//Log probability of this data
double SOGP::log_prob(const ColumnVector& in, const ColumnVector& out){
static const double ls2pi= log(sqrt(2*M_PI)); //Only compute once
double sigma;
double out2;
if(current_size == 0){ //mu = zero, sigma = kappa.
sigma=sqrt(m_params.m_kernel->kstar(in)+m_params.s20); //Is this right? V_0=kstar, v_1 = s20
out2=out.SumSquare();
}
else{
ColumnVector mu = predict(in,sigma);
mu-=out;
out2=mu.SumSquare();
}
return(-ls2pi -log(sigma) -.5*out2/(sigma*sigma));
}
// Matrix A's first n columns are orthonormal
// so A.Columns(1,n).t() * A.Columns(1,n) is the identity matrix.
// Fill out the remaining columns of A to make them orthonormal
// so A.t() * A is the identity matrix
void extend_orthonormal(Matrix& A, int n)
{
REPORT
Tracer et("extend_orthonormal");
int nr = A.nrows(); int nc = A.ncols();
if (nc > nr) Throw(IncompatibleDimensionsException(A));
if (n > nc) Throw(IncompatibleDimensionsException(A));
ColumnVector SSR;
{ Matrix A1 = A.Columns(1,n); SSR = A1.sum_square_rows(); }
for (int i = n; i < nc; ++i)
{
// pick row with smallest SSQ
int k; SSR.minimum1(k);
// orthogonalise column with 1 at element k, 0 elsewhere
// next line is rather inefficient
ColumnVector X = - A.Columns(1, i) * A.SubMatrix(k, k, 1, i).t();
X(k) += 1.0;
// normalise
X /= sqrt(X.SumSquare());
// update row sums of squares
for (k = 1; k <= nr; ++k) SSR(k) += square(X(k));
// load new column into matrix
A.Column(i+1) = X;
}
}
void test1(Real* y, Real* x1, Real* x2, int nobs, int npred)
{
cout << "\n\nTest 1 - traditional, bad\n";
// traditional sum of squares and products method of calculation
// but not adjusting means; maybe subject to round-off error
// make matrix of predictor values with 1s into col 1 of matrix
int npred1 = npred+1; // number of cols including col of ones.
Matrix X(nobs,npred1);
X.Column(1) = 1.0;
// load x1 and x2 into X
// [use << rather than = when loading arrays]
X.Column(2) << x1; X.Column(3) << x2;
// vector of Y values
ColumnVector Y(nobs); Y << y;
// form sum of squares and product matrix
// [use << rather than = for copying Matrix into SymmetricMatrix]
SymmetricMatrix SSQ; SSQ << X.t() * X;
// calculate estimate
// [bracket last two terms to force this multiplication first]
// [ .i() means inverse, but inverse is not explicity calculated]
ColumnVector A = SSQ.i() * (X.t() * Y);
// Get variances of estimates from diagonal elements of inverse of SSQ
// get inverse of SSQ - we need it for finding D
DiagonalMatrix D; D << SSQ.i();
ColumnVector V = D.AsColumn();
// Calculate fitted values and residuals
ColumnVector Fitted = X * A;
ColumnVector Residual = Y - Fitted;
Real ResVar = Residual.SumSquare() / (nobs-npred1);
// Get diagonals of Hat matrix (an expensive way of doing this)
DiagonalMatrix Hat; Hat << X * (X.t() * X).i() * X.t();
// print out answers
cout << "\nEstimates and their standard errors\n\n";
// make vector of standard errors
ColumnVector SE(npred1);
for (int i=1; i<=npred1; i++) SE(i) = sqrt(V(i)*ResVar);
// use concatenation function to form matrix and use matrix print
// to get two columns
cout << setw(11) << setprecision(5) << (A | SE) << endl;
cout << "\nObservations, fitted value, residual value, hat value\n";
// use concatenation again; select only columns 2 to 3 of X
cout << setw(9) << setprecision(3) <<
(X.Columns(2,3) | Y | Fitted | Residual | Hat.AsColumn());
cout << "\n\n";
}
/**
* Compute a distance metric between two columns of a
* matrix. <b>Note that the indexes are *1* based (not 0) as that is
* Newmat's convention</b>. Note that dist(M,i,j) must equal dist(M,j,i);
*
* @param M - Matrix whose columns represent individual items to be clustered.
* @param col1Ix - Column index to be compared (1 based).
* @param col2Ix - Column index to be compared (1 based).
*
* @return - "Distance" or "dissimilarity" metric between two columns of matrix.
*/
double GuassianRadial::dist(const Matrix &M, int col1Ix, int col2Ix) const {
double dist = 0;
if(col1Ix == col2Ix)
return 0;
ColumnVector V = M.Column(col1Ix) - M.Column(col2Ix);
dist = V.SumSquare() / (2 * m_Sigma * m_Sigma);
dist = exp(-1 * dist);
return dist;
}
void test3(Real* y, Real* x1, Real* x2, int nobs, int npred)
{
cout << "\n\nTest 3 - Cholesky\n";
// traditional sum of squares and products method of calculation
// with subtraction of means - using Cholesky decomposition
Matrix X(nobs,npred);
X.Column(1) << x1; X.Column(2) << x2;
ColumnVector Y(nobs); Y << y;
ColumnVector Ones(nobs); Ones = 1.0;
RowVector M = Ones.t() * X / nobs;
Matrix XC(nobs,npred);
XC = X - Ones * M;
ColumnVector YC(nobs);
Real m = Sum(Y) / nobs; YC = Y - Ones * m;
SymmetricMatrix SSQ; SSQ << XC.t() * XC;
// Cholesky decomposition of SSQ
LowerTriangularMatrix L = Cholesky(SSQ);
// calculate estimate
ColumnVector A = L.t().i() * (L.i() * (XC.t() * YC));
// calculate estimate of constant term
Real a = m - (M * A).AsScalar();
// Get variances of estimates from diagonal elements of invoice of SSQ
DiagonalMatrix D; D << L.t().i() * L.i();
ColumnVector V = D.AsColumn();
Real v = 1.0/nobs + (L.i() * M.t()).SumSquare();
// Calculate fitted values and residuals
int npred1 = npred+1;
ColumnVector Fitted = X * A + a;
ColumnVector Residual = Y - Fitted;
Real ResVar = Residual.SumSquare() / (nobs-npred1);
// Get diagonals of Hat matrix (an expensive way of doing this)
Matrix X1(nobs,npred1); X1.Column(1)<<Ones; X1.Columns(2,npred1)<<X;
DiagonalMatrix Hat; Hat << X1 * (X1.t() * X1).i() * X1.t();
// print out answers
cout << "\nEstimates and their standard errors\n\n";
cout.setf(ios::fixed, ios::floatfield);
cout << setw(11) << setprecision(5) << a << " ";
cout << setw(11) << setprecision(5) << sqrt(v*ResVar) << endl;
ColumnVector SE(npred);
for (int i=1; i<=npred; i++) SE(i) = sqrt(V(i)*ResVar);
cout << setw(11) << setprecision(5) << (A | SE) << endl;
cout << "\nObservations, fitted value, residual value, hat value\n";
cout << setw(9) << setprecision(3) <<
(X | Y | Fitted | Residual | Hat.AsColumn());
cout << "\n\n";
}
void test5(Real* y, Real* x1, Real* x2, int nobs, int npred)
{
cout << "\n\nTest 5 - singular value\n";
// Singular value decomposition method
// load data - 1s into col 1 of matrix
int npred1 = npred+1;
Matrix X(nobs,npred1); ColumnVector Y(nobs);
X.Column(1) = 1.0; X.Column(2) << x1; X.Column(3) << x2; Y << y;
// do SVD
Matrix U, V; DiagonalMatrix D;
SVD(X,D,U,V); // X = U * D * V.t()
ColumnVector Fitted = U.t() * Y;
ColumnVector A = V * ( D.i() * Fitted );
Fitted = U * Fitted;
ColumnVector Residual = Y - Fitted;
Real ResVar = Residual.SumSquare() / (nobs-npred1);
// get variances of estimates
D << V * (D * D).i() * V.t();
// Get diagonals of Hat matrix
DiagonalMatrix Hat; Hat << U * U.t();
// print out answers
cout << "\nEstimates and their standard errors\n\n";
ColumnVector SE(npred1);
for (int i=1; i<=npred1; i++) SE(i) = sqrt(D(i)*ResVar);
cout << setw(11) << setprecision(5) << (A | SE) << endl;
cout << "\nObservations, fitted value, residual value, hat value\n";
cout << setw(9) << setprecision(3) <<
(X.Columns(2,3) | Y | Fitted | Residual | Hat.AsColumn());
cout << "\n\n";
}
void test2(Real* y, Real* x1, Real* x2, int nobs, int npred)
{
cout << "\n\nTest 2 - traditional, OK\n";
// traditional sum of squares and products method of calculation
// with subtraction of means - less subject to round-off error
// than test1
// make matrix of predictor values
Matrix X(nobs,npred);
// load x1 and x2 into X
// [use << rather than = when loading arrays]
X.Column(1) << x1; X.Column(2) << x2;
// vector of Y values
ColumnVector Y(nobs); Y << y;
// make vector of 1s
ColumnVector Ones(nobs); Ones = 1.0;
// calculate means (averages) of x1 and x2 [ .t() takes transpose]
RowVector M = Ones.t() * X / nobs;
// and subtract means from x1 and x1
Matrix XC(nobs,npred);
XC = X - Ones * M;
// do the same to Y [use Sum to get sum of elements]
ColumnVector YC(nobs);
Real m = Sum(Y) / nobs; YC = Y - Ones * m;
// form sum of squares and product matrix
// [use << rather than = for copying Matrix into SymmetricMatrix]
SymmetricMatrix SSQ; SSQ << XC.t() * XC;
// calculate estimate
// [bracket last two terms to force this multiplication first]
// [ .i() means inverse, but inverse is not explicity calculated]
ColumnVector A = SSQ.i() * (XC.t() * YC);
// calculate estimate of constant term
// [AsScalar converts 1x1 matrix to Real]
Real a = m - (M * A).AsScalar();
// Get variances of estimates from diagonal elements of inverse of SSQ
// [ we are taking inverse of SSQ - we need it for finding D ]
Matrix ISSQ = SSQ.i(); DiagonalMatrix D; D << ISSQ;
ColumnVector V = D.AsColumn();
Real v = 1.0/nobs + (M * ISSQ * M.t()).AsScalar();
// for calc variance of const
// Calculate fitted values and residuals
int npred1 = npred+1;
ColumnVector Fitted = X * A + a;
ColumnVector Residual = Y - Fitted;
Real ResVar = Residual.SumSquare() / (nobs-npred1);
// Get diagonals of Hat matrix (an expensive way of doing this)
Matrix X1(nobs,npred1); X1.Column(1)<<Ones; X1.Columns(2,npred1)<<X;
DiagonalMatrix Hat; Hat << X1 * (X1.t() * X1).i() * X1.t();
// print out answers
cout << "\nEstimates and their standard errors\n\n";
cout.setf(ios::fixed, ios::floatfield);
cout << setw(11) << setprecision(5) << a << " ";
cout << setw(11) << setprecision(5) << sqrt(v*ResVar) << endl;
// make vector of standard errors
ColumnVector SE(npred);
for (int i=1; i<=npred; i++) SE(i) = sqrt(V(i)*ResVar);
// use concatenation function to form matrix and use matrix print
// to get two columns
cout << setw(11) << setprecision(5) << (A | SE) << endl;
cout << "\nObservations, fitted value, residual value, hat value\n";
cout << setw(9) << setprecision(3) <<
(X | Y | Fitted | Residual | Hat.AsColumn());
cout << "\n\n";
}