Discussion Paper No. 107
A Multi-generation Product Diffusion Modelwith Social Media Effects-Accerelating Effect of Social Mediaon Leapfrogs and Switches
by the iPhone 6 Battery Problem 2016–2017 Yinxing Li and Nobuhiko TeruiFebruary, 2020

Data Science and Service Research
Discussion Paper

Center for Data Science and Service Research
Graduate School of Economic and Management
Tohoku University27-1 Kawauchi, Aobaku
Sendai 980-8576, JAPANA Multi-generation Product Diffusion Model with Social Media Effects-Accerelating Effect of Social Media on Leapfrogs and Switches
by the iPhone 6 Battery Problem 2016–2017 -Yinxing Li and Nobuhiko Terui 1

Graduate School of Economics and Management
Tohoku University
Sendai, Japan

February, 2020

1

Tohoku University, Graduate School of Economics and Management, Kawauchi Aoba-ku,

Sendai, 980-8576, Japan. yinxing.li.a8@tohoku.ac.jp; terui@tohoku.ac.jp
Terui acknowledges a grant by JSPS KAKENHI Grant Number (A)17H01001. Li acknowledges
a grant by JSPS KAKENHI Grant-in-Aid for Research Activity start-up 19K23187.

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A Multi-generation Product Diffusion Model with Social Media Effects-Accerelating Effect of Social Media on Leapfrogs and Switches
by the iPhone 6 Battery Problem 2016–2017 -Abstract

This paper proposes a multi-generational model that captures the direct and indirect effects of social media on product diffusion. Direct effects appear in the adoption ratefunction as covariates, and indirect effects are imbedded in hierarchical models connecting diffusion parameters to successive generations. Unlike previous multi-generational
diffusion models, ours forecasts sales of new-generation products before launch usingsocial media as the leading indicator and a hierarchical model connecting successive
diffusion parameters. Empirical results show our model forecasts more precisely and reveals how social media influenced sales of smartphones, particularly leapfrogging andswitching to other generations and competing products, as Apple contended with defective batteries in the iPhone 6 during 2016–2017.

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1. Introduction
Frameworks of diffusion studies generally follow the Bass (1969) model that assumes asocial network is homogeneous and fully connected and that new adopters enter the market influenced by innovators and producers' marketing communications. Studies traditionally attribute the internal effect to the influence of word of mouth.
Peres et al. (2010) trace how diffusion studies since 1990 have transitioned from interpersonal communication (Mahajan, Muller, and Bass, 1990; Mahajan and Wind, 2000) to
more general interactions, including social interdependence (Goldberg and Lilien, 2010;
Van den Burte and Lilien, 2001).
Diffusion of multi-generational products also has been investigated using the Bassmodel by connecting single-generation models sequentially. Since Norton and Bass
(1987) proposed a diffusion model applicable to successive generations, many studies,including Mahajan and Muller (1996) and Jiang and Jain (2012), have extended it tomulti-generational diffusions using sales and marketing mix data during the initial period after launch.
Alongside such data, this study incorporates data from consumers communicatingover social media and proposes a multi-generational diffusion model that detects the effects of social media before and after a product launches. It advances the literature inseveral respects. First, our model uses data other than post-launch sales to track diffusion of new products. In addition to price and competitors featured in the extended Bassmodels in Von Bertalanffy (1957), Mahajan and Muller (1981), Eastingwood et al.
(1983) and Bewley and Fiebig (1988), we construct covariates by extracting text features from social media and incorporating them as social media effects into the adoption3

rate function of a multi-generational diffusion model, implying the direct effect of social
media.
Second, earlier diffusion models assume that key parameters of market potential andimitation rates are independent among generations when the innovator parameter is given.
Our model structures generation-specific parameters hierarchically to depict the mechanism underlying parameter shifts between generations. Our structural model of parameter
shift across generations facilitates forecasting sales of new-generation products before
launch. The term of social media effect is incorporated in the hierarchical models and it
affects indirectly to sales.
Third, our model measures the effects of social media on sales, particularly leapfrogs
and switches, by the defective battery problem that plagued the iPhone 6 since early November of 2016 and it became a hot topic in social media until the late of 2017. This
“Battery Problem” is explained in detail in section 5. To measure this effect, we proposea labeled dynamic topic model as a hybrid of the labeled topic model (Daniel et al., 2009)
and the dynamic topic model (Blei and Lafferty, 2006). Our model employs “battery” apriori as the specific dynamic topic that captures the shift in distribution of topics overtime. We show empirically that social media reactions to iPhone's battery problem accelerated switching and leapfrogging to later-generation iPhones and competing products.
Section 2 discusses the effects of social media on multi-generational diffusion. Section
3 explains our model. Section 4 reveals empirical results for sales of successive generations of iPhones and shows our model predicts sales of products pre-launch by recognizing social media effects. Section 5 shows the accelerating effects of the iPhone 6 battery
problem on leapfrogging and switching to later-generation iPhones and competing
phones. Section 6 concludes.

42. Social Media Effects and Diffusion
2.1 Multi-generational Diffusion Model
Research into the diffusion of product generations adopts the framework of multi-generational diffusion models. The main differences among earlier models are their assumptions about key parameters and their use of marketing mix variables. Among studies distinguished by their key parameters, Norton and Bass (1987) assume that market potential
m for a generation of product depends on innovation and imitation parameters p and q are
constant across generations. Mahajan and Muller (1996), Jun and Park (1999), Kim et al.
(2000), Danaher et al. (2001), and Jiang and Jain (2012) assume the constancy of p acrossgenerations, generation-specific market potential, and imitation parameters. Jiang (2010)
and Guo and Chen (2018) assume that all parameters vary across generations.
Among diffusion studies distinguished by marketing mix variables, Robinson and
Lakhani (1975) insert a price effect term into the adoption rate function, and Bass (1980)
introduces price effects. These studies show that incorporating price information improves model performance. Horsky and Simon (1983) incorporate an advertising variabledirectly into sales, not into adoption rate function, and Horsky (1990) incorporates jointly
advertising and price information.
The adoption rate function in Bass et. al (1994) includes two marketing mix variables:
rates of change in prices and advertising expenditures relative to expenditures at productlaunch. Doing so permits comparing the effects of marketing variables. Jiang and Jain
(2012) extend this model to a multi-generational diffusion model.
2.2 Labeled Dynamic Topics on Social Media
Marketing studies consider the effects of social media by extracting latent topics andtheir features from large-volume documents and modeling them as covariates. Natural5

language processing is noteworthy in this regard, as indicated by the Latent Dirichlet Allocation (LDA) model of Blei et al. (2003).
Recent research uses topic models to analyze text data for marketing applications. The
LDA model of Tirunillai and Tellis (2014) incorporates consumer reviews into five setsof marketing data to extract dimensions (topics) from UGC (User Generated Content) forcomparison among markets. They find that some topics resonate across multiple markets
and others only in certain markets. Through sentimental analysis they tag topics for betterinterpretation and show that multidimensional scaling via LDA captures the dynamics ofbrand positioning. Netzer et al. (2012) exploit the co-occurrence of words and semantic
network analysis to derive market structure from online consumer forums. Their study
highlights use of text mining to indicate marketing effectiveness and how marketers canaffect brand position.
Li and Terui (2018) incorporate social media effects into a diffusion model to discuss
how they influence market potential and internal parameters. Analyzing text data of SNS
on mobile phones, they extract subjective and objective features by naïve Bayes and topic
analysis respectively, then use them as covariates of time-varying market potential andimitation parameters. They show how social media affect single-generation diffusion ofmobile phones and enhance forecasts. The supervised topic model of Ansari et al. (2018)
identifies topics hidden in product reviews that reveal consumer preferences. Their stochastic variational Bayesian approach yields fast and scalable inferences from big data.
The dynamic topic model (DTM) is a generative model that reveals the evolution of
topics hidden in collections of documents. Blei and Lafferty (2006) extend the LDA
model to a dynamic model for handling sequential documents. They describe the dynamics of hyper parameters hθt ,k and hψ t ,k respectively for topic distribution with parameters6=, t 1,..., K − 1} in, k 1,..., K − 1} and vocabulary distribution with parameters {ψ
{θ=
t ,k

t ,k

a state space model so that they have a shift in the mean of Gaussian distribution when
the previous state is given. That is, they assume that parameters shift as follows:()

hθt ,k | hθt −1 ,k  N hθt −1 ,k , σ θ2 ,()

hψ t | hψ t −1  N hψ t −1 , σψ2 .

(3.1)

Corresponding to latent topics in the DTM, our study identifies topics on social media
and enters them into our multi-generational diffusion model to uncover changes in consumer’s interests.
Next, focusing on the social media coverage following iPhone's battery problem, we
extend the DTM by labeling one topic “battery.” Daniel et al. (2009) extend the LDA
model by labeling topics to be estimated in advance. We extend their labeled topic model
to a labeled dynamic topic model (LDTM), which is characterized as a hybrid of the labeled topic and DTMs. Its graphical representation appears in the Appendix.
We denote 𝜂𝜂 as the hyper parameter for the prior probability that some specific words

include “battery” among three topics. Then we define 𝜂𝜂”𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏” = (1, 0, 0) to indicate all

instances of “battery” are bound to the first topic when assigned in the MCMC procedure.
Other words are allocated as per DTM.

3. Models
3.1 Direct Effects of Social Media on Multi-generational Diffusion
We employ the sales function in Srinivasan and Mason (1986) to define our multi-generational diffusion model. Norton and Bass (1987) proposed a successive-generations
model by assuming constant 𝑝𝑝 and 𝑞𝑞, but we employ the diffusion model generalized by7

Jiang and Jain (2012) that assumes p is constant across generations but q as well as market
potential m are generation-specific.
First, we denote yG (t ) as the G-th generation adoptions at time t with launch time 𝜏𝜏𝐺𝐺 >0, and then following Jiang and Jain (2012), we define our model with additive noise for

adoption of each generation as follows: Starting with the first generation launched at time
0, we havem1 f1 (t ) + u1 (t ),t <τ2,
 y1 (t ) =(t ) m1 f1 (t ) (1 − F2 ( t − τ 2 ) ) + u1 (t ),t ≥τ2,

 y1=
τ G ≤ t < τ G +1 , 1 < G ≤ N , yG (t ) = I G (t ) + LG (t ) + SG (t ) + uG (t ),≡ mG f G (t − τ G ) FG ( t − τ G ) + yG −1 (t ) FG ( t − τ G ) + YG −1 (t ) f G (t − τ G ) + uG (t ) y (t ) = I (t ) + L (t ) + S (t ) − L (t ) + u (t ),t ≥ τ G +1 , 1 < G ≤ N ,
G
G
G
G +1
G
 G

≡ ( mG f G (t − τ G ) FG ( t − τ G ) + yG −1 (t ) FG ( t − τ G ) + YG −1 (t ) f G (t − τ G ) )
× (1 − FG +1 ( t − τ G +1 ) ) + uG (t ),(3.1)


where I G (t ) is the independent adoption of the G-th generation, YG (t ) denotes cumulative adoptions (sales) of the G-th generation at time t, fG = FG ( t ) − FG ( t − 1) , and LG (t )and SG (t ) are adoptions from leapfrogging and switching, respectively.
In Eq.(3.1), the error term uG (t ) is assumed to follow normal distributionuG (t )  N ( 0, σ 2 ) independently across generations and time. We assume constant variance across generations because our data do not identify sales by generation. The adoption
rate function FG ( t ) for the G-th generation product accommodates not only marketingmix according to Jiang and Jain (2012) but also social media effects :FG ( t ) =1−exp( −( p + qG ) X G (t ) )1+( qG / p )exp( −( p + qG ) X G (t ) ),where the covariate X G (t ) is formulated by the hierarchical model as follows:

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(3.2)

X G (t ) =t + βG log (VG (t ) VG (0) ) + αG ' LTopicG (t − 1) + ex (t ).

(3.3)

In the above, VG (t ) is the price of the G-th generation product in period t and βG is thecoefficient of the price effect. LTopicG (t − 1) is a K dimensional vector with the elementof log (TopicGi (t − 1) TopicGi (0) ) , where TopicGi (t − 1) denotes the frequency of words
in the i-th topic at t − 1 , and αG is the corresponding coefficient vector. We assume that()

error term ex (t ) follows ex (t )  N 0, σ x2 .
We use 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝐺𝐺𝐺𝐺 (𝑡𝑡 − 1) rather than 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝐺𝐺𝐺𝐺 (𝑡𝑡) to acknowledge the leading propertyof the social media effect, i.e., people chat before they act. Social media variables usually
are leading indicators of sales in related research (Li and Terui, 2018), and we show it isempirically confirmed for our data in section 4.
3.2 Indirect Effects of Social Media via Generation-Specific Parameters
We assume that diffusion in each generation persists from the previous generation. Specifically, generation-specific parameters are determined by those of previous generations,and topics 𝑻𝑻𝑮𝑮 are defined as the vector of number of words allocated to each topic after
the

G-th

generation

launchfor𝑡𝑡 < 𝜏𝜏𝐺𝐺 .

If

wedenoteTopicG (t ) =(TopicG1 (t ),..., TopicGK (t ) ) ' , then TG is represented by ∑ t <τ TopicG (t ) .
GThen we define the structural equations for 𝑚𝑚𝐺𝐺 and 𝑞𝑞𝐺𝐺 as

δ m 0 + δ m1mG −1 + δmT ' TG + ε m
mG =
q =
 G δ q 0 + δ q1qG −1 + δqT ' TG + ε qδα 0 + δ α 1 'α G −1 + δαT ' TG + ε ααG = βG =
δ β 0 + δ β 1βG −1 + δβ ' TG + ε β ,


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(3.4)

where we assume ε i  N ( 0, σ i2 ) for the i-th equation and independence across equations.
These models define prior distributions for the parameter and we constitute the posterior
distributions when combined with the likelihood function derived by Eqs. (3.1)–(3.3).
Jiang and Jain (2012) use the data of number of units and the aggregate adoption sales
with marketing variables. However, their empirical performance is undemonstrated because their marketing mix data are unavailable. We also find that estimated parametersof multi-generations are unstable when the number of generations increased. Furthermore, models by Jiang and Jain (2012) and Norton and Bass (1987) structurally do notallow forecasting next-generation sales before launch because they lack structure to connect generations for parameters. Our hierarchical model in Eq.(3.4) accommodates parameter shifts from previous generations and social media effects on parameters.
3.3 Posterior Density for Model Parameters
The joint posterior density of our model parameters is represented by(

p {mG , qG , p} , σ , {αG , β G } , σ x , {Δm ,Δq ,Δα ,Δβ } , {σ m , σ q , σ α , σ β } | { yG , t} , { LTopicG , TG ,VG }

(

)

)

=p {mG , qG , p} , { X G } | { yG , t} , {Δm ,Δq } , {α G , β G } , σ p (σ | {mG , qG , p} , { yG , t , X G } )× p ( α G | { X G , t , LTopicG ,VG } , Δα , σ x )× p ( β G | { X G , t , LTopicG ,VG } , Δβ , σ x ) p (σ x | {α G , β G } , { X G , t , LTopicG ,VG } )× p ( Δm | {mG , mG −1} , {TG } , σ m ) p (σ m | {mG , mG −1} , {TG } , Δm )× p ( Δq | {qG , qG −1} , {TG } , σ q ) p (σ q | {qG , qG −1} , {TG } , Δq )× p ( Δα | {α G , α G −1} , {TG } , σ α ) p ( σ α | {α G , α G −1} , {TG } , Δα )× p ( Δβ | {βG , βG −1} , {TG } , σ β ) p (σ β | {βG , βG −1} , {TG } , Δβ )where Δi is the coefficient of the vector of each equation in Eq. (3.4).

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(3.5)

The second line of (3.5) captures the product of conditional posterior density for parameters in the diffusion model in Eqs. (3.1) and (3.2). The third and fourth lines are joint
posterior density of parameters in the marketing mix in Eq. (3.3). The fifth to eighth lines
define joint posterior density for parameters in the hierarchical structure connecting the(G-1) generation to the G generation in Eq.(3.4).
We employ MCMC to estimate parameters because the procedure for hierarchicalmodels is well-established and some of necessary conditional posterior densities areavailable in closed form. The sampling scheme of MCMC for estimating the model is ahybrid of Metropolis-Hastings and Gibbs sampling for other parameters in hierarchicalmodels. The algorithm appears in the Appendix.
3.4 Predictive Density
According to Eqs. (3.1) and (3.2), if we simulate sales for the second-generation product,
we need at least one data point of that generation to estimate 𝑚𝑚2 and 𝑞𝑞2 even when 𝑝𝑝 is
assumed constant across generations. 𝜶𝜶𝟐𝟐 and 𝛽𝛽2 also must be estimated using data for thesecond generation. The feature of their models hampers forecasting new generations with-out numerical data for this generation.

Fig. 3.1 Role of Social Media Pre- and Post-Launch

We note that the topic’s total frequency vector (𝑻𝑻𝑮𝑮 ) in Eq. (3.4) differs from vector

𝑳𝑳𝑻𝑻𝑻𝑻𝑻𝑻𝑻𝑻𝑻𝑻𝐺𝐺 (𝑡𝑡 − 1) in Eq. (3.3), as Fig 3.1 exemplifies. Before G-th generation launches,
only social media provide information about a new generation of G-th product. Then, in
terms of hierarchical models in Eq. (3.4) implying prior information, we forecast adoptionof new generations when parameters are given. After G-th generation product launches,11the marketing mix (Eq. (3.3)) enters the adoption rate function to generate the forecast.
Two structures using social media make it possible to use text data fully and to comparesocial media effects before and after new generations launch. Forecasting is constituted
formally by the predictive density.

p ( yG (t ) | mG , qG , p, X G (t ), TG , LTopicG (t − 1), VG (t ) )∫ p ( y (t ) | m , q , p, t ) p ( m , q , p | m , q , T ) dm dq dp if t ≤ τ(3.6)= ∫ p ( y (t ) | m , q , p, X (t ) ) p ( m , q , p | m , q , T )
G

G

G

G

G

G

G

G

G −1

G

G

G

G −1

G

G

G −1

G −1

G

G

G

× p ( X G (t ) | α G , βG , t , LTopicG (t − 1), VG (t ) )× p (α G , βG | α G −1 , βG −1 ,TG ) dX G (t )dmG dqG dα G d βG dp

if t > τ G

These integrations are numerically evaluated in the MCMC iterations as were applied
to time series forecasting by Terui et al. (2010) and Terui and Ban (2014). The algorithm
for Eq.(3.6) appears in the Appendix.

4. Empirical Results
4.1 Data
We use five generations of iPhone products with the same launching times between
iPhones 5 through 7 as training data, .i.e., G1(iPhone 5), G2(iPhone 5s, 5c), G3(iPhone
6, 6 Plus), G4(IPhone 6s, 6s Plus), G5(iPhone 7, 7 Plus). The last generation G6(iPhone
8, 8 Plus, X) is reserved for test data. All data were obtained from Statista(www.statista.com).

Fig. 4.1 iPhone Sales

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Sales data are total iPhone sales shown in Fig.4.1. There are no unit sales for eachgeneration, making it hard to optimize all parameters for each generation only by total
sales.
Text data were acquired from gsmarena (http://www.gsmarena.com/). This is a wellknown BBS where users worldwide comment upon mobile phones. Through conventional preprocessing procedure taken in natural language processing, punctuations, stop
words and low frequency words less than ten times were removed. Then we use remained
124,263 comments in total for all generations.
4.2 Dynamic Topics
The number of topics needs to be fixed to construct covariate of features in social media
and we use the criterion of root mean squared error (RMSE) of forecasts for test data tochoose the number of topics. This measure is shown in Fig 4.2 when the number of topics
changes from one to five. T suggests three topics for our social media.

Fig. 4.2 Evaluating Number of Topics

LDTM lets us detect changes in each topic, implying potential changes in customer
interest. The top words among topics appear in Table 4.1. We interpret the topics as follows.
1.

Topic 1 (Property-battery): because the frequent words are “nfc” (iPhone5), “apps,”“battery” (all), “face recognition,” and “fingerprint” (iPhone X).

2.

Topic 2 (Comparison): because competitors and their product names appear continually (s3, Samsung, Android) with the word “better.”3.

Topic 3 (Discussion): because of top words “don’t,” “u” (you), “buy,” and “im” (I
13

am).
4.3 Leading Property of Topics to Sales
Fig. 4.3 plots time series for sales and frequency (number of words) under three topics
from 2012: Q4 to 2017: Q4. It shows that the frequency of each topic leads sales by one
period.
Fig. 4.3 Time Series Plot of Sales and Topics

Table 4.2 shows cross-correlation of time lags of one and two periods with sales andfrequency of three topics. They show all topics lead sales by one period with similar magnitudes of effect. There is no delayed relation after one period for Topics 1 and 2, although
two lags could exist for Topic 3.

Table 4.1 Top Words within Each Topic
Table 4.2 Cross-correlation between Topics and Sales
Table 4.3 Comparative Models

4.4 Model Comparison
We now compare 10 models by covariate, treatment of generation-specific heterogeneityof parameters, and hierarchical structure of parameters connecting current and previous
generations (Table 4.3).
Models 1–4.
Model 1 observes the formula in Norton-Bass (1987). Model 2 mirrors Jiang andJain (2012). Models 3 and 4 include topic variables. All models lack the structure to14

connect generations except as an innovator parameter (i.e., no trans-generational
memory). Thus we call them 0th-order models.
Model 5 ~ Model 8.
Models 5–8 are structured for parameter shifts to the next generation as hierarchicalmodels. Models 5 and 7 use only parameters of previous generations. Models 6 and 8
include additional variables for topics in their hierarchical models. These models can predict one step ahead for a new product even before launch. These are first-order models.
Model 9 ~ Model 10.
Models 9 and 10 also are first-order models. The difference from Models 5–8 is thehomogeneous coefficients of marketing mix variables between generations. To improvemodel performance, all previous studies including marketing mix assume that it is heterogeneous. We assume marketing variables are homogeneous for all generations toachieve parsimony.
We compare the 10 models in three measures: log of marginal likelihood (LMD), deviance information criteria (DIC), and RMSEs of forecasts for training data (G1–G5) andtest data (G6). We confirmed convergence for all models via Geweke’s test (Geweke,1992) at 95% significance in Table 4.4.

Table 4.4 Model Evaluations

0th-order models (Models 1–4) cannot forecast sales of G6 as test data because theylack structure to accommodate the shift of ( mG , qG ) . Then we evaluate all first-ordermodels by RMSE(Train), log of marginal likelihood and DIC, and RMSE (Test).

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Model 3 exhibits the best in-sample performance indicated by RMSE and log of marginal likelihood. Accompanying the marketing mix with social media information sharpens forecasting precision. RMSE of the holdout sample and DIC suggest Model 10 has
the best performance. This finding shows the effectiveness of topic models. Also, ourassumption of a homogeneous marketing mix yields better fit to the holdout sample, indicating heterogeneity may lead models to over-fit training data.
Comparing models among the first-order group shows that those with topic variables(Models 6, 8, and 10) out-perform those without topics (Models 5, 7, and 9). In general,model fit erodes for models with more restrictions, but even in the training data RMSE
and LMD of first-order models approach those of Model 3 (best among the zero-ordergroup). First-order Models 6 and 10 exhibit larger LMD and smaller DIC than Model 3.
This finding implies hierarchical models enhance model fit.
After examining RMSE of test data and DIC criteria, we chose Model 10 and examine
the results of estimation and forecasting. Model 10 features no price effect. That is consistent with iPhones having higher prices across generations than competing mobile
phones and with adopters of iPhones showing themselves insensitive to price.
4.5 Parameter Estimates
Parameter estimates of Model 10 are in Table 4.5, where number indicates posteriormeans and the posterior standard deviation is in parentheses.

Table 4.5 Parameter Estimates

16Columns 𝑚𝑚 and show that estimates of 𝑚𝑚𝐺𝐺 , 𝑞𝑞𝐺𝐺 , and p are constant across generations.

Coefficients α i , i = 1, 2,3 , are homogeneous among generations in response to covariatesfor Model 10.
Values of 𝑚𝑚𝐺𝐺 , p, and 𝑞𝑞𝐺𝐺 define the curve for the G-th generation’s diffusion, and it isinfluenced by the topic feature of log (TopicGi (t − 1) TopicGi (0) ) .

Estimates of 𝑚𝑚𝐺𝐺 imply that the market for G1 is potentially double that of other gener-

ations. The first generation faces its original potential market. After the G2 launches, themarket divides into an original and an influenced segment. As a result, its market issmaller than for the market for G1.
Estimates for imitation parameter 𝑞𝑞𝐺𝐺 increase slowly across all generations except G2.

This finding suggests that consumers become imitators rather than innovators as generations proceed and await reviews before buying new-generation products.
Quasi-t values (posterior mean divided by standard deviation) are large for

α i , i = 1, 2,3 , implying that all three topics are significant in this model. Topic 1(Propertybattery) has expected negative correlations with sales. Topics 2(Competitor) and 3 (Discussion) have significant positive correlations with sales.
Table 4.6 examines the hierarchical model’s estimates in Eq. (4.5).

Table 4.6 Estimates of Hierarchical Structure

We summarize results from Table 4.6 as follows.
(1) Estimated coefficients of 𝑚𝑚𝐺𝐺−1 and 𝑞𝑞𝐺𝐺−1 are nearly 1, indicating parameters for market size and imitation rate shift smoothly from previous generations. Estimates of 𝑚𝑚𝐺𝐺

decline gradually and estimates for 𝑞𝑞𝐺𝐺 increase across generations, expecting that17

𝑚𝑚𝐺𝐺 declines because fewer new customers remain as the market matured and sales

were more influenced by previous generations and their marketing. After several generations, 𝑞𝑞𝐺𝐺 trends upward with the popularity of products.

(2) We interpret the indirect topic effect from the estimated hierarchical structure as follows. Topic 1(Property-battery) has significant positive correlations with 𝑚𝑚𝐺𝐺 and 𝑞𝑞𝐺𝐺 .
This finding implies that market size and imitation rate swell as consumers talked

more about the property(battery) before launch. Topic 2 (Competitor) has a negative
correlation with 𝑚𝑚𝐺𝐺 and a positive correlation on 𝑞𝑞𝐺𝐺 . These findings mean that morecommunication on competitors attract imitators, but the risk to declining market size
remains. Topic 3 (Discussion) has the effect opposite that of Topic 2.
(3) Indirect effects of topics in hierarchical models differ from direct effects in the adoption rate function as marketing mix variables. This finding implies different roles forsocial media pre- and post-launch. For instance, Topic 1(Property-battery) has a positive indirect effect on parameter changes in the hierarchical structure and a negative
direct effect on adoption rate function. These findings show that customers who wereinterested in properties of new generations before launch may be disappointed withthem post-launch.
4.6 Forecasting Sales of Unlaunched New Generation
Using the estimated hierarchical structure and diffusion model parameter estimates, we
can forecast sales of new generations before launch. Fig 4.4 shows forecast (solid line) of
new-generation G6 and its actual sales (dashed line) during 2017Q4 to 2018Q3. Forecasts
are accurate even if products were launched without using social media and prior structureon the parameter shifts inferred from hierarchical model.

18

Fig. 4.4 Forecasting Sales of Unlaunched New-Generation Products

5. Influence of Social Media on the Battery Problem
5.1 What is the battery problem?
Since early November of 2016, increasing numbers of iPhone 6 owners worldwide havecomplained their phones shut down unexpectedly even when adequately charged. Apple
admitted that some phones needed their batteries replaced and offered to replace themgratis. In late 2016, Apple noted that some phones shut down "under normal conditions
in order for the iPhone to protect its electronics." Although Apple tried to solve the battery
problem by updating the iPhone OS, consumers complained on social media that theirdevices slowed after updating the OS. This problem caused by the battery appeared prominently on social media until late 2017. We focus on how social media discussion of thebattery problem affected sales, leapfrogging, and switching of iPhones to subsequent generations and competing products.
5.2 Acceleration Effect of Social Media on Leapfrogging to Competitor
Multi-generational diffusion models demonstrate the leapfrog effect, but discussions of it
differ in the literature. Without mentioning leapfrog or switch effects explicitly, Norton
and Bass (1987) distinguish independent from influenced markets. The former is the original or incremental market for current-generation products. The latter implies sales effects
from previous generations.
Mahajan and Muller (1996) extend the Norton and Bass (1987) model to incorporate
leapfrog and switch parameters into multi-generational diffusion of durable technologicalinnovations. Jiang and Jain (2012) enlarge the model to consider how marketing mix

19shapes diffusion of each generation where leapfrog and switch effects are incorporated asin the Norton-Bass model.
The blue line in the left panel of Fig 5.1 shows iPhone sales. It is hard from this figure
to identify the influence of social media on the battery problem. Thus, we collect data forsales and market share of competing Android smartphones. Sales of Androids and total
units, (iPhone plus Android) appear on the left and market shares on the right in Fig 5.1.

Fig. 5.1 Smartphone Unit Shares and Marketing Share

We seek to detect leapfrogging to a later iPhone generation and switching to Android
smartphones induced by social media. Following Jiang and Jain (2012), we calculate themas=
L2 (t ) m1 f1 (t ) F2 (t − τ 2 )

(5.1)

=
S 2 (t ) m1 F1 (t ) f 2 (t − τ 2 ) .

(5.2)

𝐿𝐿2 (𝑡𝑡) and 𝑆𝑆2 (𝑡𝑡) are leapfroggers and switchers, respectively, from the first to the second

generation. L2 (t ) is affected by the remaining fraction of first generation, and S 2 (t ) is
affected by the installed fraction of first generation. The model detects switching from aprevious to a later generation and leapfrogging after generations but not leapfrogging
from iPhones to Androids. We then discuss how social media affect leapfrogging to acompetitor using additional Android sales data.
Denote MS q (t ) as the market share of iPhones and TS (t ) as total unit sales of iPhones
and Androids at time t. Then we defineD(t ) =TS (t ) × [ MSG (t ) − MSG (t − 1)] ，20

t ≥ 2,(5.3)

where D(t ) indicates the change in unit sales of iPhones corresponding to the total
smartphone market. If we define SL(t ) as total leapfrogging from one iPhone to its later
generation in period t, we derive the leapfrog effect from competitor CL(t ) by

CL=(t ) D(t ) − SL(t ) .

(5.4)

We compare two kinds of leapfrogging from iPhones and Androids via the ratio
rate(t ) =SL(t ).
CL(t )(5.5)

From 2016:Q4 through 2017:Q4, the period of the battery problem, iPhone unit sales
did not decrease significantly (left of Fig. 5.1), but the market share of iPhones declined
during the two preceding years. To detect how social media coverage of the battery problem affected sales, we compare two models—one including topics extracted from social
media and one that ignores them. Fig. 5.2 shows a numerical example of the social media
effect when one topic is included in the model.

Fig. 5.2 Social Media Effect

The left panel of Fig. 5.2 shows topic frequency. It implies the influence on sales bycomparing the model that estimated 𝛼𝛼𝐺𝐺 and that which estimated 𝛼𝛼𝐺𝐺 = 0. The “alpha” inthe right panel of Fig. 5.2 depicts sales corresponding to three kinds of value, (𝛼𝛼𝐺𝐺 =0, 0.5,

1) when we fix the parameters as (m=10, p=0.1, q=0.5). 𝛼𝛼𝐺𝐺 = 0 indicates the model without social media effects. Social media become more influential as 𝛼𝛼𝐺𝐺 increases.

Fig. 5.3 reveals leapfrogging ( SL(t ) ) among iPhone generations with and without thesocial media effect. Social media exert greater influence for later generations, implyingthat consumers were more influenced as generations proceeded.
21Fig 5.4 plots 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟(𝑡𝑡) from Eq. (5.5). A negative value means more customers leap-

frogged to iPhones than to Androids. A positive value means more leapfrogged to Androids. To evaluate SL(t ) in the model lacking social media effects, we enforced 0 as thevalue of 𝜶𝜶𝑮𝑮 . Then we measured the effect of social media by the difference between these
two leapfrogs estimates (Fig. 5.5).

Fig. 5.3 Leapfrog Effect of iPhone

Fig. 5.4 Comparison of Leapfrogging between iPhone and Android

Fig. 5.5 Difference in Leapfrogging

In Fig 5.5, over periods with positive unit sales, which implies social media were significant with respect to the battery problem, unit sales of iPhones decline gradually. Periods with negative sales mean social media had a negative effect on iPhones. Starting with
iPhone 5, the social media effect slowly erodes unit sales of iPhones. When the battery
problem was exposed in 2016:Q4, the social media effect was distinguished by leapfrogging to Android. Social media had a significant influence on product diffusion.

6. Conclusion
This study proposed a multi-generational diffusion model with social media effects that
included a hierarchical structure connecting diffusion parameters of successive generations. Results show the effects of social media using a dynamic labeled topic model directly on the adoption rate function as effective marketing mix as well as indirectly on

22

the parameter change described by hierarchical model on the shift of diffusion parameters
to next generation.
Unlike previous multi-generational diffusion models, ours forecasts sales of new-generation products before launch using social media as the leading indicator and the hierarchical model.
Finally, our model featuring social media effects improves the precision of forecastingand shows how social media affected sales of smartphones via leapfrogging and switching
to next-generation and competing products. We examined how iPhone’s battery problem
affected sales, leapfrogging, and switching to later-generation iPhones and competing
Androids.
Further issues remain. We proposed the LDTM to extract dynamic features hidden insocial media and confirmed that “people chat before they act.” However, results of themodel depend on prior LDTM settings. Determining the full robustness of dynamic text
analysis awaits future studies.

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26

Appendix.
A. Labeled Dynamic Topic model
The DTM is base model for LDTM as detailed in Blei and Lafferty (2006). Based on
DTM, we assume the word “battery” belongs to Topic 1 in this study. The algorithm ofLDTM can be written as
For t = 1,2, …, T
1. Draw Topics
𝛽𝛽𝑡𝑡,𝑘𝑘 �𝛽𝛽𝑡𝑡−1,𝑘𝑘 ~ 𝛮𝛮�𝛽𝛽𝑡𝑡−1,𝑘𝑘 , 𝜎𝜎 2 𝐼𝐼�

2.

Draw at𝛼𝛼𝑡𝑡 |𝛼𝛼𝑡𝑡−1 ~ 𝛮𝛮(𝛼𝛼𝑡𝑡−1 , 𝛿𝛿 2 𝐼𝐼)

For each document:
3. Topic proportion𝜂𝜂𝑖𝑖,𝑑𝑑 |𝛼𝛼𝑡𝑡 ~ 𝑁𝑁(𝛼𝛼, 𝑎𝑎2 𝐈𝐈)𝜃𝜃𝑖𝑖,𝑑𝑑 |𝜂𝜂𝑖𝑖,𝑑𝑑 ~ 𝜋𝜋�𝜂𝜂𝑖𝑖,𝑑𝑑 �For each word:
if word == ‘battery’:𝑧𝑧𝑡𝑡,𝑑𝑑,𝑛𝑛 = 1

else:

4. Topic-word assignment𝑧𝑧𝑡𝑡,𝑑𝑑,𝑛𝑛 |𝜃𝜃𝑡𝑡,𝑑𝑑 ~ 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀�𝜃𝜃𝑡𝑡,𝑑𝑑 �

𝑤𝑤𝑑𝑑,𝑛𝑛 |𝑧𝑧𝑑𝑑,𝑛𝑛 �𝛽𝛽𝑡𝑡,𝑘𝑘 � ~ 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 �𝜋𝜋�𝛽𝛽𝑡𝑡,𝑧𝑧𝑑𝑑,𝑧𝑧 ��Here 𝜋𝜋(∙) is a softmax function that can fix the sum of 𝛽𝛽𝑡𝑡,𝑧𝑧𝑑𝑑,𝑧𝑧 to 1.

27

B. MCMC method for the Bass model
B.1 Prior Settings for the Bass modelParameter

Setting

𝑚𝑚𝐺𝐺 ~ 𝑁𝑁(𝜇𝜇𝑚𝑚0 , 𝜏𝜏𝑚𝑚0 −1 )𝜇𝜇𝑚𝑚0 = 0, 𝜏𝜏𝑚𝑚0 = 0.1

𝑝𝑝 ~ 𝑁𝑁�𝜇𝜇𝑝𝑝0 , 𝜏𝜏𝑝𝑝0 −1 �𝜇𝜇𝑝𝑝0 = 0, 𝜏𝜏𝑝𝑝0 = 0.1

𝑞𝑞𝐺𝐺 ~ 𝑁𝑁�𝜇𝜇𝑞𝑞0 , 𝜏𝜏𝑞𝑞0 −1 �𝜇𝜇𝑞𝑞0 = 0, 𝜏𝜏𝑞𝑞0 = 0.1

𝑋𝑋𝐺𝐺 ~ 𝑁𝑁(𝜇𝜇𝑋𝑋 , 𝜏𝜏𝑋𝑋 −1 )𝜇𝜇𝑋𝑋0 = 0, 𝜏𝜏𝑋𝑋0 = 0.1

𝛼𝛼𝐺𝐺 ~ 𝑁𝑁(𝜇𝜇α0 , 𝜏𝜏α0 −1 )𝜇𝜇𝛼𝛼0 = 0, 𝜏𝜏α0 = 0.1

σ𝑥𝑥 ~ 𝐼𝐼𝐼𝐼(𝑎𝑎, 𝑏𝑏)

a= 3, b = 10

σ~ 𝐼𝐼𝐼𝐼(𝑎𝑎, 𝑏𝑏)

a = 3, b = 10

𝜷𝜷𝑮𝑮 ~ 𝑁𝑁�𝜇𝜇β0 , 𝜏𝜏β0 −1 �𝜇𝜇𝛽𝛽0 = 0, 𝜏𝜏β0 = 0.1

Θ ~ 𝑁𝑁(𝜇𝜇Θ0 , 𝜏𝜏Θ0 −1 )𝜇𝜇Θ0 = 0, 𝜏𝜏Θ0 = 0.1

𝚵𝚵~ 𝐼𝐼𝐼𝐼(𝑎𝑎, 𝑏𝑏)

a = 3, b = 10

* 𝛅𝛅 is the vector of �Δ𝑚𝑚 , Δ𝑞𝑞 , 𝜎𝜎𝛼𝛼 , 𝜎𝜎𝛽𝛽 �, 𝚵𝚵 is the vector of �𝜎𝜎𝑚𝑚 , 𝜎𝜎𝑞𝑞 , 𝜎𝜎𝛼𝛼 , 𝜎𝜎𝛽𝛽 �.

B.2 Conditional Posterior Distributions(1) mG |{ yG , t , X G }{ p, qG }, σFor iter (=1, …, R) of MCMC iterations, we use Metropolis-Hastings with a random
walk algorithm for each generation G,(𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖)

𝑚𝑚𝐺𝐺(𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖−1)= 𝑚𝑚𝐺𝐺

where the acceptance probability is
α = min �1,(𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖)

𝑝𝑝�𝑚𝑚𝐺𝐺| {𝑦𝑦𝐺𝐺 , 𝑡𝑡, 𝑋𝑋𝐺𝐺 }, {𝑝𝑝, 𝑞𝑞𝐺𝐺 }, 𝜎𝜎�(𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖−1)

𝑝𝑝�𝑚𝑚𝐺𝐺

where t = 1, …, N and G = 1, 2, …, 5.

(B. 1)+ 𝜆𝜆𝑚𝑚 ; 𝜆𝜆𝑚𝑚 ~ 𝑁𝑁(0,0.05),| {𝑦𝑦𝐺𝐺 , 𝑡𝑡, 𝑋𝑋𝐺𝐺 }, {𝑝𝑝, 𝑞𝑞𝐺𝐺 }, 𝜎𝜎�28�,(B. 2)

(2) p |{ yG , t , X G }{mG , qG }, σFor p and 𝑞𝑞𝐺𝐺 , we also use Metropolis-Hastings sampling, which is the same as 𝑚𝑚𝐺𝐺

above.

−=p (iter ) p (iter 1) +λ p ; λ p  N ( 0, 0.05 ) .

The probability of acceptance ismin �1,

𝑝𝑝�𝑝𝑝

(𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖)| {𝑦𝑦𝐺𝐺 , 𝑡𝑡, 𝑋𝑋𝐺𝐺 }, {𝑚𝑚𝐺𝐺 , 𝑞𝑞𝐺𝐺 }, 𝜎𝜎�𝑝𝑝�𝑝𝑝

(𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖−1)(𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖)= 𝑞𝑞𝐺𝐺𝑞𝑞𝐺𝐺

The acceptation probability is
min �1,(4) X G |{ yG , t}{mG , qG , p}, σ(𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖)

𝑋𝑋𝐺𝐺(𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖−1)(𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖)

𝑝𝑝�𝑞𝑞𝐺𝐺

The acceptation probability is
min �1,(𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖)(5) σ | {𝑦𝑦𝐺𝐺 , 𝑡𝑡, 𝑿𝑿𝑮𝑮 }, {𝑚𝑚𝐺𝐺 , 𝑞𝑞𝐺𝐺 , 𝑝𝑝}

(B.5)

�.

+ 𝜆𝜆𝑋𝑋 ; 𝜆𝜆𝑋𝑋 ~ 𝑁𝑁(0,0.05).

| {𝑦𝑦𝐺𝐺 , 𝑡𝑡, }, {𝑚𝑚𝐺𝐺 , 𝑞𝑞𝐺𝐺 , 𝑝𝑝}, 𝜎𝜎�(𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖−1)

𝑝𝑝�𝑋𝑋𝐺𝐺+ 𝜆𝜆𝑞𝑞 ; 𝜆𝜆𝑞𝑞 ~ 𝑁𝑁(0,0.05).

| {𝑦𝑦𝐺𝐺 , 𝑡𝑡, 𝑋𝑋𝐺𝐺 }, {𝑚𝑚𝐺𝐺 , 𝑝𝑝}, 𝜎𝜎�(𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖−1)= 𝑋𝑋𝐺𝐺

𝑝𝑝�𝑋𝑋𝐺𝐺(B. 4)| {𝑦𝑦𝐺𝐺 , 𝑡𝑡, 𝑋𝑋𝐺𝐺 }, {𝑚𝑚𝐺𝐺 , 𝑝𝑝}, 𝜎𝜎�(𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖−1)𝑝𝑝�𝑞𝑞𝐺𝐺�.

| {𝑦𝑦𝐺𝐺 , 𝑡𝑡, 𝑋𝑋𝐺𝐺 }, {𝑚𝑚𝐺𝐺 , 𝑞𝑞𝐺𝐺 }, 𝜎𝜎�(3) qG |{ yG , t , X G }{mG , p}, σ

(B.3)

�.
| {𝑦𝑦𝐺𝐺 , 𝑡𝑡}, {𝑚𝑚𝐺𝐺 , 𝑞𝑞𝐺𝐺 , 𝑝𝑝}, 𝜎𝜎�(B. 6)(B. 7)(B. 8)

If we define estimated sales in period t for the Gth generation as

we can update σ byy� 𝐺𝐺𝐺𝐺 = 𝑓𝑓({𝑦𝑦𝐺𝐺 , 𝑡𝑡, 𝑋𝑋𝐺𝐺 }, {𝑚𝑚𝐺𝐺 , 𝑞𝑞𝐺𝐺 }),(B. 9)∑𝑛𝑛𝑡𝑡=1(y𝑡𝑡 − ∑𝑀𝑀n� 𝐺𝐺𝐺𝐺 )2
𝐺𝐺=1 y𝐼𝐼𝐼𝐼 �a + , b +�,22(B. 10)

M stands for number of generations in this equation.

29

Marketing Mix.
(6) α G | { X G , t , LTopicG ,VG } , Δα , σ xFor each topic j the posterior of coefficient 𝛼𝛼𝐺𝐺𝐺𝐺 can be derived from a normal regression

equation from𝑁𝑁 �(𝑛𝑛σ𝛼𝛼 + σ𝛼𝛼0

)−1

𝑛𝑛

𝑇𝑇≠𝑗𝑗

𝑖𝑖=1

𝑘𝑘=1

−1
�σ𝛼𝛼 � �𝑋𝑋𝐺𝐺𝐺𝐺 − 𝑽𝑽𝑮𝑮𝑖𝑖 ∙ 𝛽𝛽𝐺𝐺 − � 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝐺𝐺𝐺𝐺𝐺𝐺 ∙ 𝛼𝛼𝐺𝐺𝐺𝐺 � 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝐺𝐺𝐺𝐺+ 𝜇𝜇𝛼𝛼0 σ𝛼𝛼0 � , (𝑛𝑛σ𝛼𝛼 + σ𝛼𝛼0 )−1 �≡ mG fG (t − τ G ) FG ( t − τ G ) + yG −1 (t ) FG ( t − τ G ) + YG −1 (t ) fG (t − τ G )(𝐵𝐵. 11)(7) β G | { X G , t , LTopicG ,VG } , Δβ , σ x−1𝑁𝑁 ��𝑛𝑛σ𝛽𝛽 + σ𝛽𝛽0 �𝑛𝑛−1

�σ𝛼𝛼 �(𝑋𝑋𝐺𝐺𝐺𝐺 − 𝑻𝑻𝑻𝑻𝑻𝑻𝑻𝑻𝑻𝑻𝑮𝑮𝑮𝑮 ∙ 𝜶𝜶𝑮𝑮 )𝑉𝑉𝐺𝐺−1 + 𝜇𝜇𝛽𝛽0 σ𝛽𝛽0 � , �𝑛𝑛σ𝛽𝛽 + σ𝛽𝛽0 � � (𝐵𝐵. 12)𝑖𝑖=1(8) σ x | {α G , β G } , { X G , t , LTopicG ,VG }∑𝑛𝑛𝑖𝑖=1(X 𝐺𝐺𝐺𝐺 − 𝑻𝑻𝑻𝑻𝑻𝑻𝑻𝑻𝑻𝑻𝑮𝑮𝑮𝑮 ∙ 𝜶𝜶𝑮𝑮 − 𝑽𝑽𝑮𝑮𝑖𝑖 ∙ 𝛽𝛽𝐺𝐺 )2n
𝐼𝐼𝐼𝐼 �α + , β +�.
22(B. 13)

Hierarchical Structure.
(9) Δm | {mG , mG −1} , {TG } , σ mAssuming D as data matrix for hierarchical structure, we can derive the posterior of coefficient 𝛿𝛿𝑚𝑚𝑚𝑚 by𝑁𝑁 �(𝑛𝑛σ𝑚𝑚 + σ𝑚𝑚0

)−1

𝑀𝑀

𝐾𝐾≠𝑗𝑗

𝐺𝐺=1

𝑧𝑧=1

−1
�σ𝑚𝑚 � �𝑚𝑚𝐺𝐺 − �(𝐷𝐷𝐺𝐺𝐺𝐺 ∙ 𝛿𝛿𝑚𝑚𝑚𝑚 ) � 𝐷𝐷𝐺𝐺𝐺𝐺+ 𝜇𝜇𝑚𝑚0 σ𝑚𝑚0 � , (𝑛𝑛σ𝑚𝑚 + σ𝑚𝑚0 )−1 �.

(B. 14)(10) σ m | {mG , mG −1} , {TG } , Δm
30

2

∑𝑀𝑀
n
𝐺𝐺=1�m𝐺𝐺 − ∑𝑧𝑧=1(𝐷𝐷𝐺𝐺𝐺𝐺 ∙ 𝛿𝛿𝑚𝑚𝑚𝑚 )�
𝐼𝐼𝐼𝐼 �α + , β +�.
22(B. 15)As the sampling methods are the same among all the hierarchical structures, we cansample for other parameters (𝚫𝚫𝒒𝒒 , 𝚫𝚫𝜶𝜶 𝑎𝑎𝑎𝑎𝑎𝑎 𝚫𝚫𝜷𝜷) as well.
Forecasting(10) y (t ) | mG , qG , p, X G (t ), TG , LTopicG (t − 1), VG (t )For 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 = 1, … , ITER, after sampling all the parameters using MCMC, the forecast-ing unit sales for the G-th generation 𝑦𝑦𝐺𝐺 (𝑡𝑡) can be written ast <τ2,t ≥τ2,

 y1 (t )( iter ) m1( iter ) f1 (t ) ,t )( iter ) m1( iter ) f1 (t ) (1 − F2 ( t − τ 2 ) ) , y1 (=

=τ G ≤ t < τ G +1 , 1 < G ≤ N ,yG (t )( iter ) mG ( iter ) f G (t − τ G ) FG ( t − τ G )
+ yG −1 (t )( iter ) FG ( t − τ G ) + YG −1 (t )( iter ) f G (t − τ G ),
( iter )t ≥ τ G +1 , 1 < G ≤ N ,(mG ( iter ) f G (t − τ G ) FG ( t − τ G )G (t ) y=
+ yG −1 (t )( iter ) FG ( t − τ G ) + YG −1 (t )( iter ) f G (t − τ G )) × (1 − FG +1 ( t − τ G +1 ) ) ,(B.16)
where(

FG t | p(iter )

, qG

(iter )

) = 1+( qG(

1− exp − ( p ( iter ) + qG ( iter ) ) X G (t )( iter )( iter )/p( iter ))exp( −( p( iter )+ qG( iter ))) X (t ) )( iter ).

(B.17)

G

Note that X G (t )(iter ) t ,(iter )=
t + βG (iter ) log (VG (t ) VG (0) ) + αG (iter ) ' LTopicG (t − 1),
 X G (t )t < τGt ≥ τG(B.18)

then total sales y (t )(iter )

can be calculated byy (t )(iter ) = ∑ yG (t )G31(iter ).

(B.19)

Table 4.1 Cross-correlation between Topics and Sales

t
t-1
t-2

Topic 1 Topic 2 Topic 3-0.193-0.271-0.051
0.575
0.575
0.688
0.097
0.020
0.286

Table 4.2 Top Words of Each Topic

𝑏𝑏
𝜑1,𝑣

Topic 1

iPhone 5 iPhone 5s
apple
1 applebattery
time
2
new
ios
3
use
4 iphoneapps
5 marketproblem
6 device
7 innovation phone
nfc
update8app
9 apples
like
like
10

𝑏𝑏
𝜑2,𝑣

Topic 2

iPhone 6
phone
apple
android
ios
use
time
apps
problemdevicenewiPhone 6s
apple
battery
ios
phone
time
back
charging
new
apps
jack

iPhone 7 iPhone X/8
apple
apple
face
ios
phone
phone
id
androidcharging charging
recognition
backbattery fingerprint
fast
like
screentime
wireless innovation

iPhone 6iphone6s
6bettersamsung
cameraandroid
ram
plus
s6

iPhone 6s
iphone7plus
betterram
camerasamsung
screen6ss7iPhone 7 iPhone X/8
iphoneiphone8s8
samsung
plus
x
7
8
x
display
betterscreen5s8
betterscreenandroidram
display

iPhone 6iphone
phone
apple
dont
buy
like
peoplebest
oneuiPhone 6s
iphone
phone
buy
apple
dont
like
u
phonesandroidwant

iPhone 7 iPhone X/8
iphoneiphone
phone
phone
apple
apple
dont
x
buy
dont
like
buy
people
like
one
want
im
bettersamsungone

Comparison

iPhone 5
1 iphone5
2
3 samsung4 better
5 galaxy
4s
6
s3
7
8
ios
9 screen
10 lumia𝑏𝑏
𝜑3,𝑣

Topic 3

Property

iPhone 5s
iphone5s
android
bettersamsungs4
phone
5
cameragood

Discussion

iPhone 5
1 iPhone 5
2 apple3 phone4 iphone
5
dont
u
6
7
buy
8
like
9 people10
imiPhone 5s
iphone
phone
buy
u
dont
5s
apple
one
im
want

32

33

Table 4.3 Comparative Models

Table 4.4 Model Evaluations
Zeroth-OrderFirst-Order
8
9
10
7
Model
1
2
3
4
5
6
RMSE(Train) 3.251 3.215 2.883 3.059 4.217 3.766 3.649 3.038 3.587 2.897- 9.097 5.091 8.881 4.987 3.552 3.495
RMSE(Test)log(ml)-95.07 -90.57 -75.6 -75.9 -85.56 -74.48 -84.473 -75.993 -85.92 -75.911
258.4 250.5 207.18 208.88 269.52 211.57 271.43 205.55 241.58 196.1
DIC

Table 4.5 Parameter Estimatesm18.391(1.035)9.916(0.739)10.182(0.746)10.839(0.473)10.106(1.275)

G1
G2
G3
G4
G5

p

q

0.098
(0.012)0.900(0.186)0.285(0.112)1.002(0.103)1.061(0.137)1.108(0.204)

𝛼𝛼1

𝛼𝛼2

𝛼𝛼3

-0.018(0.001)

0.011(0.001)

0.047(0.000)

Table 4.6 Estimates of Hierarchical Structureintercept

𝛿𝛿𝑚𝑚
𝛿𝛿𝑞𝑞

0.037
(0.002)-0.014(0.000)

𝑇𝑇1

𝑇𝑇2

0.031
(0.002)0.036(0.001)-0.025(0.001)
0.011(0.002)

34

𝑇𝑇3

0.022
(0.001)-0.049(0.001)𝑚𝑚(𝑞𝑞)𝐺𝐺−1
0.976(0.001)1.010(0.001)

Fig. 3.1 Role of Social Media before and after Launch

Fig. 4.1 Sales of iPhone35

Fig. 4.2 Evaluating Number of Topics

Fig. 4.3 Time Series Plot of Sales and Topics
36

Fig. 4.4 Forecasting Sales of Unlaunched New Generation

Fig. 5.1 Smartphone Unit Shares and Marketing Share37

Fig. 5.2 Social Media EffectFig. 5.3 Leapfrog Effect of iPhone38

Fig. 5.4 Leapfrog Effect Comparison between iPhone and Android

Fig. 5.5 Difference of Leapfrogging

39