Multivariate adaptive regression splines

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Multivariate adaptive regression splines (MARS) is a form of regression analysis introduced by Jerome Friedman in 1991 [1]. It is a non-parametric regression technique and can be seen as an extension of linear models that automatically models non-linearities and interactions.

The term "MARS" is trademarked and licensed to Salford Systems.

Contents

[edit] The basics

This section introduces MARS using a few examples. We start with a set of data: a matrix of input variables x, and a vector of the observed responses y, with a response for each row in x. For example, the data could be:

x y
10.5 16.4
10.7 18.8
10.8 19.7
... ...
20.6 77.0

Here there is only one independent variable, so the x matrix is just a single column. Given these measurements, we would like to build a model which predicts the expected y for a given x.

A linear model
A linear model

A linear model for the above data is

\hat{y} = -37 + 5.1 x

The hat on the \hat{y} indicates that \hat{y} is estimated from the data. The figure on the right shows a plot of this function: a line giving the predicted \hat{y} versus x, with the original values of y shown as red dots.

The data at the extremes of x indicates that the relationship between y and x may be non-linear (look at the red dots relative to the regression line at low and high values of x). We thus turn to MARS to automatically build a model taking into account non-linearities. MARS software constructs a model from the given x and y as follows

\begin{align} \hat{y} = &\ 25 \\ & + 6.1 \max(0, x - 13) \\ & - 3.1 \max(0, 13 - x) \\ \end{align}
A simple MARS model of the same data
A simple MARS model of the same data

The figure on the right shows a plot of this function: the predicted \hat{y} versus x, with the original values of y once again shown as red dots. The predicted response is now a better fit to the original y values.

MARS has automatically produced a kink in the predicted y to take into account non-linearity. The kink is produced by hinge functions. The hinge functions are the expressions starting with max (where max(a,b) is a if a > b, else b). Hinge functions are described in more detail below.

In this simple example, we can easily see from the plot that the y has a non-linear relationship with x (and might perhaps guess that y varies with the square of x). However, in general there will be multiple independent variables, and the relationship between y and these variables will be unclear and not easily visible by plotting. We can use MARS to discover that non-linear relationship.

An example MARS expression with multiple variables is

\begin{align} ozone = &\ 5.2 \\ & + 0.93 \max(0, temp - 58) \\ & - 0.64 \max(0, temp - 68) \\ & - 0.046 \max(0, 234 - ibt) \\ & - 0.016 \max(0, wind - 7) \max(0, 200 - vis)\\ \end{align}
Variable interaction in a MARS model
Variable interaction in a MARS model

This expression models air pollution (the ozone level) as a function of the temperature and a few other variables. Note that the last term in the formula (on the last line) incorporates an interaction between wind and vis.

The figure on the right plots the predicted ozone as wind and vis vary, with the other variables fixed at their median values. The figure shows that wind does not affect the ozone level unless visibility is low. We see that MARS can build quite flexible regression surfaces by combining hinge functions.

[edit] The MARS model

MARS builds models of the form

\hat{f}(x) = \sum_{i=1}^{k} c_i B_i(x) .

The model is a weighted sum of basis functions Bi(x). Each ci is a constant coefficient. For example, each line in the formula for ozone above is one basis function multiplied by its coefficient.

Each basis function Bi(x) takes one of the following three forms:

1) a constant 1. There is just one such term, the intercept. In the ozone formula above, the intercept term is 5.2.

2) a hinge function. A hinge function has the form max(0,x − const) or max(0,const − x). MARS automatically selects variables and values of those variables for knots of the hinge functions. Examples of such basis functions can be seen in the middle three lines of the ozone formula.

3) a product of two or more hinge functions. These basis function can model interaction between two or more variables. An example is the last line of the ozone formula.

[edit] Hinge functions

A mirrored pair of hinge functions with a knot at x=3.1
A mirrored pair of hinge functions with a knot at x=3.1

Hinge functions are a key part of MARS models. A hinge function takes the form

max(0,x − c)

or

max(0,c − x)

where c is a constant, called the knot. The figure on the right shows a mirrored pair of hinge functions with a knot at 3.1.

A hinge function is zero for part of its range, so can be used to partition the data into disjoint regions, each of which can be treated independently. Thus for example a mirrored pair of hinge functions in the expression

6.1max(0,x − 13) − 3.1max(0,13 − x)

creates the piecewise linear graph shown for the simple MARS model in the previous section.

One might assume that only piecewise linear functions can be formed from hinge functions, but hinge functions can be multiplied together to form non-linear functions.

Hinge functions are also called hockey stick functions. Instead of the max notation used in this article, hinge functions are often represented by [\pm(x_i - c)]_+ where [] + means take the positive part.

Alternatively, the hinge function may be defined as \int \int \delta(x-c)dx^2 where δ(x) is the dirac delta function defined as an impulse at the origin whose integral is 1.

[edit] The model building process

MARS builds a model in two phases: the forward and the backward pass. This two stage approach is the same as that used by recursive partitioning trees.

[edit] The forward pass

MARS starts with a model which consists of just the intercept term (which is the mean of the response values).

MARS then repeatedly adds basis function in pairs to the model. At each step it finds the pair of basis functions that gives the maximum reduction in sum-of-squares residual error (it is a greedy algorithm). The two basis functions in the pair are identical except that a different side of a mirrored hinge function is used for each function. Each new basis function consists of a term already in the model (which could perhaps be the intercept i.e. a constant 1) multiplied by a new hinge function. A hinge function is defined by a variable and a knot, so to add a new basis function, MARS must search over all combinations of the following:

1) existing terms (called parent terms in this context)

2) all variables (to select one for the new basis function)

3) all values of each variable (for the knot of the new hinge function).

This process of adding terms continues until the change in residual error is too small to continue or until the maximum number of terms is reached. The maximum number of terms is specified by the user before model building starts.

The search at each step is done in a brute force fashion, but a key aspect of MARS is that because of the nature of hinge functions the search can be done relatively quickly using a fast least-squares update technique. Actually, the search is not quite brute force. The search can be sped up with a heuristic that reduces the number of parent terms to consider at each step ("Fast MARS" [2]).

[edit] The backward pass

The forward pass usually builds an overfit model. (An overfit model has a good fit to the data used to build the model but will not generalize well to new data.) To build a model with better generalization ability, the backward pass prunes the model. It removes terms one by one, deleting the least effective term at each step until it finds the best submodel. Model subsets are compared using the GCV criterion described below. The backward pass has an advantage over the forward pass: at any step it can choose any term to delete, whereas the forward pass at each step can only see the next pair of terms.

The forward pass adds terms in pairs, but the backward pass typically discards one side of the pair and so terms are often not seen in pairs in the the final model. A paired hinge can be seen in the equation for \hat{y} in the first MARS example above; there are no complete pairs retained in the ozone example.

[edit] Generalized cross validation (GCV)

The backward pass needs to compare the performance of model subsets in order to choose the best subset. It uses the GCV to do this. The GCV is a form of regularization: it trades off goodness-of-fit against model complexity. The raw residual sum-of-squares (RSS) on the training data is inadequate for comparing models, because the RSS always increases as MARS terms are dropped. In other words, if the RSS were used to compare models, the backward pass would always choose the largest model.

The formula for the GCV is

GCV = RSS / (N * (1 - EffectiveNumberOfParameters / N)^2)

where RSS is the residual sum-of-squares measured on the training data and N is the number of observations (the number of rows in the x matrix).

The EffectiveNumberOfParameters is defined in the MARS context as

EffectiveNumberOfParameters = NumberOfMarsTerms + Penalty * (NumberOfMarsTerms - 1 ) / 2

where Penalty is about 2 or 3 (the MARS software allows the user to preset Penalty).

Note that (NumberOfMarsTerms - 1 ) / 2 is the number of hinge-function knots, so the formula penalizes the addition of knots. Thus the GCV formula adjusts (i.e. increases) the training RSS to take into account the flexibility of the model.

Generalized Cross Validation is so named because it uses a formula to approximate the error that would determined by leave-one-out validation. GCVs were introduced by Craven and Wahba and extended by Friedman for MARS.

[edit] Constraints

One constraint has already been mentioned: the user can specify the maximum number of terms in the forward pass.

A further constraint can be placed on the forward pass by specifying a maximum allowable degree of interaction. Typically only one or two degrees of interaction are allowed, but higher degrees can be used when the data warrants it. The maximum degree of interaction in the first MARS example above is one (i.e. no interactions or an additive model); in the ozone example it is two.

Other constraints on the forward pass are possible. For example, the user can specify that interactions are allowed only for certain input variables. Such constraints could make sense because of knowledge of the process that generated the data.

[edit] Pros and cons of MARS

No regression modeling technique is best for all situations. The guidelines below are intended to give an idea of the pros and cons of MARS, but there will be exceptions to the guidelines. It is useful to compare MARS to recursive partitioning and this is done below. (Recursive partitioning is also commonly called regression trees, decision trees, or CART; see the recursive partioning article for details).

  • MARS models are simple to understand and interpret. Compare the equation for ozone concentration above to, say, the innards of a trained neural network or a random forest.
  • MARS can handle both numeric and categorical data [3]. MARS tends to be better than recursive partitioning for numeric data because hinges are more appropriate for numeric variables than the piecewise constant segmentation used by recursive partitioning.
  • Building MARS models often requires little or no data preparation. The hinge functions automatically partition the input data, so the effect of outliers is contained. In this respect MARS is similar to recursive partitioning which also partitions the data into disjoint regions, although using a different method.
  • MARS (like recursive partioning) does automatic variable selection (meaning it includes important variables in the model and excludes unimportant ones). However, bear in mind that variable selection is not a clean problem and there is usually some arbitrariness in the selection, especially in the presence of collinearity and 'concurvity'.
  • MARS models tend to have a good bias-variance tradeoff. The models are flexible enough to model non-linearity and variable interactions (thus MARS models have fairly low bias), yet the constrained form of MARS basis functions prevents too much flexibility (thus MARS models have fairly low variance).
  • MARS is suitable for handling fairly large datasets. It is a routine matter to build a MARS model from an x matrix with, say, 100 predictors and 105 observations. Such a model can be built in about a minute on a 1 GHz machine, assuming the maximum degree of interaction of MARS terms is limited to one (i.e. additive terms only). A degree two model with the same data on the same 1 GHz machine takes longer --- about 12 minutes. Be aware that these times are highly data dependent. Recursive partioning is much faster than MARS.
  • With MARS models, as with any non-parametric regression, parameter confidence intervals and other checks on the model cannot be calculated directly (unlike linear regression models). Cross-validation and related techniques must be used for validating the model instead.
  • MARS models do not give as good fits as boosted trees, but can be built much more quickly and are more interpretable. (An 'interpretable' model is in a form that makes it clear what the effect of each predictor is.)


[edit] See also

Linear regression

Generalized linear models (GLMs) can be incorporated into MARS models by applying a link function after the MARS model is built. Thus, for example, MARS models can incorporate logistic regression to predict probabilities.

Non-linear regression is used when the underlying form of the function is known and regression is used only to estimate the parameters of that function. MARS, on the other hand, estimates the functions themselves, albeit with severe constraints on the nature of the functions. (These constraints are necessary because discovering a model from the data is an inverse problem and is not well-posed without constraints on the model.)

Recursive partioning (commonly called CART). MARS can be seen as a generalization of recursive partioning that allows the model to better handle numerical (i.e. non-categorical) data.

Generalized additive models. From the user's perspective GAMs are similar to MARS but (a) fit smooth loess or polynomial splines instead of MARS basis functions, and (b) do not automatically model variable interactions. The fitting method used internally by GAMs is very different from that of MARS.

Polynomial and rational function modeling

Spline interpolations

[edit] References

The primary references are the Friedman papers. The 1991 paper is considered a classic.

  1. ^ Friedman, J. H. (1991) Multivariate Adaptive Regression Splines (with discussion), Annals of Statistics (19)
  2. ^ Friedman, J. H. (1993) Fast MARS, Stanford University Department of Statistics, Technical Report 110
  3. ^ Friedman, J. H. (1993) Estimating Functions of Mixed Ordinal and Categorical Variables Using Adaptive Splines, New Directions in Statistical Data Analysis and Robustness (Morgenthaler, Ronchetti, Stahel, eds.), Birkhauser

[edit] Books

The following books have sections on MARS.

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