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Sigmoid Function
\(\frac{p_1}{(1+e^{-p_2 \times (x - p_3)})}\)
p1 is the full scale from 0 to p1. For logistic regression it is 1. In the later examples it will be hardcoded as 1.
p2 is the sharpness of change.
p3 x position of the 50% height.
sigmoid = function(x, p1, p2, p3) {p1 / (1 + exp(-p2 * (x - p3)))}
curve(sigmoid(x, p1=1, p2=2, p3=-3), from=-10, to=+10)
abline(v=-3, lty=2)
Sigmoid that starts at zero.
Sometimes we want to convert a parameter that varies only as \((0,+\infty)\) instead of \((-\infty,+\infty)\). First, to convert x form \((0,+\infty)\) to \((-\infty,+\infty)\) we’ll take the log. The formular for the sigmoid/logistic curve will be transformed as…
\(\frac{1}{(1+e^{-p_2 \times (log(\frac{x}{p_3}))})}\)
sigmoid2 = function(x, p2, p3) {1 / (1 + exp(-p2 * (log(x/p3))))}
curve(sigmoid2(x, p2=20, p3=3), from=0.01, to=+10)
abline(v=3, lty=2)
Transformed Positive Sigmoid
If we move p2 inside the log followed by logarithm exponentiation we’ll get
\(\frac{1}{1+(\frac{p_3}{x})^{p_2}}\)
sigmoid3 = function(x, p2, p3) {1 / (1 + (p3/x)^p2)}
curve(sigmoid3(x, p2=20, p3=3), from=0.01, to=+10)
abline(v=3, lty=2)
# smoother
curve(sigmoid3(x, p2=4, p3=3), from=0.01, to=+10, add=T, col='navyblue')
In case we need to model an opposite dependency the trend has to be reversed. For example we want to formalize a probablity of chemical compound being missed in analysis as a function of intensity.
\(\frac{1}{1+(\frac{x}{p_3})^{p_2}}\)
sigmoid3 = function(x, p2, p3) {1 / (1 + (x/p3)^p2)}
curve(sigmoid3(x, p2=20, p3=3), from=0.01, to=+10)
abline(v=3, lty=2)
# smoother
curve(sigmoid3(x, p2=4, p3=3), from=0.01, to=+10, add=T, col='navyblue')
Arctangent
Another way of modeling sigmoid dependency.
\(\frac{2}{\pi} \times arctan((x-p3) \times p2)\)
The notation of parameters as above:
p2 - shaprness
p3 - position of 50% change
sigmoid2 = function(x, sharpness, threshold) (2/pi)*atan((x-threshold)*sharpness)
curve(sigmoid2(x, sharpness=1, threshold=2), from=-10, to=+10)
curve(sigmoid2(x, sharpness=3, threshold=2), from=-10, to=+10, add=TRUE, col='blue')
abline(v=2)
