\[ \newcommand{\Expect}[1]{\mathbb{E}\left[ #1 \right]} \newcommand{\Indicator}[1]{\mathbb{1}\left( #1 \right)} \]

Data: body weights of cats

library(MASS)
data(cats)
head(cats$Bwt)

## [1] 2.0 2.0 2.0 2.1 2.1 2.1

summary(cats$Bwt)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   2.000   2.300   2.700   2.724   3.025   3.900

The empirical cumulative distribution function

• ECDF: \[ \hat{F}_n(a) = \frac{1}{n}\sum_{i=1}^{n}{\Indicator{x_i \leq a}} \]
  • Makes a jump at each distinct value seen in the data
  • Constant in between the jumps

Computationally: ecdf

plot(ecdf(cats$Bwt), xlab = "Body weight (kg)", ylab = "Cumulative probability")

The ECDF estimates the CDF

• Suppose \(x_1, x_2, \ldots x_n\) are IID with CDF \(F\). Then for any \(a\), \[ \hat{F}_n(a) \rightarrow F(a) \] by law of large numbers
  • Also: see backup slides

The CDF is not enough

• What’s the probability of a cat weighing between \(3.21\) and \(3.22\) kg?
  • ECDF says: exactly 0
• Q from discussion: What about running a smoother (e.g., a spline) through the ECDF?
• A: You could do that, but you’d need to make sure the curve was always monotone, between 0 and 1, etc.
  • Once we estimate the density, we can recover the CDF by integrating

The CDF is not enough

• More generally, pdf is derivative of CDF, \(f(x) = dF/dx\)
• Derivative of an empirical CDF is either 0 or \(\infty\)
  • Where is it \(\infty\)?
• How do we estimate a pdf?

What about a histogram?

hist(cats$Bwt, xlab = "Body weight (kg)", ylab = "Number of cats", breaks = 20)

• Gives us an idea of where we have more or fewer data points…

Fitting a parametric model

• Pick a parametric pdf that looks sensible
• For example, the gamma distribution with shape parameter \(a\) and scale \(s\) is \[ f(x) = \frac{1}{s^a \Gamma(a)} x^{a-1} e^{-x/s} \]
  • Exponential is the special case \(a=1\)
  • \(a > 1\) is a hump with an exponential right tail

Fitting a parametric model (cont’d)

• Now maximize the likelihood
• For some (but not all) standard distributions, the function fitdistr in the package MASS has pre-programmed maximum likelihood estimates
  • You saw this in homework 3 for the \(t\) distribution
  • As it happens, fitdistr knows about gamma distributions too
• It’s also easy to maximize the log-likelihood yourself

# Define the (negative, normalized) log-likelihood function
gamma.loglike <- function(params, data = cats$Bwt) {
    -mean(log(dgamma(x = data, shape = params[1], scale = params[2])))
}
# Ask R to optimize it, with a starting guess about the parameters
cats.gamma <- optim(fn = gamma.loglike, par = c(1, 1))
# What is the best (negative, normalized) log-likelihood?
cats.gamma$value

## [1] 0.6677077

# What parameters did optim find for us?
cats.gamma$par

## [1] 32.65187332  0.08341489

Compare the fitted density to the histogram

plot(hist(cats$Bwt, plot = FALSE, breaks = 20), freq = FALSE, xlab = "Body weight (kg)")
curve(dgamma(x, shape = cats.gamma$par[1], scale = cats.gamma$par[2]), add = TRUE)

The histogram is already a density estimate

• Estimated density is constant on bins
• \(N_i =\) number of samples in bin \(i\)
• \(i(x)=\) which bin \(x\) falls into
• \(h =\) width of the bins

\[ \hat{f}_{hist}(x) \equiv \frac{1}{h} \frac{N_{i(x)}}{n} \]

Why this works

Probability of falling into a region of length \(h\) around \(x\) will be \[ \int_{x-h/2}^{x+h/2}{f(u) du} \approx f(x) h \] for very small \(h\). So \[ N_i \sim \mathrm{Binom}(n, f(x) h) \] and \[ \hat{f}_{hist}(x) = \frac{N_i}{nh} \sim \frac{1}{nh} \mathrm{Binom}(n, f(x) h) \] which \(\rightarrow f(x)\) as \(n \rightarrow \infty\)

How bad is a histogram?

Error from variance: \(\propto 1/nh\)

Error from bias: \(\propto h^2 (f^{\prime}(x))^2\)

Total error for estimating the density, \(O(h^2) + O(1/nh)\)

Best \(h = O(n^{-1/3})\)

Best total error, \(O(n^{-2/3})\)

vs. \(O(n^{-1})\) if we’ve got a correct parametric model

Q from discussion: What about variable-width bins?

A: That can help, but it’s even better to get away from fixed bins altogether

Kernels

Idea: “Smooth out” each observation into a little distribution

Kernel functions are pdfs

Put a copy of the kernel at each observation; add them up

\[ \hat{f}_{kern}(x) = \frac{1}{n} \sum_{i=1}^{n}{\frac{1}{h} K\left(\frac{x-x_i}{h}\right)} \]

How this works for cats

Say \(h=0.01\)

How do we pick the bandwidth?

• Cross-validation
• Cross-validate what?
• Integrated squared error: \(\int{(f(x) - \hat{f}(x))^2 dx}\)
• Or (negative, normalized) log-likelihood \(\int{-\log{(\hat{f}(x))} f(x) dx}\)

Evaluating the log-likelihood

• The integral is an expected value: \[ \int{-\log{(\hat{f}(x))} f(x) dx} = \Expect{-\log{\hat{f}(X)}} \]
• With \(n\) IID samples \(x_i\), \[ \Expect{-\log{\hat{f}(X)}} \approx \frac{1}{n}\sum_{i=1}^{n}{-\log{\hat{f}(x_i)}} \] by LLN
  • A similar but more complicated trick for ISE; see backup slides

Computationally: npudens

library(np)
bwt.np <- npudens(cats$Bwt)
plot(bwt.np, xlab = "Body weight (kg)")

Joint PDFs

• What about multiple variables? We want \(f(x_1, x_2, \ldots x_p) = f(\vec{x})\)

plot(Hwt ~ Bwt, data = cats, xlab = "Body weight (kg)", ylab = "Heart weight (g)")

• Parametric models work just the same way as before
• Kernel density estimates work almost the same way

Joint KDEs

• Use a product of kernels, one for each variable \[ \hat{f}_{kern}(x_1, x_2) = \frac{1}{n} \sum_{i=1}^{n}{\frac{1}{h_1 h_2} K\left(\frac{x_{1}-x_{i1}}{h_1}\right)K\left(\frac{x_{2}-x_{i2}}{h_2}\right)} \]
• Need as many bandwidths as there are variables
• Pick bandwidths for best prediction of both variables

Computationally: npudens again

cats.np <- npudens(~Bwt + Hwt, data = cats)
plot(cats.np, view = "fixed", theta = 10, phi = 45)

EXERCISE: Where do the ridges come from?

• Q1: What happened to the bandwidth on bodyweight (Bwt)?
  • It got a lot smaller
• Q2: Why did that bandwidth change?
  • There’s a strong relationship between Bwt and Hwt; if we want to get both right, we can’t average over as wide a range of Bwt

Conditional PDFs

• Conditional pdf: \[ f(x_2|x_1) = \frac{f(x_1, x_2)}{f(x_1)} \] so find the joint and marginal pdfs and divide
• Conditional KDE: \[ \hat{f}_{kern}(x_2|x_1) = \frac{\frac{1}{n} \sum_{i=1}^{n}{\frac{1}{h_1 h_2} K\left(\frac{x_{1}-x_{i1}}{h_1}\right)K\left(\frac{x_{2}-x_{i2}}{h_2}\right)}}{\frac{1}{n} \sum_{i=1}^{n}{\frac{1}{h_1} K\left(\frac{x-x_i}{h_1}\right)}} \]
• Pick bandwidths for best prediction of \(x_2\)
  • i.e., look at the conditional log-likelihood of \(x_2\) given \(x_1\)

Computationally: npcdens

cats.hwt.on.bwt <- npcdens(Hwt ~ Bwt, data = cats)
plot(cats.hwt.on.bwt, view = "fixed", theta = 10, phi = 65)

How well does kernel density estimation work?

• The same sort of analysis we did for kernel regression will work here too
  • More book-keeping — see the text
• Integrated squared error of KDE is \(O(n^{-4/5})\) in 1D
  • Worse than a well-specified parametric model, \(O(n^{-1})\)
  • Better than a mis-specified parametric model, \(O(1)\)
  • Better than a histogram, \(O(n^{-2/3})\)
• In \(p\) dimensions, ISE of joint KDE is \(O(n^{-4/(4+p)})\)
  • Nothing which is universally consistent does any better
  • Curse of dimensionality again

Summing up

• It’s easy to learn the CDF
  • use the empirical CDF
• It’s easy to estimate parametric distributions
  • maximize the log-likelihood
• Histograms are a first step to a non-parametric density estimate
  • Kind of ugly and statistically inefficient, though
• Kernel density estimation lets us learn marginal pdfs, joint pdfs, conditional pdfs…
  • The curse of dimensionality is always with us though

READING: Chapter 14

Backup: More about the ECDF

• Glivenko-Cantelli theorem: If \(X\)’s are IID with CDF \(F\), then \[ \max_{a}{|\hat{F}_n(a) - F(a)|} \rightarrow 0 \]
  • This is a stronger result than just saying “for each \(a\), \(|\hat{F}_n(a)-F(a)| \rightarrow 0\)”
• Kolmogorov-Smirnov (KS) test: To test whether \(X\)’s are IID with CDF \(F_0\), use \(d_{KS} = \max_{a}{|\hat{F}_n(a)-F_0(a)|}\)
  • If \(X \sim F_0\), then \(d_{KS} \rightarrow 0\) as \(n\rightarrow \infty\)
  • If \(X \not\sim F_0\), then \(d_{KS} \rightarrow\) some positive number
  • The sampling distribution of \(d_{KS}\) under the null turns out to be the same for all \(F_0\) (at least for big \(n\))
  • Needs modification if you estimate the \(F_0\) from data as well

Backup: Parametric density estimates

• We have a parametric model for the density \(f(x;\theta)\) with a finite-dimensional vector of parameters \(\theta\)
• We maximize the log-likelihood to get \(\hat{\theta}\)
• Equivalently we minimize the negative normalized log-likelihood \[ L(\theta) = -\frac{1}{n}\sum_{i=1}^{n}{\log{f(X_i;\theta)}} \]
• This is a sample mean so LLN says \(L(\theta) \rightarrow \Expect{L(\theta)} \equiv \lambda(\theta)\)
• You can show that if the data came from the model with some true parameter \(\theta_0\), then \(\lambda(\theta_0)\) is the unique global minimum

Backup: More about parametric density estimates

• Quite generally, the MLE \(\hat{\theta} \rightarrow \theta_0\)
• Almost as generally, for large \(n\), \(\hat{\theta} \sim \mathcal{N}(\theta_0, \mathbf{J}(\theta_0)/n)\)
  • Hence \(\hat{\theta} = \theta_0 + O(1/\sqrt{n})\)
  • Here \(\mathbf{J}(\theta) =\) matrix of 2nd partial derivatives of \(\lambda\) = Fisher information
  • Can plug in \(\mathbf{J}(\hat{\theta})\) to get approximate confidence intervals, etc.
  • Doesn’t matter if the data is Gaussian — \(L(\theta)\) is a sample average so by the CLT it’s (approximately) Gaussian, and that propagates
• Things are more complicated if the model is mis-specified

Backup: Evaluating ISE

• If we’re comparing different estimates, term (A) doesn’t matter, so we can ignore it
• Term (C) depends only on our estimate, and we can get it by (say) numerical integration
• That leaves term (B), and with \(n\) independent samples \(x_i\), \[ \int{\hat{f}(x) f(x) dx} \approx \frac{1}{n}\sum_{i=1}^{n}{\hat{f}(x_i)} \] by LLN