Jdan Jdan - 1 month ago 17
R Question

Producing 3D plot for surface of revolution (a GLM logistic curve example)

Original title (vague): How to make circular surface from x and z value only




I have data that relate to an x-axis and a z-axis similar to values of
new.data
:

mydata <- structure(list(Dist = c(82, 82, 85, 85, 126, 126, 126, 126, 178,
178, 178, 178, 178, 236, 236, 236, 236, 236, 312, 368, 368, 368,
368, 368, 425, 425, 425, 425, 425, 425, 560, 560, 560, 560, 560,
612, 612, 612, 612), pDet = c(1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0,
1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0,
0, 0, 0, 0, 1, 0, 0)), .Names = c("Dist", "pDet"), row.names = c(NA,
-39L), class = "data.frame")

model <- glm(pDet ~ Dist, data = mydata, family = binomial(link = "logit"))
new.data <- data.frame(Dist = seq(0, 650, 50))
new.data$fit <- predict(model, newdata = new.data, type="response")


I want generate a surface / matrix where values of
new.data$fit
represent the z-axis and x- and y-axis are generated from the radius being the
new.data$Dist
.

In other words I want a circle generated from radius
Dist
and cells populated by
z
value of logistic probability curve. I would like to say that I have tried some certain solutions but not even sure where to begin.

Answer

So you want to plot surface of revolution, by spinning the logistic curve around vertical line Dist = 0. Statistically I don't know why we need this, but purely from a mathematical aspect and for the sake of 3D visualization, this is sort of useful, hence I decide to answer this.

All we need is a function of the initial 2D curve f(d), where d is distance from a point to spinning centre, and f is some smooth function. As we will use outer to make surface matrix, f must be defined to be vectorized function in R. Now the surface of revolution is generated as f3d(x, y) = f((x ^ 2 + y ^ 2) ^ 0.5).

In your logistic regression setting, the above f is logistic curve, the predicted response of a GLM. It can be obtained from predict.glm, which is a vectorized function. The following code fits a model, and define such function f plus its 3D extension.

mydata <- structure(list(Dist = c(82, 82, 85, 85, 126, 126, 126, 126, 178, 
178, 178, 178, 178, 236, 236, 236, 236, 236, 312, 368, 368, 368, 
368, 368, 425, 425, 425, 425, 425, 425, 560, 560, 560, 560, 560, 
612, 612, 612, 612), pDet = c(1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 
1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 
0, 0, 0, 0, 1, 0, 0)), .Names = c("Dist", "pDet"), row.names = c(NA, 
-39L), class = "data.frame")

model <- glm(pDet ~ Dist, data = mydata, family = binomial(link = "logit"))

## original 2D curve
f <- function (d, glmObject) 
  unname(predict.glm(glmObject, newdata = list(Dist = d), type = "response"))

## 3d surface function on `(x, y)`
f3d <- function (x, y, glmObject) {
  d <- sqrt(x ^ 2 + y ^ 2)
  f(d, glmObject)
  }

Due to symmetry, we only call f3d on the 1st quadrant for a surface matrix X1, while flipping X1 for surface matrices on other quadrants.

## prediction on the 1st quadrant
x1 <- seq(0, 650, by = 50)
X1 <- outer(x1, x1, FUN = f3d, glmObject = model)

## prediction on the 2nd quadrant
X2 <- X1[nrow(X1):2, ]

## prediction on the 3rd quadrant
X3 <- X2[, ncol(X2):2]

## prediction on the 4th quadrant
X4 <- X1[, ncol(X1):2]

Finally we combine matrices from different quadrants and make 3D plot. Note the combination order is quadrant 3-4-2-1.

## combined grid
x <- c(-rev(x1), x1[-1])
# [1] -650 -600 -550 -500 -450 -400 -350 -300 -250 -200 -150 -100  -50    0   50
#[16]  100  150  200  250  300  350  400  450  500  550  600  650

## combined matrix
X <- cbind(rbind(X3, X4), rbind(X2, X1))

## make 3D surface plot
persp(x, x, X, col = "lightblue", theta = 35, phi = 40,
      xlab = "", ylab = "", zlab = "pDet")

enter image description here


Making a toy routine for plotting surface of revolution

In this part, we define a toy routine for plotting surface of revolution. As noted above, we need for this routine:

  1. a (vectorized) function of 2D curve: f;
  2. evaluation grid on the 1st quadrant {(x, y) | x >= 0, y >= 0} (due to symmetry we take y = x);
  3. possible additional arguments to f, and customized graphical parameters to persp.

The following is a simple implementation:

surfrev <- function (f, x, args.f = list(), ...) {
  ## extend `f` to 3D
  .f3d <- function (x, y) do.call(f, c(list(sqrt(x ^ 2 + y ^ 2)), args.f))
  ## surface evaluation
  X1 <- outer(x, x, FUN = .f3d)
  X2 <- X1[nrow(X1):2, ]
  X3 <- X2[, ncol(X2):2]
  X4 <- X1[, ncol(X1):2]
  xbind <- c(-rev(x), x[-1])
  X <- cbind(rbind(X3, X4), rbind(X2, X1))
  ## surface plot
  persp(xbind, xbind, X, ...)
  ## invisible return
  invisible(list(grid = xbind, z = X))
  }

Now suppose we want to spin a cosine wave on [0, pi] for a surface of revolution, we can do

surfrev(cos, seq(0, pi, by = 0.1 * pi), col = "lightblue", theta = 35, phi = 40,
        xlab = "", ylab = "", zlab = "")

enter image description here

We can also use surfrev to plot your desired logistic curve:

## `f` and `model` defined at the beginning
surfrev(f, seq(0, 650, by = 50), args.f = list(glmObject = quote(model)),
        col = "lightblue", theta = 35, phi = 40, xlab = "", ylab = "", zlab = "")