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DesignMatrix

Status: Stable

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

Description

DesignMatrix[data, {f1, ..., fn}, vars] gives the design matrix with entries f_i evaluated at the data coordinates.
Data shapes match Fit. The WorkingPrecision option converts entries to machine or n-digit reals; otherwise they are exact.

Examples

All examples below are verified against the current Mathilda build.

In[1]:= DesignMatrix[{{0,1},{1,0},{3,2},{5,4}}, {1, x}, x]
Out[1]= {{1, 0}, {1, 1}, {1, 3}, {1, 5}}

In[2]:= DesignMatrix[{{0,0,0},{1,0,1},{0,1,2}}, {1, x, y}, {x, y}]
Out[2]= {{1, 0, 0}, {1, 1, 0}, {1, 0, 1}}

Implementation notes

Algorithm. builtin_designmatrix (in fit.c, the same module as Fit) builds the design matrix m_{ij} = f_i(coords_j) for DesignMatrix[data, {f1,…,fn}, vars]. It normalises the data shape ({v1,…} → {{1,v1},…}; {{x,v},…} univariate; {{x1,…,xk,v},…} multivariate, dropping the trailing response value to obtain the coordinate vector for each row), then evaluates each basis function f_i at each data point by substituting the vars symbols with that row's coordinates (pattern/ReplaceAll-style substitution followed by evaluate). The result is a List of rows, one per data point, each the vector {f_1(coords), …, f_n(coords)} — exactly the matrix that Fit/LeastSquares solve against.

Data structures. A List of Lists of evaluated basis-function values; entry representation (exact vs. machine vs. MPFR) follows the same WorkingPrecision handling as Fit. This is the Vandermonde-like basis matrix for a polynomial basis but works for any list of basis functions.

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]:= DesignMatrix[{{0, 1}, {1, 0}, {3, 2}, {5, 4}}, {1, x}, x]
Out[1]= {{1, 0}, {1, 1}, {1, 3}, {1, 5}}

In[1]:= DesignMatrix[{{1, 1}, {2, 8}, {3, 27}}, {1, x, x^2, x^3}, x]
Out[1]= {{1, 1, 1, 1}, {1, 2, 4, 8}, {1, 3, 9, 27}}

In[1]:= DesignMatrix[{{1, 1, 5}, {2, 4, 6}, {3, 9, 2}}, {1, x, y, x*y}, {x, y}]
Out[1]= {{1, 1, 1, 1}, {1, 2, 4, 8}, {1, 3, 9, 27}}

In[1]:= DesignMatrix[{{1, 2}, {2, 5}}, {1, Sin[x]}, x, WorkingPrecision -> 40]
Out[1]= {{1.0, 0.84147098480789650665250232163029899962254}, {1.0, 0.90929742682568169539601986591174484270222}}

Notes

DesignMatrix[data, {f1, ..., fn}, vars] builds the matrix whose rows are the basis functions f_i evaluated at each data coordinate — exactly the matrix Fit assembles internally before solving the normal equations. The data shapes match Fit: with a single variable, each row is {x_k, y_k} (or just {x_k}) and only the leading coordinate(s) are substituted, so the response column is ignored when forming the design entries. A polynomial basis {1, x, x^2, x^3} therefore produces a Vandermonde matrix. For several predictors the basis may mix the variables freely ({1, x, y, x*y}). Entries are kept exact unless WorkingPrecision -> MachinePrecision or a digit count is supplied, in which case each entry is converted to an approximate number — useful when the basis contains transcendental functions such as Sin.