Datenanalyse und Stochastische Modellierung
0. Predictions



Data Analysis and Stochastic Modeling



Dr. Philipp Meyer, University of Potsdam

WS 2022/2023


What is the best prediction?

...for the next movement without further information

Chapter 1

...for a random movement in a potential

Chapter 2

...for a directed movement given two time points

Chapter 7

...for a directed movement given two time points and additional information

Chapter 13

...for a directed movement given two time points and additional information

Chapter 13

...for a dice throw?

source:wikipedia.org

Mean squared error

Evaluating predictions \[\hat{X}\] \[\mathrm{MSE}=\frac{1}{N}\sum_{n+1}^N (x_n-\hat{x}_n)^2\]
  • The mean squared error is a commonly used metric (loss function)
  • Dice throw: best guess is x=3.5

Data-driven models

  • Models can be built from first principles (bottom up) or from observations (top down)
  • Here: we start from data and construct models from observations
  • This is necessary if the system is very complex and/or most parameters of the system are unknown
  • Starting from general stochastic models encoding only essential information towards complex machine learning models with many parameters

Which models explain the intuition?

  1. Brownian motion
  2. Autokokorrelations
  3. Chaos Theory
  4. Non-Gaussian processes
  5. Time-reversal symmery
  6. The power spectral density
  7. Active motion
  8. Multiple timescales
  9. Bayesian statistics (Kalman filter)
  10. Autoregressive processes
  11. Supervised learning
  12. Causality
  13. Feature-based Learning

What data can we describe?

Air pressure and Temperature

Financial asset values

Movement of animals

... or any given time series!