Home › Stats-Regression › Generalized least squares

Fits a linear model under a given error covariance and reports what a summary table holds.

public static OlsSummary Fit(ReadOnlySpan<double> design, ReadOnlySpan<double> response, ReadOnlySpan<double> covariance, int featureCount, OlsOptions options = null)

Parameters — design is the regressors, row-major: featureCount values per row, with no constant column of your own. response is one observed value per row of design. covariance is the error covariance, row-major, one row and one column per row of design: symmetric and positive definite. featureCount is how many regressors each row carries. options chooses whether to fit an intercept, which covariance of the estimates to report, and at what confidence; null fits an intercept at 0.95.

Returns — OlsSummary: the fitted model, with its standard errors, t statistics, p-values, confidence intervals and variance inflation factors.

Exceptions — ArgumentOutOfRangeException when featureCount is not positive, or when covariance holds a value that is not finite. ArgumentException when design is not a whole number of rows, when response has a different length, when covariance is not the square of that length, is not symmetric or is not positive definite, when no residual degrees of freedom are left, or when a column of the whitened design is collinear with the columns before it.

Example — a diagonal covariance is the weighted fit with weights 1/σ, to rounding.

using Lodestar.Stats.Regression;

double[] x = [1.0, 2.0, 3.0, 4.0, 5.0, 6.0];
double[] y = [2.2, 3.9, 6.4, 7.7, 10.3, 11.8];
double[] variances = [1.0, 0.5, 2.0, 1.0, 0.25, 4.0];
double[] covariance = new double[36];
double[] weights = new double[6];
for (int i = 0; i < 6; i++)
{
    covariance[(i * 6) + i] = variances[i];
    weights[i] = 1.0 / variances[i];
}

OlsSummary generalized = GeneralizedLeastSquares.Fit(x, y, covariance, featureCount: 1);
OlsSummary weighted = WeightedLeastSquares.Fit(x, y, weights, featureCount: 1);

double gap = Math.Abs(generalized.Coefficients[1] - weighted.Coefficients[1]);
bool agree = gap < 1e-12;  // => True

Remarks — the rows are whitened by L⁻¹, where covariance = L Lᵀ is its Cholesky factorization, and fitted as OrdinaryLeastSquares.Fit would, the robust covariances on the whitened rows included. Three numbers follow the reference rather than the whitened rows:

• RSquared with an intercept is centred on the mean estimated in whitened space, (L⁻¹y)·(L⁻¹1) / (L⁻¹1)·(L⁻¹1), which reduces to the weighted mean when covariance is diagonal.
• The model's own constant stays out of the robust Wald test.
• VarianceInflationFactors are those of the design as given, which variance_inflation_factor returns.

An asymmetric covariance is refused, where the reference's Cholesky reads the lower triangle and ignores the upper one; a pair of mirrored entries may differ by 1e-12 of the larger. L⁻¹ is applied by forward substitution rather than formed, which agrees with the reference's explicit inverse to rounding.

Applies to — net10.0, netstandard2.0.

See also — OlsSummary, OlsOptions, WeightedLeastSquares.Fit, the generalized least squares index.