Estimating Functions

TL;DR

Methods that approximate a target function using known data to enable prediction or analysis.
Commonly applied in statistics and data analysis to forecast values or reveal relationships between variables.
Examples include linear regression (fits a line by minimizing the sum of the squared differences) and k-nearest neighbors (predicts class by majority among the k closest points).

Definition

Estimating functions are mathematical tools that are used to approximate the values of a given function based on a set of known data points.

Explanation

Estimating functions use observed data to produce an approximate functional relationship between variables. They are applied to make predictions about new or future data and to understand underlying relationships among variables. Different estimating functions implement different strategies: for example, some fit a parametric form to the data (such as a line), while others use local similarity to existing data points to determine predictions.

One parametric approach is linear regression, which uses a linear equation to fit a line to data by minimizing the sum of the squared differences between the observed values and the fitted line. A nonparametric, instance-based approach is the k-nearest neighbors algorithm, which predicts based on the classes or values of the k closest data points according to some similarity measure.

Examples

Linear regression

A linear regression model fits a line to a set of data points using a linear equation, aiming to minimize the sum of the squared differences between the observed data and the fitted line. For instance, with a dataset containing the weight and height of a group of people, a linear regression line can describe the relationship between weight and height and be used to predict weight from height or vice versa.

k-nearest neighbors

The k-nearest neighbors algorithm predicts the class of a new data point by finding the k data points in the dataset that are closest to the new point based on a similarity measure, then assigning the majority class among those k neighbors. For example, with a dataset containing height, weight, and gender for a group of people, k-nearest neighbors can predict the gender of a new person from their height and weight by identifying the k closest data points and using the majority gender.

Use cases

Making predictions about future or unseen data.
Understanding underlying relationships between variables in statistical and data analysis contexts.

Linear regression
k-nearest neighbors