[Nonparametric] 4. Multivariate Kernel Density Estimation

$\newcommand{\argmin}{\mathop{\mathrm{argmin}}\limits}$ $\newcommand{\argmax}{\mathop{\mathrm{argmax}}\limits}$

Until now, I only covered univariate cases. From now on, our focus is on multivariate cases where samples $X_1,\cdots,X_n\in\mathbb{R}^d$ are independently drawn from the density $p:\mathbb{R}^d\to[0,\infty)$ with respect to the Lebesgue measure.


Kernel and bandwidth settings

Kernel function

We need a $d$-variate kernel function $\kappa$ for this. There are several possible ways to define it.

  1. Product kernel:

    \[\kappa(x) := \prod_{j=1}^d K(x_j),\]

    where $K:\mathbb{R}\to\mathbb{R}$ is a univariate function with $\int K = 1.$

  2. Radially symmetric kernel:

    \[\kappa(x) := K(x^\intercal x),\]

    where $K:\mathbb{R}\to\mathbb{R}$ is a univariate function with $\int K(x^\intercal x)dx = 1.$

We will focus on the simple case of using product kernel.


Bandwidth

Now the bandwidth is also multivariate. Three choices of bandwidth are available.

  1. Scalar bandwidth:

    Use the same $h\in\mathbb{R}$ for all $d$ coordinates. i.e. \(\hat p(x) = \frac{1}{nh^d} \sum_{i=1}^n \kappa\left(\frac{x-X_i}{h}\right).\)

  2. Vector bandwidth (coordinate-wise bandwidth):

    If the density has different smoothness along different coordinates, then setting a scalar bandwidth might not provide good result. To account for the problem, vector bandwidth use $h_j\in\mathbb{R}$ for $j$-th coordinates, $j=1,\cdots,d.$

    \[\hat p(x) = \frac{1}{n\prod_{j=1}^d h_j} \sum_{i=1}^n \kappa\left(\frac{x_1-X_{i1}}{h_1}, \cdots, \frac{x_d-X_{id}}{h_d} \right).\]
  3. Bandwidth matrix:

    In order to further account for possible rotation of the density contour, we can use a bandwidth matrix $H \in \mathbb{R}^{d\times d}$ so that our estimate be

    \[\hat p(x) = \frac{1}{n|H|} \sum_{i=1}^n \kappa\left( H^{-1} (x-X_i) \right).\]

Throughout this post, we will only focus on multivariate KDE cases with product kernel function and coordinate-wise bandwidth.


Error analysis

Suppose conditions (A3') and (A4) hold. (A3'): $p$ is $2$-times partially continuously differentiable at $x,$ and $\frac{\partial^2}{\partial x_j \partial x_k} p(x)$ is integrable for all $j,k.$
(A4): $|u|^3 K(u) \to 0$ as $|u|\to\infty,$ $\int u^2 K(u) du < \infty,$ $\int K = 1,$ $K$ is symmetric.
Then, $$ E\hat p(x) - p(x) = \frac{1}{2}\mu_2(K) \sum_{j=1}^d h_j^2 \frac{\partial^2}{\partial x_j^2}p(x) + o\left( \sum_{j=1}^d h_j \right). $$


(A3’) is a multivariate version of (A3) that was discussed earlier, and (A4) is the same as before.



$$ \begin{aligned} E\hat p(x) &= \kappa_h * p(x) \\ &= \int \frac{1}{h_1}K\left( \frac{x_1-u_1}{h_1} \right) \cdots \frac{1}{h_1}K\left( \frac{x_d-u_d}{h_d} \right) p(u_1,\cdots,u_d) d\mathbf{u} \\ &= \int \prod_{j=1}^d K(w_j) p(x_1-w_1h_1,\cdots,x_d-w_dh_d)d\mathbf{w} \\ &= p(x) + \int \prod_{j=1}^d K(w_j) \sum_{j=1}^d\left[ \frac{\partial}{\partial x_j} p(x) \cdot (-h_jw_j) \right] d\mathbf{w} + R\big(p(x)\big) + o\left( \sum_{j=1}^d h_j^2 \right) \\ &= p(x) + R\big(p(x)\big) + o\left( \sum_{j=1}^d h_j^2 \right) \end{aligned} \tag{1} $$ The second equality is by substitution $w_j=\frac{x_j-u_j}{h_j},$ and the third follows from the first order Taylor expansion. Here, $R\big(p(x)\big)$ can be organized as follows, thus we get the desired result. $$ \begin{aligned} R\big(p(x)\big) &= \frac{1}{2} \sum_{j=1}^d \sum_{k=1}^d \left( \frac{\partial^2}{\partial x_j \partial x_k} p(x) h_jh_k\int w_jw_k\prod_{j=1}^d K(w_j) d\mathbf{w} \right) \\ &= \frac{1}{2} \sum_{j=1}^d \left( \frac{\partial^2}{\partial x_j^2 p(x)} h_j^2 \int w_j^2 K(w_j) dw_j \right) \\ &= \frac{1}{2} \mu_2(K) \sum_{j=1}^d h_j^2 \frac{\partial^2}{\partial x_j^2} p(x). \end{aligned} $$


Suppose (A5) and (A6) hold.
(A5): $p$ is continuous at $x.$
(A6): $|u|K(u) \to 0$ as $|u|\to\infty,$ $\int K^2 <\infty.$
Then, $$ \text{Var}(\hat p(x)) = \frac{1}{n \prod_{j=1}^d h_j} \mu_0(K^2) p(x) + o\left(\frac{1}{n \prod_{j=1}^d h_j}\right). $$


Note that $\text{Var}(\hat p(x)) = \mathcal{O}\left( \frac{1}{n\prod_{j=1}^d h_j} \right).$ What it says, intuitively, is that the variance of multivariate KDE is reciprocal to the number of data inside the region around the point $x$ that the kernel function looks for (I would like to call it the receptive field1 of the kernel) when it estimates the density at $x.$

By similar arguments from univariate cases, we can derive the mean integrated squared error.


$p$: is $2$-time partiallly continuously differentiable on $\mathbb{R}^d,$ $\frac{\partial^2}{\partial x_j \partial x_k} p(x)$ is integrable for all $j,k$ and $p$ is square integrable.
$K$: $\int u^2 K(u) du <\infty,$ $\int K = 1,$ $K$ is symmetric, $\int K^2 < \infty.$
Then, $$ \begin{aligned} \text{MISE}(\mathbf{h}) =~& \frac{1}{4} \mu_2(K)^2 \int \left( \sum_{j=1}^d h_j^2 \frac{\partial^2}{\partial x_j^2} p(x) \right)^2 dx \\ &+ \frac{1}{n \prod_{j=1}^d h_j} \mu_0(K^2) p(x) \\ &+ o\left( \sum_{j=1}^d h_j^2 + \frac{1}{n \prod_{j=1}^d h_j}\right). \end{aligned} $$


The optimal $h_\text{opt}$ is the one that makes the first and second terms of MISE to have the same order. Letting

\[n^{-1} h_j^{-d} \sim h_j^4,~ j=1,\cdots,d,\]

yields

\[h_{j, \text{opt}} \sim n^{-1/(d+4)}. \tag{2}\]


Optimal bandwidth

Coordinate-wise scaling

One method for finding optimal bandwidth is to use the result (2) and assume that

\[h_j = c_j \cdot n^{-1/(d+4)}.\]

That is, each coordinate-wise bandwidth $h_j$ is a scalar multiple of another. As before, letting the first and second term of MISE to be the same, we get the system of $d$ equations:

\[\int \left( \sum_{k=1}^d c_k^2 \frac{\partial^2}{\partial x_k^2} p(x) \right) \left( c_j^2 \frac{\partial^2}{\partial x_j^2} p(x) \right) dx = \frac{\mu_0(K^2)}{\mu_2(K)} \left(\prod_{j=1}^d c_j\right)^{-1},~ 1\le j\le d. \tag{3}\]

Solve the equation and let $h_{j, \text{opt}} = c_{j, \text{opt}}\cdot n^{-1/(d+4)}.$


Global scaling

However, the equation (3) cannot be explicitly solved. To get the idea of the constant $c,$ we fallback to the scalar bandwidth

\[h_\text{opt} = c_\text{opt} \cdot n^{-1/(d+4)}\]

where $c_\text{opt}$ is the solution of the equation (3) with $c=c_j$ for all $1\le j\le d.$ This time we can explicitly get the solution

\[c_\text{opt} = \left( \frac{d \cdot \mu_0(K^2)}{\mu_2(K)^2 \int \left( \sum_{j=1}^d \frac{\partial^2}{\partial x_j^2} p(x) \right)^2 dx} \right)^{1/(d+4)}.\]


Bandwith selection

As in the univariate cases, we need to select $\hat h$ that best acts the optimal $h.$ Recall that the LSCV selector is defined as

\[\text{LSCV}(\mathbf{h}) = \int \hat p(x)^2 dx - \frac{2}{n} \sum_{i=1}^n \hat p_{-i}(X_i).\]

However, full-dimensional minimization of LSCV is intractable. We use coordinate-wise approach to avoid the problem.

Bandwidth shrinkage

Bandwidth shrinkage scheme uses a minimizer of marginal $p_j(x_j)$ to infer $\hat h_j.$

  1. Minimize marginal.

    For each $j,$ let $\hat h_j^\text{mg} = \argmin_{h>0} \text{LSCV}_j(h_j),$ where $\text{LSCV}_j = \int \hat p_j(x_j)^2 dx_j - \frac{2}{n} \hat p_{-i,j}(X_{ij}).$

  2. Grid search a global scaler.

    Let $\hat \alpha = \argmin_{\alpha \in \mathcal{G}} \text{LSCV}(\alpha \hat h_1^\text{mg}, \cdots, \alpha \hat h_d^\text{mg}),$ where $\mathcal{G} \subset \mathbb{R}$ is a finite grid.

  3. Let $\hat h_j = \hat \alpha \cdot \hat h_j^\text{mg}$ be our bandwidth.

Coordinate-wise bandwidth selection

Coordinate-wise bandwidth selection (CBS) is simply a coordinate descent algorithm for LSCV. Instead of minimizing marginals, CBS minimizes conditional joints.

  1. Initialize.

    Choose $\left( \hat h_1^{(0)}, \cdots, \hat h_d^{(0)} \right)$ from a finite grid $\mathcal{G}_1 \times \cdots \times \mathcal{G}_d \subset \mathbb{R}^d.$

  2. Coordinate descent until convergence.

    \[\hat h_1^{(t+1)} = \argmin_{g_1 \in \mathcal{G}_1} \text{LSCV}\left( g_1, \hat h_2^{(t)}, \cdots, \hat h_d^{(t)} \right), \\ \hat h_2^{(t+1)} = \argmin_{g_2 \in \mathcal{G}_2} \text{LSCV}\left( \hat h_1^{(t+1)}, g_2, \cdots, \hat h_d^{(t)} \right), \\ \vdots \\ \hat h_d^{(t+1)} = \argmin_{g_d \in \mathcal{G}_d} \text{LSCV}\left( \hat h_1^{(t+1)}, \hat h_2^{(t+1)}, \cdots, g_d \right).\]


Curse of dimensionality

The optimal MISE under the assumption $h_j = c_j \cdot n^{-1/(d+4)}$ is

\[\inf_{\mathbf{h}>0} \text{MISE}(\mathbf{h}) \sim n^{4/(d+4)}.\]

As $d$ increases, the error deteriorate fast. This is because the receptive field $r_\kappa(x)$1 of $\kappa$ around $x$ is defined as

\[r_\kappa = \prod_{j=1}^d \left[ x_j - c_jh_j,~ x_j + c_jh_j \right],\]

the volume of it is proportional to $\prod_{j=1}^d h_j,$ which goes to zero exponentially fast as $d \to \infty.$ Since using the the concept of receptive field as in univariate cases are useless in high-dimensional setting, we shoud think of other methods.


Multiplicative model

Assume that the density is multiplicative.

\[\hat p(x) = \prod_{j=1}^d f_j(x_j),~ x \in S \subset \mathbb{R}^d.\]

This can be viewed as assuming that $p$ is a product of marginals. If $S$ is rectangular, the problem becomes solving $d$ different univariate KDE problems.


Projection pursuit density estimation

PPDE estimates the density by projecting the data into lower-dimensional space, and feeding it to a multiplicative-like model. Friedman, Stuetzle and Schroeder (1984) described that “it attempts to overcome the curse of dimensionality by extending the classical univariate density estimation methods to higher dimensions in a manner that involves only univariate estimation.”

PPDE is defined as

\[\hat p_{d_1}(x) = \hat p_0(x) \prod_{j=1}^{d_1} f_j(\beta_j^\intercal x),\]

where $\hat p_0$ is a known initial guess, $\beta_j \in \mathbb{R}^{d_1},$ $d_1 \le d$ and $f_j$ is a univariate function. Again quoting the authors, “PPDE is seen to approximate the multivariate density by an initially proposed density $\hat p_0,$ multiplied (augmented) by a product of univariate functions $f_m$ of linear combinations $\beta_j\cdot x$ of the coordinates.” $\hat p_0$ should be prespecified by the user, and PPDE’s goal is to choose the projections $\beta_j$’s and functions $f_j$’s. The model can be reorganized into recursive form:

\[\hat p_{d_1} = \hat p_{d_1 - 1} \cdot f_{d_1}(\beta_{d_1}^\intercal x).\]

Thus PPDE can be seen as a process of adjusting the initial guess $\hat p_0$ with information from each coordinate $\beta_j^\intercal x$ using the augmenting function $f_j.$


References

  • Nonparametric Function Estimation (Spring, 2021) @ Seoul National University, Republic of Korea (instructor: Prof. Byeong U. Park).
  • Friedman, Stuetzle and Schroeder. 1984. Projection Pursuit Density Estimation. Journal of the American Statistical Association. 79: 387. pp599-608.
  1. These are not formal expressions.  2