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!