1、Generalized additive models with integrated smoothness estimation 廣義加性模型與集成的平滑估計描述-Description-Fits a generalized additive model (GAM) to data, the term GAM being taken to include any quadratically penalized GLM. The degree of smoothness of model terms is estimated as part of fitting. gam can also f

2、it any GLM subject to multiple quadratic penalties (includingestimation of degree of penalization). Isotropic or scale invariant smooths of any number of variables are available as model terms, as are linear functionals of such smooths; confidence/credible intervals are readily available for any qua

3、ntity predicted using a fitted model; gam is extendable: users can add smooths.適合一個廣義相加模型（GAM）的數(shù)據(jù)，“GAM”被視為包括任何二次處罰GLM。模型計算的平滑度估計作為擬合的一部分。 gam也可以適用于任何GLM多個二次處罰（包括估計程度的處罰）。各向同性或規(guī)模不變平滑的任意數(shù)量的變量的模型計算，這樣的線性泛函平滑的信心/可信區(qū)間都是現(xiàn)成的使用擬合模型預(yù)測任何數(shù)量，“gam是可擴展的：用戶可以添加平滑。Smooth terms are represented using penalized regres

4、sion splines (or similar smoothers) with smoothing parameters selected by GCV/UBRE/AIC/REML or by regression splines with fixed degrees of freedom (mixtures of the two are permitted). Multi-dimensional smooths areavailable using penalized thin plate regression splines (isotropic) or tensor product s

5、plines(when an isotropic smooth is inappropriate). For an overview of the smooths available see smooth.terms.For more on specifying models see gam.models, random.effects and linear.functional.terms. For more on model selection see gam.selection. Do read gam.check and choose.k.平滑術(shù)語表示使用懲罰回歸花鍵（或類似的平滑）與

6、由GCV / UBRE的/ AIC / REML或由固定的自由度（兩個的混合物被允許）的的回歸花鍵與選擇的平滑化參數(shù)。多維平滑可使用懲罰薄板回歸樣條曲線（各向同性）或張量積樣條線（各向同性的光滑是不恰當(dāng)?shù)模?。的平滑的概述，請參閱smooth.terms。欲了解更多有關(guān)指定模型gam.models，random.effects和linear.functional.terms。模型選擇的更多信息，請參閱gam.selection。不要讀為gam.check和choose.k。See gam from package gam, for GAMs via the original Hastie and

7、 Tibshirani approach (see details for differences to this implementation).見GAM包gam，GAMS通過原來的Hastie和Tibshirani方法（詳情請參閱本實施方案的差異）。For very large datasets see bam, for mixed GAM see gamm and random.effects.對于非常大的數(shù)據(jù)集，請參閱bam，混合GAM看到gamm和random.effects。用法-Usage-gam(formula,family=gaussian(),data=list(),wei

8、ghts=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,.)參數(shù)-Arguments-參數(shù)：formulaA GAM formula (see formula.gam and also gam.models).This is exactly l

9、ike the formula for a GLM except that smooth terms, s and te can be addedto the right hand side to specify that the linear predictor depends on smooth functions of predictors(or linear functionals of these). 一個GAM的公式（見formula.gam和gam.models）。這是完全一樣的公式，除非GLM那光滑的條款，s和te可以被添加到指定的線性預(yù)測依賴于光滑函數(shù)的預(yù)測（或線性泛函的右手

10、邊這些）。參數(shù)：familyThis is a family object specifying the distribution and link to use in fitting etc. See glm and family for more details. A negative binomial family is provided: see negbin.quasi families actually result in the use of extended quasi-likelihoodif method is set to a RE/ML method (McCullag

11、h and Nelder, 1989, 9.6). 這是一個家庭對象指定的分配和使用鏈接配件等glm和family更多的細(xì)節(jié)。負(fù)二項分布家庭提供：看到negbin。 quasi家庭實際上導(dǎo)致在使用擴展的擬似然method設(shè)置為一個RE / ML方法（McCullagh和Nelder，1989年，9.6）。參數(shù)：dataA data frame or list containing the model response variable andcovariates required by the formula. By default the variables are takenfrom en

12、vironment(formula): typically the environment fromwhich gam is called.式所需的一個數(shù)據(jù)框或列表包含模型響應(yīng)變量，協(xié)變量。默認(rèn)情況下，變量從environment(formula)：gam被稱為典型的環(huán)境。參數(shù)：weightsprior weights on the data.現(xiàn)有的數(shù)據(jù)上的權(quán)重。參數(shù)：subsetan optional vector specifying a subset of observations to be used in the fitting process.一個可選的矢量指定的裝配過程中可以使用

13、的觀測值的一個子集。參數(shù)：na.actiona function which indicates what should happen when the data contain NAs.The default is set by the na.action setting of options, and is na.fail if that is unset.The “factory-fresh” default is na.omit.一個函數(shù)，它表示時會發(fā)生什么數(shù)據(jù)包含“NA”。默認(rèn)設(shè)置是“na.action設(shè)置選項，na.fail”如果是沒有設(shè)置的。 “工廠新鮮的

14、”默認(rèn)“na.omit。參數(shù)：offsetCan be used to supply a model offset for use in fitting. Note that this offset will always be completely ignored when predicting, unlike an offsetincluded in formula: this conforms to the behaviour of lm and glm.可以用來提供一個模型偏移量用于接頭。請注意，此偏移量總是被完全忽略當(dāng)預(yù)測，不像一個偏移量包含在formula：這符合的lm和glm的行

15、為。參數(shù)：controlA list of fit control parameters to replace defaults returned bygam.control. Values not set assume default values.一個合適的控制參數(shù)，以取代默認(rèn)值返回gam.control。未設(shè)置假設(shè)值默認(rèn)值。參數(shù)：methodThe smoothing parameter estimation method. GCV.Cp to use GCV for unknown scale parameter and Mallows Cp/UBRE/AIC for known sc

16、ale. GACV.Cp is equivalent, but using GACV in place of GCV. REMLfor REML estimation, including of unknown scale, P-REML for REML estimation, but using a Pearson estimateof the scale. ML and P-ML are similar, but using maximum likelihood in place of REML.平滑參數(shù)估計方法。 GCV.Cp使用GCV對未知的尺度參數(shù)和錦葵“的CP / UBRE /

17、AIC已知的規(guī)模。 GACV.Cp是等價的，但使用的GCV GACV的地方。 REMLREML估計，包括不明刻度，P-REMLREML估計，但使用的Pearson估計規(guī)模。 ML和P-ML是相似的，但用最大似然的地方REML。參數(shù)：optimizerAn array specifying the numerical optimization method to use to optimize the smoothingparameter estimation criterion (given by method). perf for performance iteration. outerfo

18、r the more stable direct approach. outer can use several alternative optimizers, specified in thesecond element of optimizer: newton (default), bfgs, optim, nlmand nlm.fd (the latter is based entirely on finite differenced derivatives and is very slow).一個數(shù)組，指定的數(shù)值優(yōu)化方法，使用優(yōu)化的平滑參數(shù)估計準(zhǔn)則（method）。 perf性能迭代。

19、 outer更穩(wěn)定的直接方法。 outer可以使用optimizer：newton（默認(rèn)），bfgs，optim，nlm和第二個元素中指定的幾種可供選擇的優(yōu)化， nlm.fd（后者則是完全基于上有限差分衍生工具，很慢）。參數(shù)：scaleIf this is positive then it is taken as the known scale parameter. Negative signals that thescale parameter is unknown. 0 signals that the scale parameter is 1for Poisson and binomia

20、l and unknown otherwise.Note that (RE)ML methods can only work with scale parameter 1 for the Poisson and binomial cases. 如果這是正的，那么它被當(dāng)作已知尺度參數(shù)。負(fù)信號，規(guī)模參數(shù)是未知的。 0信號泊松分布和二項分布和未知的，否則，尺度參數(shù)為1。需要注意的是（RE）的ML方法只能工作與尺度參數(shù)的泊松分布和二項式情況下。參數(shù)：selectIf this is TRUE then gam can add an extra penalty to each term sothat i

21、t can be penalized to zero.This means that the smoothing parameter estimation that ispart of fitting can completely remove terms from the model. If the correspondingsmoothing parameter is estimated as zero then the extra penalty has no effect.如果這是TRUE然后gam可以添加一個額外的處罰，以每學(xué)期，以便它可以被扣分零。這意味著平滑參數(shù)估計是擬合的一部分

22、的，可以完全除去從模型中的條款。如果相應(yīng)的平滑參數(shù)估計值為零，那么額外的罰款沒有任何效果。參數(shù)：knotsthis is an optional list containing user specified knot values to be used for basis construction.For most bases the user simply supplies the knots to be used, which must match up with the k value supplied (note that the number of knots is not alwa

23、ys just k).See tprs for what happens in the tp/ts case.Different terms can use different numbers of knots, unless they share a covariate.這是一個可選的列表，其中包含用戶指定的節(jié)點值用于基礎(chǔ)建設(shè)。對于最基礎(chǔ)的用戶只需提供要使用的節(jié)，它必須匹配的k值（附注的節(jié)點數(shù)不是永遠(yuǎn)只是k）。見tprstp/ts情況下會發(fā)生什么。不同的術(shù)語可以使用不同的節(jié)數(shù)，除非他們共享一個協(xié)。參數(shù)：spA vector of smoothing parameters can be pro

24、vided here. Smoothing parameters must be supplied in the order that the smooth terms appear in the modelformula. Negative elements indicate that the parameter should be estimated, and hence a mixtureof fixed and estimated parameters is possible. If smooths share smoothing parameters then length(sp)m

25、ust correspond to the number of underlying smoothing parameters.平滑化參數(shù)的一種向量，可以提供在這里。必須提供平滑參數(shù)的順序，順利的詞出現(xiàn)在模型公式。負(fù)性元件表明應(yīng)當(dāng)估計的參數(shù)，因此，固定和估計參數(shù)的混合物是可能的。如果平滑份額平滑參數(shù)，那么length(sp)必須符合相關(guān)的平滑參數(shù)的數(shù)量。參數(shù)：min.spLower bounds can be supplied for the smoothing parameters. Note that if this option is used then the smoothing pa

26、rameters full.sp, in thereturned object, will need to be added to what is supplied here to get thesmoothing parameters actually multiplying the penalties. length(min.sp) shouldalways be the same as the total number of penalties (so it may be longer than sp, if smooths share smoothing parameters).下界能

27、夠供給的平滑化參數(shù)。請注意，如果使用此選項，然后平滑參數(shù)full.sp，返回的對象中，將需要添加什么是這里提供的平滑參數(shù)乘以處罰。 length(min.sp)應(yīng)始終是相同的刑罰（所以它可能是長于sp，如果平滑份額平滑參數(shù)）的總?cè)藬?shù)。參數(shù)：HA user supplied fixed quadratic penalty on the parameters of theGAM can be supplied, with this as its coefficient matrix. A common use of this term isto add a ridge penalty to the

28、 parameters of the GAM in circumstances in which the model is close to un-identifiable on the scale of the linear predictor, but perfectly well defined on the response scale.用戶提供的固定二次罰的GAM的參數(shù)可以提供，這是系數(shù)矩陣。使用這一術(shù)語是一個常見的添加脊處罰，GAM的情況下，該模型是未識別的線性預(yù)測的規(guī)模，但完全定義的響應(yīng)規(guī)模的參數(shù)。參數(shù)：gammaIt is sometimes useful to inflate

29、 the model degrees offreedom in the GCV or UBRE/AIC score by a constant multiplier. This allowssuch a multiplier to be supplied. 有時它是有用的GCV或UBRE的/ AIC得分由一個常乘數(shù)充氣模型的自由度。這允許將要提供這樣一個乘法器。參數(shù)：fitIf 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

30、 an object G containing what would be required to fit is returned is returned. See argument G.如果這種說法是TRUE然后gam設(shè)置模式和適合它，但如果它是FALSE然后對模型進(jìn)行設(shè)置和對象G包含將需要，以適應(yīng)返回返回。請參閱參數(shù)G。參數(shù)：paraPenoptional list specifying any penalties to be applied to parametric model terms.gam.models explains more.可選的列表，指定參數(shù)模型計算被應(yīng)用到任何處罰。

31、 gam.models解釋更多。參數(shù)：GUsually NULL, but may contain the object returned by a previous call to gam withfit=FALSE, in which case all other arguments are ignored except for gamma, in.out, scale, control, method optimizer and fit.通常是NULL，但可能包含對象返回以前調(diào)用gam的fit=FALSE，在這種情況下，所有其它參數(shù)將被忽略，除了gamma，in.out ，scale，c

32、ontrol，methodoptimizer和fit。參數(shù)：in.outoptional list for initializing outer iteration. If supplied then this must contain two elements: sp should be an array of initialization values for all smoothing parameters (there must be a value for all smoothing parameters, whether fixed or to be estimated, but

33、those for fixed s.p.s are not used); scale is the typical scale of the GCV/UBRE function, for passing to the outer optimizer, or the the initial value of the scale parameter, if this is to be estimated by RE/ML.初始化外部循環(huán)的可選列表。如果提供，則必須包含兩個要素：sp應(yīng)該是一個數(shù)組初始化所有的平滑參數(shù)值（是固定的還是要估計，必須有所有的平滑參數(shù)的值，而固定SPS不使用的話）;scal

34、e是GCV / UBRE功能的的典型尺度，用于傳遞到外的優(yōu)化器，或尺度參數(shù)的初始值，如果這是要估計的RE / ML。參數(shù)：.further arguments forpassing on e.g. to gam.fit (such as mustart).在例如通過進(jìn)一步的論據(jù)gam.fit（如mustart）。Details-Details-A generalized additive model (GAM) is a generalized linear model (GLM) in which the linearpredictor is given by a user specifie

35、d sum of smooth functions of the covariates plus aconventional parametric component of the linear predictor. A simple example is:一個廣義相加模型（GAM）是一個廣義線性模型（GLM）的線性預(yù)測是由用戶指定的協(xié)變量的函數(shù)平滑，再加上傳統(tǒng)的參數(shù)化組件的線性預(yù)測的總和。一個簡單的例子是：where the (independent) response variables y_iPoi, and f_1 and f_2 are smooth functions of cov

36、ariates x_1 andx_2. The log is an example of a link function.（獨立的）響應(yīng)變量y_iPoi和f_1和f_2是光滑函數(shù)的協(xié)變量x_1和x_2。的log的一個例子是一個鏈接函數(shù)。If absolutely any smooth functions were allowed in model fitting then maximum likelihoodestimation of such models would invariably result in complex overfitting estimates off_1and f_

37、2. For this reason the models are usually fit bypenalized likelihoodmaximization, in which the model (negative log) likelihood is modified by the addition ofa penalty for each smooth function, penalizing its wiggliness. To control the tradeoffbetween penalizing wiggliness and penalizing badness of f

38、it each penalty is multiplied byan associated smoothing parameter: how to estimate these parameters, andhow to practically represent the smooth functions are the main statistical questionsintroduced by moving from GLMs to GAMs.如果確實被允許在任何光滑的函數(shù)模型擬合，最大似然估計這些模型往往會導(dǎo)致復(fù)雜的過擬合估計f_1和f_2。出于這個原因的模型通常是適合由懲罰的可能性最

39、大化，其中模型（負(fù)對數(shù)）的可能性被修改通過加入每個平滑函數(shù)罰款，懲罰“wiggliness。要控制，之間的的懲罰wiggliness和懲罰不良適合每個罰球乘以相關(guān)的平滑參數(shù)：如何估計這些參數(shù)的權(quán)衡，以及如何在實踐中代表順利的功能是主要的統(tǒng)計問題，介紹了從GLMS GAMS。The mgcv implementation of gam represents the smooth functions usingpenalized regression splines, and by default uses basis functions for these splines thatare des

40、igned to be optimal, given the number basis functions used. The smooth terms can befunctions of any number of covariates and the user has some control over how smoothness ofthe functions is measured.mgcvgam實施順利使用懲罰的回歸樣條曲線的功能，在默認(rèn)情況下使用這些曲線的設(shè)計是最佳的，因為數(shù)基函數(shù)的基礎(chǔ)功能。光滑的術(shù)語可以是任意數(shù)量的協(xié)變量的函數(shù)，并且用戶具有一定的控制的函數(shù)的平滑度如何測量。

41、gam in mgcv solves the smoothing parameter estimation problem by using theGeneralized Cross Validation (GCV) criteriongam在mgcv解決了平滑參數(shù)估計問題通過使用廣義交叉驗證（GCV）標(biāo)準(zhǔn)，or an Un-Biased Risk Estimator (UBRE )criterion或無偏風(fēng)險估計（UBRE）標(biāo)準(zhǔn)where D is the deviance, n the number of data, s the scale parameter andDoF the eff

42、ective degrees of freedom of the model. Notice that UBRE is effectively just AIC rescaled, but is only used when s is known.其中D是越軌行為，n數(shù)據(jù)的數(shù)量，s的尺度參數(shù)和DoF有效度模型的自由。請注意，UBRE實際上只是AIC重新調(diào)整，但只用在s被稱為。Alternatives are GACV, or a Laplace approximation to REML. There is some evidence that the latter may actually

43、be the most effective choice.替代品GACV，或Laplace逼近REML。有一些證據(jù)表明，后者實際上可能是最有效的選擇。Smoothing parameters are chosen tominimize the GCV, UBRE/AIC, GACV or REML scores for the model, and the main computational challenge solvedby the mgcv package is to do this efficiently and reliably. Various alternative numer

44、ical methods are provided which can be set by argument optimizer.平滑化參數(shù)的選擇，以盡量減少GCV，UBRE / AIC，GACV或模型REML分?jǐn)?shù)，和求解的主要計算挑戰(zhàn)mgcv包是有效和可靠地做到這一點。各種替代數(shù)值方法提供了可以設(shè)置的參數(shù)optimizer。Broadly gam works by first constructing basis functions and one or more quadratic penaltycoefficient matrices for each smooth term in th

45、e model formula, obtaining a model matrix forthe strictly parametric part of the model formula, and combining these to obtain acomplete model matrix (/design matrix) and a set of penalty matrices for the smooth terms.Some linear identifiability constraints are also obtained at this point. The model

46、isfit using gam.fit, a modification of glm.fit. The GAMpenalized likelihood maximization problem is solved by Penalized IterativelyReweightedLeast Squares (P-IRLS) (see e.g. Wood 2000).Smoothing parameter selection is integrated in one of two ways. (i) Performance iteration uses the fact that at eac

47、h P-IRLS iteration a penalizedweighted least squares problem is solved, and the smoothing parameters of that problem canestimated by GCV or UBRE. Eventually, in most cases, both model parameter estimates and smoothingparameter estimates converge. (ii) Alternatively the P-IRLS scheme is iterated to c

48、onvergence for each trial set of smoothing parameters, and GCV, UBRE or REML scores are only evaluated on convergence - optimization is then outer to the P-IRLS loop: in this case the P-IRLS iteration has to be differentiated, to facilitate optimization, and gam.fit3 is used in place of gam.fit. The

49、 default is the second method, outer iteration.廣義gam的工作原理是第一構(gòu)造的基礎(chǔ)功能和一個或多個二次罰系數(shù)矩陣中的模型公式為每個平滑內(nèi)，獲得模型矩陣模型公式為嚴(yán)格的參數(shù)的一部分，并結(jié)合這些以獲得一個完整的模型/設(shè)計矩陣（矩陣）和刑罰矩陣順利條款的一組。一些線性辨識性約束在這一點上也能獲得。該模型是適合使用gam.fit，glm.fit修改。的的GAM處罰的可能性最大化問題得到解決，由受罰迭代加權(quán)最小二乘法（P-IRLS）（如木材2000）。平滑參數(shù)的選擇是集成在以下兩種方式之一。 （I）的性能迭代“，在每個P-IRLS迭代一個懲罰加權(quán)最小二乘問

50、題的解決，這個問題可以平滑參數(shù)估計GCV或UBRE所使用的事實。最終，在大多數(shù)情況下，兩個模型參數(shù)的估計和平滑參數(shù)估計值的收斂。 （2）或者的P-IRLS計劃的迭代收斂，為每個審判平滑參數(shù)，GCV，UBRE或REML分?jǐn)?shù)的評價收斂 - “外部”是的P-IRLS循環(huán)的優(yōu)化：在本情況下，P-IRLS迭代以加以區(qū)分，以方便優(yōu)化，和gam.fit3被用于代替gam.fit。默認(rèn)的是第二種方法，外部循環(huán)。Several alternative basis-penalty typesare built in for representing model smooths, but alternatives

51、can easily be added (see smooth.termsfor an overview and smooth.construct for how to add smooth classes). In practice thedefault basis is usually the best choice, but the choice of the basis dimension (k in thes and te terms) is something that should be considered carefully (the exact value is not c

52、ritical, but it is important not to make it restrictively small, nor very large and computationally costly). The basis shouldbe chosen to be larger than is believed to be necessary to approximate the smooth function concerned.The effective degrees of freedom for the smooth will then be controlled by

53、 the smoothing penalty onthe term, and (usually) selected automatically (with an upper limit set by k-1 or occasionally k). Of coursethe k should not be made too large, or computation will be slow (or in extreme cases there will be morecoefficients to estimate than there are data).幾種可供選擇的依據(jù)，處罰類型建立模型

54、平滑，但替代品可以很容易地添加（見smooth.terms的概述和smooth.construct如何添加平滑的類）。在實踐中，默認(rèn)的基礎(chǔ)通常是最好的選擇，但選擇的基礎(chǔ)尺寸（ks和te條款）的東西，應(yīng)該仔細(xì)考慮（確切值不是關(guān)鍵的，但重要的是不要使它限定小，也不是非常大的和計算昂貴）。應(yīng)選擇的基礎(chǔ)上，要大于被認(rèn)為是必要的近似的平滑函數(shù)有關(guān)。將被控制的有效程度的自由的順利平滑的術(shù)語刑罰，和（通常情況下）自動選擇（上限設(shè)定k-1或偶爾k）。當(dāng)然，k不應(yīng)該過大，或計算將是緩慢的（或在極端的情況下，將會有更多的系數(shù)估計比有數(shù)據(jù)）。Note that gam assumes a very inclusiv

55、e definition of what counts as a GAM:basically any penalized GLM can be used: to this end gam allows the non smooth modelcomponents to be penalized via argument paraPen and allows the linear predictor to depend ongeneral linear functionals of smooths, via the summation convention mechanism described

56、 inlinear.functional.terms.請注意，gam承擔(dān)的最重要的一個GAM一個很大的包容性的定義：基本上，可用于任何處罰GLM：為此gam允許非光滑模型組件被處罰通過參數(shù)paraPen和允許的線性預(yù)測依賴上一般線性泛函的平滑，通過的求和約定機制，在linear.functional.terms。Details of the default underlying fitting methods are given in Wood (2011 and 2004). Some alternative methods are discussed in Wood (2000 and 2

57、006).相關(guān)擬合方法的默認(rèn)木材（2011年和2004年）。一些替代方法進(jìn)行了探討伍德（2000年至2006年）。gam() is not a clone of Trevor Hasties oroginal (as supplied in S-PLUS or package gam) The major differences are (i) that by default estimation of the degree of smoothness of model terms is part of model fitting, (ii) a Bayesian approach to variance estimation is employed that makes for easier confidence interval calcu