Chapter 1 Exploratory data analysis
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1 Chapter 1 Exploratory data analysis Xavier Gendre M2 SE X. Gendre (M2 SE) Data Mining 1 / 33
2 cars$dist ## [1] ## [18] ## [35] summary(cars$dist) ## Min. 1st Qu. Median Mean 3rd Qu. Max. ## X. Gendre (M2 SE) Data Mining 2 / 33
3 p <- (0:100)/100 qp <- quantile(cars$dist, probs = p) plot(p, qp, type = "l", col = "red") qp 60 Third quartile Median First quartile p X. Gendre (M2 SE) Data Mining 3 / 33
4 boxplot(cars$dist, horizontal = TRUE, col = "orange") X. Gendre (M2 SE) Data Mining 4 / 33
5 hist(cars$dist, breaks = 20 * (-1:7), freq = F, col = "orange", ylab = "") Histogram of cars$dist cars$dist X. Gendre (M2 SE) Data Mining 5 / 33
6 plot(cars$speed, cars$dist, pch = 4) cars$dist cars$speed X. Gendre (M2 SE) Data Mining 6 / 33
7 cor(cars$speed, cars$dist) ## [1] a <- cov(cars$speed, cars$dist)/var(cars$speed) b <- mean(cars$dist) - a * mean(cars$speed) X. Gendre (M2 SE) Data Mining 7 / 33
8 abline(b, a, col = "red") cars$dist cars$speed X. Gendre (M2 SE) Data Mining 8 / 33
9 Grades data X ## Math Phys Fr Eng ## Benny ## Bobby ## Brandy ## Coby ## Daisy ## Emily ## Judy ## Marty ## Sandy Xbar <- scale(x, scale = F) Sigma <- (t(xbar) %*% Xbar)/9 ## Math Phys Fr Eng ## Math ## Phys ## Fr ## Eng X. Gendre (M2 SE) Data Mining 9 / 33
10 boxplot(xbar, col = "orange") Math Phys Fr Eng X. Gendre (M2 SE) Data Mining 10 / 33
11 dg <- eigen(sigma) dg$values ## [1] cumsum(dg$values)/sum(dg$values) ## [1] X. Gendre (M2 SE) Data Mining 11 / 33
12 plot(1:4, dg$values, type = "b", col = "orange") dg$values :4 X. Gendre (M2 SE) Data Mining 12 / 33
13 Skyrim bows data X ## Weight Value Damage Speed ## Long Bow ## Hunting Bow ## Orcish Bow ## Nord Hero Bow ## Dwarven Bow ## Elven Bow ## Glass Bow ## Ebony Bow ## Daedric Bow ## Dragonbone Bow ## Crossbow ## Enhanced Crossbow ## Dwarven Crossbow ## Enhanced Dwarven Crossbow X. Gendre (M2 SE) Data Mining 13 / 33
14 Xbar <- scale(x, scale = F) Sigma <- (t(xbar) %*% Xbar)/nrow(X) ## Weight Value Damage Speed ## Weight ## Value ## Damage ## Speed M <- diag(1/diag(sigma)) ## Weight Value Damage Speed ## Weight e ## Value e ## Damage e ## Speed e X. Gendre (M2 SE) Data Mining 14 / 33
15 boxplot(xbar, col = "orange") boxplot(xbar %*% sqrt(m), col = "blue") Weight Value Damage Speed Weight Value Damage Speed X. Gendre (M2 SE) Data Mining 15 / 33
16 Mhalf <- diag(1/sqrt(diag(sigma))) dg <- eigen(mhalf %*% Sigma %*% Mhalf) dg$vectors <- diag(sqrt(diag(sigma))) %*% dg$vectors dg$values ## [1] cumsum(dg$values)/sum(dg$values) ## [1] X. Gendre (M2 SE) Data Mining 16 / 33
![## [1] 2.51105 1.25027 0.20949 0.](/docs-images/50/15063162/images/page_16.jpg "02919 cumsum(dg$values)/sum(dg$values) ## [1] 0.6278 0.9403 0.9927 1.")
17 plot(1:4, dg$values, type = "b", col = "orange") dg$values :4 X. Gendre (M2 SE) Data Mining 17 / 33
18 Grades data C <- Xbar %*% dg$vectors cos2 <- rowsums(c[, 1:2]^2)/rowSums(C^2) ## Benny Bobby Brandy Coby Daisy Emily Judy Marty Sandy ## rbind(c[, 1]^2/(nrow(X) * dg$values[1]), C[, 2]^2/(nrow(X) * dg$values[2])) ## Benny Bobby Brandy Coby Daisy Emily Judy Marty ## [1,] ## [2,] ## Sandy ## [1,] ## [2,] X. Gendre (M2 SE) Data Mining 18 / 33
![9808 rbind(c[, 1]^2/(nrow(X) * dg$values[1]), C[, 2]^2/(nrow(X) * dg$values[2])) ## Benny Bobby Brandy Coby Daisy Emily Judy Marty ##](/docs-images/50/15063162/images/page_18.jpg "[1,] 0.29187 0.05921 0.04063 0.381947 0.16152 0.0362 0.004138 0.01502 ## [2,] 0.01835 0.00233 0.11110 0.003319 0.03868 0.2237 0.375596 0.")
19 plot(c[, 1:2], pch = 2, cex = cos2, col = "orange", xlab = "", ylab = "") Emily 4 Marty 2 Daisy Benny Bobby 0 Coby -2 Sandy Brandy -4-6 Judy X. Gendre (M2 SE) Data Mining 19 / 33
20 Skyrim bows data C <- Xbar %*% M %*% dg$vectors cos2 <- rowsums(c[, 1:2]^2)/rowSums(C^2) ## Long Bow Hunting Bow ## ## Orcish Bow Nord Hero Bow ## ## Dwarven Bow Elven Bow ## ## Glass Bow Ebony Bow ## ## Daedric Bow Dragonbone Bow ## ## Crossbow Enhanced Crossbow ## ## Dwarven Crossbow Enhanced Dwarven Crossbow ## X. Gendre (M2 SE) Data Mining 20 / 33
21 plot(c[, 1:2], pch = 2, cex = cos2, col = "orange", xlab = "", ylab = "") Daedric Bow Dragonbone Bow Ebony Bow Glass Bow Elven Bow Dwarven Bow Orcish Bow Nord Hero Bow Hunting Bow Long Bow Enhanced Dwarven Crossbow Crossbow Enhanced Crossbow X. Gendre (M2 SE) Data Mining 21 / 33
22 Grades data Rho <- diag(1/sqrt(diag(sigma))) %*% dg$vectors %*% diag(sqrt(dg$values)) Rho[, 1:2] ## [,1] [,2] ## Math ## Phys ## Fr ## Eng rowsums(rho[, 1:2]^2) ## Math Phys Fr Eng ## X. Gendre (M2 SE) Data Mining 22 / 33
23 DrawCorCircle(Rho) Phys Math Eng Fr X. Gendre (M2 SE) Data Mining 23 / 33
24 DrawBiplot(C, Rho) Math Phys Marty Emily Coby Daisy Bobby Benny Eng Fr Sandy Brandy Judy X. Gendre (M2 SE) Data Mining 24 / 33
25 Skyrim bows data Rho <- diag(1/sqrt(diag(sigma))) %*% dg$vectors %*% diag(sqrt(dg$values)) rownames(rho) <- colnames(x) Rho[, 1:2] ## [,1] [,2] ## Weight ## Value ## Damage ## Speed rowsums(rho[, 1:2]^2) ## Weight Value Damage Speed ## X. Gendre (M2 SE) Data Mining 25 / 33
26 DrawCorCircle(Rho) Value Weight Damage -1.0 Speed X. Gendre (M2 SE) Data Mining 26 / 33
27 DrawBiplot(C, Rho) Ebony Bow 1 Value 0.5 Glass Bow Elven Bow Dwarven Bow Orcish Bow Dragonbone Bow Nord Hero Hunting Bow Long Bow Bow 0 Daedric Bow 0-1 Weight Damage -0.5 Crossbow Enhanced Crossbow -2 Enhanced Dwarven Crossbow Speed X. Gendre (M2 SE) Data Mining 27 / 33
28 Bourdieu data DR SCE LET SC MD PH PD IUT Total EAG SAG PT PLCS CM EMP OUV OTH Total X. Gendre (M2 SE) Data Mining 28 / 33
29 D1 <- diag(1/rowsums(t)) P1 <- D1 %*% T D2 <- diag(1/colsums(t)) P2 <- D2 %*% t(t) PCA of line profiles: M1half <- diag(sqrt(n/colsums(t))) M1halfInv <- diag(sqrt(colsums(t)/n)) dg <- eigen(m1half %*% t(p1) %*% t(p2) %*% M1halfInv) dg$values <- dg$values[2:8] # Avoid the trivial eigenvlue dg$vectors <- (M1halfInv %*% dg$vectors)[, 2:8] # Idem C1 <- P1 %*% (n * D2) %*% dg$vectors Transition formula: C2 <- P2 %*% C1 %*% diag(1/sqrt(dg$values)) X. Gendre (M2 SE) Data Mining 29 / 33
30 # Representation quality of the x's values rowsums(c1[, 1:2]^2)/rowSums(C1^2) ## EAG SAG PT PLCS CM EMP OUV OTH ## # Representation quality of the y's values rowsums(c2[, 1:2]^2)/rowSums(C2^2) ## DR SCE LET SC MD PH PD IUT ## cumsum(dg$values)/sum(dg$values) ## [1] X. Gendre (M2 SE) Data Mining 30 / 33
31 plot(dg$values, type = "b", col = "orange", xlab = "", ylab = "") X. Gendre (M2 SE) Data Mining 31 / 33
32 DrawCA(C1, C2) SAG OUV OTH LET PD EMP CM SC DR SCE PT PLCS MD IUT PH EAG X. Gendre (M2 SE) Data Mining 32 / 33
33 Bibliography The Elements of Statistical Learning : Data Mining, Inference and Prediction, J. Friedman, T. Hastie and R. Tibshirani (2009) Principal Component Analysis, I.T. Jolliffe (2002) Wiki Stat, X. Gendre (M2 SE) Data Mining 33 / 33
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