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1 Accracy verss interpretability in exible modeling: implementing a tradeo sing Gassian process models Tony A. Plate (tap@mcs.vw.ac.nz) School of Mathematical and Compting Sciences Victoria University of Wellington, Wellington, New Zealand ABSTRACT One of the widely acknowledged drawbacks of exible statistical models is that they are often extremely diclt to interpret. However, if exible models are constrained to be additive they are mch easier to interpret, as each inpt can be considered independently. The problem with additive models is that they cannot provide an accrate model if the phenomenon being modeled is not additive. This paper proposes that a tradeo between accracy and additivity can be implemented easily in a particlar type of exible model: a Gassian process model. One can bild a series of Gassian process models which begin with the completely exible and are constrained to be more and more additive, and ths interpretable. Observations of how the test error and importance of interactions change as the model becomes more additive give insight into the importance and natre of interactions. Models in the series can also be interpreted graphically with a techniqe for visalizing the eects of inpts in non-additive models that was adapted from plots for generalized additive models. This visalization techniqe shows the overall eects of dierent inpts and also shows which inpts are involved in interactions and how strong those interactions are.. Introdction The ability to provide good predictions on new data is a vital property of a statistical model. However, it is often desirable and sometimes essential to interpret the model also. This is especially tre in medical domains, where black-box methods are viewed with great sspicion [3 6]. Flexible statistical models, sch as neral networks, bagged decision trees, or Gassian processes tend to be rather diclt to interpret. When interpretation is important, simpler bt more easily interpretable methods sch as logistic regression are often sed instead. There is a risk with sch simpler methods that they may fail to discover an important relationship in the data becase they lack the exibility to model it. The standard practice in the neral network modeling commnity, and, to a lesser extent, in the exible statistical modeling commnity has been to minimize model complexity only in the service of minimizing prediction error. However, given the importance of interpretability in some domains, it cold be sefl to have a modeling techniqe in which we can trade slight decreases in predictive power for increases in interpretability. The primary case of diclty ofinterpretation of exible models is interactions rather than non-linear eects in individal variables, so the interpretability ofa model can be regarded as inversely proportional to the importance of interactions in the model. This paper combines ideas from generalized additive models [8] and from Gassian process modeling [] in an attempt to implement and graphically present the reslts of varios dierent positions in the tradeo between minimizing prediction error and maximizing interpretability (i.e., minimizing the importance of interactions). Generalized additive models (GAMs) are among the most interpretable, exible, general-prpose, statistical models. GAMs are not flly exible Classication and regression trees [6] are probably the in a GAM, variables can have non-linear eects bt do not interact. The lack ofinteraction is, in fact, what makes them so easily interpretable: the independent eect of each variable can be shown in a simple plot. These plots provide a very informative visalization of the model, and althogh they were originally designed for additive models, Plate et al. [] showed how these plots cold be adapted to visalize non-additive models. Gassian process models are an older modeling techniqe that have recently captred wider interest. They are flly exible models and can be sed for both regression and classication. Rasmssen [3], in some carefl stdies, showed that Gassian process models often ot-perform other models sch as neral networks and decision trees. The idea presented here is that of installing a \knob" on Gassian processes, which can be sed to lower, in steps, the inence of interactions in the model, which makes the model more additive and hence more interpretable. The modeling techniqe advocated in this paper is to bild a series of Gassian process models progressively constrained to be more and more additive. This additivity knobisintended to serve three prposes: () to allow one to see whether interactions are merely gratitos or are actally necessary to achieve good predictions () to verify observations abot the individal eects of variables in an nconstrained model and (3) to investigate the form of the model and how itchanges as the model is forced to be more interpretable. Additive eects plots and diagnostics oer insight into these three isses. The plot of generalization error verss degree of additivityshows how important interactions are in the model. The plots of the additiveeectsof individal variables in the series of models show the only other easily-interpretable class of general-prpose, exible, statistical models (as long as the trees are small and not sed in committees as in Breiman's [5] bagged trees).

2 overall eects of each variable and which variables are involved in interactions.. Additive models and additive plots Part of the attraction of generalized additive models (GAMs) [8] is that, althogh they are non-linear, they are easily interpreted from simple plots. A GAM can be expressed as g(x) =h((x )+(x )++ k(x k )) where x = (x x ::: x k ) is the vector of inpt vales and g(x) is the predicted otpt vale. The fnctions etc can be arbitrary non-linear fnctions. The fnction h relates the sm of the i 's to the otpt variable and depends on the distribtion of the target variable and the distribtion of errors in the target vales. For example, for continos target variables with Gassian errors of constant variance, the identity fnction is appropriate for h. For binary valed target variables, the logistic fnction is appropriate for h: h(a) ==(+e ;a ), which gives the probability of the otpt being. The possibility of dierent fnctions for h is what makes GAMs generalized: appropriate fnctions exist for a wide range of target error distribtions. The way the fnctions of the inpts combine is what makes GAMs additive. Althogh the eect of a particlar inpt can be non-linear there are no interactions between inpts: the contribtion of a particlar inpt does not depend on the vales of other inpts. A GAM can be easily interpreted via plots of the eects of inpts on the additive scale of the model, i.e., plots of i. The i are sally fnctions of one variable, and hence can be plotted easily. 3. Gassian process models and additive Gassian process models Only an informal explanation of Gassian process models will be given here for a detailed description of how Gassian process models can be applied to regression and classication tasks, the reader shold conslt Williams and Rasmssen [], Rasmssen [3], or Neal []. Gassian process models, as sed in these papers, can be thoght of as a type of smooth nearest neighbor model with an adaptive distance metric. To begin with, ignore the inpts. Sppose we have N training examples, and we want to make a prediction on new example. The otpts are assmed to be from a zero-mean N + dimensional Gassian distribtion, 3 with by ann + by N + covariance matrix. The vector of otpts can be thoght of as a sample from this Gassian distribtion. The predicted distribtion for the otpt for the new example is calclated by conditioning on the other (training) otpts. The mean (or median, or mode) of this distribtion can be sed as a point vale prediction for this otpt. For example, if there is one training case, and the covariance between the training otpt and the new otpt is high, then the new otpt will have a narrow It is possible to se fnctions of more than one variable in a component fnction in a GAM. This allows for interactions between particlar inpts. 3 Althogh the mean of the otpt distribtion is assmed to be zero, this does not imply that the mean otpt for a particlar set of examples is zero. distribtion with a mean close to the training case vale. If the covariance is low, the mean will be close to zero and the distribtion will be wide. To make this prediction, we havesedacovari- ance matrix which has (N +)(N +)= possible degrees of dom. This may seem like an enormos nmber of parameters when we only have N training cases. However, this is where the inpts enter the pictre. The elements of the covariance matrix are not bt are derived in a highly parameterized manner from the inpts. Element k j of the covariance matrix is a fnction of the distance between the inpt vectors for the kth and jth examples (here, training and testing examples are treated eqally). The distance metric is parameterized to allow the importance of dierent dimensions of the inpt space to vary. For pairs of examples which are close in inpt space according to the distance metric, the corresponding elements of the covariance matrix will be large, which means that their otpts are expected to be close. Williams and Rasmssen [], Rasmssen [3], and Neal [] se small variations on the following covariance fnction, where the inpt vectors are d dimensional and x (k) is the inpt for example k: C kj = c + x(k) x(j) + exp ; (x (k) ; x(j) )! + J kj : The above explanation of the covariance matrix in terms of distances between inpts assmes that the exponential term, which involves the distance between x (k) and x (j), dominates this formla. The parameters of the model are c, the,, the and J. Predictions can be made based on maximm likelihood vales for these parameters, or in a Bayesian techniqe sing Markov Chain Monte Carlo (MCMC) sampling of the parameter space. The pblicly available software described in Neal [] ses the latter, and this software was sed for the experiments described in this paper. A Gassian process model of the above form can easily be made to be additive bychanging the covariance fnction to be an additive fnction of distances on dierent inpt dimensions [3 ]. In fact, a general exponential part can also be inclded, which gives the model both additive and general parts. The form of the covariance fnction sed in this paper is as follows: C kj = c + exp ; + (x (k)! ; x(j) ) exp ; (x (k) ; x(j) ) + J kj : The rst exponential term in this formla, which is the same as above, is the general term which can take accont of distance in any direction. The other Neal [] allows covariance fnctions to have more than one exponential part, and allows the power in the exponential parts to vary.

3 exponential terms (in the sm) can each onlylook at distance in one inpt direction. Ths, if =, the covariance fnction is an additive fnction of the x. Since the predictions of a Gassian process model are additive with respect to the terms in the covariance, the predictions of the model will be additive (if = ). In some cases where the otpt actally is an additive fnction on the inpts, the general exponential part is atomatically fond to have zero importance (i.e, = ). The techniqe sggested in this paper for prodcing a series of models which range from a general fnction of inpts to a prely additive fnction of inpts is as follows: () Train a model with all parameters (predictions are made based on the MCMC samples from the posterior density) () Train a series of models with decreasing xed vales of. When is very small, the model will be additive. The proposal of this paper is that insight into the inpt-otpt relationship in the data can be gained by visalizing the fnction compted, and observing the qality of predictions and the importance of interactions in the eects of individals inpts, as the model is forced to be more and more additive.. Plotting non-additive fnctions with additive plots Plate et al. [] describe how additive plots [8] can be adapted for visalizing non-additive models. Only a brief description is given here. Consider a general fnction of d variables, f(x x ::: x d ), and the problem of nderstanding how f varies with the x i. If d = we can plot a srface, bt if d is higher, conventional plotting techniqes fail s. The problem is that the vale of f with x i set to some vale can vary with the remaining x j. The method advocated in Plate et al. [] is create a scatterplot of eects for each inpt variable. The scatterplot for inpt i shows what eect changing the vale for inpt i has on the otpt (on the appropriate additive scale, which for a neral network will sally be the total inpt of the otpt nit). Each scatterplot consists of a nmber of short line segments, one for each training or test example. The segments for the rst inpt are located at: (x (k) f(x(k) x(k) ::: x(k) d ) ; f(b x (k) ::: x(k) d )) where k is the example index and b i is a reference vale for the ith inpt dimension. The slope of each line segment is j x (k). Each line segment shows the eect, in the context of example k, of changing inpt i from b i to the vale actally occrring in example k. For models which are trly additive, this techniqe reprodces the standard GAM plots of the component fnctions. For non-additive models, the plots give information abot the following: (a) the overall importance of each inpt (b) the trend in the eect of each inpt (c) non-linearities in the eect of an inpt and (d) interactions between the one inpt and others. These plots give a considerable degree of insight into the fnction compted by a exible statistical model. Plate et al. [] sed this method to visalize the eects of varios variables in a MLP model of the risk of developing lng cancer. The plots revealed trends in the eects of individal variables similar to those fond in a logistic regression model, bt also showed the presence of considerable interactions. As the MLP was only slightly sperior to the logistic regression model in terms of predictive performance, the qestions arises as to whether the blk of the observed interactions in the MLP are really sefl or merely gratitos a prodct of excess exibility in the model and noise in the data. This qestion was one of the motivations for the work described in the crrent paper the development of a model in which one can trn down the degree of interaction and observe the eect on the model and its predictive power. Unfortnately, the lng-cancer data was not sitable to test the modeling techniqe as there are too many examples for se with crrent Gassian Process modeling software, and too little dierence in the performance of logistic regression and more exible models. 5. Modeling spectroscopy data Thodberg [9] reported impressive reslts sing a neral network to predict fat content in meat from near-infrared spectroscopy measrements. On this data, consisting of 7 training examples, and 3 test examples, a stepwise linear regression model has a mean sqared test error of 7.7 (.8% of the variance nexplained), and a stepwise qadratic regression model has a mean sqared test error of.6 (.%). Thodberg reports a mean sqared test error of.3 (.8%) for a committee of neral networks trained with Atomatic Relevance Determination. All of these models se the most signicant principal components of the inpts (in normalized form). It is interesting to try to nderstand what featres of the data the better models extract, especially as they are so mch better than the linear model. Gassian process models with ten additive exponential parts (one general and one for each inpt) were applied to this data, sing the nine most signicant principal components as inpts. Firstly, an nconstrained model was trained. 5 The average vale for in this nconstrained model was.8. Next, a series of models with set to dierent vales were trained and tested. Althogh the test errors are not as low as Thodberg achieved, they are qite good (the lowest is arond., or.7%) and one can make ot interesting featres discovered by the models. Figres shows diagnostics 6 and Figre shows the eects of the dierent principal components. The following conclsions can be drawn from these plots: 5 The modeling was performed sing Neal's [] software, version 5.3. The relevant specications were as follows: gp-spec 9 -. /.5:.:: /.:.: omit:- /... (repeated for nine inpts.) model-spec logfile real.5: mc-spec logfile heatbath hybrid :. One hndred Monte- Carlo samples were taken, and the last fty were sed for predictions. was set to a specic vale by changing the rst prior specication after the slash in gp-spec from.5:5 to the desired vale (i.e., a single vale, no colon). 6 The importance of interactions to the eect of x i is measred by calclating the average sqared error in a smooth t to the gradient vales in the plot for x i, i.e., the points x j x (k). The smooth ts were compted sing the BRUTO crve tter described in Hastie and Tibshirani [8].

4 The interactions present in the nconstrained model are not gratitos. In fact, a model forced to have a higher degree of interaction (i.e., a higher ) has slightly better predictions. Models with lower degrees of interaction perform worse. Components to 5 have a large eect and are involved in important interactions. Component has a strong positive trend in its eect, with a slight drop-o at high vales. This is conrmed in the plot for the nearlyadditive model ( = 65). Component has no strong trend bt is involved in signicant interactions. Components 3 and have strong negative trends, and components 5and6mildpositive trends. One nal point to note is that easy to prodce these plots for the eects of the original variables rather than the principal component terms. However, lack of space prevents presenting or discssing these plots here Test error Train error 6 8 (a) Average training and test errors PC PC PC PC3 PC5 PC6 PC7 PC8 PC9 (b) Overall importance of interactions in the effect of each variable Fig. : Model diagnostics for dierent vales of. 6. Discssion Methods for interpreting complex models have received some attention within the eld of neral networks. Three dierent approaches can be discerned. The rst, and most prominent, is that of extracting rles that describe either how the network comptes a fnction [ ] or the fnction compted by the network []. This approach is most sited to applications in which inpts are discrete featres or in which classication decisions are clear-ct, i.e., applications where smooth continity of the compted fnction is relatively nimportant. The visalization techniqes described in this paper (which can be easily applied to any kind of statistical model with continos inpts) are intended for precisely the opposite kind of application: ones in which inpts are continos and otpts vary smoothly. The second approach to interpreting neral networks is based on providing an interpretable view of the internal representational space [7 8 5]. Again, this approach ismore sited to tasks with discrete inpts and clearct otpts. The third approach is based on providing qantitative or graphical indications of the eect of inpt variables on the otpt [ 3 ]. This approach is the one most similar to that presented in this paper. However, the work of Baxt mainly concerned models with discrete inpts. One important dierence between Baxt's methods and the plots of additive eects is that the latter make clear how the magnitde of the eect varies with the vale of the inpt..65 Methods for enforcing or formlating additivity in varios families of exible models have been investigated by a nmber of researchers. Moody and Rognvaldsson [9] discss varios smoothing terms for feedforward neral networks which penalize higher order derivatives with respect to inpts. A smoothing term which penalized o-diagonal second derivatives of model otpts with respect to inpts wold psh the model towards additivity. Girosi, Jones, and Poggio [7] discss formlations of additive models in terms of reglarization networks. The smoothing-spline anova (SS- ANOVA) ofwahba, Wang, G, Klein, and Klein [] is also a Gassian process model, thogh with a dierent covariance fnction to that sed here. SS- ANOVA is also based on additive components, and is actally more general than the model sed in this paper: it starts with fnctions of single variables, then adds in fnctions of two variables, and so on p to the fll interaction term. The Gassian process model sed in this paper only has fnctions of single variables and the fll interaction term. There are two practical advantages of this type of Gassian process model over neral networks or a more general Gassian process model: (i) there is a single knob (the vale) that controls the degree of additivity, and (ii) it is simple to specify and t a series of models with range of degrees of additivity. 7. Conclsion The visalization methods described here can be sed to show the eects of inpts on a single scalar otpt of any statistical model, inclding complex models sch as committees of neral networks or bagged decision trees. These methods can also be sed to display eects of original data when the model is based on transformed data (by pertrbing data in the original inpt coordinates). The gain that comes from combining this visalization method with the general+additive-term formlation of Gassian process models is that one has a \knob" to control the degree of additivity (and ths interpretability) of the model and a way of visalizing the eects of trning the knob. The visalizations show three important aspects of the model: the overall importance of interactions, the general eects of each variable, and which variables are involved in interactions and how important those interactions are. Acknowledgments Thanks to Radford Neal and Chris Williams for very sefl discssions and comments on a draft of this paper, and to Radford for the se of his Markov-Chain Monte-Carlo Gassian process modeling software and advice on sing it. 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