Untitled Quiz

Forecasting Time Series Overview

• Course name: DSBA 6211
• Instructor: Dr. Zhao
• Topics covered: Introduction, Time Series Characteristics and Components, Forecasting Models
• Forecasting aims to predict future behavior of variables, accounting for internal structures like autocorrelation, trends, and seasonality.
• Time series data is used for forecasting.

Introduction to Forecasting

• Forecasting is a form of predictive modeling.
• Aims to predict outcome variables (e.g., sales, buy/no buy).
• Critical to account for internal time-related patterns (like autocorrelation, trends, seasonality).
• Time series data encompasses the information collected over time using equally spaced time intervals.
• Time-series data allows for the visualization and quantification of patterns over a time interval.
• Enables forecasting for future points based on past behavior.

Business Applications

• Inventory Management: Should you use shelf space for more peanut butter or salsa? Will putting an item on sale decrease demand?
• Demand Management: What time of day produces peak server demand?
• Supply Chain Management: When should you reorder raw materials?
• Pricing: What are pricing trends in the past quarter compared to the previous three years?

Caution

• Prediction is challenging, especially about the future (Nils Bohr).
• Forecasts are only as good as the included past data.
• History is not a perfect predictor of the future.

Time Series Characteristics and Components

• A statistical time series is a sequence of indexed numbers (dates or other numerical values).
• Many business time series are equally spaced (same interval between consecutive points).
• Equally spaced time series can have missing values.

The Universal Time Series Model

• Yt = f(Tt, St, Xt, Et)
• Components:
  • Trend (T): long-term movement
  • Seasonal (S): recurring patterns (e.g., monthly, quarterly)
  • Input (X): external factors influencing the time series
  • Error (E): random fluctuations

Airline Passengers 1994-1997 Series Plot

• Shows a rise in airline passengers between 1994 and 1997.

Time Series Trend

• Trend represents a deterministic function of time.
• Stochastic components are subject to random variation.
• Deterministic components exhibit no random variation and can be predicted perfectly.
• Examples of deterministic trend functions: linear, curvilinear, logarithmic, exponential.

Deterministic Trend Models

• Linear Trend: Yt = β0 + β1t
• Quadratic Trend: Yt = β0 + β1t + β2t²

Stochastic Trend Models

• Random Walk: Yt = Yt-1 + Et
• Random Walk with Drift: Yt = μ + Yt-1 + Et

Accommodating Stochastic Trend: Differencing

• First difference of Random Walk process: Yt - Yt-1 = Et
• ΔYt = Yt - Yt-1

Accommodating Seasonal Components

• Trigonometric functions (sine waves)
• Seasonal dummy variables/indicator variables
• Seasonal differences (Box-Jenkins modeling)

Dummy Variables

• Indicator variable.
• Takes value 1 for a specific time point, 0 otherwise.
• Used to represent seasonal variations.

Seasonal Dummy Variables

• For a series with S seasons, there are S dummy variables.
  • Monthly: IJAN, IFEB, ..., IDEC
  • Daily: ISUN, IMON, ..., ISAT
  • Quarterly: IQ1, IQ2, IQ3, IQ4

Stochastic Seasonal Functions: Seasonal Differencing

• Express current value as a function involving value S time units prior.
• Yt = Yt-S + Trend + Irregular

The Irregular Component

• Remains after removing trend, seasonal, and input effects.
• Represents forecast error.

Additive Decomposition of the Airline Data

• Breaks down the time series into trend, seasonal, and irregular components.

Forecasting Models

• Regression-based: Uses suitable predictors to capture trends and seasonality.
  • Examples: Linear, Quadratic trend, Additive seasonality
• Smoothing methods:
  • Moving average: Uses past t periods to predict t+1
    • Simple: average of past n periods;
    • Weighted: Past periods have different weights (more recent periods usually have higher weights).
  • Exponential smoothing: Emphasizes most recent values. Smoothing constant (α) determines the degree.
• Autoregressive Integrated Moving Average (ARIMA) Models:
  • Uses lagged values of the dependent variable and/or random disturbance term as predictors.
  • Relies heavily on autocorrelation patterns in the data.

Model Performance Evaluation

• Forecast error measures:
  • MA (mean error), RSMA (root mean squared error), MAE (mean absolute error)
  • MPE (mean percentage error)
  • MAPE (mean absolute percentage error)
  • Scaling errors based on training MAE; MASE (mean absolute scaled error)

Event Variable Improvements to Accuracy

• Event variables (like promotions, policy changes) help accommodate disruptions in time series data.
• Primarily used to modify intercepts of models.
• Expressed as binary variables (0 or 1).

Event Variable Creation

• Event variables are created by adding a 0-1 (binary) column to existing data, specifying events.