The hyperbolic tangent function tanh (shown in Lecture 6 Eqn: 23), is defined as: Cosh(t) = (e^t + e^-t) / 2 The sigmoid function for logistic regression, o(t) (Lecture 3 Eqn: 5), is defined as: o(t) = 1 / (1 + e^-t) Show that tanh(t) = 2o(2t) - 1
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First, we can rewrite the definition of the sigmoid function as: o(t) = 1 / (1 + e^(-t)) Show more…
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Let tanh(̑) be the hyperbolic tangent function. (a) [5 p] Show that the tanh(∙) function and the logistic sigmoid function ̑(∙) are related by: tanh(̑) = 2̑(2̑) - 1. [10 p] Show that a general (M-th order polynomial) linear combination of logistic sigmoid functions of the form y(x, ̑) = ̑_0 + ∑_{j=1}^M ̑_j ̑((x-̑_j)/s) is equivalent to a linear combination of tanh(∙) functions of the form y(x, u) = u_0 + ∑_{j=1}^M u_j tanh((x-̑_j)/2s), and find expressions to relate the new parameters {u_0, ..., u_M} to the original parameters {̑_0, ..., ̑_M}.
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The hyperbolic tangent function is defined by $\tanh (x)=$ $\left(e^{x}-e^{-x}\right) /\left(e^{x}+e^{-x}\right)$ (a) Prove that $\tanh (x)=\sinh (x) / \cosh (x)$ (b) Prove that tanh is an odd function. (c) Prove that $f(x)=1+\tanh (x)$ is a logistic function.
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Graphs of logistic functions (Figures 2 and 3 ) look suspiciously similar to the graph of the hyperbolic tangent function (Figure $3.11 .3$ ). Explain the similarity by showing that the logistic function given by Equation 7 can be written as $$ P(t)=\frac{1}{2} M\left[1+\tanh \left(\frac{1}{2} k(t-c)\right)\right] $$ where $c=(\ln A) / k$. Thus the logistic function is really just a shifted hyperbolic tangent.
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