Mathematics / advanced
Time series and dependent data
Model temporal dependence through stationarity, spectral structure, state-space inference, forecasting, and uncertainty.
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Module 01 advanced
Time series and dependent data
Model temporal dependence through stationarity, spectral structure, state-space inference, forecasting, and uncertainty.
- 01
Weak and strict stationarity
Separate invariance of finite-dimensional laws from invariance of second moments, and identify the extra role of joint Gaussianity.
34 min - 02
Autocovariance and positive definite sequences
Use variance positivity across every finite window to distinguish valid autocovariance sequences from plausible-looking lag tables.
35 min - 03
ARMA processes, causality and invertibility
Fix polynomial signs and mean treatment before using root locations to identify causal and invertible stationary representations.
38 min - 04
Wold decomposition and linear prediction
Separate remote-past predictable structure from orthogonal innovations and compute forecasts within the correct linear information space.
39 min - 05
Spectral density and Wiener–Khinchin
Move between covariance and frequency using an explicit radians convention, while keeping spectral atoms distinct from a density.
38 min - 06
Periodograms and spectral estimation
Normalize a finite Fourier transform, identify raw periodogram variability and explain why spectral smoothing requires more than a finer frequency grid.
39 min - 07
Likelihood and innovation algorithms
Factor a finite Gaussian likelihood through sequential prediction errors, keeping its initial distribution and estimated mean explicit.
39 min - 08
State space models and Kalman smoothing
Track scalar Gaussian states with exact filtering and backward smoothing, declaring which observations each estimate can use.
40 min - 09
Unit roots, cointegration and trend
Distinguish stochastic accumulation from a removable deterministic trend, and state the nonstandard inference and common-trend structure precisely.
41 min - 10
Confounding, nonstationarity and forecast validation
Evaluate forecasts at declared information boundaries, separate predictive association from causation and score both point accuracy and uncertainty.
40 min