Post on 05-Aug-2020
Forecasting Compositional Time Series:
A State Space Approach
Ralph Snyder (Monash University, Australia)
Keith Ord (Georgetown University, USA)
Anne Koehler (Miami University, Ohio, USA)
Keith McLaren(Monash University, Australia)
Adrian Beaumont (University of Melbourne, Australia)
Topics to be covered
• What is compositional data?
• Automobile sales in the USA
• Brief literature review
• The log-ratio model
• Compatibility and the choice of a base series
• Results for the automobiles data
• Discussion
Compositional Time Series
• Compositional time series relate to proportions
oGeochemical analysis of a rock sample
oMarket shares of competing products
oRelative employment levels in different sectors of the
economy
• May have data only on proportions, or on both proportions
and total
oEven when the total is available, a decomposition into
proportions and total may offer a more effective
approach to forecasting
• Key reference: John Aitchison (1986) The Statistical
Analysis of Compositional Data. London: Chapman and Hall
Annual Series of Automobile Sales for 1961-2013
Six major groups (grouped from 40 brands listed by Ward’s):
• American: GM, Ford, Chrysler [3 series]
• Japanese: Honda, Isuzu, Mazda, Mitsubishi, Nissan, Subaru, Suzuki
and Toyota [1 series]
• Korean: Hyundai and Kia [1 series]
• Other: BMW, Daimler, Volkswagen and Other [1 series].
Note 1: Japanese vehicles were sold in the USA from 1965 onwards and
Korean vehicles from 1986 onwards
Note 2: Data from Ward’s Auto Group http://wardsauto.com/data-center
Plots of Relative Shares, 1961 - 2013
Brief Literature Review
• The most common approach is the log-ratio method
introduced by Aitchison (1986) for cross-sectional data
• Used for time series by Quintana & West, 1988;
Brundson & Smith, 1998) among others
• Alternative approaches include the use of the Dirichlet
distribution (Grunwald, Raftery & Guttorp, 1993) and a
hyper-spherical transformation (Mills, 2010)
• We consider the log-ratio transformation integrated with
vector exponential smoothing (Hyndman, Koehler, Ord &
Snyder, 2008; De Silva, Hyndman & Snyder, 2009, 2010)
The log-ratio model
• Assume (r+1) series of equal length n [unequal lengths
considered later] denoted by: {𝑧𝑖𝑡; 𝑖 = 0,1,⋯ , 𝑟; 𝑡 = 1,2,⋯ , 𝑛}
• Specify the log-ratios 𝑦𝑖=𝑙𝑛 𝑧𝑖/𝑧0 where 𝑧0 is the base series
• Reverse transformation: 𝑧𝑖 =𝑒𝑦𝑖
1+ 𝑒𝑦𝑗
, 𝑖 = 1,2,⋯ , 𝑟;
𝑧0 = 1 − 𝑧𝑗
• With only two series, the standard logistic approach results;
with more than two series, the analysis must be invariant to
the choice of the base.
• Each of the log-ratio series may be modeled using simple
state space (or ARIMA) methods but the error terms will
clearly be dependent
Forecasting Methods
State-space model ARIMA model Forecasting method
Local Level (LLM) ARIMA(0,1,1) Simple exponential smoothing
Local Trend
(LTM)
ARIMA(0,2,2) Linear exponential smoothing
Local Momentum
(LMM, α=1)
ARIMA(0,2,2),
restricted
Simple exponential smoothing
for first difference
State space models and their (ARIMA)
reduced forms
Estimation Issues
• The state space models have a finite time starting point and do
not require and assumption of stationarity, making it possible to
accommodate series starting at different points in time.
• Use maximum likelihood to estimate both model parameters and
seed states [𝑟 series, 𝑘 seed values per series corresponding to 𝑘state variables, 𝑝 ‘process’ parameters] plus 𝑟(𝑟 + 1)/2parameters in the variance matrix
oUse the concentrated LF to reduce the dimensionality of the
parameter search space to 𝑟𝑘 + 𝑝
oHeuristic seed values may be used, albeit with some loss of
efficiency, reducing the dimensionality to 𝑝
o If the parameters were different for each series we would typically
have the number of parameters remaining 𝑝 = 𝑟𝑘. When faced with a
large number of short series, estimation problems would arise!
oBUT…
Compatibility
• Consider two formulations: one using series 0 as the base and the other using series 1 as the base.
• The two formulations are compatible if and only if their sub-models share a common structure
o [INFORMALLY] You get the same results from each formulation
oCompatibility implies that each sub-model is the same [e.g. all sub-models are LTM (local trend models)]
oCompatibility further implies common parameters in each sub-model [e.g. common (α, β) in the LTM]
oProofs of these results are given in the paper
The Implications of Compatibility
• The number of process parameters is reduced to 𝑘 for any
number of series, plus 𝑟(𝑟 + 1)/2 parameters in the variance
matrix
• The variance matrix will change with the formulation so the
model is akin to Zellner’s SUR (Seemingly Unrelated
Regression) structure
• Any subset of the total number of series can be analyzed
consistently without regard for the other series, although this
may result in some loss of efficiency
oFor example, we might examine {Coca-Cola, Pepsi Cola}
but could ignore the remainder {Dr. Pepper, Mountain
Dew, all other soft drinks}
Results for the Automobiles Data
Model Diagnostics [Using a chi-square statistic for the error terms]
Prediction Intervals for 2013(computed by simulation)
Predicted probabilities of an increase in market
share, using the Local Trend Model
Discussion
• The log-ratio approach enables time series analysis for proportions and the state space formulation can accommodate series starting at different times.
• The compatibility requirement is both necessary and sufficient to ensure that:
oThe choice of base series is irrelevant
oThe process parameters are common to each series, greatly reducing the number of parameters to be estimated
oAny subset of the series may be analyzed consistently without regard to the others
• Prediction intervals for individual ‘products’ must be developed by simulation methods
• Explanatory variables may be introduced; their coefficients will differ across series
Summary of Main Points
• Use of non-stationary rather than stationary forecasting methods
for composite data.
• Connects exponential smoothing with the log-ratio transformation.
• Shows that a common method and common parameters are
necessary for predictions to be invariant to the choice of base
series.
• Streamlines the search for maximum likelihood estimates by
concentrating the variance matrix of the errors out of the likelihood
function.
• Handles series of unequal length arising from late entrants or early
exits of market players.
• Local momentum model may have appeal as a conceptual
extension of simple exponential smoothing.
The working paper may be accessed at:
http://www.buseco.monash.edu.au/ebs/pubs/wpapers/
2015/wp11-15.pdf
THANK YOU!