A statistical method for estimating piecewise linear sales trends

Abstract

Due to the structural breaks in time series, estimated current trend of product sales differs between the case where data for the entire span are used and the case where only the most recent data are used. The purpose of this study is to establish the piecewise linear approximation (PLA) as a trend analysis method that accounts for the structural breaks. PLA uses the complete data to simultaneously estimate the breakpoints and the continuously connected trends, immediately before and after the break. Thus, PLA not only ensures the ease of interpretation of the results, but also eliminates the probability of using discretion by uniquely determining the current trend, making the estimated result reliable. The case study demonstrates the proposition that, several products’ sales trends and the necessity of determining an appropriate time span for data analysis, underwent changes at least once. The method’s validity is demonstrated by showing the changes of the sales trends immediately after analyzed store’s renovation. Data collection from the time period immediately prior to a structural break can help identify the factors changing the product sales.

Appendices

Appendix 1

The piecewise linear function’s objective variable $y$, explanatory variable $x$, and $m$ breaking points ${\psi }_{1},\dots ,{\psi }_{m}$, are given by the following Eq. (1)

$$y = \alpha + \beta_{0} x + \mathop \sum \limits_{s = 1}^{m} \beta_{s} \left( {x - \psi_{s} } \right)_{ + } ,$$

(1)

where

$$\left( z \right)_{ + } : = \left\{ \begin{gathered} z\quad z \ge 0 \hfill \\ 0\quad z < 0. \hfill \\ \end{gathered} \right.$$

(2)

Appendix 2

This study regards explanatory variable $x$ as time $t$ and applies the piecewise linear approximation to time-series data. Suppose the time interval is 7 days, $T$ is the period, the objective variable ${y}_{t}$ is on sale on the $t$-th week, and satisfies the following,

$$y_{t} = \alpha + \beta_{0} t + \mathop \sum \limits_{s = 1}^{m} \beta_{s} \left( {t - \psi_{s} } \right)_{ + } + \varepsilon_{t} ,$$

(3)

where the error term ${\varepsilon }_{t}$ is independent and identically normally distributed.

Appendix 3

The piecewise linear model’s parameters $\left(\alpha ,{\beta }_{0},\dots ,{\beta }_{m},{\psi }_{1},\dots ,{\psi }_{m}\right)$ estimated by the least-squares method, minimize the following squares error.

$$\mathop \sum \limits_{t = 1}^{T} \left\{ {y_{t} - \alpha - \beta_{0} t - \left( {\mathop \sum \limits_{s = 1}^{m} \beta_{s} \left( {t - \psi_{s} } \right)_{ + } } \right)} \right\}^{2} .$$

(4)

Number of breaks $m$ in the piecewise linear model is determined based on AIC (Akaike’s Information Criterion), which is given by

$${\text{AIC}} = - 2{\text{ln}}L + 2k,$$

(5)

where $L$ is the maximum value of the likelihood function and $k$ is the number of the model.

Appendix 4

The recent trend is defined by

$$\mathop \sum \limits_{s = 0}^{m} \beta_{s} .$$

If ${y}_{t}$ satisfies equation (3), the least-squares estimator is normally distributed. The t value is obtained from Muggeo (2008).