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Proof capsule

Micro-to-macro reconstruction of aggregate volatility

Research computation, variance identities and reproducible tests

A formal and computational proof case for decomposing weighted micro-unit fluctuations into aggregate variance and covariance contributions.

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Problem

Aggregate volatility is easy to misread when weighted units differ in size, variance and co-movement. The task is to account for how marginal fluctuations and signed covariance terms combine without replacing an identity with an informal scaling claim.

What I built

I implemented a proof-oriented computational case around an exact additive decomposition of weighted micro-unit variance into diagonal variance and signed covariance contributions. The code separates the mathematical identity, demonstrations and deterministic checks so the result can be inspected without treating it as an empirical estimate.

My role and ownership

  • Derived and encoded the weighted variance decomposition and its component terms.
  • Designed the Python demonstrations and tests around the identity rather than a fitted scaling law.
  • Kept the thesis, current manuscript and public source code as distinct evidence routes.

Key decisions

Represent aggregate variance as additive diagonal and signed covariance contributions.

Tradeoff: The decomposition is exact and auditable, but it does not by itself identify a universal empirical scaling exponent.

Test the identity deterministically before presenting numerical demonstrations.

Tradeoff: This prioritizes mathematical and implementation correctness over claims about external datasets.

Link the engineering proof case to the source repository and thesis archive.

Tradeoff: Readers can inspect the route from formal argument to code, while publication status remains explicit.

Reliability & testing

The public repository includes deterministic tests for the scaling decomposition and variance identity, plus executable demonstrations of the component terms.

  • tests/test_scaling_decomposition.py: checks the decomposition components.
  • tests/test_variance_identity.py: checks the aggregate variance identity.

Recovery: A failing identity test blocks confidence in the corresponding computational result.

What this proves

  • Implement an exact additive variance decomposition with signed covariance terms.
  • Connect formal reasoning to deterministic numerical tests and demonstrations.
  • Keep a scientific identity distinct from an empirical scaling estimate.