SSJ
V. 2.6.

## umontreal.iro.lecuyer.randvarmulti Class MultinormalPCAGen

```java.lang.Object
umontreal.iro.lecuyer.randvarmulti.RandomMultivariateGen
umontreal.iro.lecuyer.randvarmulti.MultinormalGen
umontreal.iro.lecuyer.randvarmulti.MultinormalPCAGen
```

`public class MultinormalPCAGenextends MultinormalGen`

Extends `MultinormalGen` for a multivariate normal distribution, generated via the method of principal components analysis (PCA) of the covariance matrix. The covariance matrix Σ is decomposed (by the constructor) as Σ = VΛVt where V is an orthogonal matrix and Λ is the diagonal matrix made up of the eigenvalues of Σ. Vt is the transpose matrix of V. The eigenvalues are ordered from the largest (λ1) to the smallest (λd). The random multinormal vector X is generated via

X = μ + AZ,

where A = V()1/2, and Z is a d-dimensional vector of independent standard normal random variates. The decomposition method uses the SingularValueDecomposition class in colt.

Constructor Summary
```MultinormalPCAGen(NormalGen gen1, double[] mu, double[][] sigma)```
Equivalent to `MultinormalPCAGen`(gen1, mu, new DenseDoubleMatrix2D(sigma)).
```MultinormalPCAGen(NormalGen gen1, double[] mu, cern.colt.matrix.DoubleMatrix2D sigma)```
Constructs a multinormal generator with mean vector mu and covariance matrix sigma.

Method Summary
`static cern.colt.matrix.DoubleMatrix2D` `decompPCA(double[][] sigma)`
Computes the decomposition sigma = Σ = VΛVt.
`static cern.colt.matrix.DoubleMatrix2D` `decompPCA(cern.colt.matrix.DoubleMatrix2D sigma)`
Computes the decomposition sigma = Σ = VΛVt.
` double[]` `getLambda()`
Returns the eigenvalues of Σ in decreasing order.
`static double[]` `getLambda(cern.colt.matrix.DoubleMatrix2D sigma)`
Computes and returns the eigenvalues of sigma in decreasing order.
` cern.colt.matrix.DoubleMatrix2D` `getPCADecompSigma()`
Returns the matrix A = V()1/2 of this object.
` void` `nextPoint(double[] p)`
Generates a point from this multinormal distribution.
`static void` ```nextPoint(NormalGen gen1, double[] mu, double[][] sigma, double[] p)```
Equivalent to `nextPoint`(gen1, mu, new DenseDoubleMatrix2D(sigma), p).
`static void` ```nextPoint(NormalGen gen1, double[] mu, cern.colt.matrix.DoubleMatrix2D sigma, double[] p)```
Generates a d-dimensional vector from the multinormal distribution with mean vector mu and covariance matrix sigma, using the one-dimensional normal generator gen1 to generate the coordinates of Z, and using the PCA decomposition of Σ.
` void` `setSigma(cern.colt.matrix.DoubleMatrix2D sigma)`
Sets the covariance matrix Σ of this multinormal generator to sigma (and recomputes A).

Methods inherited from class umontreal.iro.lecuyer.randvarmulti.MultinormalGen
`getMu, getMu, getSigma, setMu, setMu`

Methods inherited from class umontreal.iro.lecuyer.randvarmulti.RandomMultivariateGen
`getDimension, getStream, nextArrayOfPoints, setStream`

Methods inherited from class java.lang.Object
`equals, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait`

Constructor Detail

### MultinormalPCAGen

```public MultinormalPCAGen(NormalGen gen1,
double[] mu,
double[][] sigma)```
Equivalent to `MultinormalPCAGen`(gen1, mu, new DenseDoubleMatrix2D(sigma)).

### MultinormalPCAGen

```public MultinormalPCAGen(NormalGen gen1,
double[] mu,
cern.colt.matrix.DoubleMatrix2D sigma)```
Constructs a multinormal generator with mean vector mu and covariance matrix sigma. The mean vector must have the same length as the dimensions of the covariance matrix, which must be symmetric and positive semi-definite. If any of the above conditions is violated, an exception is thrown. The vector Z is generated by calling d times the generator gen1, which must be a standard normal 1-dimensional generator.

Parameters:
`gen1` - the one-dimensional generator
`mu` - the mean vector.
`sigma` - the covariance matrix.
Throws:
`NullPointerException` - if any argument is null.
`IllegalArgumentException` - if the length of the mean vector is incompatible with the dimensions of the covariance matrix.
Method Detail

### decompPCA

`public static cern.colt.matrix.DoubleMatrix2D decompPCA(double[][] sigma)`
Computes the decomposition sigma = Σ = VΛVt. Returns A = V()1/2.

### decompPCA

`public static cern.colt.matrix.DoubleMatrix2D decompPCA(cern.colt.matrix.DoubleMatrix2D sigma)`
Computes the decomposition sigma = Σ = VΛVt. Returns A = V()1/2.

`public cern.colt.matrix.DoubleMatrix2D getPCADecompSigma()`
Returns the matrix A = V()1/2 of this object.

Returns:
the PCA square root of the covariance matrix

### getLambda

`public static double[] getLambda(cern.colt.matrix.DoubleMatrix2D sigma)`
Computes and returns the eigenvalues of sigma in decreasing order.

### getLambda

`public double[] getLambda()`
Returns the eigenvalues of Σ in decreasing order.

### setSigma

`public void setSigma(cern.colt.matrix.DoubleMatrix2D sigma)`
Sets the covariance matrix Σ of this multinormal generator to sigma (and recomputes A).

Parameters:
`sigma` - the new covariance matrix.
Throws:
`IllegalArgumentException` - if sigma has incorrect dimensions.

### nextPoint

```public static void nextPoint(NormalGen gen1,
double[] mu,
cern.colt.matrix.DoubleMatrix2D sigma,
double[] p)```
Generates a d-dimensional vector from the multinormal distribution with mean vector mu and covariance matrix sigma, using the one-dimensional normal generator gen1 to generate the coordinates of Z, and using the PCA decomposition of Σ. The resulting vector is put into p. Note that this static method will be very slow for large dimensions, because it recomputes the singular value decomposition at every call. It is therefore recommended to use a MultinormalPCAGen object instead, if the method is to be called more than once.

Parameters:
`p` - the array to be filled with the generated point.
Throws:
`IllegalArgumentException` - if the one-dimensional normal generator uses a normal distribution with μ not equal to 0, or σ not equal to 1.
`IllegalArgumentException` - if the length of the mean vector is different from the dimensions of the covariance matrix, or if the covariance matrix is not symmetric and positive-definite.
`NullPointerException` - if any argument is null.

### nextPoint

```public static void nextPoint(NormalGen gen1,
double[] mu,
double[][] sigma,
double[] p)```
Equivalent to `nextPoint`(gen1, mu, new DenseDoubleMatrix2D(sigma), p).

### nextPoint

`public void nextPoint(double[] p)`
Generates a point from this multinormal distribution. This is much faster than the static method as it computes the singular value decomposition matrix only once in the constructor.

Overrides:
`nextPoint` in class `MultinormalGen`
Parameters:
`p` - the array to be filled with the generated point

SSJ
V. 2.6.

To submit a bug or ask questions, send an e-mail to Pierre L'Ecuyer.