This image below shows a tiling made from just two simple shapes, arranged in a pattern that never repeats. You can extend it as far as you like, and it will keep growing in complexity without ever falling into a regular cycle. That property alone makes it interesting, but the background is even better.
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Here’s a short snippet that draws the Hilbert space-filling curve using a recursive approach.
It sounds counterintuitive. A curve normally has no area, it’s just a line. But a space-filling curve is a special type of curve that gets arbitrarily close to every point in a 2D square. If you keep refining it, the curve passes so densely through the space that it effectively covers the entire square. In the mathematical limit, it touches every point.
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In some cases, we need to construct a correlation matrix with a predefined set of eigenvalues, which is not trivial since arbitrary symmetric matrices with a given set of eigenvalues may not satisfy correlation constraints (e.g., unit diagonal elements).
A practical method to generate such matrices is based on the Method of Alternating Projections (MAP), as introduced by Waller (2018). This approach iteratively adjusts a matrix between two sets until convergence. It goes like this:
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Introduction
In quantitative finance, correlation matrices are essential for portfolio optimization, risk management, and asset allocation. However, real-world data often results in correlation matrices that are invalid due to various issues:
- Merging Non-Overlapping Datasets: If correlations are estimated separately for different periods or asset subsets and then stitched together, the resulting matrix may lose its positive semidefiniteness.
- Manual Adjustments: Risk/assert managers sometimes override statistical estimates based on qualitative insights, inadvertently making the matrix inconsistent.
- Numerical Precision Issues: Finite sample sizes or noise in financial data can lead to small negative eigenvalues, making the matrix slightly non-positive semidefinite.
Calculating the median of data points within a moving window is a common task in fields like finance, real-time analytics and signal processing. The main applications are anomal- and outlier-detection / removal.
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