func TestWeightedTimeSeeded(t *testing.T) {
if !*prob {
t.Skip("probabilistic testing not requested")
}
t.Log("Note: This test is stochastic and is expected to fail with probability ≈ 0.05.")
rand.Seed(time.Now().Unix())
f := make([]float64, len(obt))
for i := 0; i < 1e6; i++ {
item, ok := newTestWeighted().Take()
if !ok {
t.Fatal("Weighted unexpectedly empty")
}
f[item]++
}
exp := newExp()
fac := floats.Sum(f) / floats.Sum(exp)
for i := range f {
exp[i] *= fac
}
// Check that our obtained values are within statistical expectations for p = 0.05.
// This will not be true approximately 1 in 20 tests.
X := chi2(f, exp)
if X >= sigChi2 {
t.Errorf("H₀: d(Sample) = d(Expect), H₁: d(S) ≠ d(Expect). df = %d, p = 0.05, X² threshold = %.2f, X² = %f", len(f)-1, sigChi2, X)
}
}
func TestWeightIncrease(t *testing.T) {
rand.Seed(0)
want := Weighted{
weights: []float64{1 << 0, 1 << 1, 1 << 2, 1 << 3, 1 << 4, 1 << 5, 1 << 9 * 2, 1 << 7, 1 << 8, 1 << 9},
heap: []float64{
exp[0] + exp[1] + exp[3] + exp[4] + exp[7] + exp[8] + exp[9] + exp[2] + exp[5] + exp[9]*2,
exp[1] + exp[3] + exp[4] + exp[7] + exp[8] + exp[9],
exp[2] + exp[5] + exp[9]*2,
exp[3] + exp[7] + exp[8],
exp[4] + exp[9],
exp[5],
exp[9] * 2,
exp[7],
exp[8],
exp[9],
},
}
ts := newTestWeighted()
ts.Reweight(6, ts.weights[len(ts.weights)-1]*2)
if !reflect.DeepEqual(ts, want) {
t.Fatalf("unexpected new Weighted value:\ngot: %#v\nwant:%#v", ts, want)
}
f := make([]float64, len(obt))
for i := 0; i < 1e6; i++ {
ts := newTestWeighted()
ts.Reweight(6, ts.weights[len(ts.weights)-1]*2)
item, ok := ts.Take()
if !ok {
t.Fatal("Weighted unexpectedly empty")
}
f[item]++
}
exp := newExp()
fac := floats.Sum(f) / floats.Sum(exp)
for i := range f {
exp[i] *= fac
}
if f[6] < f[9] {
t.Errorf("unexpected selection rate for re-weighted item: got: %v want:%v", f[6], f[9])
}
if reflect.DeepEqual(f[:6], obt[:6]) {
t.Fatal("unexpected selection: too many elements chosen in range:\ngot: %v\nwant:%v",
f[:6], obt[:6])
}
if reflect.DeepEqual(f[7:], obt[7:]) {
t.Fatal("unexpected selection: too many elements chosen in range:\ngot: %v\nwant:%v",
f[7:], obt[7:])
}
}
func TestWeightedUnseeded(t *testing.T) {
rand.Seed(0)
want := Weighted{
weights: []float64{1 << 0, 1 << 1, 1 << 2, 1 << 3, 1 << 4, 1 << 5, 1 << 6, 1 << 7, 1 << 8, 1 << 9},
heap: []float64{
exp[0] + exp[1] + exp[3] + exp[4] + exp[7] + exp[8] + exp[9] + exp[2] + exp[5] + exp[6],
exp[1] + exp[3] + exp[4] + exp[7] + exp[8] + exp[9],
exp[2] + exp[5] + exp[6],
exp[3] + exp[7] + exp[8],
exp[4] + exp[9],
exp[5],
exp[6],
exp[7],
exp[8],
exp[9],
},
}
ts := newTestWeighted()
if !reflect.DeepEqual(ts, want) {
t.Fatalf("unexpected new Weighted value:\ngot: %#v\nwant:%#v", ts, want)
}
f := make([]float64, len(obt))
for i := 0; i < 1e6; i++ {
item, ok := newTestWeighted().Take()
if !ok {
t.Fatal("Weighted unexpectedly empty")
}
f[item]++
}
exp := newExp()
fac := floats.Sum(f) / floats.Sum(exp)
for i := range f {
exp[i] *= fac
}
if !reflect.DeepEqual(f, obt) {
t.Fatalf("unexpected selection:\ngot: %#v\nwant:%#v", f, obt)
}
// Check that this is within statistical expectations - we know this is true for this set.
X := chi2(f, exp)
if X >= sigChi2 {
t.Errorf("H₀: d(Sample) = d(Expect), H₁: d(S) ≠ d(Expect). df = %d, p = 0.05, X² threshold = %.2f, X² = %f", len(f)-1, sigChi2, X)
}
}
func TestCategoricalCDF(t *testing.T) {
for _, test := range [][]float64{
{1, 2, 3, 0, 4},
} {
c := make([]float64, len(test))
copy(c, test)
floats.Scale(1/floats.Sum(c), c)
sum := make([]float64, len(test))
floats.CumSum(sum, c)
dist := NewCategorical(test, nil)
cdf := dist.CDF(-0.5)
if cdf != 0 {
t.Errorf("CDF of negative number not zero")
}
for i := range c {
cdf := dist.CDF(float64(i))
if math.Abs(cdf-sum[i]) > 1e-14 {
t.Errorf("CDF mismatch %v. Want %v, got %v.", float64(i), sum[i], cdf)
}
cdfp := dist.CDF(float64(i) + 0.5)
if cdfp != cdf {
t.Errorf("CDF mismatch for non-integer input")
}
}
}
}
func TestCategoricalProb(t *testing.T) {
for _, test := range [][]float64{
{1, 2, 3, 0},
} {
dist := NewCategorical(test, nil)
norm := make([]float64, len(test))
floats.Scale(1/floats.Sum(norm), norm)
for i, v := range norm {
p := dist.Prob(float64(i))
if math.Abs(p-v) > 1e-14 {
t.Errorf("Probability mismatch element %d", i)
}
p = dist.Prob(float64(i) + 0.5)
if p != 0 {
t.Errorf("Non-zero probability for non-integer x")
}
}
p := dist.Prob(-1)
if p != 0 {
t.Errorf("Non-zero probability for -1")
}
p = dist.Prob(float64(len(test)))
if p != 0 {
t.Errorf("Non-zero probability for len(test)")
}
}
}
// locationAsy returns the node locations and weights of a Hermite quadrature rule
// with len(x) points.
func (h Hermite) locationsAsy(x, w []float64) {
// A. Townsend, T. Trogdon, and S.Olver, Fast computation of Gauss quadrature
// nodes and weights the whole real line, IMA J. Numer. Anal.,
// 36: 337–358, 2016. http://arxiv.org/abs/1410.5286
// Find the positive locations and weights.
n := len(x)
l := n / 2
xa := x[l:]
wa := w[l:]
for i := range xa {
xa[i], wa[i] = h.locationsAsy0(i, n)
}
// Flip around zero -- copy the negative x locations with the corresponding
// weights.
if n%2 == 0 {
l--
}
for i, v := range xa {
x[l-i] = -v
}
for i, v := range wa {
w[l-i] = v
}
sumW := floats.Sum(w)
c := math.SqrtPi / sumW
floats.Scale(c, w)
}
// Estimate computes model parameters using sufficient statistics.
func (g *Model) Estimate() error {
if g.NSamples > minNumSamples {
/* Estimate the mean. */
floatx.Apply(floatx.ScaleFunc(1.0/g.NSamples), g.Sumx, g.Mean)
/*
* Estimate the variance. sigma_sq = 1/n (sumxsq - 1/n sumx^2) or
* 1/n sumxsq - mean^2.
*/
tmp := g.variance // borrow as an intermediate array.
// floatx.Apply(sq, g.Mean, g.tmpArray)
floatx.Sq(g.tmpArray, g.Mean)
floatx.Apply(floatx.ScaleFunc(1.0/g.NSamples), g.Sumxsq, tmp)
floats.SubTo(g.variance, tmp, g.tmpArray)
floatx.Apply(floatx.Floorv(smallVar), g.variance, nil)
} else {
/* Not enough training sample. */
glog.Warningf("not enough training samples, name [%s], num samples [%e]", g.ModelName, g.NSamples)
floatx.Apply(floatx.SetValueFunc(smallVar), g.variance, nil)
floatx.Apply(floatx.SetValueFunc(0), g.Mean, nil)
}
g.setVariance(g.variance) // to update varInv and stddev.
/* Update log Gaussian constant. */
floatx.Log(g.tmpArray, g.variance)
g.const2 = g.const1 - floats.Sum(g.tmpArray)/2.0
glog.V(6).Infof("gaussian reest, name:%s, mean:%v, sd:%v", g.ModelName, g.Mean, g.StdDev)
return nil
}
func fitnessRMSE(ind, targ *imgut.Image) float64 {
// Images to vector
dataInd := imgut.ToSlice(ind)
dataTarg := imgut.ToSlice(targ)
// (root mean square) error
floats.Sub(dataInd, dataTarg)
// (root mean) square error
floats.Mul(dataInd, dataInd)
// (root) mean square error
totErr := floats.Sum(dataInd)
return math.Sqrt(totErr / float64(len(dataInd)))
}
func ExampleStdErr() {
x := []float64{8, 2, -9, 15, 4}
weights := []float64{2, 2, 6, 7, 1}
mean := Mean(x, weights)
stdev := StdDev(x, weights)
nSamples := floats.Sum(weights)
stdErr := StdErr(stdev, nSamples)
fmt.Printf("The standard deviation is %.4f and there are %g samples, so the mean\nis likely %.4f ± %.4f.", stdev, nSamples, mean, stdErr)
// Output:
// The standard deviation is 10.5733 and there are 18 samples, so the mean
// is likely 4.1667 ± 2.4921.
}
func main() {
runtime.GOMAXPROCS(runtime.NumCPU() - 2)
gopath := os.Getenv("GOPATH")
path := filepath.Join(gopath, "prof", "github.com", "reggo", "reggo", "nnet")
nInputs := 10
nOutputs := 3
nLayers := 2
nNeurons := 50
nSamples := 1000000
nRuns := 50
config := &profile.Config{
CPUProfile: true,
ProfilePath: path,
}
defer profile.Start(config).Stop()
net, err := nnet.NewSimpleTrainer(nInputs, nOutputs, nLayers, nNeurons, nnet.Linear{})
if err != nil {
log.Fatal(err)
}
// Generate some random data
inputs := mat64.NewDense(nSamples, nInputs, nil)
outputs := mat64.NewDense(nSamples, nOutputs, nil)
for i := 0; i < nSamples; i++ {
for j := 0; j < nInputs; j++ {
inputs.Set(i, j, rand.Float64())
}
for j := 0; j < nOutputs; j++ {
outputs.Set(i, j, rand.Float64())
}
}
// Create trainer
prob := train.NewBatchGradBased(net, true, inputs, outputs, nil, nil, nil)
nParameters := net.NumParameters()
parameters := make([]float64, nParameters)
derivative := make([]float64, nParameters)
for i := 0; i < nRuns; i++ {
net.RandomizeParameters()
net.Parameters(parameters)
prob.ObjGrad(parameters, derivative)
fmt.Println(floats.Sum(derivative))
}
}
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)))
}
}
func MakeFitMSE(targetImage *imgut.Image) func(*imgut.Image) float64 {
dataTarg := imgut.ToSliceChans(targetImage, "R")
return func(indImage *imgut.Image) float64 {
// Get data
dataImg := imgut.ToSliceChans(indImage, "R")
// Difference (X - Y)
floats.Sub(dataImg, dataTarg)
// Squared (X - Y)^2
floats.Mul(dataImg, dataImg)
// Summation
return floats.Sum(dataImg) / float64(len(dataImg))
}
}
func sampleCategorical(t *testing.T, dist Categorical, nSamples int) []float64 {
counts := make([]float64, dist.Len())
for i := 0; i < nSamples; i++ {
v := dist.Rand()
if float64(int(v)) != v {
t.Fatalf("Random number is not an integer")
}
counts[int(v)]++
}
sum := floats.Sum(counts)
floats.Scale(1/sum, counts)
return counts
}
// 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
}
// LogDet returns the log of the determinant and the sign of the determinant
// for the matrix that has been factorized. Numerical stability in product and
// division expressions is generally improved by working in log space.
func (lu *LU) LogDet() (det float64, sign float64) {
_, n := lu.lu.Dims()
logDiag := make([]float64, n)
sign = 1.0
for i := 0; i < n; i++ {
v := lu.lu.at(i, i)
if v < 0 {
sign *= -1
}
if lu.pivot[i] != i {
sign *= -1
}
logDiag[i] = math.Log(math.Abs(v))
}
return floats.Sum(logDiag), sign
}
// Mean computes the weighted mean of the data set.
// sum_i {w_i * x_i} / sum_i {w_i}
// If weights is nil then all of the weights are 1. If weights is not nil, then
// len(x) must equal len(weights).
func Mean(x, weights []float64) float64 {
if weights == nil {
return floats.Sum(x) / float64(len(x))
}
if len(x) != len(weights) {
panic("stat: slice length mismatch")
}
var (
sumValues float64
sumWeights float64
)
for i, w := range weights {
sumValues += w * x[i]
sumWeights += w
}
return sumValues / sumWeights
}
// 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
}
// CDF returns the empirical cumulative distribution function value of x, that is
// the fraction of the samples less than or equal to q. The
// exact behavior is determined by the CumulantKind. CDF is theoretically
// the inverse of the Quantile function, though it may not be the actual inverse
// for all values q and CumulantKinds.
//
// The x data must be sorted in increasing order. If weights is nil then all
// of the weights are 1. If weights is not nil, then len(x) must equal len(weights).
//
// CumulantKind behaviors:
// - Empirical: Returns the lowest fraction for which q is greater than or equal
// to that fraction of samples
func CDF(q float64, c CumulantKind, x, weights []float64) float64 {
if weights != nil && len(x) != len(weights) {
panic("stat: slice length mismatch")
}
if floats.HasNaN(x) {
return math.NaN()
}
if !sort.Float64sAreSorted(x) {
panic("x data are not sorted")
}
if q < x[0] {
return 0
}
if q >= x[len(x)-1] {
return 1
}
var sumWeights float64
if weights == nil {
sumWeights = float64(len(x))
} else {
sumWeights = floats.Sum(weights)
}
// Calculate the index
switch c {
case Empirical:
// Find the smallest value that is greater than that percent of the samples
var w float64
for i, v := range x {
if v > q {
return w / sumWeights
}
if weights == nil {
w++
} else {
w += weights[i]
}
}
panic("impossible")
default:
panic("stat: bad cumulant kind")
}
}
// Fit sets the parameters of the probability distribution from the
// data samples x with relative weights w.
// If weights is nil, then all the weights are 1.
// If weights is not nil, then the len(weights) must equal len(samples).
//
// Note: Laplace distribution has no FitPrior because it has no sufficient
// statistics.
func (l *Laplace) Fit(samples, weights []float64) {
if len(samples) != len(weights) {
panic(badLength)
}
if len(samples) == 0 {
panic(badNoSamples)
}
if len(samples) == 1 {
l.Mu = samples[0]
l.Scale = 0
return
}
var (
sortedSamples []float64
sortedWeights []float64
)
if sort.Float64sAreSorted(samples) {
sortedSamples = samples
sortedWeights = weights
} else {
// Need to copy variables so the input variables aren't effected by the sorting
sortedSamples = make([]float64, len(samples))
copy(sortedSamples, samples)
sortedWeights := make([]float64, len(samples))
copy(sortedWeights, weights)
stat.SortWeighted(sortedSamples, sortedWeights)
}
// The (weighted) median of the samples is the maximum likelihood estimate
// of the mean parameter
// TODO: Rethink quantile type when stat has more options
l.Mu = stat.Quantile(0.5, stat.Empirical, sortedSamples, sortedWeights)
sumWeights := floats.Sum(weights)
// The scale parameter is the average absolute distance
// between the sample and the mean
absError := stat.MomentAbout(1, samples, l.Mu, weights)
l.Scale = absError / sumWeights
}
// SuffStat computes the sufficient statistics of set of samples to update
// the distribution. The sufficient statistics are stored in place, and the
// effective number of samples are returned.
//
// The exponential distribution has one sufficient statistic, the average rate
// of the samples.
//
// If weights is nil, the weights are assumed to be 1, otherwise panics if
// len(samples) != len(weights). Panics if len(suffStat) != 1.
func (Exponential) SuffStat(samples, weights, suffStat []float64) (nSamples float64) {
if len(weights) != 0 && len(samples) != len(weights) {
panic("dist: slice size mismatch")
}
if len(suffStat) != 1 {
panic("exponential: wrong suffStat length")
}
if len(weights) == 0 {
nSamples = float64(len(samples))
} else {
nSamples = floats.Sum(weights)
}
mean := stat.Mean(samples, weights)
suffStat[0] = 1 / mean
return nSamples
}
// SuffStat computes the sufficient statistics of set of samples to update
// the distribution. The sufficient statistics are stored in place, and the
// effective number of samples are returned.
//
// The exponential distribution has one sufficient statistic, the average rate
// of the samples.
//
// If weights is nil, the weights are assumed to be 1, otherwise panics if
// len(samples) != len(weights). Panics if len(suffStat) != 1.
func (Exponential) SuffStat(samples, weights, suffStat []float64) (nSamples float64) {
if len(weights) != 0 && len(samples) != len(weights) {
panic(badLength)
}
if len(suffStat) != 1 {
panic(badSuffStat)
}
if len(weights) == 0 {
nSamples = float64(len(samples))
} else {
nSamples = floats.Sum(weights)
}
mean := stat.Mean(samples, weights)
suffStat[0] = 1 / mean
return nSamples
}
// NewModel creates a new Gaussian model.
func NewModel(dim int, options ...Option) *Model {
g := &Model{
ModelName: "Gaussian",
ModelDim: dim,
Diag: true,
variance: make([]float64, dim),
varianceInv: make([]float64, dim),
tmpArray: make([]float64, dim),
}
g.Type = reflect.TypeOf(*g).String()
// Set options.
for _, option := range options {
option(g)
}
if len(g.Sumx) == 0 {
g.Sumx = make([]float64, dim)
}
if len(g.Sumxsq) == 0 {
g.Sumxsq = make([]float64, dim)
}
if g.Mean == nil {
g.Mean = make([]float64, dim)
}
if g.StdDev == nil {
g.StdDev = make([]float64, dim)
floatx.Apply(floatx.SetValueFunc(smallSD), g.StdDev, nil)
}
floatx.Sq(g.variance, g.StdDev)
// Initializes variance, varianceInv, and StdDev.
g.setVariance(g.variance)
floatx.Log(g.tmpArray, g.variance)
g.const1 = -float64(g.ModelDim) * math.Log(2.0*math.Pi) / 2.0
g.const2 = g.const1 - floats.Sum(g.tmpArray)/2.0
return g
}
// Quantile returns the sample of x such that x is greater than or
// equal to the fraction p of samples. The exact behavior is determined by the
// CumulantKind, and p should be a number between 0 and 1. Quantile is theoretically
// the inverse of the CDF function, though it may not be the actual inverse
// for all values p and CumulantKinds.
//
// The x data must be sorted in increasing order. If weights is nil then all
// of the weights are 1. If weights is not nil, then len(x) must equal len(weights).
//
// CumulantKind behaviors:
// - Empirical: Returns the lowest value q for which q is greater than or equal
// to the fraction p of samples
func Quantile(p float64, c CumulantKind, x, weights []float64) float64 {
if !(p >= 0 && p <= 1) {
panic("stat: percentile out of bounds")
}
if weights != nil && len(x) != len(weights) {
panic("stat: slice length mismatch")
}
if floats.HasNaN(x) {
return math.NaN() // This is needed because the algorithm breaks otherwise
}
if !sort.Float64sAreSorted(x) {
panic("x data are not sorted")
}
var sumWeights float64
if weights == nil {
sumWeights = float64(len(x))
} else {
sumWeights = floats.Sum(weights)
}
switch c {
case Empirical:
var cumsum float64
fidx := p * sumWeights
for i := range x {
if weights == nil {
cumsum++
} else {
cumsum += weights[i]
}
if cumsum >= fidx {
return x[i]
}
}
panic("impossible")
default:
panic("stat: bad cumulant kind")
}
}
// SuffStat computes the sufficient statistics of a set of samples to update
// the distribution. The sufficient statistics are stored in place, and the
// effective number of samples are returned.
//
// The normal distribution has two sufficient statistics, the mean of the samples
// and the standard deviation of the samples.
//
// If weights is nil, the weights are assumed to be 1, otherwise panics if
// len(samples) != len(weights). Panics if len(suffStat) != 2.
func (Normal) SuffStat(samples, weights, suffStat []float64) (nSamples float64) {
lenSamp := len(samples)
if len(weights) != 0 && len(samples) != len(weights) {
panic("dist: slice size mismatch")
}
if len(suffStat) != 2 {
panic("dist: incorrect suffStat length")
}
if len(weights) == 0 {
nSamples = float64(lenSamp)
} else {
nSamples = floats.Sum(weights)
}
mean := stat.Mean(samples, weights)
suffStat[0] = mean
// Use Moment and not StdDev because we want it to be uncorrected
variance := stat.Moment(2, samples, mean, weights)
suffStat[1] = math.Sqrt(variance)
return nSamples
}
// 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
}
func main() {
fmt.Println("Sum: ", floats.Sum(a))
fmt.Println("Product:", floats.Prod(a))
}
// KolmogorovSmirnov computes the largest distance between two empirical CDFs.
// Each dataset x and y consists of sample locations and counts, xWeights and
// yWeights, respectively.
//
// x and y may have different lengths, though len(x) must equal len(xWeights), and
// len(y) must equal len(yWeights). Both x and y must be sorted.
//
// Special cases are:
// = 0 if len(x) == len(y) == 0
// = 1 if len(x) == 0, len(y) != 0 or len(x) != 0 and len(y) == 0
func KolmogorovSmirnov(x, xWeights, y, yWeights []float64) float64 {
if xWeights != nil && len(x) != len(xWeights) {
panic("stat: slice length mismatch")
}
if yWeights != nil && len(y) != len(yWeights) {
panic("stat: slice length mismatch")
}
if len(x) == 0 || len(y) == 0 {
if len(x) == 0 && len(y) == 0 {
return 0
}
return 1
}
if floats.HasNaN(x) {
return math.NaN()
}
if floats.HasNaN(y) {
return math.NaN()
}
if !sort.Float64sAreSorted(x) {
panic("x data are not sorted")
}
if !sort.Float64sAreSorted(y) {
panic("y data are not sorted")
}
xWeightsNil := xWeights == nil
yWeightsNil := yWeights == nil
var (
maxDist float64
xSum, ySum float64
xCdf, yCdf float64
xIdx, yIdx int
)
if xWeightsNil {
xSum = float64(len(x))
} else {
xSum = floats.Sum(xWeights)
}
if yWeightsNil {
ySum = float64(len(y))
} else {
ySum = floats.Sum(yWeights)
}
xVal := x[0]
yVal := y[0]
// Algorithm description:
// The goal is to find the maximum difference in the empirical CDFs for the
// two datasets. The CDFs are piecewise-constant, and thus the distance
// between the CDFs will only change at the values themselves.
//
// To find the maximum distance, step through the data in ascending order
// of value between the two datasets. At each step, compute the empirical CDF
// and compare the local distance with the maximum distance.
// Due to some corner cases, equal data entries must be tallied simultaneously.
for {
switch {
case xVal < yVal:
xVal, xCdf, xIdx = updateKS(xIdx, xCdf, xSum, x, xWeights, xWeightsNil)
case yVal < xVal:
yVal, yCdf, yIdx = updateKS(yIdx, yCdf, ySum, y, yWeights, yWeightsNil)
case xVal == yVal:
newX := x[xIdx]
newY := y[yIdx]
if newX < newY {
xVal, xCdf, xIdx = updateKS(xIdx, xCdf, xSum, x, xWeights, xWeightsNil)
} else if newY < newX {
yVal, yCdf, yIdx = updateKS(yIdx, yCdf, ySum, y, yWeights, yWeightsNil)
} else {
// Update them both, they'll be equal next time and the right
// thing will happen
xVal, xCdf, xIdx = updateKS(xIdx, xCdf, xSum, x, xWeights, xWeightsNil)
yVal, yCdf, yIdx = updateKS(yIdx, yCdf, ySum, y, yWeights, yWeightsNil)
}
default:
panic("unreachable")
}
dist := math.Abs(xCdf - yCdf)
if dist > maxDist {
maxDist = dist
}
// Both xCdf and yCdf will equal 1 at the end, so if we have reached the
// end of either sample list, the distance is as large as it can be.
if xIdx == len(x) || yIdx == len(y) {
return maxDist
}
}
}
func (b *BatchGradient) funcGrad(params, deriv []float64) float64 {
nParameters := len(deriv)
// Send out all of the work
done := make(chan result)
sz := b.nSamples / b.Workers
sent := 0
for i := 0; i < b.Workers; i++ {
outputDim := b.outputDim
last := sent + sz
if i == b.Workers-1 {
last = b.nSamples
}
go func(sent, last int) {
lossDeriver := b.Trainable.NewLossDeriver()
predOutput := make([]float64, outputDim)
dLossDPred := make([]float64, outputDim)
dLossDParam := make([]float64, nParameters)
outputs := make([]float64, outputDim)
tmpderiv := make([]float64, nParameters)
var totalLoss float64
for i := sent; i < last; i++ {
lossDeriver.Predict(params, b.features.RawRowView(i), predOutput)
b.Outputs.Row(outputs, i)
loss := b.Losser.LossDeriv(predOutput, outputs, dLossDPred)
if b.Weights == nil {
totalLoss += loss
} else {
totalLoss += b.Weights[i] * loss
}
lossDeriver.Deriv(params, b.features.RawRowView(i), predOutput, dLossDPred, dLossDParam)
if b.Weights != nil {
floats.Scale(b.Weights[i], dLossDParam)
}
floats.Add(tmpderiv, dLossDParam)
}
done <- result{totalLoss, tmpderiv}
}(sent, last)
sent += sz
}
// Collect all the results
var totalLoss float64
for i := range deriv {
deriv[i] = 0
}
for i := 0; i < b.Workers; i++ {
w := <-done
totalLoss += w.loss
floats.Add(deriv, w.deriv)
}
// Compute the regularizer
if b.Regularizer != nil {
tmp := make([]float64, nParameters)
totalLoss += b.Regularizer.LossDeriv(params, tmp)
floats.Add(deriv, tmp)
}
sumWeights := float64(b.nSamples)
if b.Weights != nil {
sumWeights = floats.Sum(b.Weights)
}
totalLoss /= sumWeights
floats.Scale(1/sumWeights, deriv)
return totalLoss
}
// Normalize the vector of value, summing them and dividing each by the total
func normalSlice(v []float64) {
tot := floats.Sum(v)
floats.Scale(1.0/tot, v)
}
func (Linear) Func(x []float64) float64 {
return floats.Sum(x)
}