The Data Science Lab

### Probit Regression with C#

Probit (“probability unit”) regression is a classic machine learning technique that can be used for binary classification – predicting an outcome that can only be one of two discrete values. For example, you can predict a person’s job satisfaction (0 = low satisfaction, 1 = high satisfaction) based on their gender, age, job type, and income.

Probit regression is very similar to logistic regression and the two techniques usually yield similar results. Probit regression tends to be used most often with financial and economic data, but probit regression and logistic regression can be used for any type of binary classification problem.

A good way to see where this article is going is to take a look at the screenshot of a demo program in **Figure 1**. The demo sets up 40 data elements that represent employees. There are four predictor variables: sex (male = -1, female = +1), age (divided by 100), job type (mgmt. = 1 0 0, supp = 0 1 0, tech = 0 0 1) , and income (divided by $100,000). Each employee has a job satisfaction value coded as 0 = low, 1 = high. The objective is to predict job satisfaction from the four predictive values.

The demo also sets up an 8-item test dataset to evaluate the model after training. The training and testing datasets are completely artificial.

The demo program uses 40-element training data and a modified form of stochastic gradient descent (SGD) to create a probit regression prediction model. After training, the model achieves 75% accuracy on the training data (30 correct out of 40), as well as 75% accuracy on the test data (6 correct out of 8).

The demo program ends by making a prediction for a new, never-before-seen data item where gender = male, age = 24, job type = management, income = $48,500. The model output is a p-value of 0.6837. Since the p-value is greater than 0.5, the prediction is job satisfaction = high. If the p-value had been less than 0.5, the prediction would have been job satisfaction = low.

This article assumes you have intermediate or better programming skills with the C# language, but does not assume you know anything about probit regression. The complete demo code and associated data are shown in this article. Source code and data are also available in the accompanying download. All normal error checking has been removed to keep the main ideas as clear as possible.

**Understanding Probit Regression**

Suppose, as in the demo, you want to predict job satisfaction based on gender, age, job type, and income. The input values corresponding to (male, 24, management, $48,500) are (x0, x1, x2, x3, x4, x5) = (-1, 0.24, 1, 0, 0, 0.485) . Numeric values are normalized so that they are all between 0 and 1, Boolean values are coded minus one plus one, and categorical values are hot coded. There are several normalization and encoding alternatives, but the scheme used by the demo is simple and works well in practice.

Each input value has an associated numeric constant called a weight. There is an additional constant called bias. For the demo problem, the model weights are (w0, w1, w2, w3, w4, w5) = (-0.057, 0.099, 0.460, 0.041, -0.508, -0.113) and the model bias is b = – 0.0077.

The first step is to calculate a z-value as the sum of the products of the weights multiplied by the inputs, plus the bias:

z = (w0 * x0) + (w1 * x1) + (w2 * x2) + (w3 * x3) + (w4 * x4) + (w5 * x5) + b = (-0.057 * -1) + (0.099 * 0.24) + (0.460 * 1) + (0.041 * 0) + (-0.508 * 0) + (-0.113 * 0.485) + (-0.0077) = 0.0570 + 0.0238 + 0.4600 + 0.0000 + 0.0000 + (-0.0548) + (-0.0077) = 0.4783

The second step is to pass the calculated z-value to the phi() function:

p = phi(0.4783) = 0.6837

The result of the p-value will be a value between 0.0 and 1.0 where a value less than 0.5 indicates a class 0 prediction (low satisfaction in the demo) and a p-value greater than 0.5 indicates a class 1 prediction (high satisfaction). Therefore, for this example, the prediction is that the employee has high job satisfaction.

The phi() function is the cumulative density function (CDF) of the standard normal distribution. For any input z, the phi() function returns the area under the normal distribution from minus infinity to z. The graph in **Figure 2** shows the phi() function where an input of z = 0.4783 returns p = 0.6837.

The two main challenges when building a probit regression model are 1.) training the model to find the values for the weights and bias, and 2.) writing code to implement the phi() function.

**The demo program**

The complete demo program, with some minor modifications to save space, is shown in **List 1**. To create the program, I launched Visual Studio and created a new C# .NET Core console application named ProbitRegression. I used Visual Studio 2022 (free community edition) with .NET Core 6.0, but the demo has no significant dependencies, so any version of Visual Studio and the .NET library will work fine. You can also use the Visual Studio Code program.

Once the template code loaded, in the editor window, I removed all unnecessary namespace references, leaving only the reference to the top-level System namespace. In the Solution Explorer window, I right-clicked the Program.cs file, renamed it to the more descriptive ProbitProgram.cs, and allowed Visual Studio to automatically rename the class Program to ProbitProgram.

**List 1:**

Probit regression demo code

using System; // .NET 6.0 namespace ProbitRegression { class ProbitProgram { static void Main(string[] args) { Console.WriteLine("Begin probit regression demo "); Console.WriteLine("Predict job satisfaction " + "(0 = low, 1 = high) "); Random rnd = new Random(0); // 28 Console.WriteLine("Raw data: Sex, Age, Job, Income," + " Satisfaction looks like: "); Console.WriteLine("Male 66 mgmt $52,100.00 | low"); Console.WriteLine("Female 35 tech $86,300.00 | high"); Console.WriteLine("Male 27 supp $47,900.00 | high"); Console.WriteLine(" . . . "); Console.WriteLine("Encoded and normed data looks like: "); Console.WriteLine("-1 0.66 1 0 0 0.52100 | 0 "); Console.WriteLine(" 1 0.35 0 0 1 0.86300 | 1 "); Console.WriteLine("-1 0.27 0 1 0 0.47900 | 1 "); Console.WriteLine(" . . . "); double[][] trainX = new double[40][]; trainX[0] = new double[] { -1, 0.66, 1, 0, 0, 0.5210 }; trainX[1] = new double[] { 1, 0.35, 0, 0, 1, 0.8630 }; trainX[2] = new double[] { -1, 0.24, 0, 0, 1, 0.4410 }; trainX[3] = new double[] { 1, 0.43, 0, 1, 0, 0.5170 }; trainX[4] = new double[] { -1, 0.37, 1, 0, 0, 0.8860 }; trainX[5] = new double[] { 1, 0.30, 0, 1, 0, 0.8790 }; trainX[6] = new double[] { 1, 0.40, 1, 0, 0, 0.2020 }; trainX[7] = new double[] { -1, 0.58, 0, 0, 1, 0.2650 }; trainX[8] = new double[] { 1, 0.27, 1, 0, 0, 0.8480 }; trainX[9] = new double[] { -1, 0.33, 0, 1, 0, 0.5600 }; trainX[10] = new double[] { 1, 0.59, 0, 0, 1, 0.2330 }; trainX[11] = new double[] { 1, 0.52, 0, 1, 0, 0.8700 }; trainX[12] = new double[] { -1, 0.41, 1, 0, 0, 0.5170 }; trainX[13] = new double[] { 1, 0.22, 0, 1, 0, 0.3500 }; trainX[14] = new double[] { 1, 0.61, 0, 1, 0, 0.2980 }; trainX[15] = new double[] { -1, 0.46, 1, 0, 0, 0.6780 }; trainX[16] = new double[] { 1, 0.59, 1, 0, 0, 0.8430 }; trainX[17] = new double[] { 1, 0.28, 0, 0, 1, 0.7730 }; trainX[18] = new double[] { -1, 0.46, 0, 1, 0, 0.8930 }; trainX[19] = new double[] { 1, 0.48, 0, 0, 1, 0.2920 }; trainX[20] = new double[] { 1, 0.28, 1, 0, 0, 0.6690 }; trainX[21] = new double[] { -1, 0.23, 0, 1, 0, 0.8970 }; trainX[22] = new double[] { -1, 0.60, 1, 0, 0, 0.6270 }; trainX[23] = new double[] { 1, 0.29, 0, 1, 0, 0.7760 }; trainX[24] = new double[] { -1, 0.24, 0, 0, 1, 0.8750 }; trainX[25] = new double[] { 1, 0.51, 1, 0, 0, 0.4090 }; trainX[26] = new double[] { 1, 0.22, 0, 1, 0, 0.8910 }; trainX[27] = new double[] { -1, 0.19, 0, 0, 1, 0.5380 }; trainX[28] = new double[] { 1, 0.25, 0, 1, 0, 0.9000 }; trainX[29] = new double[] { 1, 0.44, 0, 0, 1, 0.8980 }; trainX[30] = new double[] { -1, 0.35, 1, 0, 0, 0.5380 }; trainX[31] = new double[] { -1, 0.29, 0, 1, 0, 0.7610 }; trainX[32] = new double[] { 1, 0.25, 1, 0, 0, 0.3450 }; trainX[33] = new double[] { 1, 0.66, 1, 0, 0, 0.2210 }; trainX[34] = new double[] { -1, 0.43, 0, 0, 1, 0.7450 }; trainX[35] = new double[] { 1, 0.42, 0, 1, 0, 0.8520 }; trainX[36] = new double[] { -1, 0.44, 1, 0, 0, 0.6580 }; trainX[37] = new double[] { 1, 0.42, 0, 1, 0, 0.6970 }; trainX[38] = new double[] { 1, 0.56, 0, 0, 1, 0.3680 }; trainX[39] = new double[] { -1, 0.38, 1, 0, 0, 0.2600 }; int[] trainY = new int[40] { 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1 }; double[][] testX = new double[8][]; testX[0] = new double[] { 1, 0.36, 0, 0, 1, 0.8670 }; testX[1] = new double[] { -1, 0.26, 0, 0, 1, 0.4310 }; testX[2] = new double[] { 1, 0.42, 0, 1, 0, 0.5190 }; testX[3] = new double[] { -1, 0.37, 1, 0, 0, 0.8260 }; testX[4] = new double[] { 1, 0.31, 0, 1, 0, 0.8890 }; testX[5] = new double[] { 1, 0.42, 1, 0, 0, 0.2040 }; testX[6] = new double[] { -1, 0.57, 0, 0, 1, 0.2750 }; testX[7] = new double[] { -1, 0.32, 0, 1, 0, 0.5500 }; int[] testY = new int[8] { 0, 0, 1, 1, 0, 0, 0, 1 }; Console.WriteLine("Creating dim = 6 probit model "); int dim = 6; // number predictors double[] wts = new double[dim]; double lo = -0.01; double hi = 0.01; for (int i = 0; i < wts.Length; ++i) wts[i] = (hi - lo) * rnd.NextDouble() + lo; double bias = 0.0; Console.WriteLine("Starting quasi-SGD training " + "with lr = 0.01 "); int maxEpochs = 100; double lr = 0.01; int N = trainX.Length; // number training items int[] indices = new int[N]; for (int i = 0; i < N; ++i) indices[i] = i; for (int epoch = 0; epoch < maxEpochs; ++epoch) { Shuffle(indices, rnd); for (int i = 0; i < N; ++i) // each data item { int ii = indices[i]; double[] x = trainX[ii]; // inputs double y = trainY[ii]; // target 0 or 1 double p = ComputeOutput(x, wts, bias); // [0.0, 1.0] for (int j = 0; j < dim; ++j) // each predictor weight wts[j] += lr * x[j] * (y - p) * p * (1 - p); // o-t error => t-o update bias += lr * (y - p) * p * (1 - p); // update bias } if (epoch % 10 == 0) { double mse = Error(trainX, trainY, wts, bias); double acc = Accuracy(trainX, trainY, wts, bias); Console.WriteLine("epoch = " + epoch.ToString().PadLeft(4) + " | loss = " + mse.ToString("F4") + " | acc = " + acc.ToString("F4")); } } Console.WriteLine("Done "); Console.WriteLine("Trained model weights: "); ShowVector(wts, 3, false); Console.WriteLine("Model bias: " + bias.ToString("F4")); double accTrain = Accuracy(trainX, trainY, wts, bias); Console.WriteLine("Accuracy on train data = " + accTrain.ToString("F4")); double accTest = Accuracy(testX, testY, wts, bias); Console.WriteLine("Accuracy on test data = " + accTest.ToString("F4")); Console.WriteLine("Predicting satisfaction for: "); Console.WriteLine("Male 24 mgmt $48,500.00 "); double[] xx = new double[] { -1, 0.24, 1, 0, 0, 0.485 }; double pp = ComputeOutput(xx, wts, bias, true); Console.WriteLine("output p = " + pp.ToString("F4")); if (pp < 0.5) Console.WriteLine("satisfaction = low"); else Console.WriteLine("satisfaction = high"); Console.WriteLine("End probit regression demo "); Console.ReadLine(); } // Main static double Phi(double z) { // cumulative density of Standard Normal // erf is Abramowitz and Stegun 7.1.26 if (z < -6.0) return 0.0; if (z > 6.0) return 1.0; double a0 = 0.3275911; double a1 = 0.254829592; double a2 = -0.284496736; double a3 = 1.421413741; double a4 = -1.453152027; double a5 = 1.061405429; int sign = 0; if (z < 0.0) sign = -1; else sign = 1; double x = Math.Abs(z) / Math.Sqrt(2.0); // inefficient double t = 1.0 / (1.0 + a0 * x); double erf = 1.0 - (((((a5 * t + a4) * t) + a3) * t + a2) * t + a1) * t * Math.Exp(-x * x); return 0.5 * (1.0 + (sign * erf)); } static double ComputeOutput(double[] x, double[] wts, double bias, bool showZ = false) { double z = 0.0; for (int i = 0; i < x.Length; ++i) z += x[i] * wts[i]; z += bias; if (showZ == true) Console.WriteLine("z = " + z.ToString("F4")); return Phi(z); // probit // return Sigmoid(z); // logistic sigmoid } static double Error(double[][] dataX, int[] dataY, double[] wts, double bias) { double sum = 0.0; int N = dataX.Length; for (int i = 0; i < N; ++i) { double[] x = dataX[i]; int y = dataY[i]; // target, 0 or 1 double p = ComputeOutput(x, wts, bias); sum += (p - y) * (p - y); // E = (o-t)^2 form } return sum / N; ; } static double Accuracy(double[][] dataX, int[] dataY, double[] wts, double bias) { int numCorrect = 0; int numWrong = 0; int N = dataX.Length; for (int i = 0; i < N; ++i) { double[] x = dataX[i]; int y = dataY[i]; // actual, 0 or 1 double p = ComputeOutput(x, wts, bias); if (y == 0 && p < 0.5 || y == 1 && p >= 0.5) ++numCorrect; else ++numWrong; } return (1.0 * numCorrect) / (numCorrect + numWrong); } static double AccuracyVerbose(double[][] dataX, int[] dataY, double[] wts, double bias) { int numCorrect = 0; int numWrong = 0; int N = dataX.Length; for (int i = 0; i < N; ++i) { Console.WriteLine("========="); double[] x = dataX[i]; int y = dataY[i]; // actual, 0 or 1 double p = ComputeOutput(x, wts, bias); Console.WriteLine(i); ShowVector(x, 1, false); Console.WriteLine("target = " + y); Console.WriteLine("computed = " + p.ToString("f4")); if (y == 0 && p < 0.5 || y == 1 && p >= 0.5) { Console.WriteLine("correct "); ++numCorrect; } else { Console.WriteLine("wrong "); ++numWrong; } Console.WriteLine("========="); Console.ReadLine(); } return (1.0 * numCorrect) / (numCorrect + numWrong); } static void Shuffle(int[] arr, Random rnd) { int n = arr.Length; for (int i = 0; i < n; ++i) { int ri = rnd.Next(i, n); int tmp = arr[ri]; arr[ri] = arr[i]; arr[i] = tmp; } } static void ShowVector(double[] vector, int decs, bool newLine) { for (int i = 0; i < vector.Length; ++i) Console.Write(vector[i].ToString("F" + decs) + " "); Console.WriteLine(""); if (newLine == true) Console.WriteLine(""); } } // Program } // ns