INVUPDT Function :: SAS/IML(R) 12.3 User's Guide

INVUPDT Function

INVUPDT (matrix, vector<, scalar>) ;

The INVUPDT function updates a matrix inverse.

The arguments to the INVUPDT function are as follows:

matrix: is an $n \times n$ nonsingular matrix. In most applications matrix is symmetric positive definite.
vector: is an $n \times 1$ or $1 \times n$ vector.
scalar: is a numeric scalar.

The Sherman-Morrison-Woodbury formula is

$(\mb {A} + \mb {U}\mb {V}^{\prime })^{-1} = \mb {A}^{-1} - \mb {A}^{-1} \mb {U} (\mb {I} + \mb {V}^{\prime }\mb {A}^{-1}\mb {U})^{-1} \mb {V}^{\prime }\mb {A}^{-1}$

where $\mb {A}$ is an $n \times n$ nonsingular matrix and $\mb {U}$ and $\mb {V}$ are $n\times k$ . The formula shows that a rank update to $\mb {A}$ corresponds to a rank update of $\mb {A}^{-1}$ .

The INVUPDT function implements the Sherman-Morrison-Woodbury formula for rank-one updates with $\mb {U} = w \mb {X}$ and $\mb {V} = \mb {X}$ , where $\mb {X}$ is an $n \times 1$ vector and is a scalar.

If $\mb {M} = \mb {A}^{-1}$ , then you can call the INVUPDT function as follows:

R = invupdt(M, X, w);

This statement computes the following matrix:

$\mb {R} = \mb {M} - w\mb {MX} (\mb {I} + w\mb {X}^{\prime }\mb {MX})^{-1} \mb {X}^{\prime }\mb {M}$

The matrix $\mb {R}$ is equivalent to $(\mb {A} + w\mb {XX}^{\prime })^{-1}$ . If $\mb {A}$ is symmetric positive definite, then so is $\mb {R}$ .

If is not specified, then it is given a default value of .

A common use of the INVUPDT function is in linear regression. If $\mb {Z}$ is a design matrix, $\mb {M} = (\mb {Z}^{\prime }\mb {Z})^{-1}$ is the associated inverse crossproduct matrix, and $\mb {v}$ is a new observation to be used in estimating the parameters of a linear model, then the inverse crossproducts matrix that includes the new observation can be updated from $\mb {M}$ by using the following statement:

M2 = invupdt(M, v);

If is , the function adds an observation to the inverse; if is , the function removes an observation from the inverse. If weighting is used, is the weight.

To perform the computation, the INVUPDT function uses about multiplications and additions, where is the row dimension of the positive definite argument matrix.

The following program demonstrates adding or removing observations from a linear fit and updating the inverse crossproduct matrix:

X = {0, 1, 1, 1, 2, 2, 3, 4, 4};
Y = {1, 1, 2, 6, 2, 3, 3, 3, 4};

/* find linear fit */
Z = j(nrow(X), 1, 1) || X;           /* design matrix */
M = inv(Z`*Z);

b = M*Z`*Y;                          /* LS estimate */
resid = Y - Z*b;                     /* residuals */
print "Original Fit", b resid;

/* residual for observation (1,6) seems too large.
   Take obs number 4 out of data set and refit. */
v = z[4,];
M = invupdt(M, v, -1);               /* update inverse crossprod */

keepObs = (1:3) || (5:nrow(X));
Z = Z[keepObs, ];
Y = Y[keepObs, ];
b = M*Z`*Y;                          /* new LS estimate */
print "After deleting observation 4", b;

/* Add a new obs (x,y) = (0,2) and refit. */
obs = {0 2};
v = 1 || obs[1];                     /* new row in design matrix */
M = invupdt(M, v);

Z = Z // v;
Y = Y // obs[2];
b = M*Z`*Y;                          /* new LS estimate */
print "After adding observation (0,2)", b;

Figure 23.148: Refitting Linear Regression Models

Original Fit

b	resid
2.0277778	-1.027778
0.375	-1.402778
	-0.402778
	3.5972222
	-0.777778
	0.2222222
	-0.152778
	-0.527778
	0.4722222

After deleting observation 4

b
1
0.6470588

After adding observation (0,2)

b
1.3
0.5470588