LinearModelFit¶

Status: Partial

implemented with documented limitations or caveats; some argument forms fall through to symbolic/unevaluated output.

Description¶

LinearModelFit[data]

fits a linear model with an intercept and returns a PredictorFunction carrying its coefficients. Wolfram's version returns a FittedModel with regression diagnostics (RSquared, standard errors, ANOVA); those are not implemented and are not approximated. The coefficients are the same ones Predict finds.

Examples (7)¶

Every input below was run against the current Mathilda build and its output recorded.

Basic examples (3)¶

In[1]:= m = LinearModelFit[{1. -> 3., 2. -> 5., 3. -> 7., 4. -> 9.}]
Out[1]= PredictorFunction["LinearRegression", <>]

In[2]:= {m[10.], m["Coefficients"]}
Out[2]= {21.0, {1.0, 2.0}}

In[3]:= LinearModelFit[{{1., 1., 6.}, {2., 1., 8.}, {1., 2., 9.}, {3., 2., 13.}, {2., 3., 14.}}]["Coefficients"]
Out[3]= {1.0, 2.0, 3.0}

Applications (4)¶

An intercept plus a slope

In[4]:= m = LinearModelFit[{1. -> 3., 2. -> 5., 3. -> 7., 4. -> 9.}]
Out[4]= PredictorFunction["LinearRegression", <>]

Apply the fitted line

In[5]:= m[10.]
Out[5]= 21.0

The same coefficients Predict finds

In[6]:= m["Coefficients"]
Out[6]= {1.0, 2.0}

Multiple regressors, last column the response

In[7]:= LinearModelFit[{{1., 1., 6.}, {2., 1., 8.}, {1., 2., 9.}, {3., 2., 13.}, {2., 3., 14.}}]["Coefficients"]
Out[7]= {1.0, 2.0, 3.0}

Implementation notes¶

Algorithm. builtin_linear_model_fit fits an ordinary least-squares line with an intercept through the same ml_ols kernel Predict uses — the normal equations (AᵀA) c = Aᵀy with A = [1 | x], solved by Gaussian elimination with partial pivoting — and returns a PredictorFunction["LinearRegression", {intercept, c₁, …}, dim, 0]. Data is read by ml_read_training, so a list of rules {features -> value, …} or a matrix whose last column is the response both work. A singular system (collinear features, or fewer observations than parameters) declines, since there is no unique fit.

The object is applied and queried exactly like any other PredictorFunction: m[x] evaluates intercept + c · x, and m["Coefficients"] reads the fitted vector — the same coefficients Predict finds.

Data structures. The shared positional, method-tagged model representation (see src/ml/predict.h); the coefficient List is the only payload.

Complexity / limits. O(n · p² + p³) with p = dim + 1. Wolfram's LinearModelFit returns a FittedModel whose properties are regression diagnostics (RSquared, standard errors, ANOVA); those are a separate piece of work and are not approximated here — only the coefficients, under a name a reader of regression code will look for.

Attributes: Protected.

References¶

See also: PredictorFunction, Predict

T. Hastie, R. Tibshirani and J. Friedman, The Elements of Statistical Learning, 2nd ed. (Springer, 2009), §3.2 (least squares).
Source: src/ml/predict.c
Specification: docs/spec/builtins/machine-learning.md
Tests: tests/test_ml_predict.c

Notes & additional examples¶

Notes¶

LinearModelFit[data] fits a linear model with an intercept and returns a PredictorFunction carrying its coefficients — the very ones Predict finds, through the same least-squares kernel. It is applied and queried exactly like any other PredictorFunction.

Wolfram's LinearModelFit returns a FittedModel whose properties are regression diagnostics — RSquared, standard errors, ANOVA. Those are a separate piece of work and are deliberately not approximated here; only the coefficients are offered, under a name a reader of regression code will look for. A singular system returns unevaluated.