STAT 3690 Homework 4
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STAT 3690 Homework 4
1. We have information on n = 138 samples of Canadian hard red spring wheat and the flour made from these samples. The 5-dimensional vector x contains standardized wheat measurements on: (X1 ) kernel texture, (X2 ) test weight, (X3 ) famaged kernels, (X4 ) foreign material and (X5 ) crude protein in the wheat. The 4-dimensional vector Y contains standardized flour measurements: (Y1 ) wheat per barrel of flour, (Y2 ) ash in flour, (Y3 ) crude protein in flour, and (Y4 ) gluten quality index. We are only given the sample correlation matrices:
一(一)
一(一)
、．
RX = ．(．) 1.000 0.323 一0.444 ．(．) ,
． 1.000 一0.334 ．
╱ 
一(一)
0一
、
RY = ．(．) 1.000 一0. 163 ．(．) ,
． ．
╱．一(一)
一(一)
一(一)
、．
RXY = ．(．) 0.737 0.361 一0.546 0.172 ．(．) .
． 0.527 0.461 一0.393 一0.019 ．
a. Use sequential tests (with the Holm-Bonferroni procedure) to determine the number of significant canonical correlations at level α = .05.
b. Compute sample canonical directions corresponding to the significant canonical correlations.
2. Consider the situation where you have two normal populations N (µ1 , σ 1(2)) and N (µ2 , σ2(2)). We observe independent samples X1 , 1 , . . . , X1,n1 ～ N (µ1 , σ1(2)) and X2 , 1 , . . . , X2,n2 ～ N (µ2 , σ2(2)) with means 
1 and 
2 , respectively. We plan to use the following rule
R : Classify a new X as coming from population 2 if X > (
1 + 
2 )/2 and population 1 otherwise.
Assuming priors Pr(X ～ N (µ1 , σ 1(2))) = Pr(X ～ N (µ2 , σ2(2))) = 1/2, please express the misclassification rate of rule R, i.e.,
err(X) = Pr(X > (
1 + 
2 )/2 and X ～ N (µ1 , σ 1(2))) + Pr(X ≤ (
1 + 
2 )/2 and X ～ N (µ2 , σ2(2))), in terms of n1 , n2 , µ 1 , µ2 , σ 1 , σ2 and the standard normal cumulative distribution function Φ(.).
3. Suppose there is a binary classification task: one would like to predict labels of n subjects, say Y1 , . . . , Yn ,
according to their independent p-dimensional observations x1 , . . . , xn . The two potential populations
are assumed to be MVNp (u1 , 五) and MVNp (u2 , 五) , i.e., xi ~ Yi = yi ～ MVNp (uyi , 五) , yi = 1 , 2 .
Meanwhile, let Pr(Yi = k) = πk for all k = 1 , 2 and i = 1, . . . , n .
a. Applying the linear discriminant analysis (LDA) to this problem, write down the mathematical expression of error rate in terms of u1 , u2 , 五, π 1 , π2 and the standard normal cumulative distribution function Φ(.).
b. There is a banknote authentication dataset (see below for the data import), where n = 1, 372 data points consisted of features extracted (via the wavelet transformation) from images that were taken from genuine and forged banknotes. Specifically, the features are “variance” (the variance of wavelet-transformed image), “skewness” (the skewness of wavelet-transformed image), “curtosis” (the curtosis of wavelet-transformed image), and “entropy” (the entropy of image), all continuous. The authentication of banknote is indicated by “class” (0 for authentic and 1 for inauthentic). Figure out a parametric estimate for error rate of LDA by plugging estimates of u1 , u2 , 五, π 1 and π2 into the expression obtained in Q3a.
c. Apply LDA to the dataset in Q3b and utilize 5 × 8-fold cross validation to estimate the resulting error rate. Report this error rate.
d. Make a comment with one single sentence after comparing estimates given by Q3b and Q3c.
bn_df = read.table(
"https://archive.ics.uci.edu/ml/machine-learning-databases/00267/data_banknote_authentication.txt " , sep = ","
)
names (bn_df) = c ("variance" , "skewness" , "curtosis" , "entropy" , "class")
2022-04-26