Solved – Adaptive GAM smooths in mgcv

generalized-additive-modelmgcvrsmoothingsplines

Simon Wood's book on GAMs and his associated R package mgcv are both highly detailed and informative when it comes to GAM theory and model-fitting to real and simulated data.

For 1D smooths, there is really not much to worry about, save for deciding whether to implement cyclic vs. adaptive basis functions, which can give very different prediction results compared to cubic, thin-plate and P-spline smooths, because, in the adaptive case, multiple GAMs are fitted to different regions along a spline. As far as I can tell, cyclic bases are common in time-series modelling, whereas adaptive smooths should be considered when data vary a lot with respect to the response variable; however, adaptive smooths should be used "sparingly and with care".

I have been investigating GAMs for some time now, and given my research question, I am finding myself changing my mind a lot when it comes to which smooth to implement. mgcv includes 17 different smooths to choose from (by my count). I have considered both cubic and P-spline smooths.

My question now is: when should adaptive smooths be considered over non-adaptive counterparts, if the eventual goal is to use fitted GAMs for prediction purposes? For my purposes, I am sticking with the default GCV smoothness criterion, even though it has the tendency to under-smooth.

The literature is growing in applied ecological GAMs, but I have yet to come across a study that implements an adaptive smooth.

Any advice is appreciated.

Best Answer

Most of the extra smooths in the mgcv toolbox are really there for specialist applications — you can largely ignore them for general GAMs, especially univariate smooths (you don't need a random effect spline, a spline on the sphere, a Markov random field, or a soap-film smoother if you have univariate data for example.)

If you can bear the setup cost, use thin-plate regression splines (TPRS).

These splines are optimal in an asymptotic MSE sense, but require one basis function per observation. What Simon does in mgcv is generate a low-rank version of the standard TPRS by taking the full TPRS basis and subjecting it to an eigendecomposition. This creates a new basis where the first k basis function in the new space retain most of the signal in the original basis, but in many fewer basis functions. This is how mgcv manages to get a TPRS that uses only a specified number of basis functions rather than one per observation. This eigendecomposition preserves much of the optimality of the classic TPRS basis, but at considerable computational effort for large data sets.

If you can't bear the setup cost of TPRS, use cubic regression splines (CRS)

This is a quick basis to generate and hence is suited to problems with a lot of data. It is knot-based however, so to some extent the user now needs to choose where those knots should be placed. For most problems there is little to be gained by going beyond the default knot placement (at the boundary of the data and spaced evenly in between), but if you have particularly uneven sampling over the range of the covariate, you may choose to place knots evenly spaced sample quantiles of the covariate, for example.

Every other smooth in mgcv is special, being used where you want isotropic smooths or two or more covariates, or are for spatial smoothing, or which implement shrinkage, or random effects and random splines, or where covariates are cyclic, or the wiggliness varies over the range of a covariate. You only need to venture this far into the smooth toolbox if you have a problem that requires special handling.

Shrinkage

There are shrinkage versions of both the TPRS and CRS in mgcv. These implement a spline where the perfectly smooth part of the basis is also subject to the smoothness penalty. This allows the smoothness selection process to shrink a smooth back beyond even a linear function essentially to zero. This allows the smoothness penalty to also perform feature selection.

Duchon splines, P splines and B splines

These splines are available for specialist applications where you need to specify the basis order and the penalty order separately. Duchon splines generalise the TPRS. I get the impression that P splines were added to mgcv to allow comparison with other penalized likelihood-based approaches, and because they are splines used by Eilers & Marx in their 1996 paper which spurred a lot of the subsequent work in GAMs. The P splines are also useful as a base for other splines, like splines with shape constraints, and adaptive splines.

B splines, as implemented in mgcv allow for a great deal of flexibility in setting up the penalty and knots for the splines, which can allow for some extrapolation beyond the range of the observed data.

Cyclic splines

If the range of values for a covariate can be thought of as on a circle where the end points of the range should actually be equivalent (month or day of year, angle of movement, aspect, wind direction), this constraint can be imposed on the basis. If you have covariates like this, then it makes sense to impose this constraint.

Adaptive smoothers

Rather than fit a separate GAM in sections of the covariate, adaptive splines use a weighted penalty matrix, where the weights are allowed to vary smoothly over the range of the covariate. For the TPRS and CRS splines, for example, they assume the same degree of smoothness across the range of the covariate. If you have a relationship where this is not the case, then you can end up using more degrees of freedom than expected to allow for the spline to adapt to the wiggly and non-wiggly parts. A classic example in the smoothing literature is the

library('ggplot2')
theme_set(theme_bw())
library('mgcv')
data(mcycle, package = 'MASS')
pdata <- with(mcycle,
              data.frame(times = seq(min(times), max(times), length = 500)))

ggplot(mcycle, aes(x = times, y = accel)) + geom_point()

These data clearly exhibit periods of different smoothness - effectively zero for the first part of the series, lots during the impact, reducing thereafter.

if we fit a standard GAM to these data,

m1 <- gam(accel ~ s(times, k = 20), data = mcycle, method = 'REML')

we get a reasonable fit but there is some extra wiggliness at the beginning and end the range of times and the fit used ~14 degrees of freedom

plot(m1, scheme = 1, residuals = TRUE, pch= 16)

To accommodate the varying wiggliness, an adaptive spline uses a weighted penalty matrix with the weights varying smoothly with the covariate. Here I refit the original model with the same basis dimension (k = 20) but now we have 5 smoothness parameters (default is m = 5) instead of the original's 1.

m2 <- gam(accel ~ s(times, k = 20, bs = 'ad'), data = mcycle, method = 'REML')

Notice that this model uses far fewer degrees of freedom (~8) and the fitted smooth is much less wiggly at the ends, whilst still being able to adequately fit the large changes in head acceleration during the impact.

What's actually going on here is that the spline has a basis for smooth and a basis for the penalty (to allow the weights to vary smoothly with the covariate). By default both of these are P splines, but you can also use the CRS basis types too (bs can only be one of 'ps', 'cr', 'cc', 'cs'.)

As illustrated here, the choice of whether to go adaptive or not really depends on the problem; if you have a relationship for which you assume the functional form is smooth, but the degree of smoothness varies over the range of the covariate in the relationship then an adaptive spline can make sense. If your series had periods of rapid change and periods of low or more gradual change, that could indicate that an adaptive smooth may be needed.

Related Solutions

Solved – Centering constraints in MGCV GAM

What that means is that each smooth created using a by term will be centred at zero. In effect this means that the data for each level of by has the same mean (i.e. the intercept or constant term in the model). What Simon Wood means is that if the data have different mean values (Site A has higher values than Site B say and by = Site) then because of the centring constraints this difference in the means of the different groups will not be taken into account in the model.

Using an example from ?gam.model I will demonstrate the difference:

## Factor `by' variable example (with a spurious covariate x0)
## simulate data...

dat <- gamSim(4)

## fit model...
b <- gam(y ~ fac + s(x2, by=fac) + s(x0), data=dat)
bb <- gam(y ~ s(x2, by=fac) + s(x0), data=dat) ## without fac
summary(b)
summary(bb)

Note how in model b we include the fac variable as a separate parametric term in the model formula as well as in the s( ) part. The intercept in this model will then represent the mean for the first group (level) of fac and the coefficients for fac in the model (and in the summary output) will represent the difference in means from the first group of the other groups.

Look closely at the output from summary(), notice how the deviance explained by b is much higher than that of bb (without the parametric fac term), and consequently b has a lower GCV score (lower is better) and a lower scale parameter. The higher scale parameter in bb is due to there being a large amount of unexplained variance (the between group effect of fac) left over in the simpler model.

R> summary(b)
....
R-sq.(adj) =  0.644   Deviance explained = 65.6%
GCV score = 3.9013  Scale est. = 3.7517    n = 400

R> summary(bb)
....
R-sq.(adj) =  0.451   Deviance explained = 47.3%
GCV score = 6.0322  Scale est. = 5.7758    n = 400

Solved – Comparing GAM smooths in mgcv R package using AIC

It's not flawed, per se, just pointless. I can think of (and have experienced) situations where the specific type of basis (of the two you mention) can result in markedly different fits. However, where this has happened to me, it has usually been solved by increasing k for one of the bases or because the wrong model was fitted. In this instances trivial differences in the basis used were magnified by the real problem (needing larger basis dimension, fitting the right model), not because of any fundamental difference in the performance of the individual basis.

In most situations you are going to see trivial differences in the fits of models fitted with different basis (among standard bases) and these are going to result in trivial differences in AIC. In most cases AIC is going to tell you, therefore, that the model fits are equivalent.

If you are planning on using SCAM models, I might suggest you use P splines in the GAMs as the splines in the scam package are all based on P splines.