SWHarden.com

The personal website of Scott W Harden

Simple Exponential Fit with C#

How to fit an exponential curve to data points using pure csharp
August 24, 2024
csharp

This article describes how to perform a simple exponential fit using pure C#. Exponential curves which pass through 0 can be described using two variables, so data can be translated into linear space, fitted using a linear least squares strategy, then translated back into exponential space.

// These are the data points we wish to fit
double[] xs = [1, 2, 3, 4, 5, 6, 7];
double[] ys = [258, 183, 127, 89, 65, 48, 35];

// Log-transform values, then perform a linear fit
double[] logYs = ys.Select(x => Math.Log(x)).ToArray();
(double slope, double intercept) = LeastSquaresFit(xs, logYs);

// Convert back to exponential form and display the result
double a = Math.Exp(intercept);
Console.WriteLine($"y = {a}*e^({slope}*x)");

Here’s the Linear Least Squares implementation used by the code sample above:

static (double slope, double intercept) LeastSquaresFit(double[] xs, double[] ys)
{
    double sumX = 0, sumY = 0, sumX2 = 0, sumXY = 0;

    for (int i = 0; i < xs.Length; i++)
    {
        sumX += xs[i];
        sumY += ys[i];
        sumX2 += xs[i] * xs[i];
        sumXY += xs[i] * ys[i];
    }

    double avgX = sumX / xs.Length;
    double avgY = sumY / xs.Length;
    double avgX2 = sumX2 / xs.Length;
    double avgXY = sumXY / xs.Length;
    double slope = (avgXY - avgX * avgY) / (avgX2 - avgX * avgX);
    double intercept = (avgX2 * avgY - avgXY * avgX) / (avgX2 - avgX * avgX);
    return (slope, intercept);
}

Plotting Data

The graphs above were made using ScottPlot.NET using the following code:

ScottPlot.Plot plot = new();

// plot the original data and the fitted curve
var dataMarkers = plot.Add.ScatterPoints(xs, ys);
dataMarkers.MarkerSize = 10;
dataMarkers.LegendText = "Raw Data";

// generate and plot the fitted curve
double[] fitXs = Enumerable.Range(0, 100).Select(x => x * .1).ToArray();
double[] fitYs = fitXs.Select(x => a * Math.Exp(slope * x)).ToArray();
var fitLine = plot.Add.ScatterLine(fitXs, fitYs);
fitLine.LineWidth = 2;
fitLine.LinePattern = ScottPlot.LinePattern.DenselyDashed;
fitLine.LegendText = "Fitted Curve";

// decorate the plot and save it
plot.Legend.Alignment = ScottPlot.Alignment.UpperRight;
plot.Title($"y = {a:N2}*e^({slope:N2}*x)");
plot.SavePng("fit.png", 400, 300);

Optimizing Curves for Other Formulas

The code above is ideal for calculating two terms to generate a fitted curve according to the formula:

Y = A * eB * x

…but what about fitting to a curve with third term describing a Y offset?

Y = C + A * eB * x

…or a forth term describing an X offset too?

Y = C + A * eB * x + D

The linear least squares fitting strategy described above is not sufficient for fitting data to these complex curves. The linear least squares strategy can be applied in higher dimensional space using gradient descent to seek optimized curves for more advanced equations. This strategy is summarized by The Math Coffeeshop in a fantastic YouTube video about using Linear Least Squares to Solve Nonlinear Problems . However, formulas representing the partial derivative of each unknown in the error function must be calculated which may be difficult. An alternative strategy is to use Particle swarm optimization to iteratively work toward an optimal solution for a fitted curve even if the derivative of the error function cannot be calculated. This strategy is described in my follow-up article, Curve Fitting in C# using Particle Swarm Optimization

Resources

  Feedback   Source   Contact