EX. 1
Create a data frame with x and y values
data <- data.frame(x = c(0.1, 0.5, 1, 1.5, 2.0, 2.5), y = c(0, 0, 1, 1, 1, 0))
Print the summary of the model
summary(model)
##
## Call:
## glm(formula = y ~ x, family = binomial(link = "logit"), data = data)
##
## Deviance Residuals:
## 1 2 3 4 5 6
## -0.8518 -0.9570 1.2583 1.1075 0.9653 -1.5650
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -0.8982 1.5811 -0.568 0.570
## x 0.7099 1.0557 0.672 0.501
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 8.3178 on 5 degrees of freedom
## Residual deviance: 7.8325 on 4 degrees of freedom
## AIC: 11.832
##
## Number of Fisher Scoring iterations: 4
EX. 2
Load the data. In order to prepare the data for PCA analysis, let’s
scale the variables to have zero mean and unit variance
data(mtcars)
View(mtcars)
summary(mtcars)
## mpg cyl disp hp
## Min. :10.40 Min. :4.000 Min. : 71.1 Min. : 52.0
## 1st Qu.:15.43 1st Qu.:4.000 1st Qu.:120.8 1st Qu.: 96.5
## Median :19.20 Median :6.000 Median :196.3 Median :123.0
## Mean :20.09 Mean :6.188 Mean :230.7 Mean :146.7
## 3rd Qu.:22.80 3rd Qu.:8.000 3rd Qu.:326.0 3rd Qu.:180.0
## Max. :33.90 Max. :8.000 Max. :472.0 Max. :335.0
## drat wt qsec vs
## Min. :2.760 Min. :1.513 Min. :14.50 Min. :0.0000
## 1st Qu.:3.080 1st Qu.:2.581 1st Qu.:16.89 1st Qu.:0.0000
## Median :3.695 Median :3.325 Median :17.71 Median :0.0000
## Mean :3.597 Mean :3.217 Mean :17.85 Mean :0.4375
## 3rd Qu.:3.920 3rd Qu.:3.610 3rd Qu.:18.90 3rd Qu.:1.0000
## Max. :4.930 Max. :5.424 Max. :22.90 Max. :1.0000
## am gear carb
## Min. :0.0000 Min. :3.000 Min. :1.000
## 1st Qu.:0.0000 1st Qu.:3.000 1st Qu.:2.000
## Median :0.0000 Median :4.000 Median :2.000
## Mean :0.4062 Mean :3.688 Mean :2.812
## 3rd Qu.:1.0000 3rd Qu.:4.000 3rd Qu.:4.000
## Max. :1.0000 Max. :5.000 Max. :8.000
scaled_mtcars <- scale(mtcars)
PCA analysis on the scaled data
pca_mtcars <- prcomp(scaled_mtcars, center = TRUE, scale. = TRUE)
Summary of PCA analysis
summary(pca_mtcars)
## Importance of components:
## PC1 PC2 PC3 PC4 PC5 PC6 PC7
## Standard deviation 2.5707 1.6280 0.79196 0.51923 0.47271 0.46000 0.3678
## Proportion of Variance 0.6008 0.2409 0.05702 0.02451 0.02031 0.01924 0.0123
## Cumulative Proportion 0.6008 0.8417 0.89873 0.92324 0.94356 0.96279 0.9751
## PC8 PC9 PC10 PC11
## Standard deviation 0.35057 0.2776 0.22811 0.1485
## Proportion of Variance 0.01117 0.0070 0.00473 0.0020
## Cumulative Proportion 0.98626 0.9933 0.99800 1.0000
Plot PCA results
biplot(pca_mtcars, scale = 0)
The resulting plot shows the scores of each car model on the first
two principal components, as well as the loadings of each variable on
these components. From this plot, we can see that the first principal
component (PC1) is primarily driven by variables related to car size and
power, such as hp, wt, qsec, disp, and cyl. The second principal
component (PC2) is primarily driven by variables related to fuel
efficiency, such as mpg, drat, and am.
In conclusion, the basic principal component analysis on the mtcars
dataset shows that the variables related to car size and power are the
main drivers of the first principal component, while variables related
to fuel efficiency are the main drivers of the second principal
component.
EX. 3
# Generate a random 4 x 5 matrix
set.seed(123) # set seed for reproducibility
A <- matrix(rnorm(20), nrow = 4, ncol = 5)
print(A)
## [,1] [,2] [,3] [,4] [,5]
## [1,] -0.56047565 0.1292877 -0.6868529 0.4007715 0.4978505
## [2,] -0.23017749 1.7150650 -0.4456620 0.1106827 -1.9666172
## [3,] 1.55870831 0.4609162 1.2240818 -0.5558411 0.7013559
## [4,] 0.07050839 -1.2650612 0.3598138 1.7869131 -0.4727914
# Find the SVD of the matrix A
SVD <- svd(A)
print(SVD)
## $d
## [1] 2.8293464 2.5685461 1.8185128 0.6833783
##
## $u
## [,1] [,2] [,3] [,4]
## [1,] -0.05111079 -0.1943683 0.4165550 0.8866175
## [2,] -0.89670108 0.1370951 -0.3888386 0.1610486
## [3,] 0.39412497 0.6455621 -0.5133608 0.4054329
## [4,] 0.19488550 -0.7257241 -0.6416753 0.1536131
##
## $v
## [,1] [,2] [,3] [,4]
## [1,] 0.305057630 0.4019612 -0.54406558 0.1591875
## [2,] -0.568821182 0.5550350 -0.02083334 0.5610044
## [3,] 0.348948041 0.2341790 -0.53455804 -0.1890518
## [4,] 0.003336294 -0.6690007 -0.40547715 0.6179494
## [5,] 0.679416249 0.1672171 0.50338314 0.4922717
Ex. 4
# Set seed for reproducibility
set.seed(123)
# Generate 100 data points for x1 and x2, uniformly distributed in [1,2]
x1 <- runif(100, min = 1, max = 2)
x2 <- runif(100, min = 1, max = 2)
# Generate 100 data points for x3 and x4, normally distributed with zero mean and unit variance
x3 <- rnorm(100, mean = 0, sd = 1)
x4 <- rnorm(100, mean = 0, sd = 1)
# Calculate y using the equation y = 5*x1 + 2*x2 + 2*x3 + x4
y <- 5*x1 + 2*x2 + 2*x3 + x4
# Print the first 100 values of x1, x2, x3, x4, and y
print(data.frame(x1 = x1[1:100], x2 = x2[1:100], x3 = x3[1:100], x4 = x4[1:100], y = y[1:100]))
## x1 x2 x3 x4 y
## 1 1.287578 1.599989 -0.71040656 2.19881035 10.415863
## 2 1.788305 1.332824 0.25688371 1.31241298 13.433353
## 3 1.408977 1.488613 -0.24669188 -0.26514506 9.263582
## 4 1.883017 1.954474 -0.34754260 0.54319406 13.172144
## 5 1.940467 1.482902 -0.95161857 -0.41433995 10.350564
## 6 1.045556 1.890350 -0.04502772 -0.47624689 8.442181
## 7 1.528105 1.914438 -0.78490447 -0.78860284 9.110992
## 8 1.892419 1.608735 -1.66794194 -0.59461727 8.749064
## 9 1.551435 1.410690 -0.38022652 1.65090747 11.469009
## 10 1.456615 1.147095 0.91899661 -0.05402813 11.361228
## 11 1.956833 1.935300 -0.57534696 0.11924524 12.623318
## 12 1.453334 1.301229 0.60796432 0.24368743 11.328745
## 13 1.677571 1.060721 -1.61788271 1.23247588 8.506005
## 14 1.572633 1.947727 -0.05556197 -0.51606383 11.131433
## 15 1.102925 1.720596 0.51940720 -0.99250715 9.002123
## 16 1.899825 1.142294 0.30115336 1.67569693 14.061717
## 17 1.246088 1.549285 0.10567619 -0.44116322 9.099197
## 18 1.042060 1.954091 -0.64070601 -0.72306597 7.114002
## 19 1.327921 1.585483 -0.84970435 -1.23627312 6.874888
## 20 1.954504 1.404510 -1.02412879 -1.28471572 9.248566
## 21 1.889539 1.647893 0.11764660 -0.57397348 12.404803
## 22 1.692803 1.319821 -0.94747461 0.61798582 9.826695
## 23 1.640507 1.307720 -0.49055744 1.10984814 10.946707
## 24 1.994270 1.219768 -0.25609219 0.70758835 12.606288
## 25 1.655706 1.369489 1.84386201 -0.36365730 14.341573
## 26 1.708530 1.984219 -0.65194990 0.05974994 11.266941
## 27 1.544066 1.154202 0.23538657 -0.70459646 9.794911
## 28 1.594142 1.091044 0.07796085 -0.71721816 9.591502
## 29 1.289160 1.141907 -0.96185663 0.88465050 7.690550
## 30 1.147114 1.690007 -0.07130809 -1.01559258 7.957374
## 31 1.963024 1.619256 1.44455086 1.95529397 17.898030
## 32 1.902299 1.891394 0.45150405 -0.09031959 14.106972
## 33 1.690705 1.672999 0.04123292 0.21453883 12.096529
## 34 1.795467 1.737078 -0.42249683 -0.73852770 10.867971
## 35 1.024614 1.521136 -2.05324722 -0.57438869 3.484457
## 36 1.477796 1.659838 1.13133721 -1.31701613 11.654315
## 37 1.758460 1.821805 -1.46064007 -0.18292539 9.331703
## 38 1.216408 1.786282 0.73994751 0.41898240 11.553480
## 39 1.318181 1.979822 1.90910357 0.32430434 14.693060
## 40 1.231626 1.439432 -1.44389316 -0.78153649 5.367669
## 41 1.142800 1.311702 0.70178434 -0.78862197 8.952351
## 42 1.414546 1.409475 -0.26219749 -0.50219872 8.865088
## 43 1.413724 1.010467 -1.57214416 1.49606067 7.441328
## 44 1.368845 1.183850 -1.51466765 -1.13730362 5.045287
## 45 1.152445 1.842729 -1.60153617 -0.17905159 6.065558
## 46 1.138806 1.231162 -0.53090652 1.90236182 8.996903
## 47 1.233034 1.239100 -1.46175558 -0.10097489 5.618884
## 48 1.465962 1.076691 0.68791677 -1.35984070 9.499187
## 49 1.265973 1.245724 2.10010894 -0.66476944 12.356759
## 50 1.857828 1.732135 -1.28703048 0.48545998 10.664808
## 51 1.045831 1.847453 0.78773885 -0.37560287 10.123937
## 52 1.442200 1.497527 0.76904224 -0.56187636 11.182263
## 53 1.798925 1.387909 0.33220258 -0.34391723 12.090930
## 54 1.121899 1.246449 -1.00837661 0.09049665 6.176138
## 55 1.560948 1.111096 -0.11945261 1.59850877 11.386536
## 56 1.206531 1.389994 -0.28039534 -0.08856511 8.163290
## 57 1.127532 1.571935 0.56298953 1.08079950 10.988307
## 58 1.753308 1.216893 -0.37243876 0.63075412 11.086201
## 59 1.895045 1.444768 0.97697339 -0.11363990 14.205070
## 60 1.374463 1.217991 -0.37458086 -1.53290200 7.026231
## 61 1.665115 1.502300 1.05271147 -0.52111732 12.914481
## 62 1.094841 1.353905 -1.04917701 -0.48987045 5.593788
## 63 1.383970 1.649985 -1.26015524 0.04715443 7.746662
## 64 1.274384 1.374714 3.24103993 1.30019868 16.903625
## 65 1.814640 1.355445 -0.41685759 2.29307897 13.243455
## 66 1.448516 1.533688 0.29822759 1.54758106 12.453994
## 67 1.810064 1.740334 0.63656967 -0.13315096 13.670979
## 68 1.812390 1.221103 -0.48378063 -1.75652740 8.780065
## 69 1.794342 1.412746 0.51686204 -0.38877986 12.442148
## 70 1.439832 1.265687 0.36896453 0.08920722 10.557668
## 71 1.754475 1.629973 -0.21538051 0.84501300 12.446574
## 72 1.629221 1.183828 0.06529303 0.96252797 11.606877
## 73 1.710182 1.863644 -0.03406725 0.68430943 12.894375
## 74 1.000625 1.746568 2.12845190 -1.39527435 11.357889
## 75 1.475317 1.668285 -0.74133610 0.84964305 10.080123
## 76 1.220119 1.618018 -1.09599627 -0.44655722 6.698080
## 77 1.379817 1.372238 0.03778840 0.17480270 9.893938
## 78 1.612771 1.529836 0.31048075 0.07455118 11.819039
## 79 1.351798 1.874682 0.43652348 0.42816676 11.809568
## 80 1.111135 1.581750 -0.45836533 0.02467498 7.827122
## 81 1.243619 1.839768 -1.06332613 -1.66747510 6.103506
## 82 1.668056 1.312448 1.26318518 0.73649596 14.228041
## 83 1.417647 1.708290 -0.34965039 0.38602657 10.191540
## 84 1.788196 1.265018 -0.86551286 -0.26565163 9.474337
## 85 1.102865 1.594343 -0.23627957 0.11814451 8.348595
## 86 1.434893 1.481290 -0.19717589 0.13403865 9.876730
## 87 1.984957 1.265033 1.10992029 0.22101947 14.895710
## 88 1.893051 1.564590 0.08473729 1.64084617 14.404757
## 89 1.886469 1.913188 0.75405379 -0.21905038 14.547779
## 90 1.175053 1.901874 -0.49929202 0.16806538 8.848493
## 91 1.130696 1.274167 0.21444531 1.16838387 9.799086
## 92 1.653102 1.321483 -0.32468591 1.05418102 11.313284
## 93 1.343516 1.985641 0.09458353 1.14526311 12.023294
## 94 1.656758 1.619993 -0.89536336 -0.57746800 9.155583
## 95 1.320373 1.937314 -1.31080153 2.00248273 9.857374
## 96 1.187691 1.466533 1.99721338 0.06670087 12.932649
## 97 1.782294 1.406833 0.60070882 1.86685184 14.793406
## 98 1.093595 1.659230 -1.25127136 -1.35090269 4.932990
## 99 1.466779 1.152347 -0.61116592 0.02098359 8.437240
## 100 1.511505 1.572867 -1.18548008 1.24991457 9.582216
# Perform principal component analysis on the simulated data
pca <- prcomp(data.frame(x1, x2, x3, x4, y), scale. = TRUE)
# Print the summary of the PCA results
summary(pca)
## Importance of components:
## PC1 PC2 PC3 PC4 PC5
## Standard deviation 1.4547 1.0701 0.9667 0.8968 4.539e-16
## Proportion of Variance 0.4232 0.2290 0.1869 0.1608 0.000e+00
## Cumulative Proportion 0.4232 0.6522 0.8391 1.0000 1.000e+00
Yes, the results from the principal component analysis (PCA) on the
simulated data are expected based on the formula used to generate the
data.