Untangling the complexity of market competition in consumer goods -A complex Hilbert PCA analysis
UUntangling the complexity of market competitionin consumer goods– A complex Hilbert PCA analysis
Makoto Mizuno † , Hideaki Aoyama , Yoshi Fujiwara School of Commerce, Meiji University, Tokyo 101-8301, Japan Research Institute of Economy, Trade and Industry, Tokyo 100-0013, Japan RIKEN iTHEMS, Wako, Saitama 351-0198, Japan Graduate School of Simulation Studies, University of Hyogo, Kobe 650-0047, Japan
August 24, 2020
Abstract
Today’s consumer goods markets are rapidly evolving with significantgrowth in the number of information media as well as the number of com-petitive products. In this environment, obtaining a quantitative grasp ofheterogeneous interactions of firms and customers, which have attractedinterest of management scientists and economists, requires the analysisof extremely high-dimensional data. Existing approaches in quantitativeresearch could not handle such data without any reliable prior knowledgenor strong assumptions. Alternatively, we propose a novel method calledcomplex Hilbert principal component analysis (CHPCA) and construct asynchronization network using Hodge decomposition. CHPCA enables usto extract significant comovements with a time lead/delay in the data,and Hodge decomposition is useful for identifying the time-structure ofcorrelations. We apply this method to the Japanese beer market dataand reveal comovement of variables related to the consumer choice pro-cess across multiple products. Furthermore, we find remarkable customerheterogeneity by calculating the coordinates of each customer in the spacederived from the results of CHPCA. Lastly, we discuss the policy and man-agerial implications, limitations, and further development of the proposedmethod.
Keywords: single-source data, purchase bahavior, media/ad exposure, complexHilbert principal component analysis, Hodge decomposition † Corresponding author: [email protected] a r X i v : . [ q -f i n . GN ] A ug Introduction
In rapidly evolving, highly competitive consumer goods markets, firms are facedwith a number of factors affecting business. These factors often veer away fromconventional rules of thumb or theory and can be mutually interacting, provid-ing the challenge of detecting the meaningful relationships between such factorswithout any strong assumptions. This challenge represents the complexity ofconsumer choice process caused by the proliferation of viable marketing instru-ments and competing products. Traditionally, consumer purchase could havebeen a function of price, radio or TV advertising. Recently, with the penetra-tion of the Internet and mobile devices the possible set of marketing instrumentshave been expanding. Some consumers might search for information using mo-bile phones while others might gain detailed information by visiting relatedwebsites after watching TV advertisements. If the strategy for mixing these in-struments is heterogeneous across competitive firms or products, the increasednumber of competitors would intensify the resulting complexity further.To grasp the time sequence of exposures to multiple marketing instruments,the concept of ’customer journey map’ is gaining popularity with practitionersand researchers [1] [2], for which some quantitative models have been proposed[3] [4]. This approach focuses on the consumer decision process for a singleproduct/firm only, ignoring competition among multiple products. Hence, itfails to reveal real consumer behaviors for the market with multiple competitiveproducts. In order to deal with dynamic market competition across products,multivariate time-series analyses are highly popular; and have been developedin econometrics and intensively applied to assess the long-term effectiveness ofmarketing instruments such as pricing or advertising. This method includes avariety of models such as the vector auto-regressive model [5], dynamic linearmodel [6], varying parameter model [7], and the Kalman filter [8]. These modelsare evaluated in terms of whether they satisfy the following conditions [6]: • Endogeneity: The recent quantitative marketing studies inspired by eco-nomics has increasingly treated variables of marketing instruments as en-dogenous variables, not as exogenous variables freely determined by firms.Suppose customers who are more likely to purchase a focal product aretargeted in an advertising campaign for the product. In this case, ob-served advertising data would reflect the potential purchases of the prod-uct. With no consideration for this endogeneity, the effects of advertisingon purchases would be overestimated. • Performance feedback: Even if firms do not foresee future performance,their marketing actions would be constrained by past performance dueto performance-based budget allocation. Hence the lagged effect of theperformance to the activity level of marketing instruments could exist. • Competitive reactions: Firms that compete in the same market, are sen-sitive to competitors’ actions, often causing retaliation. This has beenstudied as a hot issue in marketing science, using a variety of data andanalytical methods [9].Even multivariate time-series analyses often face serious difficulties in han-dling the complexity of current consumer markets, because of the limited number2f tractable parameters. Although most consumer goods markets are composedof many more firms/products and marketing instruments, the above extant mod-els have only been applied to cases with only three to five competitors and afew marketing instruments. The competition analyzed in this study consistsof 18 products and 4 marketing instruments (price, TV advertising, search viamobile devices or PCs, and web visits via mobile devices or PCs). Applying theconventional time-series analyses for these markets would cause an explosiveincrease in possible parameters to be estimated. To avoid this, researchers haverelied on their experience or research tradition and have been forced to imposestrong restrictions at the risk of losing crucial information on the complexity ofthe focal markets.Alternatively, we propose a novel method called complex Hilbert princi-pal component analysis (CHPCA) [10, 11] to unravel the complexity of theconsumer choice process across multiple competitive products. CHPCA wasdeveloped originally in econophysics as an extension of Principal Componentanalysis (PCA) to uncover temporal comovements among variables observed inthe macro economy [12] or foreign exchange markets [13]. CHPCA can han-dle massive high-dimensional time-series data without any strong assumptionsabout the phenomenon of interest. In addition, this method satisfies the above-mentioned three conditions for evaluating marketing models with multiple com-petitive products [6]: CHPCA satisfies the first condition (endogeneity) since ittreats all variables (marketing actions by instrument and performance indexes)as comoving variables. Additionally, this method satisfies the second condition(feedback from past outcomes) and the third condition (competitive reactions)since these are incorporated as part of possible comovements.Another advantage of CHPCA is its practicality. Practitioners can solve realproblems under time and effort constraints. First, as with ordinal PCA, users ofCHPCA may not need any prior knowledge or assumptions regarding the phe-nomenon. By using random rotation simulation (RRS), significant eigenmodes(principal components in PCA) can be selected automatically in a theoreticallyjustifiable manner. Second, the results of CHPCA are easily interpreted on thecomplex plane corresponding to each eigenmode following stylized procedures.Conveniently, the information obtained is integrated and visualized by a syn-chronized network with Hodge decomposition. Finally, this method provides askeleton of consumer choice process across competitive products using aggre-gate marketing data, which is widely available for the packaged consumer goodsmarkets. The model also provides information on heterogeneity in customerprofile.We recommend CHPCA for conducting exploratory analyses. Similar to thedivision of roles between exploratory and confirmatory factor analyses, CHPCAcan be complementary to traditional multivariate time-series analyses. CHPCAis used to generate hypotheses on possible causalities among a huge numberof variables while multivariate time-series analyses are used to rigorously testhypotheses. If a few critical relationships are detected as a result of the analysisusing CHPCA, we can apply quasi-experimental methods such as a regressiondiscontinuity design [14] or a propensity score method [15], which have beenused to prove causality in observational data.The remainder of this paper is organized as follows: In section 2, we describethe data for the beer market in Japan. We propose the application of CHPCAto understand the consumer choice process across multiple products in section3able 1:
Abbreviation and the description of the five kinds of time-series
Abbreviation DescriptionP Price per unit quantity (yen/m (cid:96) )Q Quantity purchased (m (cid:96) )Visit Visit to related web sites via mobile device or PC (seconds)TVAd Exposure to TV advertising (seconds)Search Search frequency via mobile device or PC (times)
In this study, we analyze the comovement of consumer purchases (quantities andprices paid) of beer and related marketing communication activities. The reasonfor our focus on this market is that it is a typical monopolistic-competitive mar-ket, where a few firms compete with differentiated products using a full rangeof marketing instruments such as TV advertising, web/mobile marketing, pricepromotion, etc., attracting attentions of economic policymakers and marketers.Hence, we use INTAGE Single-source Panel (i-SSP) data, which is the mostcomprehensive consumer database commercially operated in Japan measuringdaily purchases of a wide variety of consumer package goods and consumermarketing communication activities (exposure to TV ads, visits to web sites viamobile device/PC, and search activities via mobile device/PC. For the currentanalysis, we use these data for 365 days from April 1, 2013 to March 31, 2014(inclusive). The abbreviation and description of each time series is shown inTable 1.These data initially capture individual-level behaviors of the panel. For thisanalysis, however, we aggregated the data over all customers due to the limitedsize of the data. The potential heterogeneity among customers is represented asthe location of a few-dimensional space (see the Customer Profile Section.Thepurchase data are documented at the store-keeping unit (SKU) level while theadvertising and other communication variables are documented at the productlevel. In general, one product is composed of multiple SKUs. When mergingthe two types of data, the purchase data are summed across SKUs by theircorresponding product.There are 163 beer products from 14 firms in the original data. We selectedthe top 18 products according to the total quantity in the data during the statedperiod: Fig. 1 is the rank-size plot of some of the top products. The top 18products selected for the current analysis are those beyond the thin verticalline, Total Quantity > × . As is apparent in this plot, these products forma distinctive top group with a large gap between this group and the followers.4 ●●●●●●●●●●●●●●●●●●●●●●●●●● × × × × Total Quantity R a nk Figure 1:
Rank-size plot of the top selling products. /
04 2013 /
05 2013 /
06 2013 /
07 2013 /
08 2013 /
09 2013 /
10 2013 /
11 2013 /
12 2014 /
01 2014 /
02 2014 /
03 2014 / /
04 2013 /
05 2013 /
06 2013 /
07 2013 /
08 2013 /
09 2013 /
10 2013 /
11 2013 /
12 2014 /
01 2014 /
02 2014 /
03 2014 / Days Q u a n tit y ( × ) Figure 2:
Daily total quantities. Light gray: all products, Dark gray:Selected 18 products. Apparent periodic peaks correspond to Sun-days.
Furthermore, this top group roughly obeys the power-law indicated by the thindashed line, with [Rank] ∝ [Total Quantity] − . .The data for the top 18 products cover 64.9% of all the sales. Fig. 2 showsthe daily total quantities for both all products (in light gray) and the selected 18products (in dark gray). Periodic peaks apparent in this plot occur at weekendswhen quantity rises on Saturdays and peaks on Sundays. The high peak struc-ture at the end of the period; that is, at the end of March 2014, is explained bythe VAT hike from 5% to 8% on April 1, 2014. The products are shown with codes, the first letter of which corresponds tothe firm, and the second letter (digit) corresponds to the product: For example,the code “A1” means that the product is from firm ‘A’ and the product ‘1’. Withfive types of data for each product as listed in Table 1, we have 18 × We have data beyond this day for another several months. However, we decided to takethis one-year period to avoid bias from seasonal dependence and the strong influence from theincrease in quantity (and the downfall on and after April 1, 2014).
Top-selling 18 products and availability of data in days.
Rank Code Total Sales P Q Visit TVAd Search1 D2 1 . ×
365 365 158 328 2092 A4 7 . ×
365 365 25 287 1633 B1 7 . ×
365 365 164 306 1534 C3 6 . ×
365 365 29 173 765 B3 4 . ×
354 354 363 354 736 A3 3 . ×
357 357 29 249 347 A1 3 . ×
360 360 63 185 1178 B5 3 . ×
365 365 62 274 199 D3 2 . ×
339 339 3 0 010 D1 2 . ×
355 355 284 334 27011 B4 2 . ×
258 258 0 0 012 A6 1 . ×
330 330 0 0 013 B2 1 . ×
298 298 172 228 4114 C1 1 . ×
341 341 82 219 16815 A2 1 . ×
304 304 5 111 516 C2 1 . ×
342 342 39 216 2517 A5 1 . ×
276 276 0 0 018 C4 1 . ×
198 198 0 0 0 time series altogether. However as no communication activity was observed forsome products, we set the threshold to 51 days: we used only the time-seriesdata with 51 or more days of entry (more than or equal to once a week). Withthis threshold, we have 65 time series for purchases and related communicationactivities, which are listed in Table 2. 6able 3:
Descriptive statistics for the top-selling 18 products
Means andstandard deviations (in parentheses). The symbol ”—” corresponds to discardeddata because of sparsity (see text).
Rank Code P Q Visit TVAd Search1 D2 0.287(0.011) 27483.0(14758.6) 18.9(66.6) 4770.1(6042.2) 1.92(3.59)2 A4 0.294(0.014) 20596.9(14640.2) — 4164.6(6126.4) 1.30(3.07)3 B1 0.494(0.026) 20052.2(13850.3) 75.1(291.5) 4004.4(6704.8) 2.15(5.15)4 C3 0.287(0.015) 18064.9(13970.4) — 1271.7(3159.3) 0.54(1.76)5 B3 0.295(0.023) 11192.3(10534.8) 175.6(365.3) 4792.1(5725.9) 0.50(1.92)6 A3 0.350(0.025) 10926.4(11359.7) — 2964.1(4648.4) —7 A1 0.506(0.035) 9499.7(8786.6) 9.5(38.5) 4232.1(7763.3) 1.10(2.56)8 B5 0.301(0.023) 8424.9(8117.4) 19.3(118.3) 2861.0(3955.7) —9 D3 0.285(0.020) 8060.4(8510.2) — — —10 D1 0.567(0.039) 6205.9(5934.8) 73.7(184.8) 10051.1(10428.0) 3.81(6.15)11 B4 0.296(0.018) 5573.2(8665.4) — — —12 A6 0.304(0.024) 4975.0(6037.5) — — —13 B2 0.362(0.028) 4943.8(6515.0) 46.2(115.2) 1769.2(3712.4) —14 C1 0.555(0.046) 4823.9(5755.2) 10.6(42.9) 2486.8(4732.9) 1.46(3.41)15 A2 0.362(0.039) 4665.9(6790.5) — 1359.0(4166.9) —16 C2 0.508(0.041) 4202.3(5325.6) — 2019.6(5922.4) —17 A5 0.298(0.021) 3995.4(6606.3) — — —18 C4 0.296(0.028) 3884.6(6819.0) — — —
Table 3 summarizes the means and standard deviations for the time series. Sym-bol ”—” implies too sparse data due to no communication activity. We observethat these time series are highly volatile in temporal change. We then use thestandard method of subtracting the mean and dividing it by the standard devia-tion. Let us denote the resulting time series by x α ( t i ) where α = 1 , · · · , N (= 65)is the label for the time series, and t i = 1 , · · · ,
365 denotes the number of days.The mean and standard deviation of x α ( t i ) are 0 and 1 respectively, for whichwe apply our methodology explained in the next section.We depict, as a sample of x α ( t i ), the five types of time series for the topproduct “D2” ( α = 1 , · · · ,
5) in Fig. 3. Price per unit quantity (P) is mostlystable but has spikes of increasing or decreasing price change. Quantity (Q)has volatility due to growing and sluggish sales. The frequency of site visits viaPC or mobile device (Visit) have tranquility with sudden and short activities.The frequency of TV advertising exposure (TVAd) has weak periodic behaviorpresumably due to the TV advertising activities of the product’s firm and cor-responding exposure to customers. The frequency of searches via PC or mobiledevice (Search) have continuous activities with bursts.
Any set of real world time-series data contains information on the behaviorof individual time series and the inter correlations in the time series. In thisstudy, we are interested in inter correlations in the time series. To identify the7 QV i s it T VA d S ea r c h Figure 3:
Sample of time-series x i ( t ) for the top product “D2” for the fivekinds of time-series, namely, P, Q, Visit, TVAd, and search from top to bottom.8tructure and dynamics of the customer choice process, we extract informationon the inter correlation between price, sales, and other media approaches. Suchcomovement in the data set involves time lead and delay. Some time series followother time series because of direct and indirect causal relationships. Here, ouraim is to set up a methodology suitable for detecting inter relationships withtime delay.Principal component analysis (PCA) fulfills our goal partially. For thismethod, we calculate correlations between time series and identify the eigen-modes of the correlation matrix, which are independent comovements in thesystem. The larger the eigenvalue, the more significant the presence of theeigenmode. Some of the eigenmodes, however, are simply the result of randommovements in the system. To identify which modes are significant real comove-ments, people often apply random matrix theory, which predicts the eigenvaluesfrom the random time series. This method has several shortcomings.(i) When seeking comovements with time lead/delay, the time series is shiftedrelative to other time series to maximize the absolute value of the corre-lation coefficient. This is feasible for two time series but not so for a largenumber of time series. With 100 time series, for example, pair wise cal-culation is required for nearly 5,000 pairs. Then, there is the problem ofcombining them to obtain system-wide comovements.(ii) Random matrix theory (RMT) is practical on that the length of the timeseries ( T ) and the number of the time series ( N ) are both infinite with theirratio ( T /N ) kept finite, and all the time series has trivial auto correlation,none of which may be satisfied by the real data.To overcome these difficulties, we use CHPCA and rotational random simu-lation RRS.The former was originally introduced in [16, 17, 18, 19, 20] using the Hilberttransformation developed in [21, 22, 23, 24, 25] among others. The approachhas been successfully applied in several areas of natural science and economics[26, 27, 12, 13]. We further introduced improvements on CHPCA by [11].In CHPCA, we complexify each of the time series’ Hilbert transformationas an imaginary part and then calculate the complex correlation matrix. Weprovide a pedagogical explanation of the merits of this method. The Hilberttransformation, simply put, transforms each of the Fourier components in themanner cos ωt → − sin ωt and sin ωt → cos ωt . Therefore, the complexificationconverts cos ωt to e − iωt and sin ωt to i e − iωt ; clockwise rotation on its complexplane. Furthermore, the Hilbert transformation convertscos ω ( t + t ) = cos ωt cos ωt − sin ωt sin t → e − iω ( t + t ) . (1)We denote the complex time series obtained from x α ( t ) and standarized (sothat its means is equal to zero and its standard deviation is equal to one) as z α ( t ). Complex correlation coefficients (CCC) are defined as inner products ofone (complex and normalized) time series ( z α ( t )) with another; C αβ := (cid:88) t z α ( t ) z ∗ β ( t ) . (2) Hereafter, · ∗ denotes the complex conjugate of · .
9f the time series α and β are made of Fourier components of the same ω but with time constants t α and t β , the CCC has a phase factor proportional to t α − t β , the time-difference between the two time series. If the time series containmultiple Fourier components, the phase of the CCC gives a nonlinear mean ofthe time differences of each combination of the Fourier components. Thus,analysis of the resulting complex correlation matrix enables us to obtain a viewof comovements with system time-lag. By definition, this is one calculation thatavoids any pairwise optimization analysis required by PCA. The eigenmode e n of the complex correlation matrix C = { C αβ } is defined by the following: Ce n = λ n e n , (3)where the subscript n is defined as the eigenvalues λ n in descending order, λ ≥ λ ≥ · · · ≥ λ N . The eigenvalues satisfy an identity N (cid:88) n =1 λ n = N. (4)The time series are expanded in terms of the eigenmodes: x ( t ) = N (cid:88) n =1 s n ( t ) e n , (5)where the coefficient s n ( t ) is called the mode signal , satisfying λ n = T (cid:88) t =1 | s n ( t ) | . (6)In this sense, the eigenvalue λ n is the strength of the presence of the correspond-ing eigenmode e n .To avoid using the RMT, we employ RRS, introduced by [28]. This is done by(1) “‘rotating” each time series in time-direction (by attaching its end to the be-ginning) randomly, thus destroying the inter correlation between the time serieswhile preserving the autocorrelation; (2) calculating the CCC and its eigenval-ues several times (10 ∼ times typically); (3) comparing each distributionof the eigenvalues and the actual eigenvalue from the largest in descending or-der: identifying the eigenmodes whose eigenvalue is larger than that of the oneobtained in the step (3) as significant modes. This methodology overcomes theshortcoming of the RRS by allowing us to deal with data with nontrivial autocorrelation and T and N not so large.In this sense, the methodology of CHPCA with RRS is ideal for our purpose,which is to identify the customer choice process in our data. The eigenvalue distribution is shown in Fig. 4, where the ordinate is the cumu-lative eigenvalue L ( n ) := n (cid:88) k =1 λ n . (7)10
10 20 30 40 50 600102030405060 n L ( n ) Figure 4:
Cumulative eigenvalue L ( n ) defined in Eq. (7) . The green dots are for CHPCA and blue for PCA. As explained, the eigenvalueshows the rate of the presence of the corresponding eigenmode in the data.Therefore, this plot shows that CHPCA identifies the eigenmode more easilythan PCA. This is natural since PCA misses movements with lead/lag.The result of the RRS analysis of 10 times the RRS simulation is sum-marized in Fig. 5 for the eigenvalues n = 1 , , σ range is shown bythe horizontal error bars. Since the eigenvalues σ range and λ = 4 . , λ = 3 . , (8)respectively, these top eigenmodes take the share of (cid:112) ( λ + λ ) / (cid:39) . that is, , 35.7% of the data are due to comovements.The top and the second eigenvector components are shown in Fig. 6, whereeach component is shown by a marker specified by the product code at its topand the style shown in the legend. (The details of these eigenmodes are given inthe Appendix.) The horizontal axis is its phase, and the vertical axis its absolutevalue. The arbitrary overall phase in the eigenvector e n is chosen so that thecomponents representing purchase quantities are toward the right-hand side ofthe plots. By the definition of the complexification, the phase corresponds to thetime-variation; the components on the left move first, and the components to theright follow. We have changed the phase of prices by π to be consistent with thecommon knowledge that when the price goes down, quantity goes up. In theseplots, we also show the significance level of the absolute values of the componentsby the gray bands. Components with less absolute values have less significancein the respective eigenmode. To clarify this significance level, we add a randomtime series to the original data set and measure its absolute value in the firstand second eigenvectors. Repeating this simulation 100 times, we identify thedistribution of the absolute value. The shaded gray area bounded by a solidhorizontal line is 2 σ range. Therefore, the components above the gray zones arethe components with significant presence in the respective comovements.11 Eigenvalues, n = RR SP D F Eigenvalues, n = RR SP D F Eigenvalues, n = RR SP D F Figure 5:
The eigenvalues (thick ticks) with the corresponding RRSeigenvalue distributions (shaded bell-shapes) and their 2 σ ranges(horizontal bars) for eigenvalues n = 1 , , . The eigenvalues σ range and, therefore, are significant. The eigenvalues ● ● ●●● ●●● ● ●● ●● ●● ● ●■■ ■■ ■■ ■■■ ■ ■■■ ■ ■ ■ ■ ■◆◆ ◆ ◆◆ ◆◆◆▲ ▲ ▲▲ ▲▲ ▲ ▲▲ ▲▲▼ ▼▼▼ ▼▼ ▼▼ A1A1A2A2 A3A3 A4A4A5A5A6A6 B1B1 B2B2B3B3 B4B4 B5B5C1C1 C2C2C3C3 C4C4D1D1 D2D2 D3D3A1A1A2A2 A3A3A4A4 A5A5A6A6 B1B1B2B2B3B3 B4B4 B5B5C1C1C2C2 C3C3 C4C4 D1D1 D2D2 D3D3A1A1B1B1 B2B2 B3B3B5B5 C1C1D1D1D2D2A2A2 A3A3 A4A4B1B1 B2B2B3B3 C1C1 C2C2C3C3 D1D1D2D2A1A1 A4A4B1B1B3B3 C1C1C3C3 D1D1D2D2 π π π π Eigenvector No.2
Figure 6: The components of the first (upper) and the second (lower) eigenvec-tors. 13he comovement of marketing instruments and purchase quantities acrossproducts represented in Fig. 6 may still be too complicated for marketers tointerpret. Thus, we offer an additional method to reduce information obtainedfrom CHPCA focusing on synchronization of multiple time series.14
Hodge Decomposition and Synchronization Net-work
The complex correlation coefficient, C αβ , represents how strongly a pair of α and β are correlated possibly with lead and lag. The strength of the correlationis given by the magnitude ρ αβ := | C αβ | , (9)and the lead and lag can be measured by the phase θ αβ := arg C αβ . (10)Note that α leads β if θ αβ <
0, and α lags β if θ αβ > e − iωt (see Eq.(1)).If we consider all the pairs in the complex correlation C αβ , we have a com-plete graph in which every node α is connected to all the other nodes. It isdifficult to understand how individual α leads or lags others in a more sys-tematic way. To overcome this difficulty, we select comoving pairs with strongcorrelation in the following way, and then use the so-called Hodge decomposi-tion of a flow on a directed and weighted network, which we call synchronizationnetwork .First, we select pairs of α and β with • comovement: 0 < θ αβ < π/ • significant correlation: ρ αβ > ρ ∗ where ρ ∗ is a threshold given below.In the first condition, we consider only the region 0 < θ αβ < π/
2, because thecorrelation matrix satisfies the Hermite conjugate relation; that is, C βα = C ∗ αβ ,so that the pairs in the region − π/ < θ αβ < <θ αβ < π/
2. In the second condition, we determine the threshold ρ ∗ as follows.If ρ ∗ is too large, the number of pairs satisfying the condition is too small and,eventually, the graph becomes disconnected; if ρ ∗ is too small, the graph isalmost fully connected. In both cases, it would be difficult to understand thelead/lag relation. Therefore, we select ρ ∗ that connects the graph at its largestvalue. The resulting graph includes 65 nodes and 1,391 edges.Second, we use a mathematical method of ranking nodes according to theirlocation in terms of upstream and downstream flow in a directed network toidentify which nodes are leading and lagging in the entire relation. In our case,a flow is said to be present from α to β if 0 < θ βα < π/ ρ βα = ρ αβ > ρ ∗ with the amount of flow or weight, ρ αβ .We briefly recapitulate the method (see [29] for example), which is called Hodge decomposition . Denote the adjacency matrix of the binary and weightednetwork by A αβ = (cid:40) α to β, , (11)and B αβ = (cid:40) f αβ if there is a directed edge with a flow , , (12)15here f αβ is a flow from α to β , and it is assumed that f αβ >
0. Note thatthere can be such a pair of nodes that has both A αβ = 1 and A βα = 1 and alsothat has both f αβ > f βα > flow from α to β is defined by F αβ = B αβ − B βα . (13)Let us also define the net weight between α and β by W αβ = A αβ + A βα . (14)Note that F αβ is anti-symmetric while W αβ is symmetric.Hodge decomposition is given by F αβ = W αβ ( φ α − φ β ) + F (loop) αβ , (15)where F (loop) αβ is a loop flow; that is, divergence-free: (cid:88) β F (loop) αβ = 0 (16)by definition. φ α is called Hodge potential of node α .Rewriting Eq. (16), we have for each α = 1 , · · · , N , (cid:88) β L αβ φ β = (cid:88) β F αβ , (17)Here, L αβ is the so-called graph Laplacian defined by L αβ = δ αβ (cid:88) γ W αγ − W αβ , (18)where δ αβ = 1 if α = β and δ αβ = 0 otherwise. Given F αβ and W αβ , Eqs. (17)are simultaneous linear equations to determine the Hodge potential φ α of allthe nodes α .Note that simultaneous linear equations (17) are not independent of eachother. In fact, the summation over α of (17) is zero, as is easily shown, corre-sponding to the fact that there is a freedom to fix the origin of potential. It isnot difficult to prove that if the network is weakly connected; that is, connectedwhen considered an undirected graph, the potential can be determined uniquelyup to the choice of the origin [30]. In the following, we use the convention thatthe mean is zero: (cid:88) α φ α = 0 . (19)Thus, if we delete the loop flow, the remaining flow can be representedby a flow caused by the difference in potential between a pair of nodes. TheHodge potentials, therefore, can reveal which nodes are located in upstreamor downstream sides in the relative relationship of the directed network. Weemphasize that such information cannot be obtained simply by looking at thepairwise correlation among nodes because the entire connectivity of all the linksis required to discard the loop flow and to determine the potentials.16igure 7: The synchronization network’s graph layout. The vertical position ofeach node corresponds to its Hodge potential as a constraint in the force-directedalgorithm of the graph layout. Upstream (leading) nodes are located toward thetop while downstream (lagging) nodes are toward the bottom. The node no.9is not drawn as it has only one link and is not relevant to this visualization. Fig. 7 shows a layout of the synchronization network. The vertical positionof each node corresponds to its Hodge potential as a constraint in the force-directed algorithm of the graph layout. Upstream (leading) nodes are locatedtoward the top while downstream (lagging) nodes are toward the bottom.To depict a customer choice process covering all competitive products, Hodgepotentials are used to constitute the fundamental time sequence of the exposuresto marketing instruments (TV Ad, web/mobile site visit, search and price) andthe purchase quantity for each product. In Fig.8, the time is passing from topto bottom along the vertical axis. The distance of this axis can be expressedin a time scale such that the distance of one corresponds to 1.55 days. Thehorizontal axis is nominal, where 18 products are arrayed in an arbitrary order.For instance, for Product 1, just after its price decreases and the search behaviorincreases, both of which occur almost simultaneously, the purchase quantity(sales) increases followed by an increase in exposure to web/mobile sites andTV advertising.First, we trace the time sequence of variables within each product. For morethan half of the products (11 out of 18), a decrease (increase) in price leads to anincrease (decrease) in quantity to a certain extent. For some products, a changein price occurs almost simultaneously with a change in quantity. For thesetwo groups, the behaviors pf prices and quantities are consistent with standardeconomic theory. On the other hand, for Products 15 and 18, an increase(decrease) in quantity leads to a decrease (increase) in price. This phenomenon17igure 8:
The fundamental sequence of exposures to marketing instru-ments and purchases for each product.
The vertical axis represents theHodge potential, whose value is larger as it leads than others. Horizontally,products are arrayed in an arbitrary order. If the positions of two time seriesare closer, they are comoving almost simultaneously with each other.seems to be an anomaly as an effect of prices while it could be explained as anoutcome of rational behavior; for instance, it could emerge when the demand isexpanded by attracting new customers with lower willingness-to-pay [31].The increased exposures to TV advertising lag behind the increased pur-chases for seven of the 13 products that executed TV advertising in the ob-served period. This may suggest that TV advertising by firms is a reaction toan increase in demand, not as an upfront investment or, in the latter case, aftera significant time lag. The time sequence of web/mobile site visits or searches isless consistent across products than price or TV advertising. A possible reason isthat the exposures to these marketing instruments is less controllable for firms,reflecting the idiosyncratic nature of individual products. In that sense, thisinconsistency shows the advantage of our approach that analyzes the observeddata purely empirically without any strong assumptions.Second, we can compare the Hodge potentials horizontally between products,which indicates synchronization (comoving almost simultaneously) of marketinginstruments between different products of a firm or even between firms. Forinstance, as Fig.8 shows, a price cut and sales increase for Products 1, 2 and3, which are the main products of Firm 1, tend to be synchronized. That is,Firm 1 might coordinate price promotion consistently among their own productscompared to rivals. These variables seem to be synchronized also between Firms1 and 3, suggesting that these firms are mutually competing more intensively.As the potentials show that these firms tend to change prices before sales, theirmain weapon for competitive reaction is price promotion.It is noteworthy that the potentials for purchase quantity are relatively con-centrated within the narrow band for most products, implying that beer con-sumption is highly synchronized as a whole. The reason is easily explainedby the established knowledge that typical beer consumption increases duringhigher temperatures or special occasions such as weekends or holidays. A more18nteresting finding is the existence of a few products (Products 8 and 9 of Firm2) outside the band. These products are interpreted as satisfying some nichedemand in the market. Firm 2 seems to be differentiated since its pricing be-havior is not necessarily synchronized with Firms 1 and 3 as a whole. Anotherprominent feature of this firm is that customers visited its web/mobile site morefrequently while they seldom visited the sites of Firms 1 and 3 (or there may notbe a competitor). Customers may visit Firm 2’s site after making a purchase orexposure to TV advertising. On the other hand, customers seem to visit the sitein advance. Such differences might be due to variations in marketing strategies.
We found in the Method Section that there are two significant eigenmodes in theaggregate behavior of customers; the remaining N − individual behavior of customerscan be represented in terms of these two significant eigenmodes. Such represen-tation can provide deeper insight into how the two significant eigenmodes canbe interpreted by examining individual customer’s profiles such as their gender,age, income, other attributes, and their preferences for specific products.Let us denote individual customer’s time series by x p,α ( t ) ( p = 1 , . . . , P )where the index p denotes individual customers, and P is the total number ofcustomers in our data; P = 1 , α = 1 , . . . , N is the same index as used inthe Method Section with N = 65.We first complexify x p,α ( t ) into complex time series, denoted by z p,α ( t ), andstandardize (subtract mean and normalize by standard deviation) it preciselyin the same way as we did in the Method Section. Thus, we haveˆ z p,α ( t ) = z p,α ( t ) − (cid:104) z p,α (cid:105) σ p,α . (20)If x p,α ( t ) is identically equal to zero during all of time t for some α (e.g.,a customer was not exposed to any advertising), we use the convention thatˆ z p,α ( t ) = 0.Then, we project the time series to a space spanned by two significant eigen-vectors (we term it customer space hereafter); that is, a n,p ( t ) = N (cid:88) α =1 ( e n ) ∗ α ˆ z p,α ( t ) , (21)for the two significant modes n = 1 , e n ) α is the α -th component ofthe eigenmode e n . To locate each customer in a space spanned by the twoeigenmodes, we calculate the temporal mean of the squared magnitude of theprojected time series a n,p ( t ), namely, X n,p = 1 T (cid:88) t | a n,p ( t ) | , (22)which gives us two-dimensional coordinates for each customer p .Fig. 9 shows the resulting spatial representation of Eq. (22) for all the P customers. Recalling (6) in the Method Section, each coordinate’s value X n,p Individual customer’s projected representation for the twosignificant modes. See Eq. (22) for the representation. can be compared with the eigenvalues λ and λ , which are numerically givenby Eq. (8). We observe that there are customers whose positions are consistentwith Eq. (8) in the sense that X ,p /X ,p ∼ λ /λ . There are, however, morediversified customers in the two-dimensional space. Such diversification tellsus that the location of each customer might be related to the heterogeneity incustomer behavior.To assess how the customer space is associated with each customer’s pro-file, we conduct regression analysis where either of the coordinates in the cus-tomer space, X ,p or X ,p , is used as a criterion variable, and multiple variablesrepresenting customer profiles are used as explanatory variables, which are allavailable in our data set. First, we select age (nine-point scale from age 20 to24 to age 60 or older), gender (0 for male, 1 for female), marital status (0 forunmarried, 1 for the married), personal income (nine-point scale), and house-hold income (five-point scale) by preliminary analysis. Second, to capture eachcustomer’s beer preference, total purchase frequency and quantity (ml) over allproducts are added to the predictors. As the results of Model 1.1 and 2.1 ofTable 4 show, the estimated coefficients are significant only for age (0.1% signifi-cant), total purchase frequency (0.1% significant), and quantity (5% significant)for both X ,p or X ,p . The values of R indicate that most of the variations arenot explained by the above predictors.Alternatively, we replace total purchase quantity with each product’s pur-chase quantities to capture individual differences in product-level preference.The results are presented for Model 1.2 and 2.2 in Table 4. Compared to theabove models, the coefficients for age and total purchase frequency are consis-tently significant while the adjusted R s are slightly increased, implying separat-ing total purchase quantity into purchase quantity for products may marginallyimprove the model fit. The coefficients of a few products are significant fromthe 0.1% to 10% level for X ,p and X ,p . Thus, locations in the customer space20able 4: Regression of coordinates of customer space to customer pro-files (aa: p < . p < .
01; b: p < .
05; c: p < .
Model 1.1 Model 1.2 Model 2.1 Model 2.2Criterion Variable: coord. for 1st eigen mode: X ,p coord. for 2nd eigen mode: X ,p Explanatory Variables: coef. s.e. coef. s.e. coef. s.e. coef. s.e.Intercept . . aa . . aa . . aa . . aa Age . . aa . . aa . . aa . . aa Gender . . . . . . . . . . . . . . . . . . . . − . . − . . − . . − . . − . . − . . . . aa . . aa . . aa . . aa Total Purchase (m (cid:96) ) . . b — — . . b — —Purchase – A1 . . c − . . c A2 . . . . . . . . . . . . . . . . − . . . . . . . . . . − . . − . . − . . − . . . . . . b . . − . . − . . − . . − . . . . . . aa C4 . . . . . . aa . . − . . − . . . . aa . . . . . . . . . . could be explained to some extent by age, purchase frequency at the categorylevel, and purchases of some remarkable products. However, it should be notedthat most of the variations in the customer space remain unexplained. In otherwords, the individual locations in a customer space might reflect an infinite num-ber of factors, only some of which could be measured via customer surveys orpurchase history tracking. Hence, our proposed projection method contributesto the evaluation of individual deviations from a representative customer choiceprocess. The proliferation of marketing instruments and competitive products are ren-dering the consumer choice process increasingly complex to define. Existingmethods used for this purpose ignore the existence of competitive products orunderestimate the range of competition, even in cases where most customersconsider multiple alternative products by searching or shopping. One of themain reasons for this is the limited ability of the existing methods, such asmultivariate time series analyses, to handle the complexity caused by multipleproducts competing with multiple instruments. The market with 18 productsand 5 variables (4 marketing instruments and one purchase quantity), ana-lyzed in this study, is not tractable without strong assumptions to drasticallyreduce parameters. Our proposed CHPCA overcomes this limitation withoutany strong assumptions. Furthermore, CHPCA’s supplemental methods, syn-21hronization network and Hodge decomposition, can be used to summarize andvisualize the results to be more interpretable.This study shows that a set of our proposed methods could be used toeffectively understand the consumer choice process embedded in enormouslyhigh-dimensional time-series data. The application of our method to the beermarket in Japan derives some interesting findings. First, for most products onthe market, the increase (decrease) in a product’s price leads to the decrease(increase) in its purchase quantity (as the standard economic theory predicts)with a few exceptional cases. Second, the exposure to TV advertising increaseslater than the purchase quantity in many cases. Simply put, consumers noticea price change in a store, buy a product, and are then exposed to TV ads later.Third, the timing of consumers’ visits to product web/mobile sites or the usageof search engines varies across products. Fourth, we find synchronization acrossproducts, in particular within each firm, rather than across firms. These find-ings imply that individual firms are heterogeneous, each adopting a distinctivecoherent marketing strategy.The fourth point has an important implication for economic policy making.Synchronization of marketing strategy between firms indicates that their behav-ior could be competitive if prices are decreasing but be collusive if prices areincreasing. The latter case should attract a strong interest of anti-trust agencies.Our result might deny this possibility, while suggesting another difficulty in eco-nomic policy making. If corporate behaviors are heterogeneous than expected,policymakers must allow for such heterogeneity in evaluating the effectivenessof planned policies in advance. Firms should not be treated as an aggregationof representative agents.For managers in firms, the above-mentioned findings may be instructive toimprove their marketing practices. If TV advertising reaches customers laterthan their purchase, the timing of advertisement should be reconsidered, sinceconsumers cannot choose the timing of exposure to it. On the other hand, ifthe ad campaign intends to reinforce customer retention, the marketing practicecould be successful. Our method reveals which products could be real rivals,without any prejudice, by showing synchronization of marketing instrumentsbetween competitive products. This information can be used by marketers toinvestigate the dynamics of competition or substitution for their product.Researches in marketing science have traditionally measured brand loyalty byincorporating own-product inertia [32] and the long-term effect of advertising byincorporating ad stock into the model [33]. Our method could implicitly capturethe effect of brand loyalty that is embedded in the comovement of marketinginstruments. How to explicitly quantify the magnitude of loyalty or inertia maybe a possible challenge for us. We have already attempted incorporating adstock with exponentially-distributed weights and alternative parameters. Asthe result was not sensitive to such modification, we tentatively conclude thataccounting for the long-term effects of advertising in our method is not a priority.In our opinion, it is desirable to develop further extension of our method,aiming to offer some numerical indications for policymakers and marketers, toenable better actions, by proceeding to sensitivity analysis/simulations, usingthe synchronization structure discussed in this paper. A fluctuation-dissipationapproach [34] may be useful, assuming that the impact of external stimuli doesnot change the correlation structure, but simply excites some of the structure.In other words, this approach deals with small perturbations on the existing22tructure, which is, in general, true when promoting specific products or regu-lating specific firm’s behavior. Research in this direction, therefore, would befruitful.
Acknowledgments
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PPENDIX: The eigenmodes 1 and 2: detail
We list the components of the first eigenmode with absolute value above the 2 σ range in Fig. 5 and Fig. 6 and, similarly, for the second eigenmode in Fig. 7 andFig. 8. Brand Variable Phase Abs.A1 TVAd 2.22 0.21A1 WebV 2.57 0.07A1 MobV 2.85 0.10A1 Q 4.91 0.12A1 P 5.09 0.08A2 WebV 2.90 0.14A2 TVAd 2.93 0.09A2 MobV 3.08 0.08A2 P 5.57 0.07A3 MobV 2.59 0.18A3 WebV 2.8 0.20A3 TVAd 2.93 0.08A3 Q 4.83 0.09A3 P 5.32 0.08A4 TVAd 0.44 0.08A4 Q 5.29 0.07A6 P 3.88 0.10A6 Q 5.49 0.11A7 P 4.85 0.11A7 Q 5.99 0.14B1 WebV 1.82 0.07B1 TVAd 2.12 0.08B1 Q 5.41 0.09B1 P 5.83 0.07B2 Q 3.90 0.08B2 TVAd 4.81 0.08B3 WebS 3.27 0.07B3 TVAd 5.39 0.14B4 Q 4.91 0.09
Table 5: List of components in thefirst eigenmode with absolute valueabove the 2 σ range (to be continuedto Fig. 6) Brabd Variable Phase Abs.B5 WebV 3.52 0.08B5 TVAd 4.18 0.15B5 P 4.21 0.08B5 Q 5.03 0.07C1 WebV 3.62 0.07C1 TVAd 4.57 0.11C1 Q 5.33 0.08C2 MobS 1.86 0.10C2 MobV 1.86 0.10C2 Q 5.63 0.11C2 P 6.26 0.10C3 TVAd 5.50 0.12C3 Q 5.89 0.25C3 P 6.16 0.27C4 Q 0.96 0.11C4 WebS 2.89 0.07C4 WebV 3.37 0.09C4 TVAd 4.68 0.11C4 P 6.12 0.11C5 Q 2.88 0.22C5 P 6.01 0.33D1 Q 5.24 0.09D1 P 5.47 0.11D1 TVAd 5.84 0.15D1 WebS 6.07 0.07D2 MobV 3.05 0.06D2 WebV 3.54 0.07D2 WebS 5.04 0.08D2 P 5.52 0.09D2 Q 5.70 0.09D3 TVAd 3.87 0.19D3 Q 4.77 0.07
Table 6: -continued from Fig. 526 rand Variable Phase Abs.A1 TVAd 5.49 0.10A1 Q 5.70 0.17A1 P 5.84 0.09A2 TVAd 4.52 0.09A2 Q 5.73 0.11A3 MobS 4.77 0.07A3 TVAd 5.2 0.08A3 Q 5.54 0.18A3 P 5.92 0.08A3 MobV 6.06 0.07A4 WebS 4.09 0.08A4 TVAd 4.76 0.13A4 WebV 5.24 0.12A4 Q 5.80 0.20A4 P 6.11 0.14A6 P 0.73 0.06A7 Q 4.92 0.12B1 P 5.19 0.10B1 WebS 5.28 0.07B1 Q 5.36 0.18B1 TVAd 5.53 0.08B2 Q 4.06 0.16B2 WebS 5.02 0.07B2 P 5.27 0.09B2 TVAd 5.64 0.11B3 WebV 2.91 0.08B3 TVAd 5.21 0.09B4 Q 5.15 0.19B4 P 5.46 0.12