DATA 624 Lecture #4

Review:
Forecasting: Principles and Practice
Hyndman & Athanasopoulos

Forecasting is an important aid to effective and efficient planning

Predictability of a quantity/event depends on:

• How well the factors that contribute to it are understood
• How much data are available

• Whether forecasts affect the target of what is being forecast

• Good forecasts capture genuine patterns and relationships which exist in historical data

• Do not replicate past events that will not occur again

Depending on the application, organizations require forecasts of different lengths:

• Short-term: Scheduling personnel, production, or transportation
• Medium-term: Determine future resource requirements (raw materials)
• Long-term: Used in strategic planning
  

A forecasting task generally involves five basic steps:

1. Problem Definition
2. Information Gathering
3. Exploratory Analysis
4. Choosing & Fitting Models
   
5. Using and Evaluating a Forecast Model
   

Graphics enable features of the data to be visualized, including:

• Patterns
• Unusual observations
• Changes over time
• Relationships between variables
  

The type of data determines both the forecasting method, and the appropriate graph

Average Method:
\( \hat y_{T+h|T}=\bar{y}=(y_1 + ... + y_t) / T \)

Naive Method:
All future values set to \( y_T \), where \( y_T \) is the last observed value

Seasonal Method:
\( y_{T+h-km} \) where \( m= \) the seasonal period, and \( k=(h-1)/m + 1 \)

Drift Method:
\( y_T + \frac{h}{T-1} \displaystyle\sum_{t=2}^{T} (y_t - y_{t-1}) = y_t + h (\frac{y_T - y_1}{T-1}) \)

• Mathematical Transformations
  
  • Logarithmic,
  • Power (\( \sqrt{x}, x^3 \))
  • Inverse
    
• Calendar Adjustments
  
  • Remove variation from uneven distribution of days per month
    
• Population Adjustments
  
  • Use per-capita rather than totals
    
• Inflation Adjustments
  • Adjusted to a base year: \( x_t = y_t/z_t * z_{base-year} \)
    

• Scale-dependent Errors
  
  • MAE: Mean absolute error = mean\( (|e_i|) \)
    
  • RMSE: Root mean squared error = \( \sqrt{mean(e_{i}^2)} \)
• Percentage Errors
  • Given by \( p_i = 100e_{i}/y_{i} \)
    
  • MAPE: Mean absolute percentage error = mean\( (|p_i|) \)
    
• Scaled Errors
  
  • \( q_j = \frac{e_j}{\frac{1}{T - 1} \sum_{t=2}^T |y_t - y_{t-1}|} \)
  • MASE: Mean absolute scaled error = mean\( (|q_j|) \)
    
  • MSSE: Mean squared sclaed error = mean\( (q_{j}^2) \)
    

Time series can be thought to comprise three components:

• Seasonal
  
• Trend-Cycle
• Remainder
  

Additive Model: \( y_t = S_t + T_t + E_t \)

Mulitplicative Model: \( y_t = S_t \times T_t \times E_t \)

Moving average of order \( m \) can be written as:

\( \hat{T_t}=\frac{1}{m} \sum_{j = -k}^{k} y_{t+j} \), where \( m=2k+1 \)

• Found by averaging values of time series within \( k \) periods of \( t \)
• Elminates randomness
  
• \( m \)-MA = moving average of order \( m \)
  

X-12-ARIMA:

• Popular method for decomposing quarterly & monthly data (and only allows for it)
• Based on classical decomposition with many extra steps and features
• Uses ARIMA model providing forecasts foreward & backwards in time
• Separate software from US Census Bureau required w/R interface via x12 package
  
  
  Seasonal & Trend using Loess [STL]:
  
• Will handle any type of seasonality
  • Seasonal component can change over time
    
• Rate of change and smoothness of trend-cycle can be controlled by user
  
• Does not automatically handle calendar variations
• Main parameters are the trend window and seasonal window
  

Decomposition can be used for forecasting, as well as studying time series

Additive Decomposition:

• \( y_t = \hat{S_t} + \hat{A_t} \)
  
  • Where \( \hat{A_t} = \hat{T_t} + \hat{E_t} \)
    

Multiplicative Decomposition:

• \( y_t = \hat{S_t}\hat{A_t} \)
  
  • Where \( \hat{A_t} = \hat{S_t}\hat{E_t} \)
    

The seasonal component (\( \hat{S_t} \)) and the seasonally adjusted component (\( \hat{A_t} \)) are forecasted separately to forecast a decomposed time series

• Suitable for forecasting data with no trend or seasonal pattern
• In between naive and mean forecast methods
  
• Attaches larger weights to more recent observations
• Weights decrease exponentially for observations further in past
• Rate at which weights decrease is controlled by parameter \( \alpha \), where \( 0\leq\alpha\leq1 \)
  

\[ \hat{y}_{T+1|T} = \alpha y_T + \alpha(1-\alpha)y_{T-1} + \alpha(1-\alpha)^2y_{T-2} + \alpha(1-\alpha)^3y_{T-3} \dots \]

• Initialization required, need to specify value for level, \( \ell_0 \)
  
• Common approach is to set \( \ell_0 = y_1 \)
  
• Alternative is to use optimization to estimate value of \( \ell_0 \)
  
• ses function contains a parameter initial to calculate this value.

ses(oilprice, alpha=0.2, initial='simple', h=3)  # use  'simple' or 'optimal'

     Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
1998       21.80368 11.02002 32.58733 5.311500 38.29586
1999       21.80368 10.80646 32.80089 4.984891 38.62247
2000       21.80368 10.59697 33.01038 4.664504 38.94285

• Allows forecasting of data with a trend
• Involves a forecast equation plus two smoothing equations (one for level, one for trend)
  

Forecast: \( \hat{y}_{t+h|t} = \ell_t + hb_t \)
Level: \( \ell_t = \alpha y_t + (1 - \alpha)(\ell_{t-1} + b_{t-1}) \)
Trend: \( b_t = \beta^*(\ell_t - \ell_{t-1}) + (1 - \beta^*)b_{t-1} \)

• \( b_t \) is the estimate of trend at time \( t \) (slope)
• \( \beta^* \) is the smoothing parameter, where \( 0 \leq \beta \leq 1 \)
  

• Variation on Holt's linear trend method
  
• Allows level and slope to be multiplied rather than added
  
• The trend becomes exponential rather than linear
• Growth rate is constant, rather than a constant slope
  

Forecast: \( \hat{y}_{t+h|t} = \ell_t + hb_t \)
Level: \( \ell_t = \alpha y_t + (1 - \alpha)(\ell_{t-1}b_{t-1}) \)
Trend: \( b_t = \beta^* \frac{\ell_t}{\ell_{t-1}} + (1 - \beta^*)b_{t-1} \)

Simple Exponential Smoothing = (N,N)
Holts Linear Method = (A,N)
Exponential Trend Method = (M,N)
Additive Damped Trend Method = (Ad,N)
Multiplicative Damped Trend Method = (Md,N)
Additive Holt-Winters Method = (A,A)
Multiplicative Holt-Winters Method = (A,M)
Holt-Winters Damped Method = (Ad,M)

State Space Models:
Models which consist of:

• A measurement equation that describes the observed data
  
• Some transition equations that describe how the unobserved components or states (level, trend, seasonal) change over time.
  

Two models for each method:

• one with additive errors
  
• one with multiplicative errors
  

Third letter added to the classification, labeled ETS (Error, Trend, & Seasonal)

Possibilities for each component are:
Error = {A,M}, Trend ={N,A,Ad,M,Md} and Seasonal ={N,A,M}.

Total of 30 such state space models:

• 15 with additive errors
  
• 15 with multiplicative errors
  

ETS models can be selected using information criteria (AIC, \( AIC_c \), & BIC)

ets function from the forecast package can be used to estimate the models
- Leave blank or enter parameters

Questions?