Forecasting Methods Overview

### Expensive & Time Consuming

• A forecasting method that is very expensive and time consuming
• Practical only for long-term forecasting

### Subjective Models

• Cross Impact Analysis:
  • Assumes future events are related to the occurrence of earlier ones
  • Uses a conditional probability matrix
  • Requires expert input
• Historical Analogy
  • Assumes new services follow the introduction and growth patterns of similar services
  • Frequently used to predict market penetration and product life cycles
  • Can be unreliable and questionable

### Causal Models

• Data follows an identifiable pattern over time
• There's an identifiable relationship between the information to be forecasted and other factors
• Regression Analysis: Uses a linear relationship between independent and dependent variables to form a regression model:
  • Dependent variable (Y): the factor being forecasted
  • Independent variables (Xi): the factors that determine the value of Y
  • Formula: 𝑌 = 𝑎0 + 𝑎1 𝑋1 + 𝑎2 𝑋2 + …..+ 𝑎!𝑋𝑛
• Econometric Models:
  • A version of regression models
  • Consists of a set of simultaneous equations that express a dependent variable in terms of multiple independent variables
  • Requires extensive data collection and sophisticated analysis
  • Suitable for long-range forecasting

### Causal Models - Regression Models - Simple Linear Regression

• Given n data points (xi, yi), i = 1, … , 𝑛
• The function describing the relationship between x and y is: 𝑦" = 𝑚𝑥" + 𝑏
• The aim is to find the equation of the straight line: 𝑦 = 𝑚𝑥 + 𝑏
• Solution
  • Use 𝑥̅ = ∑𝑥/n, 𝑦1 = ∑𝑦/n, 𝑥𝑦 = ∑𝑥𝑦/n
  • 𝑚 = (∑𝑥𝑦 − 𝑥̅ 𝑦1) / (∑𝑥² − 𝑥̅²)
  • 𝑏 = 𝑦1 − 𝑚𝑥̅
• Correlation Coefficient (r):
  • A measure of the reliability of the linear relationship between x and y values
  • Values close to 1 indicate an excellent linear relationship
• Coefficient of Determination (r²):
  • A measure of the percentage of variation in y explained by x

Train Cost Model Example

• 10 weeks of data on the number of trains produced and their production cost
• Simple Linear Regression Model: cost = 17,8594 * (number of trains) + 164,8651

Time Series Models - Exponential Smoothing

• Variables:
  • St: Smoothed value at the end of period t
  • At: Actual observation for period t
  • F(t+1): Forecast for period t + 1
• Formula:
  • New value (St) = Old value (St-1) + α [observed error]
  • St = αAt + (1 - α) St-1
  • F(t+1) = St
• Benefits:
  • Smooths out blips in data
  • Older data are given progressively less weight
  • Simple calculation, requiring only recent data

Time Series Models - Exponential Smoothing - Weights of Past Demand

• Formula: St = aAt + a (1 - a ) At -1 + a (1 - a ) 2 At - 2 +.....+ a (1 - a )t -1 A1 + (1 - a )t S 0
• The weights a, a(1-a), a(1-a)², a(1-a)³... sum up to 1. This means the weight of each past demand is discounted by a factor of (1-a).
• Formula shows that a lot of past information is used to estimate the present, but older data are given progressively less weight.

Forecast Error

• Formula: 𝐴( ) − 𝐹( )
• Cumulative Forecast Error (CFE) = ∑)&'((𝐴& − 𝐹& )
  • The sum of forecast errors should tend to zero
  • Measures the sum of forecast errors
• Mean Absolute Error (MAE): Calculates the average magnitude of the error by summing the absolute difference between forecast and actual value over a period.
• Mean Squared Error (MSE): Calculates the average of the squared error which emphasizes large errors.
• Root Mean Squared Error (RMSE): The square-root of the mean squared error, which gives the error in the same unit as the original data.

This quiz covers various forecasting methods including expensive and time-consuming approaches, subjective models like Cross Impact Analysis and Historical Analogy, as well as causal models such as Regression Analysis. Each method is explored in terms of its practicality, assumptions, and application in forecasting.