When demonstrating his Alice in Wonderland example, Brad Paley showed how the words in the center of the display were located by grabbing a word with his mouse, clicking to show its connections with places in the text, and then moving the word, showing the lines of the connections stretching, then letting go to show the word bounce back. The image of the word connected to the places using rubber bands was clear.

What I want to know is, can somebody rig a robot arm to do this so I could feel the pull? Imagine a robot arm that can be moved within a 30cm x 40cm box. You could use this to feel the springiness of the connections in Brad’s diagram; the idea is that you’d say a word (for example, “Alice”); the pointer of the robot arm would move to the position of the word in the display, and then you could–with effort–move the robot arm away from this place. When you let go or relax your grip, the pulling of the virtual rubber bands would return the arm back to its original place, and you could feel the strength of the pull.

The arm could also be used to feel a curve (for example, a nonlinear regression or spline, or a mathematical function such the logarithm or the normal distribution curve), as follows: the arm would start at one end of the curve and the user could grip it and move it along, with the motion physically constrained so that the arm would trace the curve.

In displaying several curves–for example, level curves indicating indifference curves – the arm could start on one of the curves and be programmed to stay on that curve, unless the user pushes hard, in which case there would be resistance during which the arm moves between curves. It would then lock into the next curve, which the user could again trace until he or she pushes hard enough to get the arm unstuck again.

More generally, the robot arm could be used for exploring three-dimensional functions such as physical potentials, likelihood functions, and probability densities. From any point in the two-dimensional box, a “gravitational force” would pull the arm toward a local minimum (or, for a likelihood or probability density, the maximum) of the function. Then with moderate effort the user could move the arm around and, by feeling the resistance, get a sense of gradients, minima, and constraints.

(for example, a nonlinear regression or spline, or a mathematical function such the logarithm or the normal distribution curve), as follows: the arm would start at one end of the curve and the user could grip it and move it along, with the motion physically constrained so that the arm would trace the curve.

In displaying several curves–for example, level curves indicating indifference curves – the arm could start on one of the curves and be programmed to stay on that curve, unless the user pushes hard, in which case there would be resistance during which the arm moves between curves. It would then lock into the next curve, which the user could again trace until he or she pushes hard enough to get the arm unstuck again.

More generally, the robot arm could be used for exploring three-dimensional functions such as physical potentials, likelihood functions, and probability densities. From any point in the two-dimensional box, a “gravitational force” would pull the arm toward a local minimum (or, for a likelihood or probability density, the maximum) of the function. Then with moderate effort the user could move the arm around and, by feeling the resistance, get a sense of gradients, minima, and constraints.

An example of how I’d like to use the robot arm together with a visual graph of data and model fit

I was originally thinking of this as a statistical tool for blind people, but I’d actually like to have one of these myself, for example to understand the sensitivity of a model fit to changes in parameter values. I’m thinking of twp graphs next to each other: a graph of parameter space and a graph of data with fitted curves. The robot arm would be pointed to the posterior mode or maximum likelihood estimate in parameter space. As I moved the robot arm around, I’d feel resistance–it would be difficult to move far in parameter space without feeling the increase in -log(density)–and at the same time the curve would be moving in the fitted curve + data plot. The muscular resistance information on one graph and the visual information on the other graph would together give me a sense of what aspects of the data are determining the model fit.

Here’s an example of what it might look like:

I’d also like to be able to use the robot arm to pull on the fitted curve and feel the resistance as I move it away from the data.

P.S. Here’s the R code for the above graphs:


# Illustration of the graph of model parameters and data

library (arm)

n <- 50 x <- rnorm (n, 2,2) a <- -2 b <- .5 sigma <- 2 y <- rnorm (n, a+b*x, sigma) M1 <- lm (y ~ x) display (M1) loglik <- function (x, y, a, b, sigma){ return (sum (dnorm (y, a+b*x, sigma, log=TRUE))) } range.a <- coef(M1)[1] + se.coef(M1)[1]*c(-2.5,2.5) range.b <- coef(M1)[2] + se.coef(M1)[2]*c(-2.5,2.5) n.grid <- 100 a.values <- seq (range.a[1], range.a[2], length=n.grid) b.values <- seq (range.b[1], range.b[2], length=n.grid) ll <- array (NA, c(n.grid,n.grid)) for (i in 1:n.grid){ for (j in 1:n.grid){ ll[i,j] <- loglik (x, y, a.values[i], b.values[j], sigma.hat(M1)) } } ll <- ll - max(ll) lik <- exp (ll) png ("2graphs.png", height=600, width=320) par (mar=c(3,3,3,0), mgp=c(1.5,.2,0), tck=-.01, mfrow=c(2,1)) plot (x, y) n.sims <- 1000 sims <- sim (M1, n.sims)$coef for (s in 1:n.sims){ curve (sims[s,1] + sims[s,2]*x, lwd=.5, col="gray", from=-10, to=10, add=TRUE) } curve (coef(M1)[1] + coef(M1)[2]*x, from=-10, to=10, add=TRUE) points (x, y) mtext ("Data, the fitted regression line, y = a + b*x,nand uncertainty in the fitted line", line=.5) contour (a.values, b.values, lik, levels=c(seq(.05,.95,.1),1), labels="", xlab="a", ylab="b", yaxt="n") axis (2, seq(0,.6,.2)) # kluge to make the y-axis not look stupid mtext ("Contour plot of the likelihood function", line=.5) dev.off()