## Approximation of Causal Dependences by Structural Regression Equations

**Vyacheslav Teterin**

*Russia, Yandex School of Data Analysis*

Structural regression (linear) model is a system of regression equations with the graph of relations between variables a prior declared. The graph presents our knowledge (or assumption) about local dependencies among variables.

Example from metallurgy [1]. We used quality formation process of metal rolling as a base for modeling using structural regression (quality knowledge allows us to build a correspondent causal graph).

There are two approaches to estimate structural regression: statistical [2- 5] and approximating [6].

We will focus only on the second one for three reasons:

- it is richer for tools development and simpler for result interpretation,
- it requires fewer prior assumptions,
- it is not as popular as the statistical approach.

All methods in this approach focus on reaching the best approximation

of dependencies among input and output variables in a linear class of functions in the least square sense with different kinds of restricted conditions, determined by a given causality graph.

of dependencies among input and output variables in a linear class of functions in the least square sense with different kinds of restricted conditions, determined by a given causality graph.

Computational efficiency is achieved ?by tricks? by reducing? the initial hard objective function [7] to a series of simple criteria that don’t produce a solution equivalent to the original function, but a very close one in practice [8,9].

Causality analysis based on structural regression originated in psychology in the 1930s and moved into econometrics in the 1960s. Now it is ready to be considered as an abstract approach for causality study in general and applied in a wide range of areas [10,11]. In this talk we will illustrate the general approach using an example from metallurgy and demonstrate the abstract power of this approach for industrial applications.

To conclude, we will talk about the ways to solve the problem of estimating both the weights on edges of a given causal graph from data and the graph itself [12,13].

Reference list:

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- Teterin V.G. (1984), ” The Design Principles of Algorithms for the Estimation of Structural Regression Models”, Methods for the Complex Systems Studies, Proceedings of the Young Scientists Conf., Moscow, The Institute for System Studies, 52-58.
- Teterin V.G. (1985), “Identifiability Criteria of Structural Models with Linear Constraints of a General Form on the Coefficients” Automation and Remote Control, 2, 130-138.
- Teterin V.G. (1986), “Modeling of Technological Processes in Terms of Unobserved State Variables”, The Theory of Control Systems and its Ap- plication in Metallurgical Production, (MIS&S, Scientific papers), Moscow, Metallurgy, 102-107.
- Teterin V.G. “The identification of structural models with incomplete information”, Unpublished paper.