Multivariate adaptive regression spline
In statistics, multivariate adaptive regression splines is a form of regression analysis introduced by Jerome H. Friedman in 1991. It is a non-parametric regression technique and can be seen as an extension of linear models that automatically models nonlinearities and interactions between variables.
The term "MARS" is trademarked and licensed to Salford Systems. In order to avoid trademark infringements, many open-source implementations of MARS are called "Earth".
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: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 for the above data is
The hat on the indicates that is estimated from the data. The figure on the right shows a plot of this function:
a line giving the predicted 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. 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
The figure on the right shows a plot of this function: the predicted 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 . Hinge functions are described in more detail below.
In this simple example, we can easily see from the plot that y has a non-linear relationship with 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
This expression models air pollution as a function of the temperature and a few other variables. Note that the last term in the formula incorporates an interaction between and.
The figure on the right plots the predicted as and 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.
To obtain the above expression, the MARS model building procedure automatically selects which variables to use, the positions of the kinks in the hinge functions, and how the hinge functions are combined.
The MARS model
MARS builds models of the formThe model is a weighted sum of basis functions
Each 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 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 or . 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 functions can model interaction between two or more variables.
An example is the last line of the ozone formula.
Hinge functions
Hinge functions are a key part of MARS models. A hinge function takes the formor
where 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
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 ramp, hockey stick, or rectifier functions. Instead of the notation used in this article, hinge functions are often represented by where means take the positive part.
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.
The forward pass
MARS starts with a model which consists of just the intercept term.
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. 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 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
2) all variables
3) all values of each variable.
To calculate the coefficient of each term MARS applies a linear regression over the terms.
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.
The backward pass
The forward pass usually builds an overfit model. 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 final model. A paired hinge can be seen in the equation for in the first MARS example above; there are no complete pairs retained in the ozone example.
Generalized cross validation
The backward pass uses generalized cross validation to compare the performance of model subsets in order to choose the best subset: lower values of GCV are better. The GCV is a form of regularization: it trades off goodness-of-fit against model complexity.The formula for the GCV is
where RSS is the residual sum-of-squares measured on the training data and N is the number of observations.
The EffectiveNumberOfParameters is defined in
the MARS context as
where penalty is about 2 or 3.
Note that
is the number of hinge-function knots, so the formula penalizes the addition of knots. Thus the GCV formula adjusts the training RSS to take into account the flexibility of the model. We penalize flexibility because models that are too flexible will model the specific realization of noise in the data instead of just the systematic structure of the data.
Generalized cross-validation is so named because it uses a formula to approximate the error that would be determined by leave-one-out validation. It is just an approximation but works well in practice. GCVs were introduced by Craven and Wahba and extended by Friedman for MARS.
Constraints
One constraint has already been mentioned: the usercan 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 ;
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.
Pros and cons
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.
.
- MARS models are more flexible than linear regression models.
- 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 continuous and categorical data. 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 does automatic variable selection. However, there can be some arbitrariness in the selection, especially when there are correlated predictors, and this can affect interpretability
- MARS models tend to have a good bias-variance trade-off. The models are flexible enough to model non-linearity and variable interactions, yet the constrained form of MARS basis functions prevents too much flexibility.
- MARS is suitable for handling fairly large datasets. It is a routine matter to build a MARS model from an input 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. 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 partitioning 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. 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.
- The
earth,mda, andpolsplineimplementations do not allow missing values in predictors, but free implementations of regression trees do allow missing values using a technique called surrogate splits. - MARS models can make predictions quickly. The prediction function simply has to evaluate the MARS model formula. Compare that to making a prediction with say a Support Vector Machine, where every variable has to be multiplied by the corresponding element of every support vector. That can be a slow process if there are many variables and many support vectors.
- The resulting fitted function is not smooth.
Extensions and related concepts
- Generalized linear models 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.
- Recursive partitioning. MARS can be seen as a generalization of recursive partitioning that allows the model to better handle numerical data.
- Generalized additive models. From the user's perspective GAMs are similar to MARS but fit smooth loess or polynomial splines instead of MARS basis functions, and do not automatically model variable interactions. The fitting method used internally by GAMs is very different from that of MARS. For models that do not require automatic discovery of variable interactions GAMs often compete favorably with MARS.
- TSMARS. Time Series Mars is the term used when MARS models are applied in a time series context. Typically in this set up the predictors are the lagged time series values resulting in autoregressive spline models. These models and extensions to include moving average spline models are described in "Univariate Time Series Modelling and Forecasting using TSMARS: A study of threshold time series autoregressive, seasonal and moving average models using TSMARS".