Time-Series Analysis Practise Flashcards

Terminology and mathematical models for Time-Series Analysis, including decomposition, forecasting models, and smoothing techniques.

Random variable

A mapping from random events to real numbers.

Random process

A collection of random variables over time.

Time-series

A realization of a random process consisting of a collection of observations over time.

Cointegration

The study of the relationship among nonstationary time-series.

Decomposition components of $y_t$

Trend ($T_t$), Seasonal component ($S_t$), Cyclical component ($C_t$), and Random term/white noise ($\epsilon_t$).

White noise best forecast

The best forecast for white noise is its mean, which is $0$.

Constant mean model

A deterministic trend model defined by $y_t = \alpha + \epsilon_t$.

Linear trend model

A deterministic trend model defined by $y_t = \alpha + \beta t + \epsilon_t$ where forecasting involves Ordinary Least Squares (OLS) estimates.

S-shaped curves

A model where $y_t = \frac{k}{1 + e^{\alpha + \beta t}}$ and $k$ represents the long-term limit.

Moving Average estimate ($M_t$)

A non-deterministic trend model calculated as $M_t = \frac{1}{k} (y_t + y_{t-1} + \dots + y_{t-k+1})$.

Exponential Moving Average (EMA)

A weighted average estimate using geometrically declining weights, recursively computed as $M_t = \alpha y_t + (1 - \alpha) M_{t-1}$.

Smoothing parameter ($\alpha$)

A parameter in EMA that is chosen to be small if the series is very jagged and large if the series is smooth.

Holt’s linear trend

A moving average method that incorporates a trend slope coefficient ($b_t$) to forecast $y_{T+h} = L_T + hb_T$.

Holt-Winters

An extension of moving averages that includes linear trend with seasonality ($s$) and cyclicality ($c$).