Decomposition principalcomponentvariance

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PrincipalComponentVariance

How much of a dense matrix's variance each principal component explains — the numbers a scree plot is drawn from, and the ones a component count is chosen on.

public sealed class PrincipalComponentVariance

Example — three features, two of which move together.

using Lodestar.Decomposition;

// Five samples of three features; the third is nearly twice the first.
double[] matrix =
[
    1.0, 2.0, 2.1,
    2.0, 1.0, 3.9,
    3.0, 4.0, 6.2,
    4.0, 3.0, 7.9,
    5.0, 5.0, 10.1,
];

PrincipalComponentVariance variance =
    PrincipalComponentVariance.Compute(matrix, rowCount: 5, columnCount: 3);

int components = variance.ComponentCount;                            // => 3
double first = variance.ExplainedVarianceRatio[0];                   // => 0.95191178727706…
double firstTwo = variance.CumulativeExplainedVarianceRatio[1];      // => 0.99999072700625…

Remarks — this is not a PCA. There are no components to read and nothing to project onto: the projection is delegated, to ML.NET's ProjectToPrincipalComponents on any target or to NumFlat's PrincipalComponentAnalysis on net8.0 and above (decision 0004). What neither answers below net8.0 is how many of those components are worth keeping, and that is all this type does. Compute it here, choose the count off CumulativeExplainedVarianceRatio, and hand the count to whichever projection you use. Meta.Numerics, on netstandard2.0, reports it beside its own projection (decision 0004).

Dense, and the input's own scale. Nothing is standardised: a feature measured in thousands dominates one measured in units, which is PCA's behaviour and scikit-learn's. Standardise first with Lodestar.Preprocessing's StandardScaler when the features' units are not comparable. A CsrMatrix is refused by construction rather than densified, because centring fills it in (decision 0003).

Applies to — net10.0, netstandard2.0.

See also — PrincipalComponentVariance.Compute, TruncatedSvd, the Python equivalence table.

Properties

Property	What it holds
SampleCount	int — rows of the matrix, each one sample.
FeatureCount	int — columns of the matrix, each one feature.
ComponentCount	int — the smaller of the two: a block of n samples has at most n components however many features it has.
TotalVariance	double — the sum of the columns' sample variances, which is what the components share out.
ExplainedVariance	IReadOnlyList<double> — the variance along each component, largest first, over n − 1.
ExplainedVarianceRatio	IReadOnlyList<double> — each of those over TotalVariance.
CumulativeExplainedVarianceRatio	IReadOnlyList<double> — the running sum of the ratios, ending at one.

Members

Member	What it does
PrincipalComponentVariance.Compute	Computes the variance each principal component of a row-major matrix explains.