//This is a temporal function. It returns the determinant of a 3x3 matrix. Panics if the matrix is not 3x3.
//It is also defined in the chem package which is not-so-clean.
func det(A mat64.Matrix) float64 {
r, c := A.Dims()
if r != 3 || c != 3 {
panic(ErrDeterminant)
}
return (A.At(0, 0)*(A.At(1, 1)*A.At(2, 2)-A.At(2, 1)*A.At(1, 2)) - A.At(1, 0)*(A.At(0, 1)*A.At(2, 2)-A.At(2, 1)*A.At(0, 2)) + A.At(2, 0)*(A.At(0, 1)*A.At(1, 2)-A.At(1, 1)*A.At(0, 2)))
}
// MeanBatch predicts the mean at the set of locations specified by x. Stores in-place into yPred
// If yPred is nil new memory is allocated.
func (g *GP) MeanBatch(yPred []float64, x mat64.Matrix) []float64 {
rx, cx := x.Dims()
if cx != g.inputDim {
panic(badInputLength)
}
if yPred == nil {
yPred = make([]float64, rx)
}
ry := len(yPred)
if rx != ry {
panic(badOutputLength)
}
nSamples, _ := g.inputs.Dims()
covariance := mat64.NewDense(nSamples, rx, nil)
row := make([]float64, g.inputDim)
for j := 0; j < rx; j++ {
for k := 0; k < g.inputDim; k++ {
row[k] = x.At(j, k)
}
for i := 0; i < nSamples; i++ {
v := g.kernel.Distance(g.inputs.RawRowView(i), row)
covariance.Set(i, j, v)
}
}
yPredVec := mat64.NewVector(len(yPred), yPred)
yPredVec.MulVec(covariance.T(), g.sigInvY)
// Rescale the outputs
for i, v := range yPred {
yPred[i] = v*g.std + g.mean
}
return yPred
}
func columnMean(M mat.Matrix) mat.Matrix {
r, c := M.Dims()
SumMatrix := columnSum(M)
switch t := SumMatrix.(type) {
case *mat.Dense:
M := mat.NewDense(1, c, nil)
M.Scale(1/float64(r), SumMatrix)
return M
case mat.Mutable:
_ = t
V := SumMatrix.(mat.Mutable)
_, cols := V.Dims()
for i := 0; i < cols; i++ {
V.Set(0, i, SumMatrix.At(0, i)/float64(r))
}
return V
default:
panic("M is of an unknown type")
}
}
// findLinearlyIndependnt finds a set of linearly independent columns of A, and
// returns the column indexes of the linearly independent columns.
func findLinearlyIndependent(A mat64.Matrix) []int {
m, n := A.Dims()
idxs := make([]int, 0, m)
columns := mat64.NewDense(m, m, nil)
newCol := make([]float64, m)
// Walk in reverse order because slack variables are typically the last columns
// of A.
for i := n - 1; i >= 0; i-- {
if len(idxs) == m {
break
}
mat64.Col(newCol, i, A)
if len(idxs) == 0 {
// A column is linearly independent from the null set.
// This is what needs to be changed if zero columns are allowed, as
// a column of all zeros is not linearly independent from itself.
columns.SetCol(len(idxs), newCol)
idxs = append(idxs, i)
continue
}
if linearlyDependent(mat64.NewVector(m, newCol), columns.View(0, 0, m, len(idxs))) {
continue
}
columns.SetCol(len(idxs), newCol)
idxs = append(idxs, i)
}
return idxs
}
// Cov returns the covariance between a set of data points based on the current
// GP fit.
func (g *GP) Cov(m *mat64.SymDense, x mat64.Matrix) *mat64.SymDense {
if m != nil {
// TODO(btracey): Make this k**
panic("resuing m not coded")
}
// The joint covariance matrix is
// K(x_*, k_*) - k(x_*, x) k(x,x)^-1 k(x, x*)
nSamp, nDim := x.Dims()
if nDim != g.inputDim {
panic(badInputLength)
}
// Compute K(x_*, x) K(x, x)^-1 K(x, x_*)
kstar := g.formKStar(x)
var tmp mat64.Dense
tmp.SolveCholesky(g.cholK, kstar)
var tmp2 mat64.Dense
tmp2.Mul(kstar.T(), &tmp)
// Compute k(x_*, x_*) and perform the subtraction.
kstarstar := mat64.NewSymDense(nSamp, nil)
for i := 0; i < nSamp; i++ {
for j := i; j < nSamp; j++ {
v := g.kernel.Distance(mat64.Row(nil, i, x), mat64.Row(nil, j, x))
if i == j {
v += g.noise
}
kstarstar.SetSym(i, j, v-tmp2.At(i, j))
}
}
return kstarstar
}
func benchmarkCovarianceMatrixInPlace(b *testing.B, m mat64.Matrix) {
_, c := m.Dims()
res := mat64.NewDense(c, c, nil)
b.ResetTimer()
for i := 0; i < b.N; i++ {
CovarianceMatrix(res, m, nil)
}
}
func MatrixToImage(src mat64.Matrix) image.Image {
width, height := src.Dims()
img := image.NewRGBA(image.Rect(0, 0, width, height))
for x := 0; x < width; x++ {
for y := 0; y < height; y++ {
img.Set(x, y, Float64ToColor(src.At(x, y)))
}
}
return img
}
// linearlyDependent returns whether the vector is linearly dependent
// with the columns of A. It assumes that A is a full-rank matrix.
func linearlyDependent(vec *mat64.Vector, A mat64.Matrix) bool {
// Add vec to the columns of A, and see if the condition number is reasonable.
m, n := A.Dims()
aNew := mat64.NewDense(m, n+1, nil)
aNew.Copy(A)
col := mat64.Col(nil, 0, vec)
aNew.SetCol(n, col)
cond := mat64.Cond(aNew, 1)
return cond > 1e12
}
func rowSum(M mat.Matrix) mat.Matrix {
rows, _ := M.Dims()
floatRes := make([]float64, rows)
for i := 0; i < rows; i++ {
floatRes[i] = mat.Sum(getRowVector(i, M))
}
return mat.NewDense(rows, 1, floatRes)
}
func columnSum(M mat.Matrix) mat.Matrix {
_, cols := M.Dims()
floatRes := make([]float64, cols)
for i := 0; i < cols; i++ {
floatRes[i] = mat.Sum(getColumnVector(i, M))
}
return mat.NewDense(1, cols, floatRes)
}
// extractColumns creates a new matrix out of the columns of A specified by cols.
// TODO(btracey): Allow this to take a receiver.
func extractColumns(A mat64.Matrix, cols []int) *mat64.Dense {
r, _ := A.Dims()
sub := mat64.NewDense(r, len(cols), nil)
col := make([]float64, r)
for j, idx := range cols {
mat64.Col(col, idx, A)
sub.SetCol(j, col)
}
return sub
}
func verifyInputs(initialBasic []int, c []float64, A mat64.Matrix, b []float64) error {
m, n := A.Dims()
if len(c) != n {
panic("lp: c vector incorrect length")
}
if len(b) != m {
panic("lp: b vector incorrect length")
}
if len(c) != n {
panic("lp: c vector incorrect length")
}
if len(initialBasic) != 0 && len(initialBasic) != m {
panic("lp: initialBasic incorrect length")
}
// Do some sanity checks so that ab does not become singular during the
// simplex solution. If the ZeroRow checks are removed then the code for
// finding a set of linearly indepent columns must be improved.
// Check that if a row of A only has zero elements that corresponding
// element in b is zero, otherwise the problem is infeasible.
// Otherwise return ErrZeroRow.
for i := 0; i < m; i++ {
isZero := true
for j := 0; j < n; j++ {
if A.At(i, j) != 0 {
isZero = false
break
}
}
if isZero && b[i] != 0 {
// Infeasible
return ErrInfeasible
} else if isZero {
return ErrZeroRow
}
}
// Check that if a column only has zero elements that the respective C vector
// is positive (otherwise unbounded). Otherwise return ErrZeroColumn.
for j := 0; j < n; j++ {
isZero := true
for i := 0; i < m; i++ {
if A.At(i, j) != 0 {
isZero = false
break
}
}
if isZero && c[j] < 0 {
return ErrUnbounded
} else if isZero {
return ErrZeroColumn
}
}
return nil
}
// AssignCentroid assigns all of the examples in X to one of the groups
// in Mu
// X -> (m*n), Mu -> (K*n)
// returns (m*1)
func AssignCentroid(X, Mu mat.Matrix) *mat.Vector {
m, _ := X.Dims()
idx := mat.NewVector(m, nil)
for i := 0; i < m; i++ {
x := getRowVector(i, X)
idx.SetVec(i, float64(NearestCentroid(x, Mu)))
}
return idx
}
func getRowVector(index int, M mat.Matrix) *mat.Vector {
_, cols := M.Dims()
var rowData []float64
if cols == 0 {
rowData = []float64{}
} else {
rowData = mat.Row(nil, index, M)
}
return mat.NewVector(cols, rowData)
}
func J(idx *mat.Vector, X, Mu mat.Matrix) float64 {
Mux := ConstructXCentroidMatrix(idx, Mu)
xRows, xCols := X.Dims()
Diff := mat.NewDense(xRows, xCols, nil)
Diff.Sub(X, Mux)
Diff.MulElem(Diff, Diff)
Diff = rowSum(Diff).(*mat.Dense)
return columnSum(Diff).At(0, 0) / float64(xRows)
}
func benchmarkCovarianceMatrixWeighted(b *testing.B, m mat64.Matrix) {
r, _ := m.Dims()
wts := make([]float64, r)
for i := range wts {
wts[i] = 0.5
}
b.ResetTimer()
for i := 0; i < b.N; i++ {
CovarianceMatrix(nil, m, wts)
}
}
// MoveCentroid computes the averages for all the points inside each of the cluster
// centroid groups, then move the cluster centroid points to those averages.
// It then returns the new Centroids
func MoveCentroids(idx *mat.Vector, X, Mu mat.Matrix) mat.Matrix {
muRows, muCols := Mu.Dims()
NewMu := mat.NewDense(muRows, muCols, nil)
for k := 0; k < muRows; k++ {
CentroidKMean := columnMean(rowIndexIn(findIn(float64(k), idx), X))
NewMu.SetRow(k,
mat.Row(nil, 0, CentroidKMean))
}
return NewMu
}
func getColumnVector(index int, M mat.Matrix) *mat.Vector {
rows, _ := M.Dims()
var colData []float64
if rows == 0 {
colData = []float64{}
} else {
colData = mat.Col(nil, index, M)
}
return mat.NewVector(rows, colData)
}
// NewComplex returns a complex matrix constructed from r and i. At least one of
// r or i must be non-nil otherwise NewComplex will panic. If one of the inputs
// is nil, that part of the complex number will be zero when returned by At.
// If both are non-nil but differ in their sizes, NewComplex will panic.
func NewComplex(r, i mat64.Matrix) Complex {
if r == nil && i == nil {
panic("conv: no matrix")
} else if r != nil && i != nil {
rr, rc := r.Dims()
ir, ic := i.Dims()
if rr != ir || rc != ic {
panic(matrix.ErrShape)
}
}
return Complex{r: r, i: i, imagSign: 1}
}
// rowIndexIn returns a matrix contains the rows in indexes vector
func rowIndexIn(indexes *mat.Vector, M mat.Matrix) mat.Matrix {
m := indexes.Len()
_, n := M.Dims()
Res := mat.NewDense(m, n, nil)
for i := 0; i < m; i++ {
Res.SetRow(i, mat.Row(
nil,
int(indexes.At(i, 0)),
M))
}
return Res
}
// CovarianceMatrix calculates a covariance matrix (also known as a
// variance-covariance matrix) from a matrix of data, using a two-pass
// algorithm.
//
// The weights must have length equal to the number of rows in
// input data matrix x. If cov is nil, then a new matrix with appropriate size will
// be constructed. If cov is not nil, it should have the same number of columns as the
// input data matrix x, and it will be used as the destination for the covariance
// data. Weights must not be negative.
func CovarianceMatrix(cov *mat64.SymDense, x mat64.Matrix, weights []float64) *mat64.SymDense {
// This is the matrix version of the two-pass algorithm. It doesn't use the
// additional floating point error correction that the Covariance function uses
// to reduce the impact of rounding during centering.
r, c := x.Dims()
if cov == nil {
cov = mat64.NewSymDense(c, nil)
} else if n := cov.Symmetric(); n != c {
panic(matrix.ErrShape)
}
var xt mat64.Dense
xt.Clone(x.T())
// Subtract the mean of each of the columns.
for i := 0; i < c; i++ {
v := xt.RawRowView(i)
// This will panic with ErrShape if len(weights) != len(v), so
// we don't have to check the size later.
mean := Mean(v, weights)
floats.AddConst(-mean, v)
}
if weights == nil {
// Calculate the normalization factor
// scaled by the sample size.
cov.SymOuterK(1/(float64(r)-1), &xt)
return cov
}
// Multiply by the sqrt of the weights, so that multiplication is symmetric.
sqrtwts := make([]float64, r)
for i, w := range weights {
if w < 0 {
panic("stat: negative covariance matrix weights")
}
sqrtwts[i] = math.Sqrt(w)
}
// Weight the rows.
for i := 0; i < c; i++ {
v := xt.RawRowView(i)
floats.Mul(v, sqrtwts)
}
// Calculate the normalization factor
// scaled by the weighted sample size.
cov.SymOuterK(1/(floats.Sum(weights)-1), &xt)
return cov
}
func transpose(A mat64.Matrix) mat64.Matrix {
r, s := A.Dims()
var data []float64
for j := 0; j < s; j++ {
for i := 0; i < r; i++ {
data = append(data, A.At(i, j))
}
}
// Está medio chafa que regrese Dense, cómo hacemos para que regrese
// el mismo tipo de A?
return mat64.NewDense(s, r, data)
}
// NearestCentroid returns the index of the row in Mu for which its
// vector magnitude with x is the least.
// x should be (n*1) space and M (K*n) space
func NearestCentroid(x *mat.Vector, Mu mat.Matrix) (rowIndex int) {
k, _ := Mu.Dims()
rowIndex = 0
leastDistance := vectorDistance(x, getRowVector(rowIndex, Mu))
for i := 1; i < k; i++ {
distance := vectorDistance(x, getRowVector(i, Mu))
if distance < leastDistance {
leastDistance = distance
rowIndex = i
}
}
return
}
// PrintMat imprime una matriz con un formato entendible
func PrintMat(A mat64.Matrix) {
r, s := A.Dims()
fmt.Printf("(")
for i := 0; i < r; i++ {
fmt.Printf("\t")
for j := 0; j < s; j++ {
fmt.Printf("%0.1f\t", A.At(i, j))
}
if i != r-1 {
fmt.Printf("\n")
}
}
fmt.Printf(")\n")
}
//ScaleByCol scales each column of matrix A by Col, putting the result
//in the received.
func (F *Matrix) ScaleByCol(A, Col mat64.Matrix) {
ar, ac := A.Dims()
cr, cc := Col.Dims()
fr, fc := F.Dims()
if ar != cr || cc > 1 || ar != fr || ac != fc {
panic(ErrShape)
}
if F != A {
F.Copy(A)
}
for i := 0; i < ac; i++ {
temp := F.ColView(i)
temp.MulElem(temp, Col)
}
}
// VerifyInputs returns true if the number of rows in inputs is not the same
// as the number of rows in outputs and the length of weights. As a special case,
// the length of weights is allowed to be zero.
func VerifyInputs(inputs, outputs mat64.Matrix, weights []float64) error {
if inputs == nil || outputs == nil {
return NoData
}
nSamples, _ := inputs.Dims()
nOutputSamples, _ := outputs.Dims()
nWeights := len(weights)
if nSamples != nOutputSamples || (nWeights != 0 && nSamples != nWeights) {
return DataMismatch{
Input: nSamples,
Output: nOutputSamples,
Weight: nWeights,
}
}
return nil
}
// PrincipalComponents returns the principal component direction vectors and
// the column variances of the principal component scores, vecs * a, computed
// using the singular value decomposition of the input. The input a is an n×d
// matrix where each row is an observation and each column represents a variable.
//
// PrincipalComponents centers the variables but does not scale the variance.
//
// The slice weights is used to weight the observations. If weights is nil,
// each weight is considered to have a value of one, otherwise the length of
// weights must match the number of observations or PrincipalComponents will
// panic.
//
// On successful completion, the principal component direction vectors are
// returned in vecs as a d×min(n, d) matrix, and the variances are returned in
// vars as a min(n, d)-long slice in descending sort order.
//
// If no singular value decomposition is possible, vecs and vars are returned
// nil and ok is returned false.
func PrincipalComponents(a mat64.Matrix, weights []float64) (vecs *mat64.Dense, vars []float64, ok bool) {
n, d := a.Dims()
if weights != nil && len(weights) != n {
panic("stat: len(weights) != observations")
}
centered := mat64.NewDense(n, d, nil)
col := make([]float64, n)
for j := 0; j < d; j++ {
mat64.Col(col, j, a)
floats.AddConst(-Mean(col, weights), col)
centered.SetCol(j, col)
}
for i, w := range weights {
floats.Scale(math.Sqrt(w), centered.RawRowView(i))
}
kind := matrix.SVDFull
if n > d {
kind = matrix.SVDThin
}
var svd mat64.SVD
ok = svd.Factorize(centered, kind)
if !ok {
return nil, nil, false
}
vecs = &mat64.Dense{}
vecs.VFromSVD(&svd)
if n < d {
// Don't retain columns that are not valid direction vectors.
vecs.Clone(vecs.View(0, 0, d, n))
}
vars = svd.Values(nil)
var f float64
if weights == nil {
f = 1 / float64(n-1)
} else {
f = 1 / (floats.Sum(weights) - 1)
}
for i, v := range vars {
vars[i] = f * v * v
}
return vecs, vars, true
}
// formKStar forms the covariance matrix between the inputs and new points.
func (g *GP) formKStar(x mat64.Matrix) *mat64.Dense {
// TODO(btracey): Parallelize
r, c := x.Dims()
n := len(g.outputs)
kStar := mat64.NewDense(n, r, nil)
data := make([]float64, c)
for j := 0; j < r; j++ {
for k := 0; k < c; k++ {
data[k] = x.At(j, k)
}
for i := 0; i < n; i++ {
row := g.inputs.RawRowView(i)
v := g.kernel.Distance(row, data)
kStar.Set(i, j, v)
}
}
return kStar
}
// Given number of clusters K and the training set X of dimension (m*n)
// InitializeCentroids returns matrix Mu of dimension (K*n)
// The steps to initialize the centroids are as follows
// 1. Randomly pick K training examples (Make sure the selected examples are unique)
// 2. Set Mu to the K examples
func InitializeCentroids(K int, X mat.Matrix) (Mu mat.Matrix) {
m, n := X.Dims()
// panic if K >= m
if K >= m {
panic("K should be less than the size of the training set")
}
randomIndexes := randArray(0, m, K, true) // 1. pick K training examples
// 2. set Mu
Mu = mat.NewDense(K, n, nil)
for i := 0; i < K; i++ {
Mu.(*mat.Dense).SetRow(i, mat.Row(nil, randomIndexes[i], X))
}
return
}
// StdDevBatch predicts the standard deviation at a set of locations of x.
func (g *GP) StdDevBatch(std []float64, x mat64.Matrix) []float64 {
r, c := x.Dims()
if c != g.inputDim {
panic(badInputLength)
}
if std == nil {
std = make([]float64, r)
}
if len(std) != r {
panic(badStorage)
}
// For a single point, the stddev is
// sigma = k(x,x) - k_*^T * K^-1 * k_*
// where k is the vector of kernels between the input points and the output points
// For many points, the formula is:
// nu_* = k(x_*, k_*) - k_*^T * K^-1 * k_*
// This creates the full covariance matrix which is an rxr matrix. However,
// the standard deviations are just the diagonal of this matrix. Instead, be
// smart about it and compute the diagonal terms one at a time.
kStar := g.formKStar(x)
var tmp mat64.Dense
tmp.SolveCholesky(g.cholK, kStar)
// set k(x_*, x_*) into std then subtract k_*^T K^-1 k_* , computed one row at a time
var tmp2 mat64.Vector
row := make([]float64, c)
for i := range std {
for k := 0; k < c; k++ {
row[k] = x.At(i, k)
}
std[i] = g.kernel.Distance(row, row)
tmp2.MulVec(kStar.ColView(i).T(), tmp.ColView(i))
rt, ct := tmp2.Dims()
if rt != 1 && ct != 1 {
panic("bad size")
}
std[i] -= tmp2.At(0, 0)
std[i] = math.Sqrt(std[i])
}
// Need to scale the standard deviation to be in the same units as y.
floats.Scale(g.std, std)
return std
}