Sigmoid Model

Logistic regression models binary outcomes with the logistic function $p\left(x\right)=\frac{1}{1+e^{-(ax+b)}}$. Parameters $a$ and $b$ control the slope and offset. By adjusting them, the sigmoid curve can represent probabilities between 0 and 1 for any $x$ value.

Given a set of sample points $(x_i,y_i)$ with $y_i$ either 0 or 1, we determine $a$ and $b$ by minimizing the negative log-likelihood

$L=-\sum _i^{}\left(y_i,\ln ,p,\left(x_i\right),+,1,-,y_i,\ln ,1,-,p,\left(x_i\right)\right)$

The gradients of $L$ with respect to $a$ and $b$ yield simple update rules. Starting with zero parameters, gradient descent iteratively subtracts a fraction of these gradients until convergence. This approach works well for small data sets and illustrates how logistic regression learns to separate classes.

This calculator implements a basic gradient descent optimizer. Input your training points one per line, set a learning rate and iteration count, and click "Fit Model". The algorithm repeatedly computes predicted probabilities, accumulates the gradients, and updates $a$ and $b$. After training, the resulting parameters are displayed along with the probability estimate for each input point.

While simple, logistic regression remains a cornerstone of statistical modeling. It forms the basis for classification tasks from medical testing to marketing. Understanding its mechanics helps demystify more complex machine-learning algorithms that extend or generalize it. By experimenting with this tool, you can see firsthand how varying the learning rate or number of iterations affects convergence and final accuracy.