func ExampleCholesky() {
// Construct a symmetric positive definite matrix.
tmp := mat64.NewDense(4, 4, []float64{
2, 6, 8, -4,
1, 8, 7, -2,
2, 2, 1, 7,
8, -2, -2, 1,
})
var a mat64.SymDense
a.SymOuterK(1, tmp)
fmt.Printf("a = %0.4v\n", mat64.Formatted(&a, mat64.Prefix(" ")))
// Compute the cholesky factorization.
var chol mat64.Cholesky
if ok := chol.Factorize(&a); !ok {
fmt.Println("a matrix is not positive semi-definite.")
}
// Find the determinant.
fmt.Printf("\nThe determinant of a is %0.4g\n\n", chol.Det())
// Use the factorization to solve the system of equations a * x = b.
b := mat64.NewVector(4, []float64{1, 2, 3, 4})
var x mat64.Vector
if err := x.SolveCholeskyVec(&chol, b); err != nil {
fmt.Println("Matrix is near singular: ", err)
}
fmt.Println("Solve a * x = b")
fmt.Printf("x = %0.4v\n", mat64.Formatted(&x, mat64.Prefix(" ")))
// Extract the factorization and check that it equals the original matrix.
var t mat64.TriDense
t.LFromCholesky(&chol)
var test mat64.Dense
test.Mul(&t, t.T())
fmt.Println()
fmt.Printf("L * L^T = %0.4v\n", mat64.Formatted(&a, mat64.Prefix(" ")))
// Output:
// a = ⎡120 114 -4 -16⎤
// ⎢114 118 11 -24⎥
// ⎢ -4 11 58 17⎥
// ⎣-16 -24 17 73⎦
//
// The determinant of a is 1.543e+06
//
// Solve a * x = b
// x = ⎡ -0.239⎤
// ⎢ 0.2732⎥
// ⎢-0.04681⎥
// ⎣ 0.1031⎦
//
// L * L^T = ⎡120 114 -4 -16⎤
// ⎢114 118 11 -24⎥
// ⎢ -4 11 58 17⎥
// ⎣-16 -24 17 73⎦
}
func ExampleCholeskySymRankOne() {
a := mat64.NewSymDense(4, []float64{
1, 1, 1, 1,
0, 2, 3, 4,
0, 0, 6, 10,
0, 0, 0, 20,
})
fmt.Printf("A = %0.4v\n", mat64.Formatted(a, mat64.Prefix(" ")))
// Compute the Cholesky factorization.
var chol mat64.Cholesky
if ok := chol.Factorize(a); !ok {
fmt.Println("matrix a is not positive definite.")
}
x := mat64.NewVector(4, []float64{0, 0, 0, 1})
fmt.Printf("\nx = %0.4v\n", mat64.Formatted(x, mat64.Prefix(" ")))
// Rank-1 update the factorization.
chol.SymRankOne(&chol, 1, x)
// Rank-1 update the matrix a.
a.SymRankOne(a, 1, x)
var au mat64.SymDense
au.FromCholesky(&chol)
// Print the matrix that was updated directly.
fmt.Printf("\nA' = %0.4v\n", mat64.Formatted(a, mat64.Prefix(" ")))
// Print the matrix recovered from the factorization.
fmt.Printf("\nU'^T * U' = %0.4v\n", mat64.Formatted(&au, mat64.Prefix(" ")))
// Output:
// A = ⎡ 1 1 1 1⎤
// ⎢ 1 2 3 4⎥
// ⎢ 1 3 6 10⎥
// ⎣ 1 4 10 20⎦
//
// x = ⎡0⎤
// ⎢0⎥
// ⎢0⎥
// ⎣1⎦
//
// A' = ⎡ 1 1 1 1⎤
// ⎢ 1 2 3 4⎥
// ⎢ 1 3 6 10⎥
// ⎣ 1 4 10 21⎦
//
// U'^T * U' = ⎡ 1 1 1 1⎤
// ⎢ 1 2 3 4⎥
// ⎢ 1 3 6 10⎥
// ⎣ 1 4 10 21⎦
}
// NewNormalPrecision creates a new Normal distribution with the given mean and
// precision matrix (inverse of the covariance matrix). NewNormalPrecision
// panics if len(mu) is not equal to prec.Symmetric(). If the precision matrix
// is not positive-definite, NewNormalPrecision returns nil for norm and false
// for ok.
func NewNormalPrecision(mu []float64, prec *mat64.SymDense, src *rand.Rand) (norm *Normal, ok bool) {
if len(mu) == 0 {
panic(badZeroDimension)
}
dim := prec.Symmetric()
if dim != len(mu) {
panic(badSizeMismatch)
}
// TODO(btracey): Computing a matrix inverse is generally numerically instable.
// This only has to compute the inverse of a positive definite matrix, which
// is much better, but this still loses precision. It is worth considering if
// instead the precision matrix should be stored explicitly and used instead
// of the Cholesky decomposition of the covariance matrix where appropriate.
var chol mat64.Cholesky
ok = chol.Factorize(prec)
if !ok {
return nil, false
}
var sigma mat64.SymDense
sigma.InverseCholesky(&chol)
return NewNormal(mu, &sigma, src)
}
// ConditionNormal returns the Normal distribution that is the receiver conditioned
// on the input evidence. The returned multivariate normal has dimension
// n - len(observed), where n is the dimension of the original receiver. The updated
// mean and covariance are
// mu = mu_un + sigma_{ob,un}^T * sigma_{ob,ob}^-1 (v - mu_ob)
// sigma = sigma_{un,un} - sigma_{ob,un}^T * sigma_{ob,ob}^-1 * sigma_{ob,un}
// where mu_un and mu_ob are the original means of the unobserved and observed
// variables respectively, sigma_{un,un} is the unobserved subset of the covariance
// matrix, sigma_{ob,ob} is the observed subset of the covariance matrix, and
// sigma_{un,ob} are the cross terms. The elements of x_2 have been observed with
// values v. The dimension order is preserved during conditioning, so if the value
// of dimension 1 is observed, the returned normal represents dimensions {0, 2, ...}
// of the original Normal distribution.
//
// ConditionNormal returns {nil, false} if there is a failure during the update.
// Mathematically this is impossible, but can occur with finite precision arithmetic.
func (n *Normal) ConditionNormal(observed []int, values []float64, src *rand.Rand) (*Normal, bool) {
if len(observed) == 0 {
panic("normal: no observed value")
}
if len(observed) != len(values) {
panic("normal: input slice length mismatch")
}
for _, v := range observed {
if v < 0 || v >= n.Dim() {
panic("normal: observed value out of bounds")
}
}
ob := len(observed)
unob := n.Dim() - ob
obMap := make(map[int]struct{})
for _, v := range observed {
if _, ok := obMap[v]; ok {
panic("normal: observed dimension occurs twice")
}
obMap[v] = struct{}{}
}
if len(observed) == n.Dim() {
panic("normal: all dimensions observed")
}
unobserved := make([]int, 0, unob)
for i := 0; i < n.Dim(); i++ {
if _, ok := obMap[i]; !ok {
unobserved = append(unobserved, i)
}
}
mu1 := make([]float64, unob)
for i, v := range unobserved {
mu1[i] = n.mu[v]
}
mu2 := make([]float64, ob) // really v - mu2
for i, v := range observed {
mu2[i] = values[i] - n.mu[v]
}
n.setSigma()
var sigma11, sigma22 mat64.SymDense
sigma11.SubsetSym(n.sigma, unobserved)
sigma22.SubsetSym(n.sigma, observed)
sigma21 := mat64.NewDense(ob, unob, nil)
for i, r := range observed {
for j, c := range unobserved {
v := n.sigma.At(r, c)
sigma21.Set(i, j, v)
}
}
var chol mat64.Cholesky
ok := chol.Factorize(&sigma22)
if !ok {
return nil, ok
}
// Compute sigma_{2,1}^T * sigma_{2,2}^-1 (v - mu_2).
v := mat64.NewVector(ob, mu2)
var tmp, tmp2 mat64.Vector
err := tmp.SolveCholeskyVec(&chol, v)
if err != nil {
return nil, false
}
tmp2.MulVec(sigma21.T(), &tmp)
// Compute sigma_{2,1}^T * sigma_{2,2}^-1 * sigma_{2,1}.
// TODO(btracey): Should this be a method of SymDense?
var tmp3, tmp4 mat64.Dense
err = tmp3.SolveCholesky(&chol, sigma21)
if err != nil {
return nil, false
}
tmp4.Mul(sigma21.T(), &tmp3)
for i := range mu1 {
mu1[i] += tmp2.At(i, 0)
}
// TODO(btracey): If tmp2 can constructed with a method, then this can be
// replaced with SubSym.
for i := 0; i < len(unobserved); i++ {
for j := i; j < len(unobserved); j++ {
v := sigma11.At(i, j)
sigma11.SetSym(i, j, v-tmp4.At(i, j))
}
}
return NewNormal(mu1, &sigma11, src)
}
// AddBatch adds a set training points to the Gp. This call updates internal
// values needed for prediction, so it is more efficient to add samples
// as a batch.
func (g *GP) AddBatch(x mat64.Matrix, y []float64) error {
// Note: The outputs are stored scaled to have a mean of zero and a variance
// of 1.
// Verify input parameters
rx, cx := x.Dims()
ry := len(y)
if rx != ry {
panic(badInOut)
}
if cx != g.inputDim {
panic(badInputLength)
}
nSamples := len(g.outputs)
// Append the new data to the list of stored data.
inputs := mat64.NewDense(rx+nSamples, g.inputDim, nil)
inputs.Copy(g.inputs)
inputs.View(nSamples, 0, rx, g.inputDim).(*mat64.Dense).Copy(x)
g.inputs = inputs
// Rescale the output data to its original value, append the new data, and
// then rescale to have mean 0 and variance of 1.
for i, v := range g.outputs {
g.outputs[i] = v*g.std + g.mean
}
g.outputs = append(g.outputs, y...)
g.mean = stat.Mean(g.outputs, nil)
g.std = stat.StdDev(g.outputs, nil)
for i, v := range g.outputs {
g.outputs[i] = (v - g.mean) / g.std
}
// Add to the kernel matrix.
k := mat64.NewSymDense(rx+nSamples, nil)
k.CopySym(g.k)
g.k = k
// Compute the kernel with the new points and the old points
for i := 0; i < nSamples; i++ {
for j := nSamples; j < rx+nSamples; j++ {
v := g.kernel.Distance(g.inputs.RawRowView(i), g.inputs.RawRowView(j))
g.k.SetSym(i, j, v)
}
}
// Compute the kernel with the new points and themselves
for i := nSamples; i < rx+nSamples; i++ {
for j := i; j < nSamples+rx; j++ {
v := g.kernel.Distance(g.inputs.RawRowView(i), g.inputs.RawRowView(j))
if i == j {
v += g.noise
}
g.k.SetSym(i, j, v)
}
}
// Cache necessary matrix results for computing predictions.
var chol mat64.Cholesky
ok := chol.Factorize(g.k)
if !ok {
return ErrSingular
}
g.cholK = &chol
g.sigInvY.Reset()
v := mat64.NewVector(len(g.outputs), g.outputs)
g.sigInvY.SolveCholeskyVec(g.cholK, v)
return nil
}