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<\/a><\/p>\nWe want to do $y = b_0 + b_1x $ but obviously a linear line does not fit well. We can do something called polynomial expansion, i.e. add more predictors which are simple transformations of the predictor $x$. i.e. $y = b_0 + b_1x + b_2x^2 + b_3x^3$ The trick comes here: We can interpret the new $x^3$ basis function as a stretching of the x-axis. I.e. the further we move out on the x-axis, the longer we need to stretch the parts (exactly by x^3 times) This can be shown also for other functions:<\/p>\n Note that the logarithm is not defined for negative numbers Note how the stretching can be negative, i.e. the original negative values are stretching\/transformed to positive values One can interprete **logistic regression** with the same trick: Usually we would have some non-linear dependency on a probabilty of e.g. success. That means, with a low value of x, your success-chance are low. To model this kind of data, one can transform the y-axis using $g$ above.<\/p>\n
\n<\/a><\/p>\n
\n<\/a><\/p>\nExponential<\/h3>\n
<\/a><\/p>\nLogarithmic<\/h3>\n
\n<\/a><\/p>\nQuadratic<\/h3>\n
\n<\/a><\/p>\nUsing the trick on the y-axis<\/h2>\n
\n$$ g{-1}(y) = b_0 + b_1*x <=> y = g(b_0+b_1x)$$
\nwith $g$, the logistic (logit) function and $g^{-1}$ the inverse logistic function (invlogit)
\n$$ g^{-1} = \\ln\\frac{p}{1-p} <=> g = \\frac{1}{1+e^{-x}}$$<\/p>\n