# Are contours $h^{-1}(y)$ interesting features of a function $h:X\to \mathbb R^n$ obtained by regression?

I assume a general setup of regression, that is, a continuous function $h_\theta:X\to \mathbb R^n$ is chosen from a family $\{h_\theta\}_\theta$ to fit given data $(x_i,y_i)\in X\times \mathbb R^n, i=1,\ldots, k$ ($X$ can be any space such as cube $[0,1]^m$ or in fact any reasonable topological space) according to some natural criteria.

Are there applications of regression where one is interested in a contour $h^{-1}(y)$ of $h$ for some point $y\in \mathbb R^n$ – for instance the zero set $h^{-1}(0)$?

The explanation of my interest is the following:
Since in many situation there is uncertainty attached to the learned $h_\theta$ (imprecision or lack of the data), one might want to analyse the zero set $h^{-1}(0)$ “robustly”. Namely, study the features of the zero set that are common to all “perturbations” of $h$. A very good understanding has been developed recently in a very general setting where the perturbations $f$ can be arbitrary continuous maps close to $h$ in the $\ell_\infty$ norm. Or, essentially equivalently, $f$ is arbitrary continuous such that for every $x\in X$ we have $|f(x)-h(x)|\le c(x)$ where $c:X\to\mathbb R$ gives some confidence value at every $x$.

Our main motivation for developing the theory and the algorithms has been the exciting mathematics behind (essentially all the problems/questions get reduced to the homotopy theory). However, at the current stage, for further development and implementation of the algorithms, we need to choose more specific settings and goals.

Economists are frequently interested in this. Often we estimate consumers’ utility functions $u: \mathbb R^n \rightarrow \mathbb R$, where the domain describes how much of each good a consumer consumes and the range is how “happy” the consumption bundle makes him. We call the level sets of utility functions “indifference curves.” Often we estimate firms’ cost functions $c: \mathbb R^n \times \mathbb R^k \rightarrow \mathbb R$, where the two parts of the domain are quantities of each output the firm produces and prices for each input the firm uses in production. Level sets of $c$ are called iso-cost curves.

Most commonly, the properties of the level sets we are interested in are the slopes of the boundaries. The slope of an indifference curve tells you at what rate consumers trade-off different goods: “How many apricots would you be willing to give up for one more apple?” The slope of an iso-cost curve tells you (depending on which part of the domain), how substitutable in production different outputs are (at the same cost, if you produced 10 fewer razor blades, then how many more pins could you make), or how substitutable different inputs are.

Economists are completely obsessed with ratios of first partial derivatives because we are obsessed with trade-offs. These, I guess, can be (always?) thought of as slopes of boundaries of level sets.

Another application is the calculation of economic equilibria. The simplest example is the supply and demand system. The supply curve represents how much producers are willing to supply at each price: $q=s(p)$. The demand curve represents how much consumers are willing to demand at each price: $q=d(p)$. Take an arbitrary price, $p$, and define excess demand as $e(p)=d(p)-s(p)$. Equilibrium prices are $e^{-1}(0)$ — i.e. these are the prices at which markets clear. $q$ and $p$ can be vectors, and $d$ and $s$ are normally non-linear.

What I’m describing in the previous paragraph (demand and supply) is just an example. The general set-up is extremely common. In Game Theory, maybe we are interested in calculating the Nash Equilibria of a game. To do this you define, for player $i$, a function (the best response function) which gives their best strategy as the range and what strategies all the other players are playing as the domain: $s_i=br(s^{-i})$. Stack these all up into a vector best response function: $s=BR(s)$. If $s$ can be represented as real numbers, then you can define a function giving the distance from equilibrium: $d(s)=BR(s)-s$. Then $d^{-1}(0)$ is the set of equilibria of the game.

Whether Economists usually estimate these relationships with regression depends on how broad your definition of regression is. Commonly, we use instrumental variables regression. Also, in the case of utility functions, utility is not observed, so we have various latent variable methods for estimating those.