如何使正态分布的累积分布函数适合数据点？

Question

I have data which needs to be fitted to the formula

y = a*CDF[NormalDistribution[m, s], x]

Where I need to find a,m and s

I have tested the fit with Mathematica, and it finds the fit fairly fast and it fits the data nicely.

However, I need to implement this is c#.

Currently I've implemented a way to estimate params by going down the gradient hill, but my implementation is very slow (around .5s per estimation)

What is the best way to go about it?

Answer 1

There is an implementation by John Cook: https://www.johndcook.com/blog/csharp_phi/

and I will replicate it below:

static double Phi(double x)
{
    // constants
    double a1 = 0.254829592;
    double a2 = -0.284496736;
    double a3 = 1.421413741;
    double a4 = -1.453152027;
    double a5 = 1.061405429;
    double p = 0.3275911;

    // Save the sign of x
    int sign = 1;
    if (x < 0)
        sign = -1;
    x = Math.Abs(x) / Math.Sqrt(2.0);

    // A&S formula 7.1.26
    double t = 1.0 / (1.0 + p*x);
    double y = 1.0 - (((((a5*t + a4)*t) + a3)*t + a2)*t + a1)*t * Math.Exp(-x*x);

    return 0.5 * (1.0 + sign*y);
}

static void TestPhi()
{
    // Select a few input values
    double[] x = 
    {
        -3, 
        -1, 
        0.0, 
        0.5, 
        2.1 
    };

    // Output computed by Mathematica
    // y = Phi[x]
    double[] y = 
    { 
        0.00134989803163, 
        0.158655253931, 
        0.5, 
        0.691462461274, 
        0.982135579437 
    };

    double maxError = 0.0;
    for (int i = 0; i < x.Length; ++i)
    {
        double error = Math.Abs(y[i] - Phi(x[i]));
        if (error > maxError)
            maxError = error;
    }

        Console.WriteLine("Maximum error: {0}", maxError);
}