Notes on the Elements of Statistical Learning, Ch2

(E of SL) Ch. 1~2 笔记

• (E of SL) Ch. 1~2 笔记
  • Dots
  • Statistical decision theory
    • Concepts: loss, risk (expected prediction error, EPE)
    • The loss function
      • the squared loss
      • 0-1 loss in classification
      • Remark
  • miscellanea
    • Mean square error and residual sum of square:
  • P.F
    • Ch1 - P.F 1.1 - P.F 1.3 - P.F. 1.4

Dots

• The pair $X \in \mathbf{R}^p, y\in \mathbf{R}^K$ ($k=1$ if scalar function), SL studies the function $f$ as estimation of $y$, i.e. $\hat{y} = f(x)$.
  • and thus you need a loss [ $l = l(f,y)$ ] and risk $R = E_{x,y}(l)$ To evaluate the estimates of functions. By minimizing the risk, under constraints on space of $f, \mathcal{F}$, we have the "learned" estimates, the problem has the form:

$\min_{f} \mathbf{R}$

$s.t.\;\; f\in\mathcal{F}=\{f|f:X \mapsto Y\}$

• Three main classes in SL
  • Parametric，$f(x) = f(x|\theta)$. Since the model has an analytic form or a group parameters, you limit the space of $f$
    • examples: linear regression, logistic, LDA/QDA
  • non-parametric: local/neighborhood, this class is highly dependent on the data, the samples caught in real time.
  • Mixture of both.

Statistical decision theory

Concepts: loss, risk (expected prediction error, EPE)

• @true fact, EPE notation used in "The Elements in Statistical Learning" is basically a frequentist's risk.

The loss function

the squared loss

• $f(x) = \mathbf E(y|x);\;\forall x$, if $f$ is unconstrained, $f \in \mathbb{C}_0(\mathbb{R}^m)$
• if $f \in \mathbb{C}_0$, we have the estimation:
  • $\bf{\beta} = \bf{\mathbf{E}}(xx')^{-1}\mathbf{E}(x'y)$
    • P.F see P.F 1.1
• if $f$ is a neighborhood-typed method:
  • the k-nearest-neighbor is very similar to simple functions, where $f=\mathbf{E}(y|x\in \bf{N}(x))$
  • trees, at each node you compute the mean of response.

0-1 loss in classification

• for bayesian classifier if discrete y (categorical)
• where:
  • $\displaystyle \bf{L = 11' - I_n}$, we have:
  • $f(x) = e_k$, where $k = \text{argmax}_k(P(y_k=1|x))$
    • P.F see P.F 1.3
  • K-classification: equivalence of $\mathcal{L}_2$ for regression and 0-1 loss for bayesian classifition
    • P.F see P.F 1.4

Remark

• if $f$ additive model:
  • It turns out that the optimal estimate for the additive model uses techniques such as k-nearest neighbors to approximate univariate conditional expectations simultaneously for each of the coordinate functions
    • P.F 1.4
• if $l$ = absolute-loss
  • $f(x) = \text{median}(y|x)$

miscellanea

Mean square error and residual sum of square:

• https://stats.stackexchange.com/questions/73540/mean-squared-error-and-residual-sum-of-squares
• MSE :->> (usually frequentist) risk function: estimator to population quantities e.g. $\bar{x} - \mu$
• RSS/SSE :->> not a risk function: estimator to real samples, e.g. $\bar{x} - x_i$
• https://en.wikipedia.org/wiki/Errors_and_residuals

P.F

Ch1

P.F 1.1

$\displaystyle \mathbf{R} = \int_{y,x}(y-f(x))^2\;d{\bf P}(x,y)$, pointwise minimization CANNOT BE USED:
$\displaystyle \mathbf{R} = \int_{y,x}(y-\beta'x)^2\;d{\bf P}(x,y)$ convex in $\beta$
$\displaystyle \partial \mathbf{R} /\partial \beta = \int_{y|x}2x(y-x'\beta)\;d{\bf P}(x,y) = 0\Rightarrow \mathbf{E}(xy) = \mathbf{E}(xx')\beta$
$\beta = \mathbf{E}(xx')^{-1}\mathbf{E}(xy)$

注：$\mathbf{E}(xx')$ may not be invertible

P.F 1.3

The Bayesian classifier:
$\displaystyle \mathbf{R} = \int_{y,x}y'Lf(x)\;d{\bf P}(x,y)$, pointwise minimization, $\displaystyle \forall x$:
$\displaystyle {\bf R_{x}} = \int_{y|x} y_x'L\cdot fd{\bf P}(y|x)$
$\displaystyle \;\;\; = \int_{y|x} y_x'd{\bf P}(y|x)\cdot{\bf L}\cdot f= \mathbf{E}(y|x)' \cdot({\bf 11' - I_n})\cdot f\cdot$
$\displaystyle \;\;\; =1 - \mathbf{E}\big(y|x\big)'f$, thus we have $\forall x$ in the support:

$f = \text{arg}\max_k\big(\mathbf{E}(y_k)|x\big)=\text{arg}\max_k\big(\mathbf{P}(y_k=1)|x\big)$

P.F. 1.4

Consider regression model for k-classification using $\mathcal{L}_2$ loss:
$\displaystyle \mathbf{R} = \int_{y,x}||y-f(x)||\; d{\bf P}(x,y)$
In this case, function $f$ maps $\mathcal{X} \subseteq \mathbb{R}^m$ to set of $y \in \{\mathbf e_k\}_{k=1}^{K}$
$f=\mathbf E(y|x)$, the result is exact