pca_components

std.stats.pca_components · Level L4

The principal axes of a data set, as columns, largest variance first: the right singular vectors of the centred data, each signed so its largest component is positive.

X − X̄ = U·S·Vᵀ, axes = V

Signature

pca_components(X: f64[p+k, p]) → f64[p, p]

Structure

The function as NOVA stores it: one box per input, operation and output, and arrows that carry values. A double border marks another library function this one runs — called once, or by Scan once per element; select it to open that function.

input
operation
constant
call
output

Verification

Signature proven by NOVA’s shape solver, for every size.
Agrees with the reference eigh(Xcᵀ·Xc) vectors, largest first, signs fixed to 80 digits (100-digit arithmetic), on all 40 test cases.
All 1,018 float64 results inside the running error bound; the closest uses 10% of it.
Interpreter and NumPy backend return bit-identical results.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 7%
bit-equal to the NumPy formula in float64: 6%
largest error, in units in the last place: 1.3e+4

Large ulp counts appear where a result is tiny next to the numbers it is computed from (after cancellation, for example), so one unit in the last place is tiny too; the absolute error is still inside the bound. Results within their own error of zero are not counted.

Note

The axes come from the SVD of the centred data; the reference takes the eigenvectors of its scatter matrix instead, so two different routes have to agree. Too-close variances, or a sign that could flip, are refused.

Identity

Calls

—

Called by

—

sha256:afa30853e74e67344203c65c539f903117a39fd5c9f84a2348b7c43e9e3103a1

The semantic hash of the graph. It changes when the program changes, and never when only its documentation does.

Control handle

Symbol: Ω:std.stats.pca_components · Ω:pca_components
Pins: sha256:619ec37827d00cbf4c1bb9cfffa38728ce18cd01427f8aa5172480b4318f0b87this graph alone
Evidence: sha256:df857618b3c239e6b32dfbafb23983b9b37f893d3d15184a814a9b26cd3a5f69the hash of its verification record
Needs: no capability: a pure function

Through NOVA’s control layer, the symbol launches this function only while the program still matches what it pins: a change to this graph, or to any graph it reaches, needs a migration first.