Table 1.
Abbreviation and the description of the five kinds of time-series.
Fig 1.
Rank-size plot of the top selling products.
Each dot represents each products with the quantity Q of sales (horizontal) and the rank in descending order of Q (vertical). Filled dots denote the selected 18 brands. Note that both axes are in logarithmic scales.
Fig 2.
Light gray: all products, Dark gray: Selected 18 products. Apparent periodic peaks correspond to Sundays.
Table 2.
Top-selling 18 products and availability of data in days.
Fig 3.
Time-series for the top product “D2” and for the five kinds of time-series, namely, P, Q, Visit, TVAd, and search are shown from top to bottom.
Table 3.
Descriptive statistics for the top-selling 18 products.
Fig 4.
Cumulative eigenvalue L(n) defined in Eq (7).
This shows how the first n eigenmodes can explain the time-series with respect to the cumulative eigenvalue L(n), in other words, the strength of eigenmode (see also Eq (6)).
Fig 5.
The top three eigenvalues and the corresponding RRS eigenvalue distributions.
The eigenvalues are shown with thick ticks and the RRS eigenvalue distributions are shown with shaded bell-shaped curves. Horizontal bars show the 2σ ranges. The eigenvalues #1 and #2 are above the RRS 2σ range and, therefore, are significant. The eigenvalues #3 and below are not.
Fig 6.
The components of the first (upper) and the second (lower) eigenvectors.
Each component is indicated by a marker with the label of product code and style. Horizontal position is each component’s phase, and vertical position is its absolute magnitude. See more in the main text.
Fig 7.
The synchronization network’s graph layout.
The vertical position of each node corresponds to its Hodge potential as a constraint in the force-directed algorithm of the graph layout. Upstream (leading) nodes are located toward the top while downstream (lagging) nodes are toward the bottom. The node no.9 is not drawn as it has only one link and is not relevant to this visualization.
Fig 8.
The fundamental sequence of exposures to marketing instruments and purchases for each product.
The vertical axis represents the Hodge potential, whose value is larger as it leads than others. Horizontally, products are arrayed in an arbitrary order. If the positions of two time series are closer, they are comoving almost simultaneously with each other.
Fig 9.
Individual customer’s projected representation for the two significant modes.
Each location is calculated by the temporal mean of the squared magnitude of the projected time series (see Eq (24)).
Table 4.
Regression of coordinates of customer space to customer profiles.