PCA Exercises in R: 20 Principal Component Analysis Practice Problems

Twenty graded problems on principal component analysis in R, covering prcomp() fits, scaling decisions, scree plots, loadings, biplots, regression on PC scores, reconstruction error, and the SVD equivalence. Each exercise hides a full solution and a short explanation behind a click-to-reveal block so you can attempt the problem first.

Section 1. Fitting PCA and reading variance (4 problems)

Exercise 1.1: Fit prcomp on the four numeric columns of iris with scaling

Task: Fit a principal component analysis on the four numeric columns of the built-in iris dataset (Sepal.Length, Sepal.Width, Petal.Length, Petal.Width) with scale = TRUE so each column contributes the same variance to the fit. Save the result to ex_1_1 and print summary(ex_1_1) to inspect the standard deviation, proportion of variance, and cumulative proportion for all four components.

Expected result:

#> Importance of components:
#>                           PC1    PC2     PC3     PC4
#> Standard deviation     1.7084 0.9560 0.38309 0.14393
#> Proportion of Variance 0.7296 0.2285 0.03669 0.00518
#> Cumulative Proportion  0.7296 0.9581 0.99482 1.00000

Difficulty: Beginner

Exercise 1.2: Pull the proportion of variance explained by PC2

Task: From the ex_1_1 PCA fit above, extract just the proportion of variance explained by PC2 as a single number (not cumulative, not standard deviation). Read it directly from the importance matrix that summary() produces rather than recomputing it. Save the scalar to ex_1_2 and print it. This is the fastest sanity-check you can make on an existing fit.

Expected result:

#> [1] 0.2285

Difficulty: Beginner

Exercise 1.3: Compare scaled vs unscaled PCA on mtcars

Task: A code reviewer pushes back on an unscaled PCA of mtcars (columns mpg, disp, hp, wt) because the four variables have wildly different ranges. Fit two PCAs, one with scale = FALSE and one with scale = TRUE, on those four columns, and report the proportion of variance carried by PC1 in each. Save the two numbers as a length-2 named vector ex_1_3 with names unscaled and scaled.

Expected result:

#> unscaled   scaled
#>  0.92744  0.84305

Difficulty: Intermediate

Exercise 1.4: Extract the centre and scale vectors used by prcomp

Task: From ex_1_1 (the scaled iris fit), pull the centring and scaling vectors that prcomp() stored on the fit object. These are the column means and column standard deviations of the original data. Save a tibble with columns variable, center, scale to ex_1_4. Knowing where these live on the fit object is essential when you later project new observations onto an existing PCA.

Expected result:

#> # A tibble: 4 x 3
#>   variable     center scale
#>   <chr>         <dbl> <dbl>
#> 1 Sepal.Length   5.84 0.828
#> 2 Sepal.Width    3.06 0.436
#> 3 Petal.Length   3.76 1.77
#> 4 Petal.Width    1.20 0.762

Difficulty: Beginner

Section 2. Scree, eigenvalues, and component selection (4 problems)

Exercise 2.1: Compute cumulative variance and find PCs needed for 90%

Task: A quality team auditing a sensor pipeline wants to know the minimum number of principal components needed to retain at least 90% of the variance of USArrests (scaled). Fit the PCA, compute cumulative variance from sdev^2, then find the smallest k such that the cumulative proportion is at least 0.9. Save k as an integer to ex_2_1. This is the standard "elbow shortcut" you would write into a feature-engineering function.

Expected result:

# Smallest k such that cumulative variance >= 0.9
#> [1] 3

Difficulty: Intermediate

Exercise 2.2: Build a scree plot with ggplot2

Task: Build a scree plot for the scaled USArrests PCA showing the proportion of variance on the y-axis and the component number on the x-axis as a connected line with points. Use geom_line() and geom_point(). Label the axes "Component" and "Proportion of variance" and save the ggplot object to ex_2_2. A scree plot is the first chart most reviewers will ask to see in a PCA writeup.

Expected result:

# A ggplot object: scree plot, 4 points (PC1..PC4) connected by a line.
# Aesthetics: x = component index (1..4), y = proportion of variance.
# Approx values: 0.62, 0.247, 0.089, 0.043 (descending).
# Axes: "Component", "Proportion of variance".

Difficulty: Intermediate

Exercise 2.3: Apply the Kaiser criterion on scaled USArrests

Task: The Kaiser rule says retain every component whose eigenvalue exceeds 1, on the grounds that any such component explains more than a single original (standardised) variable would on its own. Compute eigenvalues from sdev^2 for the scaled USArrests PCA and report how many components pass the Kaiser threshold. Save the count to ex_2_3 as an integer.

Expected result:

# Count of components with eigenvalue > 1 (Kaiser rule)
#> [1] 1

Difficulty: Intermediate

Exercise 2.4: Apply the broken-stick rule to decide component retention

Task: The broken-stick rule retains the k-th component only if its observed proportion of variance exceeds the expected proportion under a uniform null where total variance is split randomly between p components. The expected proportions are b_k = (1/p) * sum(1/k:p). For the scaled USArrests PCA compute the per-component expected proportions, compare them to the observed proportions, and save the count of components passing the rule as ex_2_4.

Expected result:

# Count of components passing the broken-stick threshold
#> [1] 1

Difficulty: Advanced

Section 3. Loadings and interpretation (4 problems)

Exercise 3.1: Extract the rotation matrix and find the dominant variable for PC1

Task: Loadings live in the rotation slot of a prcomp fit. For the scaled USArrests PCA, extract the rotation matrix, then identify which original variable has the largest absolute loading on PC1. Save just the variable name (as a character string) to ex_3_1. This tells you in one line what PC1 "is mostly about".

Expected result:

#> [1] "Assault"

Difficulty: Intermediate

Exercise 3.2: Identify the top three variables by absolute loading on PC2

Task: A marketing analyst studying urbanisation patterns wants the three variables most responsible for PC2 of the scaled USArrests PCA, ranked by absolute loading. Pull the PC2 column from the rotation matrix, sort by absolute value descending, and save a tibble with columns variable and loading_pc2 containing the top three rows. Save the tibble to ex_3_2.

Expected result:

#> # A tibble: 3 x 2
#>   variable loading_pc2
#>   <chr>          <dbl>
#> 1 UrbanPop      -0.873
#> 2 Murder         0.418
#> 3 Assault        0.188

Difficulty: Intermediate

Exercise 3.3: Sign-flip a principal component for readable plots

Task: PCA component signs are arbitrary, which makes plots awkward when "more crime" comes out as negative scores. Refit the scaled USArrests PCA, then multiply the PC1 column of the rotation matrix and the PC1 column of the scores matrix by -1 so positive PC1 means more crime. Save the modified fit object (a list with the flipped rotation and x matrices) to ex_3_3.

Expected result:

#> head(ex_3_3$x[, 1:2])
#>             PC1        PC2
#> Alabama   0.976  1.122
#> Alaska    1.931  1.062
#> Arizona   1.745 -0.738
#> Arkansas -0.140  1.109
#> California 2.499 -1.527
#> Colorado  1.499 -0.978

Difficulty: Advanced

Exercise 3.4: Compute correlations between variables and components

Task: The "variable factor map" used by factoextra is just the correlation between each original variable and each principal component. For the scaled USArrests PCA, compute the full matrix of correlations (4 variables by 4 PCs) and save it to ex_3_4. For a scaled PCA this equals rotation %*% diag(sdev), so check that closed form against cor() on the raw data and the scores.

Expected result:

#>             PC1         PC2         PC3         PC4
#> Murder    -0.842  0.4163554  -0.20347 -0.270491
#> Assault   -0.918  0.1870032  -0.16089  0.309337
#> UrbanPop  -0.438 -0.8682710  -0.22631 -0.054955
#> Rape      -0.855  0.1664909   0.48386 -0.043124

Difficulty: Intermediate

Section 4. Scores and visualization (3 problems)

Exercise 4.1: Build a tidy score tibble joined to a label column

Task: Working with PC scores in base R quickly becomes painful because fit$x is a matrix without the original grouping variable. For the scaled iris PCA from ex_1_1, build a tibble with columns PC1, PC2, Species so it is ready for plotting. Save the tibble to ex_4_1 and print the first six rows.

Expected result:

#> # A tibble: 6 x 3
#>      PC1     PC2 Species
#>    <dbl>   <dbl> <fct>
#> 1 -2.26  -0.478  setosa
#> 2 -2.07   0.672  setosa
#> 3 -2.36   0.341  setosa
#> 4 -2.29   0.595  setosa
#> 5 -2.38  -0.645  setosa
#> 6 -2.07  -1.48   setosa

Difficulty: Intermediate

Exercise 4.2: Plot PC1 vs PC2 coloured by species

Task: A junior analyst onboarding to the team needs the canonical iris PCA scatter to put in a slide deck. Using the score tibble ex_4_1 from the previous exercise, build a ggplot with PC1 on the x-axis, PC2 on the y-axis, points coloured by Species, and reasonable axis labels including the proportion of variance explained. Save the ggplot to ex_4_2.

Expected result:

# A ggplot scatter, 150 points, three coloured clusters.
# setosa forms a tight cluster on the left (PC1 near -2 to -2.5).
# versicolor and virginica overlap on the right (PC1 between 0 and 3).
# Axes: "PC1 (73%)", "PC2 (23%)".

Difficulty: Intermediate

Exercise 4.3: Build a biplot of the scaled USArrests PCA

Task: Build a biplot of the scaled USArrests PCA showing both observations (state names as points) and the four variable loading vectors as arrows. The base biplot() function does this in one call; use it on the fit object and save the result as a recorded plot to ex_4_3 by wrapping the call in recordPlot() after producing it. Biplots compress two layers of PCA output into a single chart.

Expected result:

# A base R biplot:
# - State names plotted as text at their PC1/PC2 score positions.
# - Four arrows labelled Murder, Assault, UrbanPop, Rape pointing from the origin.
# - Arrow directions encode loadings: Murder/Assault/Rape cluster (similar direction);
#   UrbanPop points roughly perpendicular to the violent-crime cluster.

Difficulty: Intermediate

Section 5. Downstream use, reconstruction, and SVD (5 problems)

Exercise 5.1: Fit a regression on the first two PC scores

Task: A statistician wants a quick principal component regression on mtcars predicting mpg from the first two PC scores of disp, hp, wt, qsec. Fit a scaled PCA on those four predictors, bind PC1 and PC2 into a data frame with mpg, then run lm(mpg ~ PC1 + PC2). Save the fitted lm object to ex_5_1 and inspect the coefficients.

Expected result:

#> Call:
#> lm(formula = mpg ~ PC1 + PC2, data = pcr_df)
#>
#> Coefficients:
#> (Intercept)          PC1          PC2
#>     20.0906      -2.6178      -0.9216

Difficulty: Advanced

Exercise 5.2: Project new observations onto an existing PCA

Task: The audit team needs scores for three new car records using the PCA fit from ex_5_1 so the new cars can be plotted alongside the original mtcars rows. Build a tibble of three new observations with the same four columns as the fit, then use predict() on the existing fit to project them. Save the resulting score matrix (3 rows, 4 PCs) to ex_5_2.

Expected result:

#>        PC1         PC2          PC3        PC4
#> [1,] -2.13   0.7250     -0.0124     -0.158
#> [2,]  1.87  -0.6122      0.2840      0.071
#> [3,]  0.42   1.4520     -0.7320      0.205

Difficulty: Advanced

Exercise 5.3: Reconstruct the original data from the first k components

Task: Reconstruction error is the gold-standard measure of how lossy a k-component PCA is. For the scaled iris PCA, reconstruct the original four-column matrix using only PC1 and PC2 via scores[, 1:2] %*% t(rotation[, 1:2]), then de-scale and un-centre to get back to the original units. Compute the root-mean-square error between the original and reconstructed matrices and save it as a single number to ex_5_3.

Expected result:

#> [1] 0.1591

Difficulty: Advanced

Exercise 5.4: Show prcomp matches svd up to a sign convention

Task: prcomp() is implemented on top of svd(). Show this directly by running svd() on the centred and scaled iris matrix, then verifying the singular values relate to prcomp standard deviations via sdev = d / sqrt(n - 1). Save the largest absolute difference between the recomputed sdev and the prcomp sdev to ex_5_4.

Expected result:

# Largest absolute deviation between sdev recomputed from svd() and prcomp $sdev
#> [1] 0

Difficulty: Advanced

Exercise 5.5: Flag outliers via Mahalanobis distance on PC scores

Task: A fraud team uses PCA score space to spot outliers in USArrests. Compute Mahalanobis distance on the first two PC scores of the scaled USArrests PCA (the score-space covariance is diagonal with entries equal to the leading two eigenvalues, by construction). Flag states whose distance exceeds the 95% chi-square cutoff with 2 degrees of freedom. Save a character vector of flagged state names to ex_5_5.

Expected result:

#> [1] "Alaska"     "California" "Florida"    "Mississippi"
#> [5] "Nevada"     "North Carolina"

Difficulty: Advanced

What to do next

• PCA in R: A Complete Guide to prcomp and princomp: the parent tutorial covering theory, fit objects, and a step-by-step iris walk-through.
• k-Means Exercises in R: PCA pairs naturally with clustering; this hub drills the other half of the unsupervised toolkit.
• Linear Regression Exercises in R: exercises 5.1 and 5.2 above lean on this; if regression diagnostics feel rusty, work through that hub first.
• Multivariate Analysis in R: broader context for where PCA sits among MDS, factor analysis, and discriminant analysis.