Two-part logistic

my_curve.py

import altair as alt
import numpy as np
import polars as pl
import streamlit as st


def logistic(x, y0, scale, x0, k):
    return y0 + scale / (1 + np.exp(-k * (x - x0)))


def app():
    st.title("My curve")

    k_left = st.slider("k_left", 0.0, 5.0, 1.0)
    scale_left = st.slider("scale_left", 0.0, 1.0, 0.25)
    scale_right = st.slider("scale_right", 0.0, 3.0, 0.75)

    x0 = 0.0
    a1 = 0.0
    a2 = (scale_left - scale_right) / 2.0
    k_right = k_left * scale_left / scale_right

    x = np.linspace(-10, 10, 1001)
    y_left = logistic(x, y0=a1, scale=scale_left, x0=x0, k=k_left)
    y_right = logistic(x, y0=a2, scale=scale_right, x0=x0, k=k_right)
    y = np.where(x < x0, y_left, y_right)
    df = pl.DataFrame({"x": x, "y": y})

    chart = (
        alt.Chart(df)
        .mark_line()
        .encode(
            x=alt.X("x", title="x"),
            y=alt.Y("y", title="y", scale=alt.Scale(domain=(0.0, 1.0))),
        )
    )

    st.altair_chart(chart, use_container_width=True)

    st.header("Math")
    st.markdown(
        r"""
        Each half $i \in \{1, 2\}$ of the curve follows this form:

        $$
        f_i(x; a_i, L_i, x_0, k_i) = a_i + \frac{L_i}{1 + e^{-k_i (x - x_0)}}
        $$

        with a combined curve:

        $$
        f(x) = \begin{cases}
            f_1(x; a_1, L_1, x_0, k_1) & \text{if } x < x_0 \\
            f_2(x; a_2, L_2, x_0, k_2) & \text{if } x \geq x_0
        \end{cases}
        $$

        We immediately set $a_1 = 0$. To ensure continuity, we require:

        $$
        \begin{align*}
        f_1(x_0) &= f_2(x_0) \\
        a_1 + \frac{1}{2} L_1 &= a_2 + \frac{1}{2} L_2 \\
        \frac{1}{2} (L_1 - L_2) &= a_2
        \end{align*}
        $$

        And also:

        $$
        \begin{align*}
        f_1'(x_0) &= f_2'(x_0) \\
        \frac{1}{4} k_1 L_1 &= \frac{1}{4} k_2 L_2 \\
        k_1 \frac{L_1}{L_2} &= k_2
        \end{align*}
        $$

        This leaves free params: $x_0$, $L_1$, $L_2$, and $k_1$.
        """
    )


if __name__ == "__main__":
    app()