func MakeFitLinScale(targetImage *imgut.Image) func(*imgut.Image) float64 {
// Pre-compute image to slice of floats
dataTarg := imgut.ToSlice(targetImage)
// Pre-compute average
avgt := floats.Sum(dataTarg) / float64(len(dataTarg))
return func(indImage *imgut.Image) float64 {
// Images to vector
dataInd := imgut.ToSlice(indImage)
// Compute average pixels
avgy := floats.Sum(dataInd) / float64(len(dataInd))
// Difference y - avgy
y_avgy := make([]float64, len(dataInd))
copy(y_avgy, dataInd)
floats.AddConst(-avgy, y_avgy)
// Difference t - avgt
t_avgt := make([]float64, len(dataTarg))
copy(t_avgt, dataTarg)
floats.AddConst(-avgt, t_avgt)
// Multuplication (t - avgt)(y - avgy)
floats.Mul(t_avgt, y_avgy)
// Summation
numerator := floats.Sum(t_avgt)
// Square (y - avgy)^2
floats.Mul(y_avgy, y_avgy)
denomin := floats.Sum(y_avgy)
// Compute b-value
b := numerator / denomin
// Compute a-value
a := avgt - b*avgy
// Compute now the scaled RMSE, using y' = a + b*y
floats.Scale(b, dataInd) // b*y
floats.AddConst(a, dataInd) // a + b*y
floats.Sub(dataInd, dataTarg) // (a + b * y - t)
floats.Mul(dataInd, dataInd) // (a + b * y - t)^2
total := floats.Sum(dataInd) // Sum(...)
return math.Sqrt(total / float64(len(dataInd)))
}
}
// 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
}
// 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
}
// ConjugateUpdate updates the parameters of the distribution from the sufficient
// statistics of a set of samples. The sufficient statistics, suffStat, have been
// observed with nSamples observations. The prior values of the distribution are those
// currently in the distribution, and have been observed with priorStrength samples.
//
// For the normal distribution, the sufficient statistics are the mean and
// uncorrected standard deviation of the samples.
// The prior is having seen strength[0] samples with mean Normal.Mu
// and strength[1] samples with standard deviation Normal.Sigma. As a result of
// this function, Normal.Mu and Normal.Sigma are updated based on the weighted
// samples, and strength is modified to include the new number of samples observed.
//
// This function panics if len(suffStat) != 2 or len(priorStrength) != 2.
func (n *Normal) ConjugateUpdate(suffStat []float64, nSamples float64, priorStrength []float64) {
// TODO: Support prior strength with math.Inf(1) to allow updating with
// a known mean/standard deviation
totalMeanSamples := nSamples + priorStrength[0]
totalSum := suffStat[0]*nSamples + n.Mu*priorStrength[0]
totalVarianceSamples := nSamples + priorStrength[1]
// sample variance
totalVariance := nSamples * suffStat[1] * suffStat[1]
// add prior variance
totalVariance += priorStrength[1] * n.Sigma * n.Sigma
// add cross variance from the difference of the means
meanDiff := (suffStat[0] - n.Mu)
totalVariance += priorStrength[0] * nSamples * meanDiff * meanDiff / totalMeanSamples
n.Mu = totalSum / totalMeanSamples
n.Sigma = math.Sqrt(totalVariance / totalVarianceSamples)
floats.AddConst(nSamples, priorStrength)
}
// CovarianceMatrix calculates a covariance matrix (also known as a
// variance-covariance matrix) from a matrix of data, using a two-pass
// algorithm. The matrix returned will be symmetric and square.
//
// The weights wts should have the length equal to the number of rows in
// input data matrix x. If c is nil, then a new matrix with appropriate size will
// be constructed. If c is not nil, it should be a square matrix with the same
// number of columns as the input data matrix x, and it will be used as the receiver
// for the covariance data. Weights cannot be negative.
func CovarianceMatrix(cov *mat64.Dense, x mat64.Matrix, wts []float64) *mat64.Dense {
// 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.
// TODO(jonlawlor): indicate that the resulting matrix is symmetric, and change
// the returned type from a *mat.Dense to a *mat.Symmetric.
r, c := x.Dims()
if cov == nil {
cov = mat64.NewDense(c, c, nil)
} else if covr, covc := cov.Dims(); covr != covc || covc != c {
panic(mat64.ErrShape)
}
var xt mat64.Dense
xt.TCopy(x)
// 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(wts) != len(v), so
// we don't have to check the size later.
mean := Mean(v, wts)
floats.AddConst(-mean, v)
}
var n float64
if wts == nil {
n = float64(r)
cov.MulTrans(&xt, false, &xt, true)
// Scale by the sample size.
cov.Scale(1/(n-1), cov)
return cov
}
// Multiply by the sqrt of the weights, so that multiplication is symmetric.
sqrtwts := make([]float64, r)
for i, w := range wts {
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.
n = floats.Sum(wts)
cov.MulTrans(&xt, false, &xt, true)
// Scale by the sample size.
cov.Scale(1/(n-1), cov)
return cov
}
// PageRankSparse returns the PageRank weights for nodes of the sparse directed
// graph g using the given damping factor and terminating when the 2-norm of the
// vector difference between iterations is below tol. The returned map is
// keyed on the graph node IDs.
func PageRankSparse(g graph.Directed, damp, tol float64) map[int]float64 {
// PageRankSparse is implemented according to "How Google Finds Your Needle
// in the Web's Haystack".
//
// G.I^k = alpha.H.I^k + alpha.A.I^k + (1-alpha).1/n.1.I^k
//
// http://www.ams.org/samplings/feature-column/fcarc-pagerank
nodes := g.Nodes()
indexOf := make(map[int]int, len(nodes))
for i, n := range nodes {
indexOf[n.ID()] = i
}
m := make(rowCompressedMatrix, len(nodes))
var dangling compressedRow
df := damp / float64(len(nodes))
for j, u := range nodes {
to := g.From(u)
f := damp / float64(len(to))
for _, v := range to {
m.addTo(indexOf[v.ID()], j, f)
}
if len(to) == 0 {
dangling.addTo(j, df)
}
}
last := make([]float64, len(nodes))
for i := range last {
last[i] = 1
}
lastV := mat64.NewVector(len(nodes), last)
vec := make([]float64, len(nodes))
var sum float64
for i := range vec {
r := rand.NormFloat64()
sum += r
vec[i] = r
}
f := 1 / sum
for i := range vec {
vec[i] *= f
}
v := mat64.NewVector(len(nodes), vec)
dt := (1 - damp) / float64(len(nodes))
for {
lastV, v = v, lastV
m.mulVecUnitary(v, lastV) // First term of the G matrix equation;
with := dangling.dotUnitary(lastV) // Second term;
away := onesDotUnitary(dt, lastV) // Last term.
floats.AddConst(with+away, v.RawVector().Data)
if normDiff(vec, last) < tol {
break
}
}
ranks := make(map[int]float64, len(nodes))
for i, r := range v.RawVector().Data {
ranks[nodes[i].ID()] = r
}
return ranks
}
func (r *Rosenbrock) OptLoc() []float64 {
ans := make([]float64, r.nDim)
floats.AddConst(1, ans)
return ans
}
func (n None) Linear() (shift, scale []float64) {
shift = make([]float64, n.Dim)
scale = make([]float64, n.Dim)
floats.AddConst(1, scale)
return shift, scale
}