science

Pattern theory

Pattern theory is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Pattern theory rather than just read about it. In short: Pattern theory, formulated by Ulf Grenander, is a mathematical formalism to describe knowledge of the world as patterns. It differs from other approaches to artificial intelligence in that it does not begin by prescribing algorithms and machinery to recognize and classify patterns; rather, it prescribes a vocabulary to articulate and recast the pattern concepts in precise language.

Key takeaways

Pattern theory belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Pattern theory to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Pattern theory from memory before moving on to harder problems.

Reference excerpt

Pattern theory, formulated by Ulf Grenander, is a mathematical formalism to describe knowledge of the world as patterns. It differs from other approaches to artificial intelligence in that it does not begin by prescribing algorithms and machinery to recognize and classify patterns; rather, it prescribes a vocabulary to articulate and recast the pattern concepts in precise language. Broad in its mathematical coverage, Pattern Theory spans algebra and statistics, as well as local topological and global entropic properties. In addition to the new algebraic vocabulary, its statistical approach is novel in its aim to:

Identify the hidden variables of a data set using real world data rather than artificial stimuli, which was previously commonplace. Formulate prior distributions for hidden variables and models for the observed variables that form the vertices of a Gibbs-like graph. Study the randomness and variability of these graphs. Create the basic classes of stochastic models applied by listing the deformations of the patterns. Synthesize (sample) from the models, not just analyze signals with them. The Brown University Pattern Theory Group was formed in 1972 by Ulf Grenander. Many mathematicians are currently working in this group, noteworthy among them being the Fields Medalist David Mumford. Mumford regards Grenander as his "guru" in Pattern Theory.

See also Abductive reasoning Algebraic statistics Computational anatomy Formal concept analysis Grammar induction Image analysis Induction Lattice theory Spatial statistics

References

Further reading 2007. Ulf Grenander and Michael Miller Pattern Theory: From Representation to Inference. Oxford University Press. Paperback. (ISBN 9780199297061) 1994. Ulf Grenander General Pattern Theory. Oxford Science Publications. (ISBN 978-0198536710) 1996. Ulf Grenander Elements of Pattern Theory. Johns Hopkins University Press. (ISBN 978-0801851889)

External links Pattern Theory Group at Brown University Pattern Theory: Grenander's Ideas and Examples - a video lecture by David Mumford Pattern Theory and Applications - graduate course page with material by a Brown University alumnus

Worked examples

Example 1 — a first encounter with Pattern theory

Start with the simplest possible case. Write down what Pattern theory claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Pattern theory before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Pattern theory ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Pattern theory

In research: Pattern theory appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Pattern theory in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Pattern theory is common in secondary-school and first-year university syllabi. It links to neighbouring topics Artificial intelligence, Formalism (philosophy), so understanding it makes those chapters shorter.
In everyday life: Look for Pattern theory outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Pattern theory in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Pattern theory means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Pattern theory out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Pattern theory in simple terms?

Why does Pattern theory matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Pattern theory?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Pattern theory.