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Principal Component Analysis

Principal component analysis re-expresses correlated predictors as a smaller set of uncorrelated components, each one a linear combination of the originals, ordered so the first component captures as much of the total variance as possible, the second the most of what remains, and so on.

R equivalent

set.seed(1)
n <- 200
amount <- rnorm(n, mean = 50, sd = 15)
velocity <- rpois(n, 2)
chargebacks <- pmax(0, round(0.3 * velocity + rnorm(n, 0, 1)))
dispute_amount <- amount * 0.4 + rnorm(n, 0, 5)
cases <- data.frame(amount, velocity, chargebacks, dispute_amount)

model <- prcomp(cases, center = TRUE, scale. = TRUE)
print(summary(model))
print(model$rotation[, 1:2])

Console output:

Importance of components:
                         PC1    PC2    PC3     PC4
Standard deviation     1.337 1.1373 0.7914 0.54079
Proportion of Variance 0.447 0.3233 0.1566 0.07311
Cumulative Proportion  0.447 0.7703 0.9269 1.00000
                     PC1        PC2
amount         0.6422482  0.3008156
velocity       0.2891763 -0.6442941
chargebacks    0.2829960 -0.6486442
dispute_amount 0.6510051  0.2713958

Tool equivalent: Jube.Service.Repository.AnalyticsTrainingRun.AdvancedAnalyticsPrincipalComponentTrainingService / tool SubmitPrincipalComponentTraining — Kind = DimensionReducer, recall class PrincipalComponentRecall. Standardise (default true) is center = TRUE, scale. = TRUE above; instead of the caller picking a component count up front, VarianceThreshold (default 0.95) keeps just enough leading components to explain that share of total variance — reading straight off the Cumulative Proportion row printed above, 0.95 would keep the first three components here — with MaximumComponents as an optional hard cap on top of that rule. Recall for this family is a genuine double[] -> double[] transform (a row of raw predictors in, a row of component scores out), unlike every clusterer in this guide, which only recalls a cluster id or membership vector.

Code audit

Covariance or correlation matrix

Jube.Engine/AdvancedAnalytics/DimensionReduction/PrincipalComponentTrainer.cs:142 constructs new Accord.Statistics.Analysis.PrincipalComponentAnalysis() with no arguments. Accord’s own parameterless constructor defaults Method to PrincipalComponentMethod.Center (Accord.Statistics/Analysis/PrincipalComponentAnalysis.cs:171-177), which only mean-centres the data before computing its covariance matrix — it never asks Accord to divide by each column’s standard deviation (PrincipalComponentMethod.Standardize is never selected). Taken alone, this would make the analysis operate on the raw covariance matrix, which is scale-sensitive.

However, the trainer pre-processes the design matrix itself before Accord ever sees it: Standardised() (PrincipalComponentTrainer.cs:122-133) z-scores every column, using the train-split mean and standard deviation, whenever options.Standardise is true — which is the default (PrincipalComponentOptions.cs:21). Only when a caller explicitly sets Standardise = false does it fall back to value - means[j] (line 129), a mean-centre only, leaving each variable’s original scale in place. So the actual matrix Accord’s Center-mode covariance calculation operates on is, by default, the sample correlation matrix of the raw variables (every column already has unit variance going in) — the product is effectively correlation-matrix PCA by default, and only becomes true, scale-sensitive covariance-matrix PCA if Standardise is turned off. This distinction is real and does change results when variables have different scales, exactly as the task description anticipates; the trainer resolves it upstream of Accord rather than through Accord’s own Method switch.

Components retained by default

VarianceThreshold defaults to 0.95 (PrincipalComponentOptions.cs:23). The trainer reads analysis.ComponentProportions (Accord’s per-component explained-variance share), builds a running cumulative sum (PrincipalComponentTrainer.cs:176-183), and retains the smallest component count whose cumulative proportion first reaches or exceeds the threshold (lines 185-192) — clamped to at least 1 and at most the number of components Accord actually produced, and optionally capped further by MaximumComponents (default unset, i.e. no cap, PrincipalComponentOptions.cs:25, applied at PrincipalComponentTrainer.cs:194-197). If that cap forces fewer components than the variance threshold would otherwise need, a warning records the shortfall (PrincipalComponentTrainer.cs:199-201).

Other findings

  • Every eigenvector’s sign is normalised so its largest-magnitude loading is positive (PrincipalComponentTrainer.cs:159-174) — a deliberate post-processing step the trainer performs and Accord does not; Accord’s own documentation for this class even remarks that users often ask “why my matrices have inverted signs” (Accord.Statistics/Analysis/PrincipalComponentAnalysis.cs:69-73), which is precisely the ambiguity this sign-flip resolves for stored models.
  • Held-out quality is a hand-computed RMS reconstruction error (ReconstructionError, PrincipalComponentTrainer.cs:210-244): each row is projected onto the retained components and reconstructed, and the root-mean-square residual per row is averaged over the split. This diagnostic is not something Accord exposes directly; it is computed entirely in the training service.
  • Missing values are median-imputed per variable on the training split, and constant or near-constant columns (one value in at least 99.5% of training rows) are flagged with a warning but not excluded from the analysis (PrincipalComponentTrainer.cs:100-118), unlike Mahalanobis and Gaussian mixture density, which both drop exactly-constant columns from their active variable set.
  • At least two training rows are required (PrincipalComponentTrainer.cs:78-79,343-344), and a warning (not a hard failure) is raised if the training split has no more rows than variables, since fewer components are then identifiable than variables (PrincipalComponentTrainer.cs:203-205).