stats
::correlationMatrix
Compute the correlation matrix associated with a covariance matrix
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stats::correlationMatrix(cov
)
stats::correlationMatrix(cov)
returns to
correlation matrix cor
of the variancecovariance
matrix cov
. It is given by:
.
A covariance matrix C should be positive (semi)definite and hence satisfies C_{ij}^{2} ≤ C_{ii} C_{jj} for all indices i, j. Consequently, the absolute values of the entries of the corresponding correlation matrix do not exceed 1.
With the option CovarianceMatrix
, the routine stats::reg
returns
the variancecovariance matrix of the fit parameters in a regression
analysis. The corresponding correlation matrix of the fit parameters
is computed conveniently by applying stats::correlationMatrix
to
this matrix. Cf. Example 2.
We generate a positive definite matrix that may serve as a covariance matrix:
A := matrix([[4, 3, 2], [1, 2, 1], [0, 1, 1]]): cov := A*linalg::transpose(A)
The corresponding correlation matrix is:
stats::correlationMatrix(cov)
If the input matrix consists of floatingpoint data, the result is a matrix of floats:
stats::correlationMatrix(float(cov))
delete A, cov:
We consider a covariance matrix arising in a nonlinear regression problem. The model function y = a + b cos(x  c) is to be fit to the following randomized data:
r := stats::uniformRandom(0.1, 0.1): xdata := [i $i = 1..100]: ydata := [1 + 2*cos(x  3) + r() $ x in xdata]:
By construction, the variance of the y values
is the variance of the uniformly distributed random data on the interval [
0.1, 0.1] generated by the random generator r.
This variance is . We use stats::reg
to
obtain estimates of the fit parameters a, b, c of
the model. Appropriate weights for the regression are given by . The option CovarianceMatrix
makes stats::reg
include
the covariance matrix cov
of the fit parameters
in its return list:
weights := [300 $ i = 1..100]: DIGITS:= 4: [abc, chisquare, cov] := stats::reg(xdata, ydata, weights, a + b*cos(x  c), [x], [a, b, c], StartingValues = [1, 2, 3], CovarianceMatrix)
The correlation matrix of the parameters a, b, c is
obtained via stats::correlationMatrix
applied to
the covariance matrix cov
returned by stats::reg
:
stats::correlationMatrix(cov)
delete r, xdata, ydata, weights, DIGITS, abc, chisquare, cov:

The covariance matrix: a square matrix of category 
Matrix of the same dimension and type as the input matrix cov
. FAIL
is
returned if at least one of the diagonal elements of the input matrix cov
is
zero.