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23 The inla' software implements a fully Bayesian approach based on Markov random field generalized additive models research paper representations exploiting sparse matrix methods. Kim,.J.; Gu,. Fjdisplaystyle f_j could also be a simple parametric function as might be used in any generalized linear model. "Modelling and smoothing parameter estimation with multiple quadratic penalties". Writing gdisplaystyle g for the inverse of displaystyle Phi, this is traditionally written as g(f(x)ifi(xi)displaystyle g(f(vec x)sum _if_i(x_i).
A related effect of penalization is that the notion of degrees of freedom of a model has to be modified to account for the penalties' action in reducing generalized additive models research paper the coefficients' freedom to vary. BayesX' and its R interface provides GAMs and extensions via mcmc and penalized likelihood methods. Straight lines are unpenalized by the spline derivative penalty given above). N.; Pya,.; Saefken,. This flexibility to allow non-parametric fits with relaxed assumptions on the actual relationship between response and predictor, provides the potential for better fits to data than purely parametric models, but arguably with some loss of interpretability. The recommended package in R for GAMs is mgcv, which stands for mixed GAM computational vehicle, 10 which is based on the reduced rank approach with automatic smoothing parameter selection. A b Wood,. Statistical Models. Having replaced all the fjdisplaystyle f_j in the model with such basis expansions we have turned the GAM into a Generalized linear model (GLM with a model matrix that simply contains the basis functions evaluated at the observed xjdisplaystyle x_j values.
For example, consider the situation in which all the smooths are univariate functions. By insisting that the sum of each the fjdisplaystyle f_j evaluated at its observed covariate values should be zero. One approach is to take a fully Bayesian approach, defining priors on the (log) smoothing parameters, and using stochastic simulation or high order approximation methods to obtain information about the posterior of the model coefficients. "Minimizing GCV/GML scores with multiple smoothing parameters via the Newton method". Many other smoothing penalties can be written in the same way, and given the smoothing parameters the model fitting problem now becomes argminD jjTSjdisplaystyle hat beta textargmin_beta D(beta )sum _jlambda _jbeta TS_jbeta, which can be found using a penalized version. Gam sets up bases and penalties for generalized additive models research paper the smooth terms, estimates the model including its smoothing parameters and, in standard R fashion, returns a fitted model object, which can then be interrogated using various helper functions, such as summary, plot, predict, and AIC. Given the basis expansion for each fjdisplaystyle f_j the wiggliness penalties can be expressed as quadratic forms in the model coefficients. A b Umlauf, Nikolaus; Adler, Daniel; Kneib, Thomas; Lang, Stefan; Zeileis, Achim.
The SAS proc gampl is an alternative implementation. "Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations (with discussion. Estimating the degree of smoothness via reml can be viewed as an empirical Bayes method. 8 12 An alternative is to select the smoothing parameters to optimize a prediction error criterion such as Generalized cross validation (GCV) or the Akaike information criterion (AIC). 10 In fact summing just the diagonal elements of Fdisplaystyle F corresponding to the coefficients of fjdisplaystyle f_j gives the effective degrees of freedom for the estimate of fjdisplaystyle f_j. " Generalized additive models for location, scale and shape for high dimensional data - a flexible approach based on boosting". Note that since GLMs and GAMs can be estimated using Quasi-likelihood, it follows that details of the distribution of the residuals beyond the mean-variance relationship are of relatively minor importance. A b c Yee, Thomas (2015). A b c Fahrmeier,.; Lang,. Furthermore, to obtain the derivatives of the GCV or laml, required for optimization, involves implicit differentiation to obtain the derivatives of displaystyle hat beta.r.t. It was then shown 1 that the backfitting algorithm will always converge for these functions. 25 In high dimensional settings then it may make more sense to attempt this task using the Lasso (statistics) or Elastic net regularization.
At some level smoothing penalties generalized additive models research paper are imposed because we believe smooth functions to be more probable than wiggly ones, and if that is true then we might as well formalize this notion by placing a prior on model wiggliness. 19 Software edit Backfit GAMs were originally provided by the gam function in S, 21 now ported to the R language as the gam package. Greven, Sonja; Kneib, Thomas (2010). For example, we might use p-values for testing each term for equality to zero to decide on candidate terms for removal from a model, and we might compare Akaike information criterion (AIC) values for alternative models. This is also the default method when smoothing parameters are not estimated as part of fitting, in which case each smooth term is usually allowed to take one of a small set of pre-defined smoothness levels within the.
Smoothing parameter estimation edit So far we have treated estimation and inference given the smoothing parameters, displaystyle lambda, but these also need to be estimated. Journal of Statistical Software. Using reml smoothing parameter selection, and we expect f1displaystyle f_1 to be a relatively complicated function which we would like to model with a penalized cubic regression spline. For f2displaystyle f_2 we also have to decide whether vdisplaystyle v and wdisplaystyle w are naturally on the same scale so that an isotropic smoother such as thin plate spline is appropriate (specified via s(v,w or whether they. Examples include the R packages mboost 13, which implements a boosting approach; gss, which provides the full spline smoothing methods; 22 vgam which provides vector GAMs; 3 and gamlss, which provides Generalized additive model for location, scale and shape. An alternative approach with particular advantages in high dimensional settings is to use boosting (machine learning), although this typically requires bootstrapping for uncertainty quantification. The basis dimension Kjdisplaystyle K_j is chosen to be sufficiently large that we expect it to overfit the data to hand (thereby avoiding bias from model over-simplification but small enough to retain computational efficiency. Fractal) functions, and thus are not suitable for modeling approaches. So the question of whether a term should generalized additive models research paper be in the model at all remains. Note that Gaussian random effects can also be added to the linear predictor. If the fj(xj)displaystyle f_j(x_j) are represented using smoothing splines 5 then the degree of smoothness can be estimated as part of model fitting using generalized cross validation, or by reml (sometimes known as 'GML which exploits the duality between spline smoothers and Gaussian random effects. "Smoothing parameter selection for a class of semiparametric linear models ".
A second example illustrates how we can control these things. However smoothing parameter estimation does not typically remove a smooth term from the model altogether, because of the fact that most penalties leave some functions un-penalized (e.g. 18 Understanding this Bayesian view of smoothing also helps to understand the reml and full Bayes approaches to smoothing parameter estimation. The preceding integral is usually analytically intractable but can be approximated to quite high accuracy using Laplace's method. F(x p1np(xp)displaystyle f(vec x)Phi left(sum _p1nphi _p(x_p)right) where displaystyle Phi is a smooth monotonic function. The log smoothing parameters, and this requires some care is efficiency and numerical stability are to be maintained. See also edit References edit a b c d e f Hastie,. In the example jdisplaystyle lambda _jto infty would ensure that the estimate of fj(xj)displaystyle f_j(x_j) would be a straight line in xjdisplaystyle x_j. Or if xj(t)displaystyle x_j(t) is itself an observation of a function, we might include a term such as fj(t)xj(t)dtdisplaystyle int f_j(t)x_j(t)dt (sometimes known as a signal regression term). Marra,.; Wood,.N. A b c Wood,.N. If pjKjdisplaystyle psum _jK_j then the computational cost of model estimation this way will be O(np2)displaystyle O(np2).
In any case, checking fj(xj)displaystyle f_j(x_j) is based on examining pattern in the residuals with respect to xjdisplaystyle x_j. Writing all the parameters in one vector, displaystyle beta, suppose that D displaystyle D(beta ) is the deviance (twice the difference between saturated log likelihood and the model log likelihood) for the model. Since the penalty allows some functions through unpenalized (straight lines, given the example penalties Sdisplaystyle S_lambda is rank deficient, and the prior is actually improper, with a covariance matrix given by the Moore-Penrose pseudoinverse of Sdisplaystyle S_lambda (the impropriety corresponds. Suppose that our R workspace contains vectors y, x and z and we want to estimate the model yi0f1(xi)f2(zi)i where iN(0,2).displaystyle y_ibeta _0f_1(x_i)f_2(z_i)epsilon _itext where epsilon _isim N(0,sigma 2). 20 Finally we may choose to maximize the Marginal Likelihood (reml) obtained by integrating the model coefficients, displaystyle beta out of the joint density of,ydisplaystyle beta,y, argmaxf(y ddisplaystyle hat lambda textargmax_lambda int f(ybeta,lambda )pi (beta lambda )dbeta. 19 Smoothing parameter inference is the most computationally taxing part of model estimation/inference.
Y to the predictor variables via a structure such as g(operatorname E (Y)beta _0f_1(x_1)f_2(x_2)cdots f_m(x_m).! The KolmogorovArnold representation theorem ) that any multivariate function could be represented as sums and compositions of univariate functions. 2 Caveats edit Overfitting can be a problem with GAMs, 20 especially if there is un-modelled residual auto-correlation or un-modelled overdispersion. F(vec x)sum _q02nPhi _qleft(sum _p1nphi _q,p(x_p)right) Unfortunately, though the KolmogorovArnold representation theorem asserts the existence of a function of this form, it gives no mechanism whereby one could be constructed. Minimizing the deviance by the usual iteratively re-weighted least squares would result in overfit, so we seek displaystyle beta to minimize D jjfj(x)2dxdisplaystyle D(beta )sum _jlambda _jint f_jprime prime (x)2dx where the integrated square second derivative penalties serve to penalize. Stepwise methods operate by iteratively comparing models with or without particular model terms (or possibly with different levels of term complexity and require measures of model fit or term significance in order to decide which model to select at each stage. "Fast stable direct fitting and smoothness selection for generalized additive models ". A b Gu, Chong (2013). Estimating very large numbers of smoothing parameters is also likely to be statistically challenging, and there are known tendencies for prediction error criteria (GCV, AIC etc.) to occasionally undersmooth substantially, particularly at moderate sample sizes, with reml being somewhat less problematic in this regard.
Within R we could issue the commands library(mgcv) # load the package b gam(y s(x) s(z) In common with most R modelling functions gam expects a model formula to be supplied, specifying the model structure to fit. A b c Wood, Simon. A disadvantage of backfitting is that it is difficult to integrate with the estimation of the degree of smoothness of the model terms, so that in practice the user must set these, or select between a modest set of pre-defined smoothing levels. Bayesian smoothing priors edit Smoothing bias complicates interval estimation for these models, and the simplest approach turns out to involve a Bayesian approach. At its simplest the idea is to replace the unknown smooth functions in the model with basis expansions f_j(x_j)sum _k1K_jbeta _jkb_jk(x_j) where the bjk(xj)displaystyle b_jk(x_j) are known basis functions, usually chosen for good approximation theoretic properties (for example. That is deviance residuals (or other standardized residuals) should be examined for patterns that might suggest a substantial violation of the independence or mean-variance assumptions of the model. Such linear constraints generalized additive models research paper can most easily be imposed by reparametrization at the basis setup stage, 10 so below it is assumed that this has been done. Certain constructive proofs exist, but they tend to require highly complicated (i.e. Mayr,.; Fenske,.; Hofner,.; Kneib,.; Schmid,. Suppose that we want to estimate the model yiPoi(i) where y_isim textPoi(mu _i)text where log mu _ibeta _0beta _1x_if_1(t_i)f_2(v_i,w_i). Spline Models for Observational Data. " Generalized additive models for location, scale and shape (with discussion. One issue that is more common with GAMs than with other GLMs is a danger of falsely concluding that data are zero inflated.
Also the smooth terms were represented using penalized thin plate regression splines and the basis dimension for each was set to 10 (implying a maximum of 9 degrees of freedom after identifiability constraints have been imposed). This can be done using partial residuals overlaid on the plot of fj(xj)displaystyle hat f_j(x_j), or using permutation of the residuals to construct tests for residual pattern (as in the eck' function in R package mgcv. Each extra penalty has its own smoothing parameter and estimation then proceeds as before, but now with the possibility that terms will be completely penalized to zero. "Structured Additive Regression Models : An R Interface to BayesX". In this case the AIC penalty is based on the number of smoothing parameters (and any variance parameters) in the model. The difficulty arises when data contain many zeroes that can be modelled by a Poisson or binomial with a very low expected value: the flexibility of the GAM structure will often allow representation of a very low mean. 1 20 Naive versions of the conditional AIC have been shown to be much too likely to select larger models in some circumstances, a difficulty attributable to neglect of smoothing parameter uncertainty when computing the effective degrees of freedom,.
For one thing the generalized additive models research paper estimates are subject to some smoothing bias, which is the price that must be paid for limiting estimator variance by penalization. In statistics, a generalized additive model (GAM) is a generalized linear model in which the linear predictor depends linearly on unknown smooth functions of some predictor variables, and interest focuses on inference about these smooth functions. A b Ruppert,.; Wand,.P.; Carroll,.J. Therefore, the Generalized Additive Model 1 drops the outer sum, and demands instead that the function belong to a simpler class. This simple example has used several default settings which it is important to be aware. "Coverage properties of confidence intervals for generalized additive model components". Contents Theoretical background edit It had been known since the 1950s (via. Given these problems GAMs are often compared using the conditional AIC, in which the model likelihood (not marginal likelihood) is used in the AIC, and the parameter count is taken as the effective degrees of freedom of the model. Basing AIC on the marginal likelihood in which only the penalized effects are integrated out is possible (the number of un-penalized coefficients now gets added to the parameter count for the AIC penalty but this version of the marginal likelihood. GAMs were originally developed.
Journal of the Royal Statistical Society, Series. The specification of distribution and link function uses the family' objects that are standard when fitting GLMs in R. Model selection edit When smoothing parameters are estimated as part of model fitting then much of what would traditionally count as model selection has been absorbed into the fitting process: the smoothing parameters estimation has already selected between. Journal of the Royal Statistical Society. Scandinavian Journal of Statistics. "Smoothing spline Gaussian regression: more scalable computation via efficient approximation".
This will usually involve plotting the standardized residuals against fitted values and covariates to look for mean-variance problems or missing pattern, and may also involve examining Correlograms (ACFs) and/or Variograms of the residuals to check for violation of independence. "On quantile quantile plots for generalized linear models ". Which can be used to produce confidence/credible intervals for the smooth components, fjdisplaystyle f_j. Which is the standard formulation of a Generalized Additive Model. 28 Where appropriate, simpler models such as GLMs may be preferable to GAMs unless GAMs improve predictive ability substantially (in validation sets) for the application in question. 17 10 Furthermore, we have the large sample result that yN xtwxs)1).displaystyle beta ysim N(hat beta xtwxs_lambda )-1phi ). Siam Journal on Scientific and Statistical Computing. For example, a covariate xjdisplaystyle x_j may be multivariate and the corresponding fjdisplaystyle f_j a smooth function of several variables, or fjdisplaystyle f_j might be the function mapping the level of a factor to the value of a random effect. However, because of the well known fact that reml is not comparable between models with different fixed effects structures, we can not usually use such an AIC to compare models with different smooth terms (since their un-penalized components act like fixed effects). Penalization has several effects on inference, relative to a regular GLM. 12 These more computationally efficient methods use GCV (or AIC or similar) or reml or take a fully Bayesian approach for inference about the degree of smoothness of the model components. The marginal AIC is based on the Mariginal Likelihood (see above) with the model coefficients integrated out. A b c d e f g h Wood,.
Computational Statistics and Data Analysis. Gu,.; Wahba,. "Bayesian Confidence Intervals for the Cross Validated Smoothing Spline". A b Chambers,.M.; Hastie,. " Additive models and cross-validation" (PDF). P-value computation for smooths is not straightforward, because of the effects of penalization, but approximations are available. The solution to this problem is to penalize departure from smoothness in the model fitting process, controlling the weight given to the smoothing penalties using smoothing parameters. Generality edit The GAM model class is quite broad, given that smooth function is a rather broad category. Smoothing Spline anova Models (2nd.). Another example is a varying coefficient (geographic regression) term such as zjfj(xj)displaystyle z_jf_j(x_j) where zjdisplaystyle z_j and xjdisplaystyle x_j are both covariates.
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