Saturating functions and sigmoid functions.sagews

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A “saturating” function

It increases at first, but only up to a maximum (saturation) level. $$f(X) = m \cdot \frac{X}{h + X}$$

Parameters:

• $m =$ the saturation level, the value that $f(X)$ approaches as $X$ gets large (think $m$ for maximum)

• $h =$ the half-saturation point, the $X$ value at which $f(X) = \frac{1}{2}m$

---

Example with $m = 6$ and $h = 2$:

# An interactive allowing you to manipulate the values of m and h:

xmax=15
@interact
def saturating(
        m=slider(1, 8, 1, default=1, label="$m$"),
        h=slider(1, 10, 1, default=1, label="$h$"),
        show_m=checkbox(False, label="Show $m$"),
        show_h=checkbox(False, label="Show $h$"),
):
    f(X) = m * X / (h + X)
    label = r"%s \cdot \frac{X}{%s + X}" % (m, h)
    p = plot(f(X), (X, 0, xmax), thickness=2)
    if show_m:
        p += line(((0, m), (xmax, m)), color="red", linestyle="dashed")
        p += text(r"$m$", (0.2, m + 0.3), color="red", fontsize="larger", horizontal_alignment="left")
    if show_h:
        p += line(((0, m/2), (h, m/2), (h, 0)), color="green", linestyle="dotted")
        p += text(r"$X = h$", (h + 0.1, 0.4), color="green", fontsize="larger", horizontal_alignment="left")
        p += text(r"$\frac{m}{2}$", (0.2, m/2 + 0.5), color="green", fontsize="larger", horizontal_alignment="left")
    p.show(ymin=0, ymax=10, axes_labels=("$X$", "$f(X) = {}$".format(label)))

Interact: please open in CoCalc

Variation on the above: a sigmoid function

It has an “S” shape (sort of), which is where the term “sigmoid” comes from.

It also increases only up to a maximum (saturation) level, but it starts out flat (horizontal) for low $X$ values, then curves upward, before leveling off. The exponent $n$ controls the steepness of the transition from low values to high values. $$f(X) = m \cdot \frac{X^n}{h^n + X^n}$$ Note: Often, we just use $h = 1$, so that the function simplifies to $$f(X) = m \cdot \frac{X^n}{1 + X^n}$$

Parameters:

• $m =$ the saturation level, the value that $f(X)$ approaches as $X$ gets large (think $m$ for maximum)

• $h =$ the half-saturation point, the $X$ value at which $f(X) = \frac{1}{2}m$

• $n =$ a parameter that controls the steepness

---

Example with $m = 4$ and $n = 15$ (and $h = 1$):

# An interactive allowing you to manipulate the values of n, m and h:

xmax=15
@interact
def sigmoid(
        n=slider(2, 20, 1, default=2, label="$n$"),
        m=slider(1, 8, 1, default=1, label="$m$"),
        h=slider(1, 10, 1, default=1, label="$h$"),
        show_m=checkbox(False, label="Show $m$"),
        show_h=checkbox(False, label="Show $h$"),
):
    f(X) = m * X^n / (h^n + X^n)
    label = r"%s \cdot \frac{X^{%s}}{%s + X^{%s}}" % (m, n, h if n == 1 or h == 1 else "%s^{%s}" % (h, n), n)
    p = plot(f(X), (X, 0, xmax), thickness=2)
    if show_m:
        p += line(((0, m), (xmax, m)), color="red", linestyle="dashed")
        p += text(r"$m$", (0.2, m + 0.3), color="red", fontsize="larger", horizontal_alignment="left")
    if show_h:
        p += line(((0, m/2), (h, m/2), (h, 0)), color="green", linestyle="dotted")
        p += text(r"$X = h$", (h + 0.1, 0.4), color="green", fontsize="larger", horizontal_alignment="left")
        p += text(r"$\frac{m}{2}$", (0.2, m/2 + 0.5), color="green", fontsize="larger", horizontal_alignment="left")
    p.show(ymin=0, ymax=10, axes_labels=("$X$", "$f(X) = {}$".format(label)))

Interact: please open in CoCalc

# A simpler version of the above interactive, with m = h = 1, allowing you to play with the steepness (n):

xmax=3
@interact
def sigmoid_simple(
        n=slider(2, 40, 1, default=2, label="$n$"),
        show_m=checkbox(False, label="Show $m$"),
):
    m = 1
    f(X) = m * X^n / (1 + X^n)
    label = r"\frac{X^{%s}}{1 + X^{%s}}" % (n, n)
    p = plot(f(X), (X, 0, xmax), thickness=2)
    if show_m:
        p += line(((0, m), (xmax, m)), color="red", linestyle="dashed")
        p += text(r"$m$", (0.05, m + 0.05), color="red", fontsize="larger", horizontal_alignment="left")
    p.show(ymin=0, ymax=1.5, axes_labels=("$X$", "$f(X) = {}$".format(label)))

Interact: please open in CoCalc

A “de-saturating” function

It decreases, from its initial (maximum) level, down to near zero. $$f(X) = m \cdot \frac{h}{h + X}$$

Parameters:

• $m =$ the initial (maximum) level, the value of $f(X)$ at $X = 0$

• $h =$ the half-saturation point, the $X$ value at which $f(X) = \frac{1}{2}m$

# An interactive allowing you to manipulate the values of m and h:

xmax=15
@interact
def desaturating(
        m=slider(1, 8, 1, default=1, label="$m$"),
        h=slider(1, 10, 1, default=1, label="$h$"),
        show_h=checkbox(False, label="Show $h$"),
):
    f(X) = m * h / (h + X)
    label = r"%s \cdot \frac{%s}{%s + X}" % (m, h, h)
    p = plot(f(X), (X, 0, xmax), thickness=2)
    p += text(r"$m$", (0.1, m + 0.3), color="red", fontsize="larger", horizontal_alignment="left")
    if show_h:
        p += line(((0, m/2), (h, m/2), (h, 0)), color="green", linestyle="dotted")
        p += text(r"$X = h$", (h + 0.05, 0.4), color="green", fontsize="larger", horizontal_alignment="left")
        p += text(r"$\frac{m}{2}$", (0.1, m/2 + 0.5), color="green", fontsize="larger", horizontal_alignment="left")
    p.show(ymin=0, ymax=10, axes_labels=("$X$", "$f(X) = {}$".format(label)))

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A decreasing sigmoid function

This one has a backwards “S” shape. Like the de-saturating function, it decreases from its initial (maximum) level down to near zero, but it starts out flat (horizontal) for low $X$ values, then curves downward, before leveling off. The exponent $n$ controls the steepness of the transition from low values to high values. $$f(X) = m \cdot \frac{h^n}{h^n + X^n}$$ Note: Often, we just use $h = 1$, so that the function simplifies to $$f(X) = m \cdot \frac{1}{1 + X^n}$$

Parameters:

• $m =$ the initial (maximum) level, the value of $f(X)$ at $X = 0$

• $h =$ the half-saturation point, the $X$ value at which $f(X) = \frac{1}{2}m$

• $n =$ a parameter that controls the steepness

# An interactive allowing you to manipulate the values of n, m and h:

xmax=15
@interact
def decreasing_sigmoid(
        n=slider(2, 20, 1, default=2, label="$n$"),
        m=slider(1, 8, 1, default=1, label="$m$"),
        h=slider(1, 10, 1, default=1, label="$h$"),
        show_h=checkbox(False, label="Show $h$"),
):
    f(X) = m * h^n / (h^n + X^n)
    h_tothe_n = h if n == 1 or h == 1 else "%s^{%s}" % (h, n)
    label = r"%s \cdot \frac{%s}{%s + X^{%s}}" % (m, h_tothe_n, h_tothe_n, n)
    p = plot(f(X), (X, 0, xmax), thickness=2)
    p += text(r"$m$", (0.2, m + 0.4), color="red", fontsize="larger", horizontal_alignment="left")
    if show_h:
        p += line(((0, m/2), (h, m/2), (h, 0)), color="green", linestyle="dotted")
        p += text(r"$X = h$", (h - 0.1, 0.4), color="green", fontsize="larger", horizontal_alignment="right")
        p += text(r"$\frac{m}{2}$", (0.2, m/2 + 0.5), color="green", fontsize="larger", horizontal_alignment="left")
    p.show(ymin=0, ymax=10, axes_labels=("$X$", "$f(X) = {}$".format(label)))

Interact: please open in CoCalc

# A simpler version of the above interactive, with m = h = 1, allowing you to play with the steepness (n):

xmax=3
@interact
def decreasing_sigmoid_simple(
        n=slider(2, 40, 1, default=2, label="$n$"),
):
    m = 1
    f(X) = m * 1 / (1 + X^n)
    label = r"\frac{1}{1 + X^{%s}}" % (n, )
    p = plot(f(X), (X, 0, xmax), thickness=2)
    p.show(ymin=0, ymax=1.5, axes_labels=("$X$", "$f(X) = {}$".format(label)))

Interact: please open in CoCalc