Functional decomposition: Difference between revisions

Revision as of 16:58, 14 June 2023

Revision as of 14:15, 25 September 2023

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In [[engineering]], '''functional decomposition''' is the process of resolving a [[Function (mathematics)|functional]] relationship into its constituent parts in such a way that the original function can be reconstructed (i.e., recomposed) from those parts.

This process of decomposition may be undertaken to gain insight into the identity of the constituent components, which may reflect individual physical processes of interest. Also, functional decomposition may result in a compressed representation of the global function, a task which is feasible only when the constituent processes possess a certain level of ''modularity'' (i.e., independence or non-interaction).

{{clarify span|Interactions|reason=The notion of 'interaction' of mathematical functions is undefined, likewise for 'observable', 'perception' etc. I guess this paragraph confuses mathematical notions (like 'function') with physical intuitions (like 'process'); this should be fixed.|date=September 2020}} between the components are critical to the function of the collection. All interactions may not be {{clarify span|observable|date=September 2020}}, but possibly deduced through repetitive {{clarify span|perception|date=September 2020}}, synthesis, validation and verification of composite behavior.

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==Motivation for decomposition==

[[Image:West-side-highway traffic.png|thumb|400px|Causal influences on West Side Highway traffic. Weather and GW Bridge traffic ''screen off'' other influences]]

Decomposition of a function into non-interacting components generally permits more economical representations of the function. Intuitively, this reduction in representation size is achieved simply because each variable depends only on a subset of the other variables. Thus, variable <math>x_1</math> only depends directly on variable <math>x_2</math>, rather than depending on the ''entire set'' of variables. We would say that variable <math>x_2</math> ''screens off'' variable <math>x_1</math> from the rest of the world. Practical examples of this phenomenon surround us. Consider the particular case of "northbound traffic on the [[West Side Highway]]." Let us assume this variable (<math>{x_1}</math>) takes on three possible values of {"moving slow", "moving deadly slow", "not moving at all"}. Now, let's say the variable <math>{x_1}</math> depends on two other variables, "weather" with values of {"sun", "rain", "snow"}, and "[[GW Bridge]] traffic" with values {"10mph", "5mph", "1mph"}. The point here is that while there are certainly many secondary variables that affect the weather variable (e.g., low pressure system over Canada, [[Butterfly Effect|butterfly flapping]] in Japan, etc.) and the Bridge traffic variable (e.g., an accident on [[Interstate 95 in New York|I-95]], presidential motorcade, etc.) all these other secondary variables are not directly relevant to the West Side Highway traffic. All we need (hypothetically) in order to predict the West Side Highway traffic is the weather and the GW Bridge traffic, because these two variables ''screen off'' West Side Highway traffic from all other potential influences. That is, all other influences act ''through'' them.

==Applications==

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===Machine learning===

In practical scientific applications, it is almost never possible to achieve perfect functional decomposition because of the incredible complexity of the systems under study. This complexity is manifested in the presence of "noise," which is just a designation for all the unwanted and untraceable influences on our observations.

However, while perfect functional decomposition is usually impossible, the spirit lives on in a large number of statistical methods that are equipped to deal with noisy systems. When a natural or artificial system is intrinsically hierarchical, the [[joint distribution]] on system variables should provide evidence of this hierarchical structure. The task of an observer who seeks to understand the system is then to infer the hierarchical structure from observations of these variables. This is the notion behind the hierarchical decomposition of a joint distribution, the attempt to recover something of the intrinsic hierarchical structure which generated that joint distribution.