Introduciton to Time-Series

Introduction

Definition

• A stochastic process $$\{X_{t}\}_{t\in T}$$
  • if T is set of real numbers, a continous-time stochastic process
  • if T is set of integers, a stochastic process in discrete time

Time-Series Decomposition

• decomposes time-series data into the following four components
  • trend component, $T_t$
    • describes the long-term progression of the time series
    • such as a long-term increase or decrease in a sequence
  • cyclical component, $C_t$
    • reflects repeated but non-periodic fluctuations
    • not seasonal, changes at the scale of miliseconds or longer than a year
  • seasonal component, $S_t$
    • a variation that occurs at specific regular intervals of less than a year
    • such as daily, weekly, monthly, or yearly
  • irregular component, $I_t$, represents random and irregular influences
  • stationarity
    • not change its distribution over time
    • no trend, no deterministic seasonal variablility, although other cyclical variability is permitted
• forecast future values for each of the four components and then add these predictions together to find the most accurate overall forecast
• Additive decomposition $$y_t = T_t + C_t + S_t + I_t$$
• Multiplicative decompositionI $$y_t = T_t * C_t * S_t * I_t$$

Prominent Libraries in Time-Series

• prophet
• sktime
• gluon-ts
• tslearn
• pyts
• seglearn
• darts
• cesium
• pmdarima

Time-Series Regression Models

• Linear regression
• Non-linear regression

Exponential Smoothing

• a time series forecasting method for univariate data
• a rule-of-thumb technique for smoothing time-series data using the exponential window function
• simple exponential smoothing and triple exponential smoothing

AutoRegressive Integrated Moving Average (ARIMA) Models

• autogression model, uses observations from previous time steps to predict future values using a regression equation $$y_t = c + \phi_{1}y_{t-1} + \phi_{2}y_{t−2} + ⋯ + \phi_{p}y_{t−p} + ε_{t}$$
• moving average model, a linear combination of forecast errors $$yt = c + \epsilon t + \theta_{1}\epsilon_{t−1} + \theta_{2}\epsilon_{t−2} + ⋯ + \theta_{p}\epsilon_{t−p}$$
• ARIMA model $$y^{'}_{t} = c + \phi_{1}y^{'}_{t−1} + ⋯ + \phi_{p}y^{'}_{t−p} + \epsilon_{1}\theta_{t−1} + ⋯ + \epsilon_{q}\theta_{t−q} + \theta_{t}$$
• Seasonal ARIMA (SARIMA) model, add a linear combination of seasonal past values and forecast errors to ARIMA

Neural Networks

• Robust to noise
• Nonlinear support
• Multivariate inputs
• Multi-step forecasts

TBATS

• can deal with complex and multiple seasonal patterns
• T: Trigonometric seasonality
• B: Box-Cox transformation
• A: ARIMA errors
• T: Trend
• S: Seasonal components

Long Short-Term Memory (LSTM)

Evaluation

• Mean Square Error (MSE) $$MSE = \frac{1}{n}\sum_{t=1}^{n}(Y_{t}-F_{t})^2$$
• Mean Percentage Error (MPE) $$MPE = \frac{1}{n}\sum_{t=1}^{n}\frac{Y_{t}-F_{t}}{Y_{t}}*100$$
• Mean Absolute Error (MAE) $$MAE = \frac{1}{n}\sum_{t=1}^{n}|Y_{t}-F_{t}|$$
• Mean Absolute Percentage Error (MAPE) $$MAPE = \frac{1}{n}\sum_{t=1}^{n}\frac{|Y_{t}-F_{t}|}{Y_{t}}*100$$
• Theil’s U-Statistic $$U1 = \frac{\sqrt{\sum_{t=1}^{n}(Y_{t}-F_{t})^2}}{\sqrt{\sum_{t=1}^{n}Y_{t}^2}+\sqrt{\sum_{t=1}^{n}F_{t}^2}}$$
  • U1, 0~1, closer to 0 indicates greater forecasting accuracy $$U2 = \sqrt{\frac{\sum_{t=1}^{n-1}(\frac{Y_{t+1}-F_{t+1}}{Y_{t}})^2}{\sum_{t=1}^{n-1}(\frac{Y_{t+1}-Y_{t}}{Y_{t}})^2}}$$
  • Naive forecast, forecast values for all future periods equal to the value from the current period
  • U2 = 1, no difference between navie forecast and the forecast technique
  • U2 < 1, the forecast technique is better than navie forecast
  • U2 > 1, the forecast technique is no better than navie forecast

Libraries

• prophet
• sktime
• gluon-ts
• tslearn
• pyts
• seglearn
• darts
• cesium
• pmdarima

Reference

• Exponential Smoothing Methods
• What Is Time-Series Forecasting?