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Questions and Answers
What is the primary purpose of forecasting in the context of time series analysis?
What is the primary purpose of forecasting in the context of time series analysis?
Which characteristic defines a time series as equally spaced?
Which characteristic defines a time series as equally spaced?
What is a common application of time series forecasting in supply chain management?
What is a common application of time series forecasting in supply chain management?
Which of the following is NOT a component typically accounted for in time series data?
Which of the following is NOT a component typically accounted for in time series data?
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Why is it important to account for patterns like autocorrelation in forecasting?
Why is it important to account for patterns like autocorrelation in forecasting?
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What does 'prediction is very difficult, especially if it's about the future' imply about forecasting?
What does 'prediction is very difficult, especially if it's about the future' imply about forecasting?
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Which aspect of time series data allows for the quantification of patterns over time?
Which aspect of time series data allows for the quantification of patterns over time?
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What is a critical limitation of relying solely on past data for forecasts?
What is a critical limitation of relying solely on past data for forecasts?
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What defines the seasonality in a time series?
What defines the seasonality in a time series?
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In the context of seasonal dummy variables, what does a dummy variable represent?
In the context of seasonal dummy variables, what does a dummy variable represent?
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Which equation represents seasonal differencing in a time series?
Which equation represents seasonal differencing in a time series?
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In a time series with $S$ seasons, how many dummy variables are there?
In a time series with $S$ seasons, how many dummy variables are there?
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What is represented by the equation $Y_t = f(T_t, S_t, X_t, E_t)$ in the Universal Time Series Model?
What is represented by the equation $Y_t = f(T_t, S_t, X_t, E_t)$ in the Universal Time Series Model?
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What type of data would use the dummy variable notation $I_{MON}$ for January?
What type of data would use the dummy variable notation $I_{MON}$ for January?
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Which of the following is NOT a component removed when analyzing the irregular component of a time series?
Which of the following is NOT a component removed when analyzing the irregular component of a time series?
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When performing seasonal differencing, which of the following expressions is correct for a monthly time series?
When performing seasonal differencing, which of the following expressions is correct for a monthly time series?
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What does the symbol $Y_t$ in the Universal Time Series Model represent?
What does the symbol $Y_t$ in the Universal Time Series Model represent?
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Which of the following is a characteristic of a deterministic trend?
Which of the following is a characteristic of a deterministic trend?
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What is the formula for a linear trend in a time series?
What is the formula for a linear trend in a time series?
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What is the purpose of differencing in time series analysis?
What is the purpose of differencing in time series analysis?
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Which type of trend component includes random variations and cannot be predicted perfectly?
Which type of trend component includes random variations and cannot be predicted perfectly?
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In the context of time series, what does seasonality refer to?
In the context of time series, what does seasonality refer to?
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What is the equation for a random walk with drift?
What is the equation for a random walk with drift?
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Which of the following statements about first differences is true?
Which of the following statements about first differences is true?
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Study Notes
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.
-
Moving average: Uses past t periods to predict t+1
-
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.
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