Polars OLS

Least squares extension in Polars

Supports linear model estimation in Polars.

This package provides efficient rust implementations of common linear regression variants (OLS, WLS, Ridge, Elastic Net, Non-negative least squares, Recursive least squares) and exposes them as simple polars expressions which can easily be integrated into your workflow.

Why?

1. High Performance: implementations are written in rust and make use of optimized rust linear-algebra crates & LAPACK routines. See benchmark section.
2. Polars Integration: avoids unnecessary conversions from lazy to eager mode and to external libraries (e.g. numpy, sklearn) to do simple linear regressions. Chain least squares formulae like any other expression in polars.
3. Efficient Implementations:
   • Numerically stable algorithms are chosen where appropriate (e.g. QR, Cholesky).
   • Flexible model specification allows arbitrary combination of sample weighting, L1/L2 regularization, & non-negativity constraints on parameters.
   • Efficient rank-1 update algorithms used for moving window regressions.
4. Easy Parallelism: Computing OLS predictions, in parallel, across groups can not be easier: call .over() or group_by just like any other polars' expression and benefit from full Rust parallelism.
5. Formula API: supports building models via patsy syntax: y ~ x1 + x2 + x3:x4 -1 (like statsmodels) which automatically converts to equivalent polars expressions.

Installation

First, you need to install Polars. Then run the below to install the polars-ols extension:

pip install polars-ols

API & Examples

Importing polars_ols will register the namespace least_squares provided by this package. You can build models either by either specifying polars expressions (e.g. pl.col(...)) for your targets and features or using the formula api (patsy syntax). All models support the following general (optional) arguments:

• mode - a literal which determines the type of output produced by the model
• null_policy - a literal which determines how to deal with missing data
• add_intercept - a boolean specifying if an intercept feature should be added to the features
• sample_weights - a column or expression providing non-negative weights applied to the samples

Remaining parameters are model specific, for example alpha penalty parameter used by regularized least squares models.

See below for basic usage examples. Please refer to the tests or demo notebook for detailed examples.

import polars as pl
import polars_ols as pls  # registers 'least_squares' namespace

df = pl.DataFrame({"y": [1.16, -2.16, -1.57, 0.21, 0.22, 1.6, -2.11, -2.92, -0.86, 0.47],
                   "x1": [0.72, -2.43, -0.63, 0.05, -0.07, 0.65, -0.02, -1.64, -0.92, -0.27],
                   "x2": [0.24, 0.18, -0.95, 0.23, 0.44, 1.01, -2.08, -1.36, 0.01, 0.75],
                   "group": [1, 1, 1, 1, 1, 2, 2, 2, 2, 2],
                   "weights": [0.34, 0.97, 0.39, 0.8, 0.57, 0.41, 0.19, 0.87, 0.06, 0.34],
                   })

lasso_expr = pl.col("y").least_squares.lasso("x1", "x2", alpha=0.0001, add_intercept=True).over("group")
wls_expr = pls.compute_least_squares_from_formula("y ~ x1 + x2 -1", sample_weights=pl.col("weights"))

predictions = df.with_columns(lasso_expr.round(2).alias("predictions_lasso"),
                              wls_expr.round(2).alias("predictions_wls"))

print(predictions.head(5))

shape: (5, 7)
┌───────┬───────┬───────┬───────┬─────────┬───────────────────┬─────────────────┐
│ y     ┆ x1    ┆ x2    ┆ group ┆ weights ┆ predictions_lasso ┆ predictions_wls │
│ ---   ┆ ---   ┆ ---   ┆ ---   ┆ ---     ┆ ---               ┆ ---             │
│ f64   ┆ f64   ┆ f64   ┆ i64   ┆ f64     ┆ f64               ┆ f64             │
╞═══════╪═══════╪═══════╪═══════╪═════════╪═══════════════════╪═════════════════╡
│ 1.16  ┆ 0.72  ┆ 0.24  ┆ 1     ┆ 0.34    ┆ 0.97              ┆ 0.93            │
│ -2.16 ┆ -2.43 ┆ 0.18  ┆ 1     ┆ 0.97    ┆ -2.23             ┆ -2.18           │
│ -1.57 ┆ -0.63 ┆ -0.95 ┆ 1     ┆ 0.39    ┆ -1.54             ┆ -1.54           │
│ 0.21  ┆ 0.05  ┆ 0.23  ┆ 1     ┆ 0.8     ┆ 0.29              ┆ 0.27            │
│ 0.22  ┆ -0.07 ┆ 0.44  ┆ 1     ┆ 0.57    ┆ 0.37              ┆ 0.36            │
└───────┴───────┴───────┴───────┴─────────┴───────────────────┴─────────────────┘

The mode parameter is used to set the type of the output returned by all methods ("predictions", "residuals", "coefficients", "statistics"). It defaults to returning predictions matching the input's length. Note that "statistics" is currently only supported for OLS/WLS/Ridge models.

In case "coefficients" is set the output is a polars Struct with coefficients as values and feature names as fields. It's output shape 'broadcasts' depending on context, see below:

coefficients = df.select(pl.col("y").least_squares.from_formula("x1 + x2", mode="coefficients")
                         .alias("coefficients"))

coefficients_group = df.select("group", pl.col("y").least_squares.from_formula("x1 + x2", mode="coefficients").over("group")
                        .alias("coefficients_group")).unique(maintain_order=True)

print(coefficients)
print(coefficients_group)

shape: (1, 1)
┌──────────────────────────────┐
│ coefficients                 │
│ ---                          │
│ struct[3]                    │
╞══════════════════════════════╡
│ {0.977375,0.987413,0.000757} │  # <--- coef for x1, x2, and intercept added by formula API
└──────────────────────────────┘
shape: (2, 2)
┌───────┬───────────────────────────────┐
│ group ┆ coefficients_group            │
│ ---   ┆ ---                           │
│ i64   ┆ struct[3]                     │
╞═══════╪═══════════════════════════════╡
│ 1     ┆ {0.995157,0.977495,0.014344}  │
│ 2     ┆ {0.939217,0.997441,-0.017599} │  # <--- (unique) coefficients per group
└───────┴───────────────────────────────┘

For dynamic models (like rolling_ols) or if in a .over, .group_by, or .with_columns context, the coefficients will take the shape of the data it is applied on. For example:

coefficients = df.with_columns(pl.col("y").least_squares.rls(pl.col("x1"), pl.col("x2"), mode="coefficients")
                         .over("group").alias("coefficients"))

print(coefficients.head())

shape: (5, 6)
┌───────┬───────┬───────┬───────┬─────────┬─────────────────────┐
│ y     ┆ x1    ┆ x2    ┆ group ┆ weights ┆ coefficients        │
│ ---   ┆ ---   ┆ ---   ┆ ---   ┆ ---     ┆ ---                 │
│ f64   ┆ f64   ┆ f64   ┆ i64   ┆ f64     ┆ struct[2]           │
╞═══════╪═══════╪═══════╪═══════╪═════════╪═════════════════════╡
│ 1.16  ┆ 0.72  ┆ 0.24  ┆ 1     ┆ 0.34    ┆ {1.235503,0.411834} │
│ -2.16 ┆ -2.43 ┆ 0.18  ┆ 1     ┆ 0.97    ┆ {0.963515,0.760769} │
│ -1.57 ┆ -0.63 ┆ -0.95 ┆ 1     ┆ 0.39    ┆ {0.975484,0.966029} │
│ 0.21  ┆ 0.05  ┆ 0.23  ┆ 1     ┆ 0.8     ┆ {0.975657,0.953735} │
│ 0.22  ┆ -0.07 ┆ 0.44  ┆ 1     ┆ 0.57    ┆ {0.97898,0.909793}  │
└───────┴───────┴───────┴───────┴─────────┴─────────────────────┘

For plain OLS/WLS and Ridge models, support has been recently added for producing a simple statistical significance report. It can be used as such:

statistics = (df.select(
   pl.col("y").least_squares.ols(pl.col("x1", "x2"), mode="statistics", add_intercept=True)
)
.unnest("statistics")  # results stored in a nested series by default
.explode(["feature_names", "coefficients", "standard_errors", "t_values", "p_values"])
)

print(statistics)

shape: (3, 8)
┌─────────┬──────────┬─────────┬──────────────┬──────────────┬─────────────┬───────────┬───────────┐
│ r2      ┆ mae      ┆ mse     ┆ feature_name ┆ coefficients ┆ standard_er ┆ t_values  ┆ p_values  │
│ ---     ┆ ---      ┆ ---     ┆ s            ┆ ---          ┆ rors        ┆ ---       ┆ ---       │
│ f64     ┆ f64      ┆ f64     ┆ ---          ┆ f64          ┆ ---         ┆ f64       ┆ f64       │
│         ┆          ┆         ┆ str          ┆              ┆ f64         ┆           ┆           │
╞═════════╪══════════╪═════════╪══════════════╪══════════════╪═════════════╪═══════════╪═══════════╡
│ 0.99631 ┆ 0.061732 ┆ 0.00794 ┆ x1           ┆ 0.977375     ┆ 0.037286    ┆ 26.212765 ┆ 3.0095e-8 │
│ 0.99631 ┆ 0.061732 ┆ 0.00794 ┆ x2           ┆ 0.987413     ┆ 0.037321    ┆ 26.457169 ┆ 2.8218e-8 │
│ 0.99631 ┆ 0.061732 ┆ 0.00794 ┆ const        ┆ 0.000757     ┆ 0.037474    ┆ 0.02021   ┆ 0.98444   │
└─────────┴──────────┴─────────┴──────────────┴──────────────┴─────────────┴───────────┴───────────┘

Finally, for convenience, in order to compute out-of-sample predictions you can use: least_squares.{predict, predict_from_formula}. This saves you the effort of un-nesting the coefficients and doing the dot product in python and instead does this in Rust, as an expression. Usage is as follows:

df_test.select(pl.col("coefficients_train").least_squares.predict(pl.col("x1"), pl.col("x2")).alias("predictions_test"))

Supported Models

Currently, this extension package supports the following variants:

• Ordinary Least Squares: least_squares.ols
• Weighted Least Squares: least_squares.wls
• Regularized Least Squares (Lasso / Ridge / Elastic Net) least_squares.{lasso, ridge, elastic_net}
• Non-negative Least Squares: least_squares.nnls
• Multi-target Least Squares: least_squares.multi_target_ols

As well as efficient implementations of moving window models:

• Recursive Least Squares: least_squares.rls
• Rolling / Expanding Window OLS: least_squares.{rolling_ols, expanding_ols}

An arbitrary combination of sample_weights, L1/L2 penalties, and non-negativity constraints can be specified with the least_squares.from_formula and least_squares.least_squares entry-points.

Solve Methods

polars-ols provides a choice over multiple supported numerical approaches per model (via solve_method flag), with implications on performance vs numerical accuracy. These choices are exposed to the user for full control, however, if left unspecified the package will choose a reasonable default depending on context.

For example, if you know you are dealing with highly collinear data, with unregularized OLS model, you may want to explicitly set solve_method="svd" so that the minimum norm solution is obtained.

Benchmark

The usual caveats of benchmarks apply here, but the below should still be indicative of the type of performance improvements to expect when using this package.

This benchmark was run on randomly generated data with pyperf on my Apple M2 Max macbook (32GB RAM, MacOS Sonoma 14.2.1). See benchmark.py for implementation.

n_samples=2_000, n_features=5

n_samples=10_000, n_features=100

• Numpy's lstsq (uses divide-and-conquer SVD) is already a highly optimized call into LAPACK and so the scope for speed-up is relatively limited, and the same applies to simple approaches like directly solving normal equations with Cholesky.
• However, even in such problems polars-ols Rust implementations for matching numerical algorithms tend to outperform by ~2-3x
• More substantial speed-up is achieved for the more complex models by working entirely in rust and avoiding overhead from back and forth into python.
• Expect a large additional relative order-of-magnitude speed up to your workflow if it involved repeated re-estimation of models in (python) loops.

Credits & Related Projects

• Rust linear algebra libraries faer and ndarray support the implementations provided by this extension package
• This package was templated around the very helpful: polars-plugin-tutorial
• The python package patsy is used for (optionally) building models from formulae
• Please check out the extension package polars-ds for general data-science functionality in polars

Future Work / TODOs

• Support generic types, in rust implementations, so that both f32 and f64 types are recognized. Right now data is cast to f64 prior to estimation
• Add docs explaining supported models, signatures, and API