added about huber cost function
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@@ -814,6 +814,14 @@ the relative error (why would we prefer the MSE instead of the relative error?)
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\epsilon_{\mathrm{relative}}= \frac{\vert \hat{y} -\hat{\tilde{y}}\vert}{\vert \hat{y}\vert}.
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\]
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!et
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The squared cost function results in an arithmetic mean-unbiased
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estimator, and the absolute-value cost function results in a
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median-unbiased estimator (in the one-dimensional case, and a
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geometric median-unbiased estimator for the multi-dimensional
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case). The squared cost function has the disadvantage that it has the tendency
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to be dominated by outliers.
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We can modify easily the above Python code and plot the relative error instead
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!bc pycod
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import numpy as np
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@@ -915,7 +923,7 @@ The MAE is defined as follows
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\text{MAE}(\hat{y}, \hat{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n-1} \left| y_i - \tilde{y}_i \right|.
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\]
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!et
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Finally we present the
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We present the
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squared logarithmic (quadratic) error
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!bt
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\[
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@@ -928,6 +936,13 @@ estimate is best to use when targets having exponential growth, such
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as population counts, average sales of a commodity over a span of
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years etc.
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Finally, another cost function is the Huber cost function used in robust regression.
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It is less sensitive to outliers in data than the squared error cost function.
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A variant for classification is also sometimes used, a quantity we will meet later.
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We will discuss in more
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detail these and other functions in the various lectures. We conclude this part with another example. Instead of
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a linear $x$-dependence we study now a cubic polynomial and use the polynomial regression analysis tools of scikit-learn.
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