Draft:Fréchet regression

Motivation

In the era of data science, data types are becoming increasingly complex. One setting that is frequently encountered involves a random element taking values in a general metric space $({\mathcal {M}},d)$, where $d$ is a distance metric. Object data analysis provides a comprehensive framework for the statistical analysis of such data, where the fundamental units are complex objects—such as shapes, images, or networks—rather than traditional scalar or vector observations. In this context, there is growing interest in regression frameworks where the response random element $Y$ lies in a general metric space.

Conditional Mean

In a basic regression setting when $X\in \mathbb {R} ^{p}$ and $Y\in \mathbb {R}$, the population target is defined as a conditional expectation of $Y$ given $X=x$ by

${\textstyle m(x)=\mathbb {E} (Y|X=x),}$

where $\mathbb {E}$ is denoted as a expectation.

Definition

Fréchet regression is a natural generalization of classical regression, extending the setting where $Y$ is a real-valued random variable to the case where $Y\in {\mathcal {M}}$. This approach enables the analysis of complex data types—such as manifold-valued data, networks, and distributional data—in a more statistically rigorous and interpretable manner.

Let $({\mathcal {M}},d)$ be a metric space. We consider regression setting, where predictors and responses pairs $(X,Y)\in \mathbb {R} ^{p}\times {\mathcal {M}}$ be a stochastic process with a joint distribution ${\mathcal {F}}$.

However, as there is no basic vector operation in general metric space ${\mathcal {M}}$, such as addition, subtraction, multiplication, and division does not exist, we need another approach to define the mean of responses $Y$ using distance $d$. The Fréchet mean is given by ${\textstyle w_{\oplus }={\text{argmin}}_{w\in {\mathcal {M}}}\mathbb {E} d^{2}(Y,w).}$

Then, the Fréchet Regression is defined as a conditional Fréchet mean

${\textstyle m_{\oplus }(x)={\text{argmin}}_{w\in {\mathcal {M}}}\mathbb {E} (d^{2}(Y,w)|X=x).}$

Models

Parametric Fréchet Regression

The most popular parametric regression model is linear regression, which is a statistical method used to model the relationship between a dependent variable and one or more independent variables by fitting a linear equation to observed data. It is widely used for prediction and inference in both the natural and social sciences. Consider a pair of random elements $(X,Y)\in \mathbb {R} ^{p}\times {\mathcal {M}}$, where ${\mathcal {M}}$ is a general metric space. Let $\mu =\mathbb {E} (X)$, and $\Sigma ={\text{Var}}(X)$. Global Fréchet regression is a natural generalization of linear regression and is defined as:

${\textstyle m_{G,\oplus }(x)={\text{argmin}}_{w\in {\mathcal {M}}}\mathbb {E} (s(X,x)d^{2}(Y,w)),}$

where the weight function ${\textstyle s(z,x)}$ is given by; ${\textstyle s(z,x)=1+(z-x)^{T}\Sigma ^{-1}(x-\mu )}$.

Nonparametric Fréchet Regression

The most widely used nonparametric regression model is local regression, which is a flexible statistical technique that models the relationship between variables without assuming a specific functional form for the regression function, allowing the data to determine the shape of the curve. It is particularly useful when the true relationship is complex or unknown. The local (nonparametric) Fréchet Regression is defined as:

$m_{L,\oplus }(x)={\text{argmin}}_{w\in {\mathcal {M}}}\mathbb {E} (s(X,x,h)d^{2}(Y,w)),$

where the weight function${\textstyle s(z,x,h)=\sigma _{0}^{-2}[K_{h}(z-x)\{\mu _{2}-\mu _{1}(z-x)\}],}$ ${\textstyle \mu _{j}=\mathbb {E} [K_{h}(X-x)(X-x)^{j}]}$, $j=0,1,2$, and $\sigma _{0}^{2}=\mu _{0}\mu _{2}-\mu _{1}^{2}$.