TI 30Xa Algorithm: The Sigmoid Function and its Integral
The Sigmoid Function
The sigmoid function is defined as:
S = 1 / (1 + e^(-x))
If we multiply the last equation by e^(x) / e^(x), we get:
S = e^(x) / (e^(x) + 1) = e^(x) / (1 + e^(x))
The sigmoid function takes functions from the real numbers and maps them to the interval (0,1). If x → -∞, S → 0. If x → +∞, S → 1.
The inverse of the sigmoid function is the logit function, from which we can derive:
S = 1 / (1 + e^(-x))
1 / S = 1 + e^(-x)
1 / S – 1 = e^(-x)
ln( 1 / S – 1 ) = -x
ln( (1 – S) / S ) = -x (Note: 1 / S – 1 = 1 / S – S / S = (1 – S) / S )
-ln( (1 – S) / S ) = x
ln( S / (1 – S)) = x (Note: For any x, -ln(x) = ln(x^(-1)) = ln(1 / x))
The rest of the blog will focus on the sigmoid function.
The Integral of the Sigmoid Function
Funding the area of the curve under the sigmoid function is pretty straight forward.
∫ 1 / (1 + e^(-x)) dx
= ∫ e^(x) / (e^(x) + 1) dx
Let u = e^(x) + 1. The du = e^(x) dx and:
∫ du / u
= ln | u | + C
= ln (e^(x) +1) + C (since e^(x) + 1 > 0 for all real x)
The definite integral (area) can be calculated as:
x = b
∫ e^(x) / (e^(x) + 1) dx
x = a
= ln (1 + e^b) – ln (1 + e^a)
What if we are given the area, A (capital A), from negative infinity to a value x? Here we are finding the lower tail area.
t = x
∫ e^(t) / (e^(t) + 1) dx = A
t = -∞
ln(1 + e^(x)) – ln(1 + e^(-∞)) = A
Note that e^(-∞) → 0 as t → -∞
We can estimate that:
ln(1 + e^(x)) – ln( 1 ) = A
ln(1 + e^(x)) – 0 = A
1 + e^(x) = e^(A)
e^(x) = e^(A) – 1
x = ln(e^(A) – 1)
TI-30Xa Algorithms
Sigmoid function:
S = 1 / (1 + e^(-x)) = e^(x) / (e^(x) + 1)
Algorithm:
[ ( ] x [ +/- ] [ 2nd ] [ LN ] {e^x} [ + ] 1 [ ) ] [ 1/x ] [ = ]
Example:
S(1.5):
[ ( ] 1.5 [ +/- ] [ 2nd ] [ LN ] {e^x} [ + ] 1 [ ) ] [ 1/x ] [ = ]
Result: 0.817574476
Integral of the sigmoid function:
x = b
∫ e^(x) / (e^(x) + 1) dx
x = a
Algorithm:
[ ( ] b [ 2nd ] [ LN ] {e^x} [ + ] 1 [ ) ] [ LN ]
[ - ] [ ( ] a [ 2nd ] [ LN ] {e^x} [ + ] 1 [ ) ] [ LN ] [ = ]
Example:
x = 3
∫ e^(x) / (e^(x) + 1) dx
x = 0
(a = 0, b = 3)
[ ( ] 3 [ 2nd ] [ LN ] {e^x} [ + ] 1 [ ) ] [ LN ]
[ - ] [ ( ] 0 [ 2nd ] [ LN ] {e^x} [ + ] 1 [ ) ] [ LN ] [ = ]
Result: 2.355440171
Find x given lower tail area:
x = ln(e^(A) – 1)
Algorithm:
[ ( ] A [ 2nd ] [ LN ] {e^x} [ - ] 1 [ ) ] [ LN ] [ = ]
Example:
Area: A = 0.5
[ ( ] 0.5 [ 2nd ] [ LN ] {e^x} [ - ] 1 [ ) ] [ LN ] [ = ]
Result: -0.4327521296
Until next time,
Eddie
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