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mhjensen
2020-09-18 05:48:55 +02:00
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doc/src/week38/week38.do.txt
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@@ -1126,15 +1126,25 @@ f(y_i\vert x_i)=\beta_0+\beta_1 x_i.
\]
!et
This expression implies however that $f(y_i\vert x_i)$ could take any value from minus infinity to plus infinity. If we however let $f(y\vert y)$ be represented by the mean value, the above example shows us that we can constain to be between zero and one, that is we have $0 \le f(y_i\vert x_i) \le 1$. Looking at our last curve we see also that it has an S-shaped form. This leads us to a very popular model for the function $f$, namely the so-called Sigmoid function or logistic model. We will consider this function as representing the probability for finding a value of $y_i$ with a given $x_i$.
This expression implies however that $f(y_i\vert x_i)$ could take any
value from minus infinity to plus infinity. If we however let
$f(y\vert y)$ be represented by the mean value, the above example
shows us that we can constrain the function to take values between
zero and one, that is we have $0 \le f(y_i\vert x_i) \le 1$. Looking
at our last curve we see also that it has an S-shaped form. This leads
us to a very popular model for the function $f$, namely the so-called
Sigmoid function or logistic model. We will consider this function as
representing the probability for finding a value of $y_i$ with a given
$x_i$.
!split
===== The logistic function =====
The perceptron is an example of a ``hard classification'' model. We
Another widely studied model, is the so-called
perceptron model, which is an example of a ``hard classification'' model. We
will encounter this model when we discuss neural networks as
well. Each datapoint is deterministically assigned to a category (i.e
$y_i=0$ or $y_i=1$). In many cases, it is favorable to have a ``soft''
$y_i=0$ or $y_i=1$). In many cases, and the coronary heart disease data forms one of many such examples, it is favorable to have a ``soft''
classifier that outputs the probability of a given category rather
than a single value. For example, given $x_i$, the classifier
outputs the probability of being in a category $k$. Logistic regression
@@ -1344,14 +1354,14 @@ p(\hat{\beta}\hat{x})=\frac{ \exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_
Till now we have mainly focused on two classes, the so-called binary
system. Suppose we wish to extend to $K$ classes. Let us for the sake
of simplicity assume we have only two predictors. We have then
following model
of simplicity assume we have only two predictors. We have then following model
!bt
\[
\log{\frac{p(C=1\vert x)}{p(K\vert x)}} = \beta_{10}+\beta_{11}x_1,
\]
!et
and
!bt
\[
\log{\frac{p(C=2\vert x)}{p(K\vert x)}} = \beta_{20}+\beta_{21}x_1,
@@ -1405,14 +1415,14 @@ descent method. Newton's method and gradient descent methods are
discussed in the material on "optimization
methods":"https://compphysics.github.io/MachineLearning/doc/pub/Splines/html/Splines-bs.html".
This will be discussed next week.
This will be discussed next week. Before we develop our own codes for logistic regression, we end this lecture by studying the functionality that _Scikit-learn_ offers.
!split
===== Cancer Data again now with Decision Trees and other Methods =====
===== Wisconsin Cancer Data =====
!bc pycod
import matplotlib.pyplot as plt
import numpy as np