Voronograms

Voronograms are a primitive form of nonparametric regression intended to explore edge detection methods.  Given scattered data on the plane, the idea is to fit  piecewise constant functions  defined on the Voronoi tessellation.  I was motivated to revive an 2004 R package for this by some recent conversations about total variation smoothing with Ryan Tibshirani and one of his students.  The package is linked here. The idea was an off-shoot of work with Ivan Mizera on triograms which were intended to fit quantile surfaces using a roughness penalty that corresponded to total variation of the gradient of the fitted function  on the Delone triangulation of a sample of scattered points.  Triograms are piecewise linear with breaks in the gradient along the edges of the triangulation.  In contrast Voronograms are piecewise constant with breaks in the function itself across edges of the Voronoi tessellation.  Examples appear in the figure below which can be reproduced by the command demo(tseg) in the R package.  As usual there is a lambda parameter that controls the strength of the penalization.

Brezhnev Fone

My DIY project over the Covid Containment has been putting together this Brezhnev Fone:

 It is (almost) ready to be moved back to its new home, the desk of the Director of the School of Slavonic and East European Studies at UCL, who happens to be my spouse.  It can't (yet) get Leonid Ilyich on the line but it can talk with the Google assistant via a raspberry pi and Google voice kit lodged inside.  It provided an entertaining reintroduction to the wonderful world of simple electronics and a chance to learn a tiny bit of python.  Gory details are available  here.

Cauchy Priors Redux

 Victor Chernozhukov posed the following question on twitter a few days ago:  "Suppose that X~ N(0,1), and Y ~ Cauchy, X and Y are independent. What is   E[X | X+Y]?"

Thien An replied "isn't it the mean of the density proportional to exp(-x^2/2)/[1+(z-x)^2]?" and she included this nice plot

illustrating the behavior of E(X | X+Y).

This recalled some rather ancient suggestions by Jim Berger about the utility of Cauchy priors.Reformulating Victor's question slightly, suppose that X ~ N(t,1) and you have prior t ~ C, Thien's nice plot can be interpreted as showing the posterior mean of t as a function of X. For small values of |X| there is moderate shrinkage back to 0, and as |X| grows the posterior mean does too.  However, for large values of X, this tendency is reversed and eventually very large |X| values are ignored  entirely.  This might seem odd, why is the data being ignored?

I like to think about this in terms of the comedian Richard Pryor's famous question:  "Who are you going to believe, me, or your lying eyes*."  If the observed X is far from the center of the prior distribution, you decide that it is just an aberration that is totally consistent with your prior belief that t could be quite extreme since the prior has such heavy tails, but you stick with your prior belief that it is much more likely that t is near 0.

In contrast if  the prior were Gaussian, say, t ~ N(0,1),  then the posterior mean is determined by linear shrinkage, so E(t|X) is midway between 0 and X, and for large |X|, we end up with a posterior mean in a place where neither the prior or likelihood have any mass.

* In this case "me" should be interpreted as your prior, and X as what you see with your lying eyes.

Convexification and Potato Peeling

 You may have wondered, as I have, about the convexifying effect of peeling a potato, so it comes as a great relief that this is a well studied subject.  The opening paragraph of Goodman (1981) is priceless:

A swivel-type potato peeler, when used in the usual (and somewhat wasteful) way, produces a convex peeled potato, which is a subset of the original unpeeled and non-convex potato. This suggests, in an obvious way, the problem of maximizing the volume of a convex body contained in a given non-convex body. That such a largest convex subset exists follows at once from the Blaschke selection theorem (see Proposition 1 below). The problems then become: (1) What proportion of the original volume are we assured of saving? and (2) How do we determine the largest convex 'peeled potato' inside a given non-convex one ?

A swivel-type potato peeler, when used in the usual (and somewhat wasteful) way, produces a convex peeled potato, which is a subset of the original unpeeled and non-convex potato. This suggests, in an obvious way, the problem of maximizing the volume of a convex body contained in a given non-convex body. That such a largest convex subset exists follows at once from the Blaschke selection theorem (see Proposition 1 below). The problems then become: (1) What proportion of the original volume are we assured of saving? and (2) How do we determine the largest convex 'peeled potato' inside a given non-convex one ?

A swivel-type potato peeler, when used in the usual (and somewhat wasteful) way, produces a convex peeled potato, which is a subset of the original unpeeled and non-convex potato. This suggests, in an obvious way, the problem of maximizing the volume of a convex body contained in a given non-convex body. That such a largest convex subset exists follows at once from the Blaschke selection theorem (see Proposition 1 below). The problems then become: (1) What proportion of the original volume are we assured of saving? and (2) How do we determine the largest convex 'peeled potato' inside a given non-convex one ?A swivel-type potato peeler, when used in the usual (and somewhat wasteful) way, produces a convex peeled potato, which is a subset of the original unpeeled and non-convex potato. This suggests, in an obvious way, the problem of maximizing the volume of a convex body contained in a given non-convex body. That such a largest convex subset exists follows at once from the Blaschke selection theorem (see Proposition 1 below). The problems then become: (1) What proportion of the original volume are we assured of saving? and (2) How do we determine the largest convex 'peeled potato' inside a given non-convex one ?A swivel-type potato peeler, when used in the usual (and somewhat wasteful) way, produces a convex peeled potato, which is a subset of the original unpeeled and non-convex potato. This suggests, in an obvious way, the problem of maximizing the volume of a convex body contained in a given non-convex body. That such a largest convex subset exists follows at once from the Blaschke selection theorem (see Proposition 1 below). The problems then become: (1) What proportion of the original volume are we assured of saving? and (2) How do we determine the largest convex 'peeled potato' inside a given non-convex one ?

Two new vinaigrettes

 I've added two new vinaigrettes to the growing list at http://www.econ.uiuc.edu/~roger/research/vinaigrettes/vinaigrette.html

One is a survey of currently available methods for estimation and inference for quantile regression, the other is about computational methods for univariate quantiles.  Both incorporate some new methods in my quantreg package.  As usual there is an element of samo-kritika  to these notes, since they try to reflect several drawbacks of the current state of affairs, not just the successes.

Economics of Writing

 I've been wondering lately about the economics of writing since I find myself spending quite a lot of time writing open source software and writing about open source software.  There seem to be two extreme positions on this subject:

    o  Ben Smith quotes the Substack author Heather Cox Richardson as saying, "if you start doing things for money, they stop being authentic."

    o  Or there is Samuel Johnson:  "No man, but a blockhead ever wrote, except for money."

A Chicago price theory question might be:  Reconcile these two statements.

Conformal Quantile Regression

 I've written yet another vinaigrette, this time about a simple R implementation of the conformal quantile regression method introduced by Romano, Patterson and Candes  the pdf version of the vinaigrette is available here.