adolgert/glissa.jl

Spline functions for unequal coordinates in pure Julia

• Stochastic processes for epidemiology
• Computational demography
• Distributed, parallel, high-performance computing
• Provenance systems

glissa.jl's Issues

shape constraints on P-splines

Wood and Pya (https://link.springer.com/article/10.1007/s11222-013-9448-7) present a method to set up constraints on P-splines (monotonic, convex/concave, etc). These kinds of constraints, along with max/min constraints would be very helpful for using these to fill in unknown functions in mechanistic models! For example, mosquito feeding rate should increase with increasing host density, but cannot exceed some maximum rate given by digestion/handling time.

penalized splines?

hi @adolgert!

How hard would it be to introduce penalty matrices (penalizing, for example second derivatives) to these splines? Basically, if $X$ is the model matrix of the spline, $y$ the data, $\beta$ the spline coefficients, $\lambda$ a scalar parameter controlling the amount of smoothing, and $S$ a penalty matrix, we'd want to solve a penalized least squares problem of the form:

$$ \lVert y - X\beta \rVert^{2} + \lambda \beta^{\top} S \beta $$

I have an example here (https://gist.github.com/slwu89/50f81e6bbc296261e29851d3585ae554) of using JuMP to solve this kind of penalized least squares problem using a penalized cubic spline where the $X$ and $S$ matrices are produced by the "mgcv" R package, but it would be nice for the whole thing to be in Julia.

possible API

In mgcv, the API is set up around the idea of generic "smooths" that are incorporated into a model (including P, cubic, thin plate, and other flavors of splines).

If we were to take inspiration from this, the main interface functions for users are:

• https://rdrr.io/cran/mgcv/man/smooth.construct.html to set up the basis and penalty matrices
• https://rdrr.io/cran/mgcv/man/Predict.matrix.html to produce a model matrix $X$ (where $\mu = X \cdot \beta$).

Facilities for imposing shape constraints in mgcv are more limited, and seem to only exist for cubic splines (see https://rdrr.io/cran/mgcv/man/mono.con.html).