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Sem 5 Prep
← BI & Data SciencePrincipal component analysis

Formulas for this chapter

Principal component score

PCk score = w1 x z1 + w2 x z2 + ... + wp x zp where zj is the standardised value of variable j

Placing one observation on a component. The values must be standardised with the same means and standard deviations the loadings were computed from.

wj
Loading of variable j on this component; sign and size say how it contributes
zj
Standardised value: (raw value minus mean) over standard deviation
score
Centred at zero, so positive means above average for this dataset

Variance explained

eigenvalue(k) = ( standard deviation of PCk )^2 % variance = eigenvalue(k) / sum of all eigenvalues = eigenvalue(k) / p on standardised data cumulative % = running total

Reading a summary(prcomp) or an eigenvalue table. Check the eigenvalues sum to p before computing anything.

eigenvalue
Variance captured by that component
p
Number of variables; the total variance on standardised data

How many components to keep

Method 1: keep components until cumulative variance reaches 90 % Method 2: Kaiser, keep every component with eigenvalue >= 1 Scree plot: keep the components before the elbow (fviz_eig)

After the eigenvalue table. Use both, and if they disagree say which you took and what variance was lost.

90 %
Section B's stated threshold; other thresholds are used, so state yours
eigenvalue >= 1
A standardised variable has variance 1, so a lesser component explains less than one raw column

PCA in R

pca_result <- prcomp(data, scale = TRUE, center = TRUE) summary(pca_result) # sd, proportion and cumulative per PC pca_result$x # the scores, one row per observation fviz_eig(pca_result) # scree plot fviz_pca_biplot(pca_result, repel = TRUE)

Always with scale and center TRUE when the variables are in different units, which is nearly always.

scale = TRUE
Standardises spread, so a large-unit variable cannot dominate
center = TRUE
Subtracts each mean
$x
Scores: where each observation sits on each component
Step 1 of 21
The ideaTheory

Eleven columns, two real ideas

A car has eleven specifications: mileage, cylinders, displacement, horsepower, weight, gear ratio and so on. Comparing cars on eleven numbers at once is impossible.

But most of those columns say the same thing twice. Big engines have more cylinders, more horsepower, more weight and worse mileage. That is one idea wearing four hats.

PCA finds the ideas. On the class's car data two of them explain 84 % of everything that differs between cars, so the whole market fits on a flat page.

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