Additive models

Data Mining - CdL CLAMSES

Author
Affiliation

Tommaso Rigon

Università degli Studi di Milano-Bicocca

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  • In this unit we will cover the following topics:

    • Generalized additive models (GAMs)
    • Multivariate Adaptive Regression Splines (MARS)

  • We have seen that fully nonparametric methods are plagued by the curse of dimensionality.

  • GAMs and MARS are semi-parametric approaches that keep the model complexity under control so that:

    • they are more flexible than linear models;
    • they are not hugely impacted by the curse of dimensionality.

  • The running example is about trawl data from the Great Barrier Reef.

An ecological application

The trawl dataset

  • We consider the trawl dataset, which refers to a survey of the fauna on the sea bed lying between the coast of northern Queensland and the Great Barrier Reef.

  • The response variable is Score, which is a standardized numeric quantity measuring the amount of fishes caught on a given location.

  • We want to predict the catch score, as a function of a few covariates:

    • the Latitude and Longitude of the sampling position. The longitude can be seen as a proxy of the distance from the coast in this specific experiment;
    • the Depth of the sea on the sampling position;
    • the Zone of the sampling region, either open or closed to commercial fishing;
    • the Year of the sampling, which can be either 1992 or 1993.

  • Having remove a few observations due to missingness, we split the data into training (119 obs.) and test set (30 obs.). The full trawl dataset is available in the sm R package.

The trawl dataset

Getting started: linear models

  • Let begin our analysis by trying to predict the Score using a linear model of the form y_i = \beta_0 + \beta_1 x_{i1} + \cdots + \beta_px_{ip}, \qquad i=1,\dots,n,

  • The above values correspond to the variables of the trawl dataset, so that \begin{aligned} \texttt{Score}_i = \beta_0 &+ \beta_1 \texttt{Latitude}_i + \beta_2 \texttt{Longitude}_i + \\ &+ \beta_3\texttt{Depth}_i + \beta_4 I(\texttt{Zone}_i = \texttt{Closed}) + \beta_5 I(\texttt{Year}_i = \texttt{1993}). \end{aligned}

  • Such a model can be estimated using ordinary least squares, resulting in:

term estimate std.error statistic p.value
(Intercept) 297.690 26.821 11.099 0.000
Latitude 0.256 0.222 1.151 0.252
Longitude -2.054 0.187 -10.955 0.000
Depth 0.020 0.007 3.003 0.003
Zone_Closed -0.116 0.102 -1.143 0.255
Year_1993 0.127 0.103 1.242 0.217

Scatterplot with loess estimate

Comments and cricism of linear models

  • Is this a good model?
  • Granted that every model is just an approximation of reality, it is undeniable that there are some problematic aspects.

  • By simple graphical inspection, it seems that the relationship between Score and Longitude is non-linear.

  • Also, an interaction effect between Year and Longitude could be present.

  • These considerations support the idea that a nonparametric approach might be more appropriate.

  • However, the number of covariates is p = 5 and therefore a fully nonparametric estimation would not be feasible, because of the curse of dimensionality.

  • We need a simplified modelling strategy, that accounts for non-linearities but at the same time is not fully flexible.

Generalized additive models (GAM)

The ANOVA decomposition of a function

  • We seek for an estimate of (a suitable transformation of) the mean function, namely g^{-1}\{\mathbb{E}(Y_i)\} = f(x_{i1},\dots,x_{ip}), where g^{-1}(\cdot) is the so-called link function.
  • The unknown multivariate function f(\bm{x}) = f(x_1,\dots,x_p) : \mathbb{R}^p \rightarrow \mathbb{R} is too complex. However, the following decomposition holds f(\bm{x}) = \beta_0 + \underbrace{\sum_{j=1}^p f_j(x_j)}_{\text{Main effect}} + \underbrace{\sum_{j=1}^p\sum_{k < j}f_{jk}(x_j, x_k)}_{\text{Interaction effect}} + \underbrace{\sum_{j=1}^p\sum_{k < j}\sum_{h < k < j}f_{jkh}(x_j, x_k, x_h)}_{\text{Higher order interaction}} + \cdots.

  • By imposing suitable constraints, this decomposition can be made unique.

  • More importantly, this decomposition gives us an intuition on how to build non-linear models with a simplified structure.

Generalized additive models (GAM)

  • A generalized additive model (GAM) presumes a representation of the following type: f(\bm{x}_i) = \beta_0 + f_1(x_{i1}) + \cdots + f_p(x_{ip}) = \beta_0 + \sum_{j=1}^pf_j(x_{ij}), \qquad i=1,\dots,n, where f_1,\dots,f_p are smooth univariate functions with a potentially non-linear behavior.

  • In GAMs we include only the main effects and we exclude the interactions terms.

  • Generalized linear models (GLMs) are a special case of GAMs, in which f_j(x_{ij}) = \beta_j x_{ij}.

  • To avoid what is essentially a problem of model identifiability, it is necessary for the various f_j to be centered around 0, that is \sum_{i=1}^n f_j(x_{ij}) = 0, \qquad j=1,\dots,p.

The backfitting algorithm I

  • There exist several strategies for estimating the unknown functions f_1,\dots,f_p. One of them, called backfitting, is particularly appealing because of its elegance and generality.

  • Suppose we model each f_j(x) = \sum_{m = 1}^{M_j}\beta_{mj} h_{mj}(x) with a basis expansion, for example using regression splines.

  • In a regression problem we need to minimize, over the unknown \beta parameters, the loss \sum_{i=1}^n\left\{y_i - \beta_0 - \sum_{j=1}^pf_j(x_{ij})\right\}^2 subject to the constraint \sum_{i=1}^n f_j(x_{ij}) = 0.

  • When f_j are regression splines, the above loss can be minimized using least squares. The identifiability issue could be handled by removing the intercept term from each spline basis.

  • However, here we consider an alternative and iterative minimization method, which is similar to the coordinate descent algorithm we employed for the elastic-net.

The backfitting algorithm II

  • Now, let us re-arrange the term in the squared loss as follows: \sum_{i=1}^n\left\{\textcolor{red}{y_i - \beta_0 - \sum_{k\neq j}f_k(x_{ik})} - f_j(x_{ij})\right\}^2, where the highlighted terms are sometimes called partial residuals.
  • Hence, we can repeatedly and iteratively fit a univariate smoothing model for f_j using the partial residuals as response, keeping fixed the value of the other functions f_k, for k \neq j.
  • This algorithm produces the same fit of least squares when f_j are regression splines, but the idea is appealing because it can be used with any generic smoothers \mathcal{S}_j.
  • Finally, note that under the constraint \sum_{i=1}^n f_j(x_{ij}) = 0 the least square estimate for the intercept term is \hat{\beta}_0 = \bar{y}, i.e. the arithmetic mean.

The backfitting algorithm (regression)

The backfitting algorithm for additive regression models

  1. Initialize \hat{\beta}_0 = \bar{y} and set f_j(x_j) = 0, for j=1,\dots,p.

  2. Cycle j =1,\dots,p, j =1,\dots,p, \dots, until convergence:

    1. Update the kth function by smoothing via \mathcal{S}_j the partial residuals, so that \hat{f}_j(x) \leftarrow \mathcal{S}_j\left[\left\{x_{ij}, y_i - \hat{\beta}_0 - \sum_{k \neq j} \hat{f}_k(x_{ik})\right\}_{i=1}^n\right].

    2. Center the function by subtracting its mean \hat{f}_j(x) \leftarrow \hat{f}_j(x) - \frac{1}{n}\sum_{i=1}^n\hat{f}_j(x_{ij}).

Backfitting: comments and considerations

  • The backfitting algorithm, when f_j are modeled as regression splines, is known as “Gauss-Seidel”. The convergence is guaranteed under standard conditions.

  • Interestingly, even when \mathcal{S}_j are smoothing splines the convergence of backfitting is guaranteed; the proof for this statement is less straightforward.

  • In general, however, there is no theoretical guarantee that the algorithm will ever converge, even though the practical experience suggest that this is not a big concern.
  • When \mathcal{S}_j is a linear smoother with smoothing matrix \bm{S}_j, then by analogy with the previous unit we can define the effective degrees of freedom of \hat{f}_j as \text{df}_j = \text{tr}(\bm{S}_j). The number of degrees of the whole model therefore is \text{df} = 1 + \sum_{j=1}^p \text{df}_j.
  • A variant of backfitting for classification problems is available. Once again, relying on quadratic approximations of the log-likelihood allows for a generalization to GLMs.

The backfitting algorithm (classification)

Local scoring algorithm for additive logistic regression

  1. Initialize \hat{\beta}_0 = \text{logit}(\bar{y}) and set f_j(x_j) = 0, for j=1,\dots,p.

  2. Iterate until convergence:

    1. Define the quantities \hat{\eta}_i = \hat{\beta}_0 + \sum_{j=1}^p\hat{f}_j(x_{ij}) and \hat{\pi}_i = \{1 + \exp(-\hat{\eta}_i)\}^{-1}.

    2. Construct the working response z_i = \hat{\eta}_i + \frac{y_i - \hat{\pi}_i}{\hat{\pi}_i(1 - \hat{\pi}_i)}, \qquad i=1,\dots,n.

    3. Construct the weights w_i = \hat{\pi}_i(1 - \hat{\pi}_i), for i=1,\dots,n.

    4. Use a weighted backfitting algorithm using the z_i as responses, which produces a new set of estimates \hat{f}_1,\dots,\hat{f}_p.

GAM using penalized splines

  • A common special instance of GAM occurs when smoothing splines are employed. In the regression case, the backfitting algorithm implicitly minimizes the following penalized loss \mathscr{L}(f_1,\dots,f_p; \lambda) = \sum_{i=1}^n\left\{y_i - \beta_0 - \sum_{j=1}^pf_j(x_j)\right\}^2 + \sum_{j=1}^p\lambda_j \int_{a_j}^{b_j}\{f''_j(t)\}^2\mathrm{d}t, where \lambda = (\lambda_1,\dots,\lambda_p) is a vector of smoothing parameters.

  • Each f_j(x;\beta) is a natural cubic spline, therefore the penalized least squares criterion is \mathscr{L}(\beta; \lambda) = \sum_{i=1}^n\left\{y_i - \beta_0 - \sum_{j=1}^pf_j(x_j; \beta_j)\right\}^2 + \sum_{j=1}^p\lambda_j \beta_j^T\bm{\Omega}_j\beta_j, whose joint minimization over \beta is available in closed form.

  • Hence, a direct algorithm that minimizes \mathscr{L}(\beta; \lambda) is used instead of backfitting.

On the choice of smoothing parameters

  • In GAMs there are p smoothing parameters \lambda_1,\dots,\lambda_p that must be selected. We can proceed in the usual way, e.g. considering the generalized cross-validation criteria: \text{GCV}(\lambda_1,\dots,\lambda_p) = \frac{1}{n}\sum_{i=1}^n \left(\frac{y_i - \hat{y}_i}{1 - \text{df}/n}\right)^2.

  • An alternative criterion in this context is the REML (Restricted Maximum Likelihood), which is the marginal likelihood of the corresponding Bayesian model.

  • It is not possible to construct a grid of values for all the combinations of smoothing parameters \lambda_1,\dots,\lambda_p, because the number of terms increases exponentially in p.

  • Hence, many software packages numerically optimize the \text{GCV}(\lambda_1,\dots,\lambda_p), or other information criteria, as a function of \lambda_1,\dots,\lambda_p, using e.g. the Newton-Raphson method.

  • Such an approach is particularly convenient in combination with smoothing splines, because the derivatives needed for Newton’s method are available in closed form.

GAM and variable selection

  • When p is large there is need to remove the potentially irrelevant variables. There exist several variable selection ideas for GAMs, but we will not cover the details here.

  • Option 1. Stepwise regression. Perhaps the simplest method, although it is not as efficient as in linear models because we cannot exploit the same computational tricks.

  • Option 2. COSSO: Component Selection and Smoothing Operator (Lin and Zhang, 2006). It’s an idea based on combining lasso-type penalties and GAMs.

  • Option 3. SpAM: Sparse Additive Models (Ravikumar, Liu, Lafferty and Wasserman, 2009). Similar to the above, but it exploits a variation of the non-negative garrote.

  • Option 4. Double-penalty and shrinkage (Marra and Wood, 2011). It acts on the penalty term of smoothing splines so that high-values of \lambda_1,\dots,\lambda_p leads to constant functions.

  • Option X. Fancy name. Yet another method for variable selection with GAMs.

GAM modeling of trawl data

  • Let us get back to the trawl data. A specification based on GAM could be \begin{aligned} \texttt{Score}_i = \beta_0 &+ f_1(\texttt{Longitude}_i)+ f_2(\texttt{Latitude}_i) + f_3(\texttt{Depth}_i) +\\ &+ \beta_1 I(\texttt{Zone}_i = \texttt{Closed}) + \beta_2 I(\texttt{Year}_i = \texttt{1993}). \end{aligned}

  • In GAMs the predictors are not necessarily modeled using nonparametric methods. Indeed, it is common to have a combination of smooth functions and linear terms.

  • Besides, it does not make sense to “smooth” a dummy variable.

term estimate std.error df
(Intercept) 0.849 0.088 1
Zone_Closed -0.075 0.099 1
Year_1993 0.149 0.093 1
s(Longitude) - - 4.694
s(Latitude) - - 1
s(Depth) - - 2.447

Partial effect of GAMs (Longitude)

Warning: The `smooth` argument of `smooth_estimates()` is deprecated as of gratia
0.8.9.9.
ℹ Please use the `select` argument instead.

Partial effect of GAMs (Latitude)

Partial effect of GAMs (Depth)

Comments and criticism (trawl data)

  • The fitted GAM model highlights some interesting aspects of the trawl data.

  • In the first place, it seems confirmed that the Longitude has a marked non-linear impact on the catch score, as the initial analysis was suggesting.

  • In particular, the catch score is high when the sampling location is close to the coast (but not too close!), and then it suddenly decreases.

  • The effective degrees of freedom of Latitude is \text{df}_2 = 1, meaning that the estimated \hat{f}_2 collapsed to a linear term. The corresponding shrinkage parameter \lambda_2 is very high.

  • Overall, the effect due to the Latitude looks small or not present at all.

  • The Depth seems to have a relevant effect on the Score, but this is likely due to a few leverage points at the right extreme of the Depth range.
  • Finally, we note that both Zone and Year seem to have a minor effect.

☠️ - Naïve Bayes classifier and GAMs

  • The naïve Bayes classifier expresses the binary classification probability \text{pr}(y = 1 \mid \bm{x}) as \text{pr}(y = 1 \mid \bm{x}) = \frac{\pi_1 \prod_{j=1}^p p_{j1}(x_j)}{\pi_0\prod_{j=1}^p p_{j0}(x_j) + \pi_1\prod_{j=1}^p p_{j1}(x_j)} = \frac{\pi_1\prod_{j=1}^p p_{j1}(x_j)}{p(\bm{x})}.

  • Hence, using class 0 as a baseline, we can derive the following expression: \log\frac{\text{pr}(y = 1 \mid \bm{x})}{\text{pr}(y = 0 \mid \bm{x})} = \log\frac{\pi_1\prod_{j=1}^p p_{j1}(x_j)}{\pi_0\prod_{j=1}^p p_{j0}(x_j)} = \log\frac{\pi_1}{\pi_0} + \sum_{j=1}^p\log\frac{p_{j1}(x_j)}{p_{j0}(x_j)} = \beta_0 + \sum_{j=1}^pf_j(x_j).

  • Therefore, although naïve Bayes and GAMs are fitted in a quite different way, there is a tight connection among the two methods.

  • Naïve Bayes has a generalized additive model structure. This also suggests that the “additive assumption” is linked to the notion of independence among the covariates.

☠️ - The mgcv R package

  • GAMs were invented by Hastie and Tibshirani in 1986, including the backfitting algorithm.

  • Simon Wood (2003) described thin-plate regression splines and their estimation (no backfitting).

  • Simon Wood (2004, 2011) invented methods for estimating \lambda_1,\dots,\lambda_p in an efficient and stable manner.

  • Marra and Wood (2011) discussed many methods for practical variable selection for GAMs.

  • For further details, there is a recent and advanced book by Simon Wood (2017) entitled “Generalized Additive Models: An Introduction with R”.

  • The mgcv package in R (by Simon Wood) implements everything mentioned here.

Pros and cons of generalized additive models (GAMs)

Pros

  • GAMs can automatically model non-linear relationships. This can potentially make more accurate predictions for the response.

  • GAMs, as linear models, are interpretable: the variation of the fitted response, holding all but one predictor fixed, does not depend on the values of the other predictors.

  • In practice, this means that we can plot the fitted functions \hat{f}_j separately to examine the roles of the predictors in modelling the response.

  • Additive assumption is quite strong, but it is still possible to manually add interactions as in the linear regression case.

Cons
  • Especially when p is large, it is almost impossible to manually model all the interactions among covariates. GAMs do not take second-order effects (or higher) into account.

MARS

Multivariate Adaptive Regression Splines

  • MARS are a generalization of GAMs that avoid the additivity assumption.

  • MARS allow modeling of non–linear interactions and not just non–linear marginal effects.

  • MARS are at the same time:

    • A generalization of stepwise regression;
    • A method based on multi-dimensional tensor splines;
    • A modification of classification and regression trees (CART).

  • MARS combine many of the techniques we have seen in this course into a single sophisticated algorithm.

MARS additive representation

  • MARS is an additive model of the form: f(\bm{x}; \beta) = \beta_0 + \sum_{m=1}^M \beta_m h_m(\bm{x}), where h_m(\bm{x}) are basis functions and \beta = (\beta_1,\dots,\beta_M)^T are regression coefficients.

  • Once the basis functions are specified, the estimate for \hat{\beta} is straightforward, using for example least squares or the IWLS algorithm in the classification case.

  • The main distinction with GAMs is that in MARS the basis functions are estimated from the data and therefore they are not pre-specified in advance.

  • MARS is essentially a smart heuristic algorithm for selecting a collection of basis functions h_1(\bm{x}),\dots, h_M(\bm{x}) that hopefully does not incur in the curse of dimensionality.

Basis functions for MARS (reflected pairs)

  • The MARS algorithm begins by including just the intercept term, i.e. f(\bm{x}; \beta) = \beta_0. Then, we proceed by iteratively adding basis functions.
  • In MARS the basis functions are always coupled (or reflected), meaning that we always add them in pairs to the additive specification.

  • Let us consider the following set of pairs of basis functions (linear splines): \mathcal{C} = \{(x_j - \xi)_+, (\xi - x_j)_+ : \xi \in \{x_{1j},\dots,x_{nj}\}, \ j=1,\dots,p \}. For example, two basis functions could be h_1(\bm{x}) = (x_1 - 0.5)_+ and h_2(\bm{x}) = (0.5 - x_1)_+.

  • The knots are placed in correspondence of the observed data. Hence, there are in total 2 n p possible basis functions among which we can choose.

  • In the first step of the MARS algorithm, we identify the pair h_1(\bm{x}) = (x_j - \xi)_+ and h_2(\bm{x}) = (\xi - x_j)_+ that, together with the intercept, minimize the loss function.

An example of reflected pair basis

  • The function h_1(x) = (x - 0.5)_+ (blue) and its reflection h_2(x) = (0.5 - x)_+ (orange).

A stepwise construction

  • Hence, after the first step of the MARS algorithm, our model for example could be f(\bm{x}; \beta) = \beta_0 + \sum_{m=1}^2\beta_m h_m(\bm{x}) = \beta_0 + \beta_1 (x_1 - 0.5)_+ + \beta_2(0.5 - x_1)_+.

  • In the subsequent step, we consider a new pair of basis functions (x_j - \xi)_+, (\xi - x_j)_+ in \mathcal{C}, but this time we are allowed to perform two kind of operations:

    1. We can include the new pair to the predictor in an additive way, obtaining for example f(\bm{x}; \beta) = \beta_0 + \beta_1(x_1 - 0.5)_+ + \beta_2(0.5 - x_1)_+ + \beta_3\textcolor{red}{ (x_2 - 0.75)_+} + \beta_4\textcolor{red}{(0.75 - x_2)_+}.
    2. We can include the new pair in a multiplicative way, by considering the products between the new basis and one of the old bases of the model, obtaining for instance \begin{aligned} f(\bm{x}; \beta) = \beta_0 &+ \beta_1\textcolor{darkblue}{(x_1 - 0.5)_+} + \beta_2(0.5 - x_1)_+ \\ &+ \beta_3 \textcolor{red}{(x_1 - 0.5)_+(x_2 - 0.75)_+} + \beta_4 \textcolor{red}{(x_1 - 0.5)_+(0.75 - x_2)_+}. \end{aligned}

An example of tensor product basis

  • The product function h(\bm{x}) = (x_1 - 0.5)_+ (x_2 - 0.75)_+ in the range (0, 1)^2.

MARS algorithm (degree d)

  1. Initialize f(\bm{x}; \beta) = \beta_0 and let K be maximum number of pairs, so that M = 2K.

  2. Identify the initial pair of basis functions h_1 and h_2 in \mathcal{C} that minimize the loss function. Then, let h_0(\bm{x}) = 1 and set \mathcal{M}_1 = \{h_0, h_1, h_2\}.

  3. For k = 1,\dots,K - 1, do:

    1. Let \mathcal{M}_k = \{h_0, h_1,\dots,h_{2k}\} be the basis functions already present in the model.

    2. Consider a novel pair of bases \tilde{h}_1,\tilde{h}_2 \in \mathcal{C} \setminus \mathcal{M}_k. A candidate pair is obtained by multiplying \tilde{h}_1,\tilde{h}_2 with one of the bases in \mathcal{M}_k. Note that h_0 \in \mathcal{M}_k.

    3. A valid candidate basis does not contain the same variable x_j more than once in the product, and it must involve at most d product terms.

    4. Identify the optimal pair among the candidates at steps (ii)-(iii) that reduces the loss function the most. This results in a new pair of bases h_{2k+1}, h_{2k+2}.

    5. Set \mathcal{M}_{k+1} \leftarrow \mathcal{M}_k \cup \{h_{2k+1}, h_{2k+2}\}.

  4. Return the collection of models \mathcal{M}_1,\dots,\mathcal{M}_K.

Basis selection and backward regression

  • The degree d of the MARS algorithms allow to control the number order of interactions of the model. Note that when d = 1 it corresponds to a GAM (no interactions).

  • The final model with M terms is very likely overfitting the data. Hence, it is important to remove some of the bases using backward regression or best subset.

  • The optimal reduced model can be selected via cross-validation, but generalized cross-validation is often preferred due to computational reasons.

  • Unfortunately, it is not clear how to compute the effective degrees of freedom that are needed in the \text{GCV} formula.

  • In MARS, however, we do not have any miraculous simple formula like in LAR or ridge.

  • Simulation studies suggest that, for every knot placed, we should pay a price of about 3 degrees of freedom. However, this result is quite heuristic.

Heuristics behind MARS

  • The basis functions used in MARS have the advantage of operating locally.

  • When the basis functions in \mathcal{C} are multiplied together, the result is nonzero only over the small part of the feature space where both component functions are nonzero.

  • Hence, the estimated function is built up parsimoniously, by making small local modifications to the fit obtained at the previous step.

  • This is important, since one should “spend” degrees of freedom carefully in high dimensions, to avoid incurring into the curse of dimensionality.

  • The constructional logic of the model is hierarchical. We can multiply new basis functions that involve new variables only to the basis functions already in the model.

  • Hence, an interaction of a higher order can only be introduced when interactions of a lower order are present.

  • This constraint, introduced for computational reasons, does not necessarily reflect the real behavior of the data, but it often helps in interpreting the results.

MARS modeling of trawl data (d = 1)

  • We fit a MARS model with d = 1 (no interactions), using M = 20. Then, the model was simplified using best subset selection and \text{GCV}. The results are:
Term Basis function Coefficient
h_0(\bm{x}) 1 1.382
h_1(\bm{x}) (\texttt{Longitude} - 143.28)_+ -4.275
h_2(\bm{x}) (\texttt{Longitude} - 143.58)_+ 3.984
  • To clarify, this specification corresponds to the following estimated regression function: f(\bm{x}_i; \hat{\beta}) = 1.382 - 4.275(\texttt{Longitude}_i - 143.28)_+ + 3.984 (\texttt{Longitude}_i - 143.58)_+, which has the structure of a GAM. However, the estimation procedure is different.

  • The estimated function f(\bm{x}_i; \hat{\beta}) is remarkably simple and it involves only the Longitude. Moreover, the relationship between Score and Longitude is non-linear.

  • Both these considerations are consistent with the previous findings, obtained using GAMs.

MARS modeling of trawl data (d = 2)

  • We fit a MARS model with d = 2 (first order interactions), using M = 20. As before, the model was simplified using best subset selection and \text{GCV}. The results are:
Term Basis function Coefficient
h_0(\bm{x}) 1 1.318
h_1(\bm{x}) (\texttt{Longitude} - 143.28)_+ -5.388
h_2(\bm{x}) (\texttt{Longitude} - 143.58)_+ 4.172
h_3(\bm{x}) I(\texttt{Year} = \texttt{1993})(\texttt{Longitude} - 143.05)_+ 0.679
h_4(\bm{x}) [\texttt{Latitude} - (-11.72)]_+ (\texttt{Longitude} - 143.05)_+ 1.489

  • As expected, a degree 2 MARS lead to a more sophisticated fit involving interactions between Year and Longitude as well as between Latitude and Longitude.

  • We can explore these effects using partial plots.

Partial effects

Results on the test set

  • We can now check the predictive performance on the test set, to see which is model scored the best in terms of MAE and RMSE.
Null model Linear model GAM MARS (degree 1) MARS (degree 2)
MAE 0.611 0.361 0.315 0.305 0.334
RMSE 0.718 0.463 0.408 0.390 0.407
  • It is quite clear that including a non-linear specification for the Longitude was indeed a good idea that improved the predictive performance.

  • Somewhat surprisingly, the best model is the extremely simple (yet effective) MARS of degree 1.

  • The weak interactions effects captured by the MARS of degree 2 led an increased variance of the estimates, slightly deteriorating the fit.

Pros and cons of MARS

Pros
  • MARS constructs a sophisticated model by sequentially refining the previous fit.
  • MARS can capture interaction effects.
  • The MARS algorithm can be employed for both regression and classification problems.
  • The final model is often interpretable. The order of the interactions can be controlled by fixing d.
Cons

  • MARS can quickly overfit the data if the model complexity is not kept under control.

  • Unfortunately, this is not always easy to do as there is limited theoretical support that guides us in the choice of the effective degree of freedom.

References

References I

Specialized references I

  • Simon Wood’s contributions
    • Wood, S. N. (2003). Thin plate regression splines. Journal of the Royal Statistical Society. Series B: Statistical Methodology 65, 95-114.
    • Wood, S. N. (2004). Stable and efficient multiple smoothing parameter estimation for generalized additive models. Journal of the American Statistical Association 99, 673-86.
    • Wood, S. N. (2011). Fast stable restricted maximum likelihood and marginal likelihood estimation of semiparametric generalized linear models. Journal of the Royal Statistical Society. Series B: Statistical Methodology 73(1), 3-36.
    • Marra, G. and Wood, S. N. (2011). Practical variable selection for generalized additive models. Computational Statistics and Data Analysis 55(7), 2372–87.
    • Wood, S. N. (2017). Generalized Additive Models: An Introduction with R. CRC Press.