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Fit

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

Fit[data, {f1, ..., fn}, vars] finds a least-squares fit a1 f1 + ... + an fn to data for functions of the variables vars (a symbol or list of symbols).
Fit[{m, v}] gives the coefficient vector minimizing ||m.a - v|| for design matrix m and response vector v.
Data may be a list of values {v1, ...} (coordinates 1, 2, ...), univariate pairs {{x, v}, ...}, or multivariate rows {{x, y, ..., v}, ...}.
Options: WorkingPrecision (Automatic | n | Infinity), FitRegularization ({"Tikhonov"|"L2"|"RidgeRegression"|"LASSO"|"L1", lambda}), NormFunction.

Examples

No verified examples yet for this function.

Implementation notes

Algorithm. builtin_fit fits a linear combination a_1 f_1 + … + a_n f_n of basis functions to data and returns the symbolic fit expression. Fit[data, {f1,…,fn}, vars] first forms the design matrix m (the same construction as DesignMatrix) and response vector v, solves the linear least-squares problem for the coefficient vector a, and reassembles Σ a_i f_i. Fit[{m, v}] solves directly given a design matrix and response. The least-squares solve is reuse-first:

  • Plain L2 and ridge/Tikhonov (FitRegularization -> {"L2"|"Tikhonov"|"RidgeRegression", λ}) route through the LeastSquares builtin (PseudoInverse . v); ridge is reduced to ordinary least squares on the augmented system [m; √λ I], [v; 0].
  • LASSO ({"L1"|"LASSO", λ}) uses cyclic coordinate descent with soft-thresholding (machine precision).
  • NormFunction -> Norm[#,1] (least absolute deviations) uses iteratively reweighted least squares (IRLS).
  • Any other norm, or a norm combined with regularisation, falls back to FindMinimum, warm-started from the L2 solution.

WorkingPrecision selects exact rational (Infinity), machine (Automatic), or n-digit MPFR arithmetic.

Data structures. Design matrix and response are Lists of Lists / List; coefficients come back as a vector that is recombined with the basis-function expressions. Data shapes are normalised exactly as in DesignMatrix (implicit {1,2,…} abscissae, univariate, or multivariate coordinate rows).

Attributes: Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

Notes & additional examples

Worked examples

In[1]:= Fit[{1, 2, 1.3, 3.75, 2.25}, {1, x}, x]
Out[1]= 0.785 + 0.425 x

In[1]:= Fit[{1, 4, 9, 16}, {1, x, x^2}, x]
Out[1]= 0.0 + 0.0 x + 1.0 x^2

In[1]:= Fit[{{0, 1}, {1, 2.7}, {2, 7.4}, {3, 20.1}}, {1, x, x^2}, x]
Out[1]= 1.25 - 2.05 x + 2.75 x^2

In[1]:= Fit[{{0, 0, 1}, {1, 0, 2}, {0, 1, 3}, {1, 1, 5}}, {1, x, y}, {x, y}]
Out[1]= 0.75 + 1.5 x + 2.5 y

Notes

Fit[data, {f1, ..., fn}, x] returns the least-squares linear combination a1 f1 + ... + an fn of the basis functions. Plain {v1, v2, ...} data is taken at abscissae 1, 2, ..., while {{x, v}, ...} pairs supply explicit abscissae; the perfect-square data recovers x^2 exactly. The third example fits a quadratic trend to noisy data, and the last shows a multivariate fit in two predictors {x, y} from {x, y, v} rows — the design matrix is solved by normal equations in either case.