Pearson correlation coefficient

Question

I have written the following C# code to find the Pearson correlation coefficient between two Images. The complete source code is here in the DotNetFiddle.

Correlation Source Code:

public sealed class PearsonCorrelation 
{
    public static double GetSimilarityScore(double[,] p, double[,] q)
    {
        int Width = p.GetLength(0);
        int Height = p.GetLength(1);

        if (Width != q.GetLength(0) || Height != q.GetLength(1))
        {
            throw new ArgumentException("Input vectors must be of the same dimension.");
        }

        double pSum = 0, qSum = 0, pSumSq = 0, qSumSq = 0, productSum = 0;
        double pValue, qValue;

        for (int y = 0; y < Height; y++)
        {
            for (int x = 0; x < Width; x++)
            {
                pValue = p[y, x];
                qValue = q[y, x];

                pSum += pValue;
                qSum += qValue;
                pSumSq += pValue * pValue;
                qSumSq += qValue * qValue;
                productSum += pValue * qValue;
            }
        }

        double numerator = productSum - ((pSum * qSum) / (double)Height);
        double denominator = Math.Sqrt((pSumSq - (pSum * pSum) / (double)Height) * (qSumSq - (qSum * qSum) / (double)Height));

        return (denominator == 0) ? 0 : numerator / denominator;
    }
}

Result:

Same picture is loaded in two picture boxes.

The value of their correlation coefficient has become, -1.

Is this the correct result?

If no, what should I do to correct it?

As far as I can see in the final formula for the numerator and denominator you should divide by n - number of points - which is Height * Width, not just Height — Dmitry Bychenko
It would be very helpful if you could provide us with a reference to the formula you use. However, I believe that the result must be 1 since the two images are the same. Moreover, since the denominator is always positive, there must be something wrong with the numerator. — Mahdi
@Mahdi, github.com/cureos/aforge/blob/master/Sources/Math/Metrics/… — user366312

Dmitry Bychenko Dmitry Bychenko · Accepted Answer · 2016-08-17T20:11:45

The correlation of the identical sequences must be 1

It seems that you have a problem at the very end of the routine; instead of

 double numerator = productSum - ((pSum * qSum) / (double)Height);
 double denominator = Math.Sqrt((pSumSq - (pSum * pSum) / (double)Height) * (qSumSq - (qSum * qSum) / (double)Height));

you should put

 double n = ((double) Width) * Height;

 double numerator = productSum - ((pSum * qSum) / n);
 double denominator = 
   Math.Sqrt((pSumSq - (pSum * pSum) / n) * (qSumSq - (qSum * qSum) / n));

see

https://en.wikipedia.org/wiki/Pearson_product-moment_correlation_coefficient

Pearson correlation coefficient

1 Answers