Time-Varying Transition Probability Matrix Estimation and Its Application to Brand Share Analysis

<div><p>In a product market or stock market, different products or stocks compete for the same consumers or purchasers. We propose a method to estimate the time-varying transition matrix of the product share using a multivariate time series of the product share. The method is based on the assumption that each of the observed time series of shares is a stationary distribution of the underlying Markov processes characterized by transition probability matrices. We estimate transition probability matrices for every observation under natural assumptions. We demonstrate, on a real-world dataset of the share of automobiles, that the proposed method can find intrinsic transition of shares. The resulting transition matrices reveal interesting phenomena, for example, the change in flows between TOYOTA group and GM group for the fiscal year where TOYOTA group’s sales beat GM’s sales, which is a reasonable scenario.</p></div

The estimated directed graph for “2007-1Q”.

<p>The estimated directed graph for “2007-1Q”.</p

Quarterly sales share of manufacturers from the year 2007 to the year 2015.

<p>Quarterly sales share of manufacturers from the year 2007 to the year 2015.</p

Averages and standard deviations of the sales shares.

<p>Averages and standard deviations of the sales shares.</p

Visualization of estimated consumers’ flow from a series of data in 2008-4Q and 2009-1Q.

<p>Visualization of estimated consumers’ flow from a series of data in 2008-4Q and 2009-1Q.</p

Example of the transition of distribution <i>π</i> through the actions of the transition matrix.

<p>Example of the transition of distribution <i>π</i> through the actions of the transition matrix.</p

Observed sales share data of manufacturers from the year 2007 to the year 2015.

<p>Observed sales share data of manufacturers from the year 2007 to the year 2015.</p