概念

回归模型中部分或全部的自变量采用平滑函数，降低线性设定带来的模型风险，对模型的假定不严，如不需要假定自变量线性相关于因变量（线性或非线性都可以）。

解决logistic回归当解释变量个数较多时容易引起维度灾难（Curse of dimensionality）。

光滑函数如应用到连续型解释变量。

* http://plantecology.syr.edu/fridley/bio793/gam.html

Equation

g is a link function, y independent, f_i(x_i)为光滑函数（未知），代替经典线性回归中的x_i，对样本要求少，适用性广。（unspecified nonparametric function replaces a single coefficient）

估计方法

最小二乘法、likelyhood

检验

残差Pseduo系数(PCf)估计，PCf = 1 - RD / ND (RD残差偏差，ND 无效偏差)

分类

可加/非参数（Additive/Nonparametric）：

参数（Parametric）：

半参数/部分线性（Semiparametric/Partial Linear）：

薄板样条（Thin-plate spline）：, allow for interactions between two predictor

前提

如x1和x2并非独立而存在交互作用，则应设为Thin-plate spline: f(x1, x2)

模型中不必每一项都是非线性的，如都非线性会出现计算量大、过拟合等问题，通过查看xi与y的是否存在线性关系来判断是否使用平滑函数。

Should follow statistical and operational considerations.

光滑函数

见“样条函数”

缺点

样条函数不定参使之不能直接用于预估新的数据（Lack of parametric functional form makes it difficult to score the new data directly）

Q&A

How to define smooth.terms in R.mgcv.GAM?

competing philosophies: from "Try everything and go with the one that produces the best fit" (as measured by something like AIC) to "Write the one model that best reflects your understanding of the data-generating process and use it."

Y的分布

联系函数名称

f(Y)

正态分布（normal）

Identity

Y

二项分布（binomial）

Logit

Logit（Y）

Poisson分布

Log

Log（Y）

γ 分布（gamma）

inverse

1/（Y^-1）

负二项分布（negative binomial）

Log

Log（Y）

Concept

Separate cubic polynomials are fit at each section, and then joined at the knots to create a continuous curve.

effective degrees of freedom, or edf. In typical OLS regression the model degrees of freedom is equivalent to the number of predictors/terms in the model.

s(Girth,Height)  #Girth 和 Height 不独立，存在相互影响

gam(Overall ~ Income + Edu + Health, data = d)  # 此时与glm一样

smooth terms: 其实就是应用了光滑函数的自变量e.g. s(agecont), te(Month,Age)

l  http://www.rdocumentation.org/packages/mgcv/functions/gam

gam syntax

gam(y~s(x,k = , bs =)) / gam(y~te(x,k = , bs =))

Choose.k: sets up the dimensionality of the smoothing matrix for each term. Penalized regression smoothers. Using a substantially increased k to see if there is pattern in the residuals that could potentially be explained by increasing k. Default任意数字（normally 10 degree of freedom）。

bs: See smooth.terms for the full list. tp – DEFAULT, thin plate regression spline,cr – penalized cubic regression spline三次样条, cs – shrinkage version of cr,cc – cyclic cubic regression spline, ps – P-spline,cp – cyclic p-spline, ad – adaptive smoothing, fs – factor smooth interaction.

s: smooth s(covariate, edf); te: tensor product smooth

gam(formula,family=gaussian(),data=list(),weights=NULL,subset=NULL,

na.action,offset=NULL,method="GCV.Cp",

optimizer=c("outer","newton"),control=list(),scale=0,

select=FALSE,knots=NULL,sp=NULL,min.sp=NULL,H=NULL,gamma=1,

fit=TRUE,paraPen=NULL,G=NULL,in.out,...)

offset: Can be used to supply a model offset for use in fitting. Note that this offset will always be completely ignored when predicting, unlike an offset included in formula.

control: A list of fit control parameters to replace defaults returned by gam.control.

method: smoothing parameter estimation method. e.g. "GCV.Cp", "GACV.Cp", "REML", "P-REML", "ML", "P-ML" (ML = maximum likelihood, REML = 约束性最大似然法 restricted maximum likelihood)

fit: If this argument is TRUE then gam sets up the model and fits it, but if it is FALSE then the model is set up and an object G containing what would be required to fit is returned is returned.

Gamma: multiplier to inflate the degrees of freedom in the GCV/UBRE/AIC score.

Select: TRUE means adding an extra penalty to each term so that it can be penalized to zero.

s(x1, by=x2)

e.g. Loc = America, Doy = as.numeric(format(Date,format = "%j")), s(Doy,by = Loc)

test

gam.check(b)  # k' = k - 1

summary(gammodel)

(1) GCV, with lower being better. (2) R-sq.(adj) near to 1 is better.

AIC(mod_1d, mod_2d)

(3) with lower being better.

anova(b)  # Wald like tests

anova(mod_1d, mod_2d, test = "Chisq")  #取lower resid.deviance

anova(b,b1,test="F")

(4) select the significant one

plot

plot(mod_gam2, pages=1, residuals=T, shade=T, col='#FF8000')

vis.gam(mod_gam2, type = "response", plot.type = "contour")

vis.gam(mod_gam2, type = "response", plot.type = "persp", border=NA, phi=30, theta=30)

* If the graph looks noise, then the smooth function may be not suitable.

* http://stats.stackexchange.com/questions/14746/what-does-the-dashed-bounds-mean-when-plotting-a-contour-plot-with-r-gam

Q&A

Err: - not meaningful for factors in: Ops.factor(xx, shift[i])

A: smoothing a factor, which isn't supported (`smooth' means that f(x_1) must be close to f(x_2), e.g. if a factor has levels "brick", "sky" and "purple", how far

is it from "brick" to "purple"?)

Err: A term has fewer unique covariate combinations than specified maximum degrees of freedom / basis dimension is larger than number of unique covariates

A: for smoothing function, one independent variables portfolio cannot match to different response variable values.

Q: how to choose a proper smoothing spline (bs='?')

A: 1) use the default; 2) use a tensor product of "cr" smooths for bivariate smoothing, ie. te=(x,bs=”cr”)

Summary

Formula:

LN_Brutto ~ s(agecont, by = Sex) + factor(Sex) + te(Month, Age) +

s(Month, by = Sex)

Parametric coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept)   4.32057    0.01071  403.34   <2e-16 ***

factor(Sex)m  0.27708    0.01376   20.14   <2e-16 ***

---

Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Approximate significance of smooth terms:

edf  Ref.df      F  p-value

s(agecont):Sexf  8.1611  8.7526 20.170  < 2e-16 ***

s(agecont):Sexm  6.6695  7.5523 32.689  < 2e-16 ***

te(Month,Age)   10.3651 12.7201  6.784 2.19e-12 ***

s(Month):Sexf    0.9701  0.9701  0.641    0.430

s(Month):Sexm    1.3750  1.6855  0.193    0.787

---

Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Rank: 60/62

R-sq.(adj) =  0.781   Deviance explained = 78.7%

GCV = 0.048221  Scale est. = 0.046918  n = 1093