Weighted Variance and Weighted Standard Deviation in C++ - c++

Question

Hi I'm trying to calculate the weighted variance and weighted standard deviation of a series of ints or floats. I found these links:
http://math.tutorvista.com/statistics/standard-deviation.html#weighted-standard-deviation
http://www.itl.nist.gov/div898/software/dataplot/refman2/ch2/weightsd.pdf (warning pdf)
Here are my template functions so far. Variance and standard deviation work fine but for the life of me I can't get the weighted versions to match the test case at the bottom of the pdf:
template <class T>
inline float Mean( T samples[], int count )
{
float mean = 0.0f;
if( count >= 1 )
{
for( int i = 0; i < count; i++ )
mean += samples[i];
mean /= (float) count;
}
return mean;
}
template <class T>
inline float Variance( T samples[], int count )
{
float variance = 0.0f;
if( count > 1 )
{
float mean = 0.0f;
for( int i = 0; i < count; i++ )
mean += samples[i];
mean /= (float) count;
for( int i = 0; i < count; i++ )
{
float sum = (float) samples[i] - mean;
variance += sum*sum;
}
variance /= (float) count - 1.0f;
}
return variance;
}
template <class T>
inline float StdDev( T samples[], int count )
{
return sqrtf( Variance( samples, count ) );
}
template <class T>
inline float VarianceWeighted( T samples[], T weights[], int count )
{
float varianceWeighted = 0.0f;
if( count > 1 )
{
float sumWeights = 0.0f, meanWeighted = 0.0f;
int numNonzero = 0;
for( int i = 0; i < count; i++ )
{
meanWeighted += samples[i]*weights[i];
sumWeights += weights[i];
if( ((float) weights[i]) != 0.0f ) numNonzero++;
}
if( sumWeights != 0.0f && numNonzero > 1 )
{
meanWeighted /= sumWeights;
for( int i = 0; i < count; i++ )
{
float sum = samples[i] - meanWeighted;
varianceWeighted += weights[i]*sum*sum;
}
varianceWeighted *= ((float) numNonzero)/((float) count*(numNonzero - 1.0f)*sumWeights); // this should be right but isn't?!
}
}
return varianceWeighted;
}
template <class T>
inline float StdDevWeighted( T samples[], T weights[], int count )
{
return sqrtf( VarianceWeighted( samples, weights, count ) );
}
Test case:
int samples[] = { 2, 3, 5, 7, 11, 13, 17, 19, 23 };
printf( "%.2f\n", StdDev( samples, 9 ) );
int weights[] = { 1, 1, 0, 0, 4, 1, 2, 1, 0 };
printf( "%.2f\n", StdDevWeighted( samples, weights, 9 ) );
Result:
7.46
1.94
Should be:
7.46
5.82
I think the problem is that weighted variance has a few different interpretations and I don't know which one is standard. I found this variation:
http://en.wikipedia.org/wiki/Algorithms_for_calculating_variance#Weighted_incremental_algorithm
template <class T>
inline float VarianceWeighted( T samples[], T weights[], int count )
{
float varianceWeighted = 0.0f;
if( count > 1 )
{
float sumWeights = 0.0f, meanWeighted = 0.0f, m2 = 0.0f;
for( int i = 0; i < count; i++ )
{
float temp = weights[i] + sumWeights,
delta = samples[i] - meanWeighted,
r = delta*weights[i]/temp;
meanWeighted += r;
m2 += sumWeights*delta*r; // Alternatively, m2 += weights[i] * delta * (samples[i]−meanWeighted)
sumWeights = temp;
}
varianceWeighted = (m2/sumWeights)*((float) count/(count - 1));
}
return varianceWeighted;
}
Result:
7.46
5.64
I also tried looking at boost and esutil but they didn't help much:
http://www.boost.org/doc/libs/1_48_0/boost/accumulators/statistics/weighted_variance.hpp
http://esutil.googlecode.com/svn-history/r269/trunk/esutil/stat/util.py
I don't need an entire statistics library, and more importantly, I want to understand the implementation.
Can someone please post functions to calculate these correctly?
Bonus points if your functions can do it in a single pass.
Also, does anyone know if weighted variance gives the same result as ordinary variance with repeated values? For example, would the variance of samples[] = { 1, 2, 3, 3 } be the same as weighted variance of samples[] = { 1, 2, 3 }, weights[] = { 1, 1, 2 }?
Update: here is a google docs spreadsheet I have set up to explore the problem. Unfortunately my answers are nowhere close to the NIST pdf. I think the problem is in the unbias step, but I can't see how to fix it.
https://docs.google.com/spreadsheet/ccc?key=0ApzPh5nRin0ldGNNYjhCUTlWTks2TGJrZW4wQUcyZnc&usp=sharing
The result is a weighted variance of 3.77, which is the square of the weighted standard deviation of 1.94 I got in my c++ code.
I am attempting to install octave on my Mac OS X setup so that I can run their var() function with weights, but it is taking forever to install it with brew. I am deeply into yak shaving now.

Answer 1

float mean(uint16_t* x, uint16_t n) {
uint16_t sum_xi = 0;
int i;
for (i = 0; i < n; i++) {
sum_xi += x[i];
}
return (float) sum_xi / n;
}
/**
* http://www.itl.nist.gov/div898/software/dataplot/refman2/ch2/weigmean.pdf
*/
float weighted_mean(uint16_t* x, uint16_t* w, uint16_t n) {
int sum_wixi = 0;
int sum_wi = 0;
int i;
for (i = 0; i < n; i++) {
sum_wixi += w[i] * x[i];
sum_wi += w[i];
}
return (float) sum_wixi / (float) sum_wi;
}
float variance(uint16_t* x, uint16_t n) {
float mean_x = mean(x, n);
float dist, dist2;
float sum_dist2 = 0;
int i;
for (i = 0; i < n; i++) {
dist = x[i] - mean_x;
dist2 = dist * dist;
sum_dist2 += dist2;
}
return sum_dist2 / (n - 1);
}
/**
* http://www.itl.nist.gov/div898/software/dataplot/refman2/ch2/weighvar.pdf
*/
float weighted_variance(uint16_t* x, uint16_t* w, uint16_t n) {
float xw = weighted_mean(x, w, n);
float dist, dist2;
float sum_wi_times_dist2 = 0;
int sum_wi = 0;
int n_prime = 0;
int i;
for (i = 0; i < n; i++) {
dist = x[i] - xw;
dist2 = dist * dist;
sum_wi_times_dist2 += w[i] * dist2;
sum_wi += w[i];
if (w[i] > 0)
n_prime++;
}
if (n_prime > 1) {
return sum_wi_times_dist2 / ((float) ((n_prime - 1) * sum_wi) / n_prime);
} else {
return 0.0f;
}
}
/**
* http://en.wikipedia.org/wiki/Algorithms_for_calculating_variance#Weighted_incremental_algorithm
*/
float weighted_incremental_variance(uint16_t* x, uint16_t* w, uint16_t n) {
uint16_t sumweight = 0;
float mean = 0;
float M2 = 0;
int n_prime = 0;
uint16_t temp;
float delta;
float R;
int i;
for (i = 0; i < n; i++) {
if (w[i] == 0)
continue;
temp = w[i] + sumweight;
delta = x[i] - mean;
R = delta * w[i] / temp;
mean += R;
M2 += sumweight * delta * R;
sumweight = temp;
n_prime++;
}
if (n_prime > 1) {
float variance_n = M2 / sumweight;
return variance_n * n_prime / (n_prime - 1);
} else {
return 0.0f;
}
}
void main(void) {
uint16_t n = 9;
uint16_t x[] = { 2, 3, 5, 7, 11, 13, 17, 19, 23 };
uint16_t w[] = { 1, 1, 0, 0, 4, 1, 2, 1, 0 };
printf("%f\n", weighted_variance(x, w, n)); /* outputs: 33.900002 */
printf("%f\n", weighted_incremental_variance(x, w, n)); /* outputs: 33.900005 */
}

Answer 2

Solution
You accidentally added an extra term "count" in the denominator of the variance update term.
When using the correction below I get your expected answer of
5.82
FYI, one way to pick up on things like this when you are doing a code review is to do a "dimensional analysis". The "units" of the equation were wrong. You were effectively dividing by an order N squared term when it should be an order N term.
Before
template <class T>
inline float VarianceWeighted( T samples[], T weights[], int count )
{
...
varianceWeighted *= ((float) numNonzero)/((float) count*(numNonzero - 1.0f)*sumWeights); // this should be right but isn't?!
...
}
After
Removing "count" this line should be replaced by
template <class T>
inline float VarianceWeighted( T samples[], T weights[], int count )
{
...
varianceWeighted *= ((float) numNonzero)/((float) (numNonzero - 1.0f)*sumWeights); // removed count term
...
}

Answer 3

Here's a much shorter version with a working Demo :
#include <iostream>
#include <vector>
#include <boost/accumulators/accumulators.hpp>
#include <boost/accumulators/statistics/stats.hpp>
#include <boost/accumulators/statistics/weighted_variance.hpp>
#include <boost/accumulators/statistics/variance.hpp>
namespace ba = boost::accumulators;
int main() {
std::vector<double> numbers{2, 3, 5, 7, 11, 13, 17, 19, 23};
std::vector<double> weights{1, 1, 0, 0, 4, 1, 2, 1, 0 };
ba::accumulator_set<double, ba::stats<ba::tag::variance > > acc;
ba::accumulator_set<double, ba::stats<ba::tag::weighted_variance > , double > acc_weighted;
double n = numbers.size();
double N = n;
for(size_t i = 0 ; i<numbers.size() ; i++ ) {
acc ( numbers[i] );
acc_weighted( numbers[i] , ba::weight = weights[i] );
if(weights[i] == 0) {
n=n-1;
}
};
std::cout << "Sample Standard Deviation, s: " << std::sqrt(ba::variance(acc) *N/(N-1)) << std::endl;
std::cout << "Weighted Sample Standard Deviation, s: " << std::sqrt(ba::weighted_variance(acc_weighted)*n/(n-1)) << std::endl;
}
Make note that n must reflect the number of samples with nonzero weights, hence extra n=n-1; line.
Sample Standard Deviation, s: 7.45729
Weighted Sample Standard Deviation, s: 5.82237