概念

迴歸模型中部分或全部的自變數採用平滑函式，降低線性設定帶來的模型風險，對模型的假定不嚴，如不需要假定自變數線性相關於因變數（線性或非線性都可以）。

解決logistic迴歸當解釋變數個數較多時容易引起維度災難（Curse of dimensionality）。

光滑函式如應用到連續型解釋變數。

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

Equation

g is a link function, y independent, f_i(x_i)為光滑函式（未知），代替經典線性迴歸中的x_i，對樣本要求少，適用性廣。（unspecified nonparametric function replaces a single coefficient）

估計方法

最小二乘法、likelyhood

檢驗

殘差Pseduo係數(PCf)估計，PCf = 1 - RD / ND (RD殘差偏差，ND 無效偏差)

分類

可加/非引數（Additive/Nonparametric）：

引數（Parametric）：

半引數/部分線性（Semiparametric/Partial Linear）：

薄板樣條（Thin-plate spline）：, allow for interactions between two predictor

前提

如x1和x2並非獨立而存在互動作用，則應設為Thin-plate spline: f(x1, x2)

模型中不必每一項都是非線性的，如都非線性會出現計算量大、過擬合等問題，通過檢視xi與y的是否存線上性關係來判斷是否使用平滑函式。

Should follow statistical and operational considerations.

光滑函式

見“樣條函式”

缺點

樣條函式不定參使之不能直接用於預估新的資料（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的分佈

聯絡函式名稱

f(Y)

正態分佈（normal）

Identity

Y

二項分佈（binomial）

Logit

Logit（Y）

Poisson分佈

Log

Log（Y）

γ 分佈（gamma）

inverse

1/（Y^-1）

負二項分佈（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 不獨立，存在相互影響

gam(Overall ~ Income + Edu + Health, data = d)  # 此時與glm一樣

smooth terms: 其實就是應用了光滑函式的自變數e.g. s(agecont), te(Month,Age)

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任意數字（normally 10 degree of freedom）。

bs: See smooth.terms for the full list. tp – DEFAULT, thin plate regression spline,cr – penalized cubic regression spline三次樣條, 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 = 約束性最大似然法 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