Nonparametric models

K-Nearest Neighbors

The k-nearest neighbors (KNN) model is a nonparametric supervised learning approach that can be applied to classification or regression problems. In a classification context, the KNN model assigns a class label for a new datapoint by taking a majority vote amongst the labels for the k closest points (“neighbors”) in the training data. Similarly, in a regression context, the KNN model predicts the target value associated with a new datapoint by taking the average of the targets associated with the k closes points in the training data.

Models

• KNN

Gaussian Process Regression

A Gaussian process defines a prior distribution over functions mapping \(X \rightarrow \mathbb{R}\), where X can be any finite (or infinite!)-dimensional set.

Let \(f(x_k)\) be the random variable corresponding to the value of a function f at a point \(x_k \in X\). Define a random variable \(z = [f(x_1), \ldots, f(x_N)]\) for any finite set of points \(\{x_1, \ldots, x_N\} \subset X\). If f is distributed according to a Gaussian Process, it is the case that

\[z \sim \mathcal{N}(\mu, K)\]

for

\[\begin{split}\mu &= [\text{mean}(x_1), \ldots, \text{mean}(x_N)] \\ K_{ij} &= \text{kernel}(x_i, x_j)\end{split}\]

where mean is the mean function (in Gaussian process regression it is common to define mean(x) = 0), and kernel is a kernel / covariance function that determines the general shape of the GP prior over functions, p(f).

In Gaussian process regression (AKA simple Kriging [2] [3]), a Gaussian process is used as a prior on functions and is combined with the Gaussian likelihood from the linear model via Bayes’ rule to compute a posterior over functions f:

\[\begin{split}y \mid X, f &\sim \mathcal{N}( [f(x_1), \ldots, f(x_n)], \alpha I ) \\ f \mid X &\sim \text{GP}(0, K)\end{split}\]

Due to the conjugacy of the Gaussian Process prior with the regression model’s Gaussian likelihood, the posterior will also be Gaussian and can be computed in closed form.

Models

• GPRegression

References

[1]	Rasmussen, C. E., & Williams, C. K. I. (2006). Gaussian Processes for Machine Learning. MIT Press, Cambridge, MA.

[2]	Krige, D. G., (1951). “A statistical approach to some mine valuations and allied problems at the Witwatersrand”, Master’s thesis of the University of Witwatersrand.

[3]	Matheron, G., (1963). “Principles of geostatistics”, Economic Geology, 58, 1246-1266.

Kernel Regression

Kernel regression is another nonparametric approach to nonlinear regression. Like the Gaussian Process regression approach (or, more generally, all regression models), kernel regression attempts to learn a function f which captures the conditional expectation of some targets y given the data X, under the assumption that

\[y_i = f(x_i) + \epsilon_i \ \ \ \ \text{where } \mathbb{E}[\epsilon | \mathbf{x}] = \mathbb{E}[\epsilon] = 0\]

Unlike the Gaussian Process regression approach, however, kernel regression does not place a prior over f. Instead, it models \(f = \mathbb{E}[y | X] = \int_y \frac{p(X, y)}{p(X)} y \ \text{d}y\) using a kernel function, k, to estimate the smoothed data probabilities. For example, the Nadaraya-Watson estimator [4] [5] uses the following probability estimates:

\[\begin{split}\hat{p}(X) &= \prod_{i=1}^N \hat{p}(x_i) = \prod_{i=1}^N \sum_{j=1}^N \frac{k(x_i - x_j)}{N} \\ \hat{p}(X, y) & \prod_{i=1}^N \hat{p}(x_i, y_i) = \prod_{i=1}^N \sum_{j=1}^N \frac{k(x_i - x_j) k(y_i - y_j)}{N}\end{split}\]

Models

• KernelRegression

References

[4]	Nadaraya, E. A. (1964). “On estimating regression”. Theory of Probability and Its Applications, 9 (1), 141-2.

[5]	Watson, G. S. (1964). “Smooth regression analysis”. Sankhyā: The Indian Journal of Statistics, Series A. 26 (4), 359–372.

See Also

The trees module contains other classic nonparametric approaches, including decision trees, random forests, and gradient boosted decision trees.