,     \n             legend.title = \"Contribui\u00e7\u00e3o\"\n)<\/code><\/pre>\n\n\n\n
<\/figure><\/div>\n\n\n\n
# Gr\u00e1fico das vari\u00e1veis e indiv\u00edduos\nfviz_pca_biplot(pca_corr, repel = TRUE,\n                col.var = \"#2E9FDF\", # cor das vari\u00e1veis\n                col.ind = \"#696969\"  # cor dos automoveis\n)<\/code><\/pre>\n\n\n\n
<\/figure><\/div>\n\n\n\n
# Trocando os nomes das categorias\nmtcars <- mtcars %>% mutate(\n  forma_motor = case_when(forma_motor == 0 ~ \"Forma V\",\n                          TRUE ~ \"Forma reta\"),\n  transmissao = case_when(transmissao == 0 ~ \"Autom\u00e1tica\",\n                          TRUE ~ \"Manual\")\n)\n\nfviz_pca_ind(pca_corr,\n             col.ind = mtcars$forma_motor, # cor por forma\n             palette = c(\"#00AFBB\",  \"#FC4E07\"),\n             addEllipses = TRUE, \n             ellipse.type = \"confidence\",\n             legend.title = \"Forma do motor\",\n             repel = TRUE\n)<\/code><\/pre>\n\n\n\n
<\/figure><\/div>\n\n\n\n
fviz_pca_ind(pca_corr,\n             col.ind = as.factor(mtcars$transmissao), # cor por transmissao\n             palette = c(\"#00AFBB\",  \"#FC4E07\"),\n             addEllipses = TRUE, \n             ellipse.type = \"confidence\",\n             legend.title = \"Transmiss\u00e3o\",\n             repel = TRUE\n)<\/code><\/pre>\n\n\n\n
<\/figure>\n\n\n\n
# Gr\u00e1fico das vari\u00e1veis e indiv\u00edduos\nfviz_pca_biplot(pca_corr, repel = TRUE,\n                col.var = \"black\", # cor das vari\u00e1veis\n                col.ind = as.factor(mtcars$transmissao),  # cor dos automoveis\n                addEllipses = TRUE,\n                legend.title = \"Transmiss\u00e3o\"\n)<\/code><\/pre>\n\n\n\n
<\/figure><\/div>\n\n\n\n
Gostou desse artigo? Siga-nos nas redes sociais para saber sempre que lan\u00e7armos conte\u00fados novos. <\/p>\n","protected":false},"excerpt":{"rendered":"
A PCA \u00e9 um m\u00e9todo estat\u00edstico multivariado que foi introduzido por Karl Pearson em 1901 e surgiu da necessidade de analisar vari\u00e1veis correlacionadas.<\/p>\n","protected":false},"author":8,"featured_media":7162,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"","site-content-layout":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","spay_email":"","footnotes":""},"categories":[442],"tags":[235,308,478,115],"class_list":["post-7139","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-fundamentos-de-estatistica","tag-rstats","tag-analise-de-componentes-principais","tag-karl-pearson","tag-pca"],"yoast_head":"\n