This statement:
As far as I'm aware, the next step is to transpose the matrix, and multiply the origin together, take the sum and finally divide by the dimensions X - 1..
And this implementation:
cov += d[i][j] * d[j][i] / (d[i].size() - 1);
Don't say the same thing. Based on the definition here:
void outer_product(vector<double> row, vector<double> col, vector<vector<double>>& dst) {
for(unsigned i = 0; i < row.size(); i++) {
for(unsigned j = 0; j < col.size(); i++) {
dst[i][j] = row[i] * col[j];
}
}
}
//computes row[i] - val for all i;
void subtract(vector<double> row, double val, vector<double>& dst) {
for(unsigned i = 0; i < row.size(); i++) {
dst[i] = row[i] - val;
}
}
//computes m[i][j] + m2[i][j]
void add(vector<vector<double>> m, vector<vector<double>> m2, vector<vector<double>>& dst) {
for(unsigned i = 0; i < m.size(); i++) {
for(unsigned j = 0; j < m[i].size(); j++) {
dst[i][j] = m[i][j] + m2[i][j];
}
}
}
double mean(std::vector<double> &data) {
double mean = 0.0;
for(unsigned i=0; (i < data.size());i++) {
mean += data[i];
}
mean /= data.size();
return mean;
}
void scale(vector<vector<double>> & d, double alpha) {
for(unsigned i = 0; i < d.size(); i++) {
for(unsigned j = 0; j < d[i].size(); j++) {
d[i][j] *= alpha;
}
}
}
So, given these definitions, we can compute the value for the covariance matrix.
void compute_covariance_matrix(vector<vector<double>> & d, vector<vector<double>> & dst) {
for(unsigned i = 0; i < d.size(); i++) {
double y_bar = mean(d[i]);
vector<double> d_d_bar(d[i].size());
subtract(d[i], y_bar, d_d_bar);
vector<vector<double>> t(d.size());
outer_product(d_d_bar, d_d_bar, t);
add(dst, t, dst);
}
scale(dst, 1/(d.size() - 1));
}