22. Time Series Data
● Stock values
● Economic variables
● Weather
● Sensor: Internet-of-Things
● Energy demand
● Signal processing
● Sales forecasting
23.
24.
25. Problems on Time Series Data
● Standard Supervised Learning
○ IID assumption
○ Same distribution for training and test data
○ Distributions fixed over time (stationarity)
● Time Series
○ Not applicable
26. Models for Time Series Analysis
● Time Series Analysis
● Models for Time Series Analysis: AR, MA, ARMA, ARIMA,
Recurrent Neural Networks
● TensorFlow TimeSeries API (TFTS)
27. Autoregressive (AR) Models
● AR(p) model
: Linear generative model based on the pth order Markov assumption
○ : zero mean uncorrelated random variables with variance
○ : autoregressive coefficients
○ : observed stochastic process
28. Moving Average (MA)
● MA(q) model
: Linear generative model for noise term on the qth order Markov
assumption
○ : moving average coefficients
29. ARMA Model
● ARMA(p,q) model
: generative linear model that combines AR(p) and MA(q) models
30. Stationarity
● Definition: a sequence of random variables is stationary if its
distribution is invariant to shifting in time.
31. Lag Operator
● Definition: Lag operator is defined by
● ARMA model in terms of the lag operator:
● Characteristic polynomial
can be used to study properties of this stochastic process.
32. ARIMA Model
● Definition: Non-stationary processes can be modeled using processes
whose characteristic polynomial has unit roots.
● Characteristic polynomial with unit roots can be factored:
● ARIMA(p, D, q) model is an ARMA(p,q) model for
33. Other Extensions
● Further variants:
○ Models with seasonal components (SARIMA)
○ Models with side information (ARIMAX)
○ Models with long-memory (ARFIMA)
○ Multi-variate time series model (VAR)
○ Models with time-varing coefficients
○ other non-linear models
