Random Projects

New York Conditional Taxi Dropoff Probabilities

I fit a twenty component mixture of multivariate normals, using scikit-learn, to the four dimensional new york taxi pickup/dropoffs dataset.

The dimensions look like (pickup_lat, pickup_lon, dropoff_lat, dropoff_lon). The aim is to predict the distribution of (dropoff_lat, dropoff_lon) by conditioning on (pickup_lat, pickup_lon).

Fancy ways to do this might include fitting a neural net or some kind of a gp to the conditional density, but here, I literally just fit a 4d mvn to the whole dataset. To condition, we just plug in the pickup position and renormalize (Bayes rule).

Envelope Modelling

Google’s Quick Draw dataset contains multiple observations of quickly drawn envelopes. I fit a 256-component restricted boltzmann machine (heavily overparameterised; not a great model - I know) to the data, which represents a big nasty distribution over the random field that represents an envelope image. Now, starting off with a completely random image, using Gibbs sampling, we can make our way to the typical set of the distribution, which hopefully looks like an envelope. Here’s what the burn in looks like:

Inferring the Extent of Differentiability

Let’s say that we have an observation of a noiseless function but we don’t know how smooth it is. You could probably fit a Matern GP with different smoothness parameters to see which parameter maximises the log marginal likelihood (the matern parameter corresponds to the number of times one can differentiate a sample from the gp).

Below, I’ve simulated a Matern GP with a particular parameter, and fit it using parameters ranging from \(\{0.5, …, 5\}\). The color & label correspond to the parameter while sampling.

Modelling my d20 dice

I fit a spline on a sphere representing my d20. The color represents the model probabilities. The distance from the centre represents the proportion of times my dice fell on a particular number during six hundred trials.

Code to help with this (messy).
library(data.table)

# I came up with a coordinate system to label the dice
# The params phi and theta are reversed w.r.t. polar convention
# phi is pi - phi w.r.t. the polar convention
vertices <- data.table(
	r = 1, theta = c(0, 0, 2 * pi * (0:9)/10),
	phi = c(pi/2, -pi/2, rep(c(-atan(0.5), atan(0.5)), 5)) + pi/2,
	label = letters[1:12])

dice_map <- list(
	"1" = "ihj", "2" = "dfe", "3" = "kjl", "4" = "beg",
	"5" = "gfh", "6" = "bck", "7" = "ahj", "8" = "cdl",
	"9" = "bki", "10" = "adl", "11" = "bgi", "12" = "adf",
	"13" = "gih", "14" = "bce", "15" = "afh", "16" = "ckl",
	"17" = "ajl", "18" = "egf", "19" = "kij", "20" = "ced")

cart <- function(polar) {
	# theta phi
	return(c(sin(polar[1]) * cos(polar[2]),
		     sin(polar[1]) * sin(polar[2]),
		     cos(polar[1])))
}

polar <- function(cart) {
	return(c(acos(cart[3]), atan2(cart[2], cart[1])))
}

my_coords <- function(pol_mine) {
	return(c(pi - pol_mine[2], pol_mine[1]))
}

arc_length <- function(polar_a, polar_b) {
	return(acos(sum(cart(polar_a) * cart(polar_b))))
}

query <- function(verts) {
	Reduce(`&`, lapply(verts, function(v) grepl(v, dice_map)))
}

rd20 <- function(n = 1) replicate(n, {
	z <- rnorm(3); z <- z/sqrt(sum(z^2))
	qd20(polar(z))
})

qd20 <- function(polar_coords) {
	dists <- sapply(1:12, function(i) arc_length(
		polar_coords, vertices[i, my_coords(c(theta, phi))]))
	min_three <- sort(dists)[1:3]
	verts <- vertices[which(dists %in% min_three), label]
	as.integer(names(dice_map))[query(verts)]
}
 
 


2019

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2018

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