PLS (Partial Least Squares)

Partial Least Squares regression for high-dimensional data and multicollinearity using the SIMPLS algorithm.

Functions

Function	Type	Description
pls_fit	Scalar	Process complete arrays in a single call
pls_fit_agg	Aggregate	Streaming row-by-row accumulation
pls_fit_predict_agg	Aggregate	Fit and predict with GROUP BY support

anofox_stats_pls_fit / pls_fit

PLS regression using the SIMPLS algorithm to find latent components that maximize covariance between X scores and y.

Signature:

anofox_stats_pls_fit(
    y LIST(DOUBLE),
    x LIST(LIST(DOUBLE)),
    [options MAP]
) -> STRUCT

Options MAP:

Key	Type	Default	Description
n_components	INTEGER	2	Number of latent components to extract
fit_intercept	BOOLEAN	true	Include intercept term

Returns:

STRUCT(
    coefficients LIST(DOUBLE),  -- Regression coefficients
    intercept DOUBLE,           -- Intercept term (if fitted)
    r_squared DOUBLE,           -- Coefficient of determination
    n_components INTEGER,       -- Number of components used
    n_observations BIGINT,      -- Number of observations
    n_features INTEGER          -- Number of features
)

Example:

-- PLS with 3 components for high-dimensional data
SELECT pls_fit(
    [y1, y2, y3, y4, y5],
    [[x1_1, x1_2, x1_3, x1_4, x1_5],
     [x2_1, x2_2, x2_3, x2_4, x2_5],
     [x3_1, x3_2, x3_3, x3_4, x3_5]],
    {'n_components': 2}
);

-- Per-group PLS regression
SELECT
    category,
    (pls_fit_agg(y, [x1, x2, x3, x4, x5], {'n_components': 2})).r_squared
FROM high_dim_data
GROUP BY category;

anofox_stats_pls_fit_agg / pls_fit_agg

Streaming PLS regression aggregate function.

SELECT pls_fit_agg(y, [x1, x2, x3], {'n_components': 2}) FROM data;

Choosing n_components

• n_components = 1: Maximum explained variance in one direction
• n_components = 2-3: Typical for moderate-dimensional data
• n_components = min(n, p): Maximum extractable components

Use cross-validation to select optimal number of components.

Use Cases

• High-dimensional data: More features than observations
• Multicollinearity: Correlated predictors
• Chemometrics and spectroscopy: NIR, Raman spectral analysis
• Genomics and bioinformatics: Gene expression data
• Dimension reduction: When features outnumber samples

See Also

• OLS - Standard regression
• Ridge - L2 regularization alternative
• Elastic Net - L1+L2 regularization