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What's the most advanced math you ever used in data science projects?

I have used quite a bit of advanced math, especially to solve problems in experimental mathematics using data science methods. For instance, solving stochastic integral equations. You can find the techniques I used in my recent book Free Book: Applied Stochastic Processes (full title: Applied Stochastic Processes, Chaos Modeling, and Probabilistic Properties of Numeration Systems.)

Some of the material used or created include:

  • Integration and differentiation of stochastic processes
  • Stochastic integral equations
  • Pseudo-logistic map and its probabilistic properties
  • Central limit theorem, stable distributions, and convergence to non-normal distribution, see here. It also involved Fourier transforms, the convolution theorem, and infinite products.
  • Constrained random walks and their equilibrium distribution
  • Computing special integrals using statistical methods (see here), such as

I also read some interesting use of advanced math in data science by Colleen Farrelly.  She mentions the following:

  • Metric geometry in the creation of new nonparametric tests for strange data structures--Hausdorff and Gromov-Hausdorff distances, Wasserstein distances.
  • Theoretical topology and geometry to extend existing statistical frameworks (GLMs, factor analysis/structural equation models, Bayesian adaptive trial designs), machine learning methods, graph/network analytics, and partial differential equation models of biological/industrial systems. These more general models allow for more flexible modeling and can accommodate diverse data structures. Lately, I've entered into some research related to quantum machine learning and topology.

Laura Knight mentions measure theory and linear spaces leading to Hilbert space, Lebesque integration and the geometry of multivariate statistics.

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