Abstract
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.
Highlights
Multivariate time series recording of actual phenomenon may have dynamics based on an intrinsic variable structure
It is impossible to uniquely determine the transition matrix from sales share data only, by making some natural assumptions on the transition matrix, we propose a method to estimate all of the transition matrices for every observation
For the sake of simplicity in analysis and visualization, among all manufacturers, the top 14 sellers (BMW Group, Chrysler Group, Daimler Group, FCA (Fiat Chrysler Automobiles), Ford Group, GM Group, PSA (Peugeot Societe Anonyme), Renault-Nissan, VW Group, Suzuki, Toyota Group, Honda, Mazda, and Hyundai-Kia Group) are used singly, and other manufacturers are grouped and named “Others.” the row sales data are transformed to the form of “sales share,” namely, the amount of sales is normalized to the ratio, which is regarded as a stationary distribution at a certain quarterly unit
Summary
Multivariate time series recording of actual phenomenon may have dynamics based on an intrinsic variable structure. The transition matrix characterizes the shifts in consumers’ preferences towards different products in terms of probability. We assume that the (i, j)-element of the transition matrix is the probability that a consumer who previously bought the i-th product purchases the j-th product instead. Information on changes in consumers’ product purchases is necessary for understanding competition between products in the market. The intrinsic structure of transition matrices, cannot be directly observed because individuals’ purchasing data are typically difficult to obtain. Individuals’ data are difficult to handle because of privacy issues. Owing to these issues, marketing data are often limited to product sales amounts such as point-of-sales
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