ISLR(7)- 非线性回归分析

多项式回归和阶梯函数

Note Summary：
0.从理想的线性到现实的非线性
1.多项式回归
2.Step Function
3.参考

0. Moving Beyond Linearity

相较于其他模型, 线性模型更易于描述和实现

• 解释性能和推断理论更有优势

However, standard linear regression can have significant limitations in terms of predictive power

• Since the linearity assumption is always an poor approximation
• Recall that Least Squares can be improved by Ridge Regression, LASSO, PCR... to reduce the complexity of the linear model
  • reduce the variance of the estimates

Goals Beyond Linearity：

Relax the Linearity assumption while

• still maintaining interpretability as much as possible

Extensions of linear models

1. Polynomial Regression(7.1)
2. Step Function(7.2)
3. Regression Spline(7.4)
4. Smoothing Spline(7.5)
5. Local Regression(7.6)

• above approaches are for modeling the relationship between a response Y and a single predictor X in a flexible way.

1. Generalized Additive Model (GAM)

• above approaches can be seamlessly integrated to model
  and several

1. Polynomial Regression

❝ Polynomial Regression extends the linear model by adding extra predictors, obtained by raising each of the original predictors to a power:

• A Cubic regression uses three variables,
  , as predictors to

  provide a non-linear fit to data

❞

Standard Linear Model to Polynomial

• for large enough 「degree d」, polynomial regression produces an extremely non-linear curve
• the coefficients
  are still estimated by Least Squre
• Genearlly, 「
  」

  since large d will lead polynomial curve overly flexible and take strange shapes

Wage & Age Non-Linear Relation

Fitting a degree-4 polynomial using least squares

• the individual coefficients are not of particular interest (black box???)
• Let
  be the value of

  age, to predict wage:

「Variance」
Compute Variance of the fit,

, we need:

• Variance Estimates for each of the fitted coefficients
  from Least Squares
• The Covariances between pairs of coefficient estimates,
  • Let
    be the 5x5 covariance matrix of the
• Let
  ：

is the

estimated pointwise standard error of

• As EACH reference point
  , this computation is repeated and get the fitted curve and twice the standard error

The pair of dotted curves at both sides of the fit are (2x) standard error curves

• Since this (2x) quantity corresponds to an approximate 95% CI, for normally distributed error terms

「Logsitic Regression」
We can treat Wage as a binary variable by splitting it into 「high/low earners」

• logistic regression can be fitted to predict binary response:

Although the sample size is n = 3000, there are only 79 high earners,

• this results in a high variance in the estimated coefficients and therefore fairly wide confidence intervals

2. Step Function

Using polynomial functions in a linear model imposes a 「global structure」 on the non-linear function of X

• use step function to avoid such global structure

❝ Step Function cut the range of a variable into K distinct regions to produce a qualitative variable

• this has the effect of fitting a piecewise constant function in each bin
• and convert a continuous variable into ordered categorical variable

❞

Create cutpoints

in the range of X, and then construct

new variables:

Since

must be in exactly one of the

intervals,

• Use Least Squares to fit a linear model by using
  as predictors:

• can be interpreted as the mean value of
  for
• can represent the average increase in the response for
  in
  relative to

Fit the Logistic Regression Model to predict the probability:

「Disadvantages:」
Unless there are natural breakpoints in the predictors, piecewise-constant functions can 「miss the action」

• age from 20 to 30

「Advantages:」
Step functions are more likely used in biostatistics and epidemiology,

• 5-year age groups are often used to define the bins

3. 参考：

• 《Introduction to Statistical Learning》
  • Section 7.1, 7.2

TOGO: (7) Basis Functions and Splines!