Discussion Paper No. 63

Social Media and the Diffusion of an Information Technology
Product
Yinxing Li and Nobuhiko Terui
Tohoku University
August 2016

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Social Media and the Diffusion of an Information Technology
Product

Yinxing Li and Nobuhiko Terui 1
Tohoku University

August 2016

Abstract
The expansion of the Internet has led to a huge amount of information posted byconsumers online through social media platforms such as forums, blogs, and product reviews.
These text data are useful especially when numeric sales data are not enough, as is typicallythe case with new product diffusion. This study proposes a diffusion model that
accommodates pre-launch social media information and combines it with post-launch sales
information in the Bass model to improve the accuracy of sales forecasts. The model ischaracterized as the extended Bass model, with time varying parameters whose evolutions areaffected by the consumer’s communications in social media.
Specifically, we first extract information from social media to build variables, such as thenumber of positive and negative comments, and also latent topics. These data are fed as key
parameters in the diffusion model’s evolution process for the purpose of plugging the gap
between the time-invariant key parameter model and that of observed sales.
We examine several models using text analysis techniques, e.g., sentiment analysis forcounting numbers of positive and negative comments and topic analysis by topic model toextract relevant topics. These results are then compared with the conventional Bass modelusing only post-launch sales data.
An empirical study of the first-generation iPhone during 2006 and 2007 shows that themodel using additional variables extracted from sentiment and topic analysis on BBS
performs best based on several criteria, including DIC (Deviance Information Criteria),
marginal likelihood, and forecasting errors of holdout samples. We discuss the role of social
media information in the diffusion process for this study.
Keywords: Bass Model, Diffusion, Hierarchical Bayes Model, Predictive Density, Social
Media Data, Text Analysis, Sentiment Analysis, Time Varying Parameter, Topic Model

1Terui acknowledges the grant by JSPS KAKENHI Grant Number (A)25245054.

1

1. Introduction
The expansion of the Internet has led to massive information posted by consumers online
through social media such as forums, blogs, and product reviews.

This provides anopportunity for firms to know consumers’ product expectations and evaluations without theneed for a direct survey.

Using text mining, Grimes (2008) found that 80% ofbusiness-relevant information originates primarily as unstructured text.
A growing number of studies have examined the influence of user-generated content inmarketing.

Lee and Bradlow (2011) have proved that customer reviews can complement

existing methods for generating attributes used in marketing analysis by comparing expert
guides and consumer surveys.

Netzer et al. (2012) have utilized large-scale,consumer-generated data on the Web to understand consumers’ top-of-mind associative
network of products and the implied market structure insights. Moe and Trusov (2011)
showed that when studying the effect of consumer’s ratings, the potentially endogenous
relation between sales and ratings must be considered. Tirunillai and Tellis (2012) used anaïve Bayes classifier and support vector machine to classify user-generated online content topositive news and negative news and incorporated this information into a financial
econometric model to forecast stock returns.
In recent years, online product reviews have taken on a larger role in the consumer
decision process. Not only do consumers prefer buying products online but they also rely
increasingly on others’ online comments. Considering that only a limited number of samples
are available for conventional new product diffusion models, online conversations, such as
SNS, blogs, and BBS, are becoming very popular and could have complementary roles.
Combined with word-of-mouth (WOM) data, these could improve forecasting performancethrough a deeper understanding of the market structure.
In this study, we use not only sales data but also user-generated online content (or online
comments) to describe and forecast the diffusion process of a new product, where online
WOM data is plugged into the model as covariates for affecting the change of key parameters
over time. From the modeling perspective, our model is characterized as a diffusion modelwith a time-varying parameter. This parameter variation of the diffusion model has been
discussed for several reasons; for example, as a competitive activity, changes in marketing2practice, different segments adopting products at different times (Eliashberg and Chatterjee,1986), specification and measurement errors (Putsis, 1998), and aggregation and omitted
variables (Sarris, 1973; Judge et al., 1985). By capturing changes in consumer expectationand evaluation before and after launch, our study incorporates an additional reason for the
diffusion model of an IT product where the WOM effect by BBS would be significantly
present by reflecting the change of consumer expectation and evaluation before and after
launch.

In particular, we show the information extracted from BBS text data leads to sales,thus motivating the development of a systematic variation model with covariates constituted
from BBS text. This is distinguished from stochastic process models using a Kalman filter(Bretschneider and Mahajan, 1980; Judge et al., 1985; Putsis, 1998; Xie et al., 1997), where
the sources of parameter variation are not always obvious.
Our proposed models belong to the class of systematic variation models (Mahajan et al.,2000, Ch. 11) and they share the advantages with other time-varying parameter models in
producing fewer forecasting errors, as was shown by Putsis (1998) and Xie et al. (1997). In
addition, our models provide insights into the time variation of parameters to guide thetransition.
We evaluate parameter estimates using a Bayesian approach, and our inference is exact
in the sense of not relying on asymptotic theory.

The predictive density is numericallyevaluated to reflect the uncertainty of point forecasts in decisions, as recently discussed by
Terui and Ban (2014) and Takada et al. (2015). This characteristic of inference is intrinsic to
the new product diffusion process as it uses a limited number of data points.
In the next section, we briefly introduce the text analysis used in our study, i.e.,
sentiment analysis using a naïve Bayes classifier and topic analysis by LDA. In Section 3,
we propose the models and explain the estimation procedure. The empirical application isreported in Section 4. We apply our model to the diffusion process of the first-generation
iPhone by augmenting the information set with user-generated content from the BBS of thisproduct. We show that temporal variables extracted from social media contain useful
information not only for improving forecasts than the original Bass model but also forunderstanding changes in the diffusion process due to interactions among potential andactual purchasers through WOM in the BBS environment. The sentiment analysis gathers
3the subjective emotional responses of consumers, and the topic analysis gathers rather
subjective information, including the effects of marketing, reviews, and discussions, namedby the characterization of extracted topics. We conclude our study in Section 5.

2. Text Analysis of Social Media
2.1 Sentiment Analysis
Our model utilizes user-generated information from social media on a new product. We use
two methods to analyze this text data: sentiment analysis and topic analysis.

We first

generate the numeric information from text data by classifying users’ comments into one ofthree comment categories: positive, negative, or neutral (no relation).

In particular, thenumber of positive comments before launch reflects the expectations of potential customers
that can lead to after-launch sales.
We use the naïve Bayes classifier for text analysis, which has been effectively applied tothe marketing problem (Tirunillai and Tellis, 2012). It is a simple probabilistic model basedon the Bayes theorem, with independence assumptions between words, and it iswell-recognized to show good performance in text analysis.
When the vector of words x in a comment is given, the posterior probability p ( Ck | x )of classifying it to Ck (category k, i.e., positive, negative, or neutral) is calculated by Bayes’sp ( Ck ) is the prior probability and can betheorem as p ( Ck | x ) ∝ p ( Ck ) p ( x | Ck ) .

defined by calculating the share of positive comments among all comments in the trainingdata.

p ( x | Ck ) is the likelihood, implying the probability that this comment with thevector of words x happens when it belongs to Ck under the assumption of independence of
word, i.e., p ( x | Ck ) =n∏ p ( x | C ) . Then, we classify the comments using the value of
i =1𝑦𝑦� from the function below:

i

k

n

yˆ = arg max k p ( Ck ) ∏ p ( xi | Ck ) .
i =1

4

(1)

2.2 Topic Analysis
Next, we extract the “topics” from a collection of documents in social media using the latent
Dirichlet allocation (LDA) model (Blei et al., 2003), which is well-established in natural
language processing and applied in a variety of disciplines. The LDA model is based on theassumption that each document can be viewed as a mixture of various latent topics, where
topics follow a multinomial distribution over words. Contrary to the fact that the naïve
Bayes classifier assumes that one document only has one topic, LDA assumes that eachdocument is a mixture of various topics.
More specifically, LDA is a generative model allowing sets of observations to be
explained by unobserved groups, explaining why some parts of the data are similar. Denote
𝑤𝑤𝑑𝑑,𝑖𝑖 as the i-th word in document d and 𝑧𝑧𝑑𝑑,𝑖𝑖 the (latent) topic of the i-th word in document d.

The model assumes that 𝑤𝑤𝑑𝑑,𝑖𝑖 has a vocabulary (v) distribution in topic k that follows amultinomial distribution ( 𝑤𝑤𝑑𝑑,𝑖𝑖 ~ 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀(𝜙𝜙𝑧𝑧𝑑𝑑 ,𝑖𝑖 ) ) and 𝜃𝜃𝑘𝑘,𝑑𝑑 follows topic distribution( 𝑧𝑧𝑑𝑑,𝑖𝑖 ~ 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀(𝜃𝜃𝑑𝑑 ) ) in document d.

Then, the model describes the probability thatvocabulary v appears in document d and is represented as the sum of the products of topic
distribution and vocabulary distribution over possible K ways:=p (v | d )KKp (v | k )p ( k | d ) ∑φ θ
∑=

=
k 1=
k 1

v ,k k ,d.

(2)

In the LDA model, the most common method to estimate latent parameter z is to use
Gibbs sampling.

However, when there is a large volume of text data like in our study, Gibbs

sampling requires a lot of time to sample the parameters. Then we employ a popular way
known as “collapsed Gibbs sampling,” which analytically uses the natural conjugate of priordistribution to integrate out 𝜃𝜃𝑘𝑘|𝑑𝑑 and 𝜙𝜙𝑤𝑤|𝑘𝑘 . Details of the MCMC procedure are given in
the Appendix.

3. Models
3.1 Diffusion Model with Social Media Information
We use the new product diffusion model by Bass (1969) as the base model and extend it in theway of incorporating social media information. Then, we assume that the potential market
size (m), the innovator ratio (p), and imitator ratio (q) are changing over time and their5dynamics are partially driven by temporal communications among potential users.
We expect different roles for sentiment analysis and topic models. The sentiment analysis
extracts emotional and rather subjective feelings of “like” (positive) or “dislike” (negative)
from consumers’ BBS communications.

On the other hand, topic analysis involves objective

factors based on consumers’ expectations and evaluations before and after the launch of a new
product and their responses to marketing activity.
We employ the empirical model of Srinivasan and Mason (1986), which uses acontinuous form of expression for the difference of cumulative sales ( xt − xt −1 ) to define themodel as=yt mt  F ( t | pt , qt ) − F ( t − 1| pt −1 , qt −1 )  + ε t(3)where the cumulative density is written by

F ( t | pt , qt ) =1 − exp {− ( pt + qt ) t}
q
1 + t exp {− ( pt + qt ) t}
pt

 
 
1
1
t
1 − exp − 
+
*
*


1
exp
p
1
exp
q
+
−
+
−
{
}
{
}
t
t  
 

=
*
 
 1 + exp {− pt } 
 
1
1


t
1+ 
exp
−
+
 
*
*
 1 + exp {−qt*} 
 
1
+
exp
−
1
+
exp
−
p
q
}
{
{
}
t
t  


 

(4)

and ε t is assumed to follow a normal distribution ε t  N ( 0,1/ t ) .
We assume that the dynamics of parameters are partly explained by extracted variablesfrom social media on the grounds that they contain changes in consumers’ emotions,
expectations, and evaluations. We describe this mechanism using a hierarchical model forthe parameters in addition to diffusion model (4). More specifically, for the appropriately()

 pt ,
 1 − pt *transformed parameter vector qt = mt* , pt* , qt* ' , where mt* = log mt , pt = log  qt 
 , and covariate vector zt (including constants and variables) by 1 − qt and qt* = log analyzing social media data.

We define the hierarchical model as=
θt Gzt −1 + ηt
6(5)

where zt −1 is a covariate vector constituted from social media data, ηt is thethree-dimensional vector of error terms and assumed to follow a normal distribution−1−1−1
ηt  N 3 ( 0, Σ ) , where Σ =diag (τττ
1 , 2 , 3 ) . That is, the models are canonically

represented by hierarchical nonlinear regression models.
We denote the static Bass model as Model 1, where we set the covariate as zt −1 = 1 .
Then, the first model (Model 2) uses three quantities to describe the comments: total numberof comments and numbers of positive and negative comments. We define the covariate aszt −1 = (1, st −1 , pt −1 , nt −1 ) ' , where st −1 means the number of comments in t−1. pt −1 and nt −1
are, respectively, the numbers of positive and negative comments in t−1.
The second model (Model 3) is defined when the covariate comprises constant terms andextracted topics as zt −1 = (1, T1t −1 , T2t −1 , T3t −1 ) ' , where Tit −1 is the number of i-th topics at t−1.
Although there are some approaches about how to select the number of topics, we assume three
topics for simplicity. The third proposed model (Model 4) combines Models 2 and 3 bysetting zt −1 = (1, st −1 , pt −1 ,, nt −1 , T1t −1 , T2t −1 , T3t −1 ) ' .
The proposed models are characterized as the hierarchical regression model whoseparameters evolve over time, synchronizing with temporal changes of variables constructedfrom social media communications at a previous time. Since the first column of coefficientmatrix G is the vector of parameters of the static Bass model (Model 1), these models arenested and include the original Bass model as a special case when additional text information
has no information on parameter evolutions in (5).

3.2 Posterior Density for Model Parameters
In terms of (4) and (5), the model is canonically described as a hierarchical nonlinear
regression model with time-varying parameters. We use a Bayesian MCMC method toestimate parameters since the procedure of hierarchical regression models has beenwell-established and the necessary conditional posterior densities are available in closed form,
except the time-varying parameter {θt } . Then we can proceed with relatively efficient
7

computational steps by combining Metropolis–Hasting sampling for three key parameters,with Gibbs sampling for the other parameters.
In fact, the joint posterior density of model parameters is formulated byp ({θt } ,t , G, Σ | { yt , t} , { zt } ) ∝ p ({θt } | { yt , t} ,t ) p (t | { yt , t} , {θt } )× p ( G | {θt } , { zt } , Σ ) p ( Σ | {θt } , { zt } , G )(6)

where the right-hand side of first line of (6) means the product of conditional posterior density
for parameters in the diffusion model (4) and the second line means those for hierarchicalmodel (5).
The sampling scheme of MCMC for this model is as follows.

Starting from the initial

parameter values, once {θt } is generated, the posterior density of hierarchical modelsp ( G | {θt } , { zt } , Σ ) and p ( Σ | {θt } , { zt } , G ) are available in closed forms, i.e., normal
and inverted gamma distributions with given hyper parameters.
likelihoodp ({ yt , t} | {θt } ,t )function

in

(4)

is

On the other hand, thecombined

with

prior

densityp ({θt } | G , { zt } , Σ ) from hierarchical model (5) to evaluate the conditional posterior

density asp ({θt } | { yt , t} ,t , G, { zt } , Σ ) ∝ p ({ yt , t} | {θt } ,t ) p ({θt } | G, { zt } , Σ ) .

(7)

We employ Metropolis–Hasting sampling for this posterior density. When {θt } is given,(

the conditional posterior density p t | { yt , t} , {θt }) of the right-hand side of (6) is known

as an inverted gamma distribution.
Finally, the posterior density of key parameters of the Bass model at the original scale isobtainedbyinversemt = exp ( mt * ) , pt =transformation

qt = ( m* , p* , q* )ofto( mt , pt , qt ) ,

i.e.,

1
1
and qt =; then, we can evaluate the joint*
1 + exp ( qt * )1 + exp ( pt )posterior density asp ({mt , pt , qt } , G | { yt , t} , { zt } ) .

(8)

The details of this algorithm, including the setting of a prior distribution, are given in the
Appendix.
8

4. Empirical Application
4.1 Data
We use the numbers for quarterly global sales of first-generation iPhones from June 2007 to
September, 2008. These data were obtained from www.statista.com and are displayed in
Figure 4.1.
-------------------------------------------------------------------Figure 4.1: iPhone Sales (June 2007–September 2008)-------------------------------------------------------------------As for social media information, corresponding to global sales data, we use “gsmarena”(http://www.gsmarena.com/), a well-known BBS for mobile phones, where users from all over
the world put their comments regarding mobile phones in which they are interested. Users of
this BBS can access information on topics for all phones and provide their own comments or
discuss topics with other users. We extract social media text data on the first-generation
iPhone and collect its sales data until the next-generation iPhone (iPhone 3G) is released. In
the BBS of gsmarena, a new topic for a mobile phone is usually created when this phone is firstannounced to the public by the company. On January 9, 2007, Steve Jobs gave a presentationon the iPhone and a thread was created the following day. Each comment has three elements:
user, date, and comment text. We extract date and comment text only because user
information is not used in this study. A total of 8,121 comments uploaded between January
10, 2007 and November 24, 2007 are divided into two groups: 1,500 comments for training
data and 500 comments for test data. The daily text data are converted to quarterly data andwe use the first four quarters for estimating models while the last two quarters are kept forholdout samples.

4.2 Sentiment Analysis
A conventional sentiment analysis uses two categories—positive and negative comments—toclassify comments.

However, this BBS usually has many unrelated comments such asquestions and discussions. Then we classify training data into three groups: positive, negative,
and no relation.

We confirmed that the no relation group improves the accuracy ofclassification.
9

We classify all 8,121 comments of training data and then test the accuracy using test datacomprising 500 comments.

The prior distribution and accuracy are given in Table 4.1. The

prior distributions 𝑝𝑝(𝐶𝐶𝑘𝑘 ) for three categories are calculated by counting the number of
positive comments in training data by interpreting each comment manually, i.e., by making a

dictionary: 39% for positive, 33.1% for negative, and 27.8% for neutral. The accuracy isdefined as the ratio of the number of hits over the number of comments in the test data of 500
comments. We found 94.2% of positively predicted comments in test data to be truly positive,
with hit rates of 90% and 84.7%, respectively, for negative and neutral predicted comments.
This shows the high precision of our dictionary for sentiment analysis.
----------------------------------------------------------------------------Table 4.1: Summary of Naïve Bayes Classifier----------------------------------------------------------------------------Figure 4.2 shows time-series plots for the numbers of positive and negative comments
used in our study.

The movements of these numbers are synchronized with those of sales

with the lag of one period; thus, these can be leading indicators for sales.
----------------------------------------------------------------------------------Figure 4.2 Time-series Plots of Positive and Negative Comments----------------------------------------------------------------------------------

4.3 Topic Analysis
In the topic model, we set the number of topics as three and the Bayesian Collapse Gibbs
sampling algorithm is used to estimate the model (see the Appendix for details).

The

number of M = 4,000 samples is used to evaluate posterior probability after discarding theprevious 1,000 samples as a burn-in period. This computation needed a tremendously longtime of about one week. Table 4.2 shows the top words for each topic.
----------------------------------------------------------------------------Table 4.2: Top Words for each Topic----------------------------------------------------------------------------First, the topic number in the table means the estimate of probability that topic k is in all10

D documents, i.e., the posterior mean E  θ k ,  =



∑θ
d

k ,d


/ D  . Topic 1 has the largest


probability of 0.554 and is dominant compared with other topics with almost half theprobability.

Next, the top twelve words for each topic are given in their order of frequency.

The number next to each word refers to its frequency in the document. According to theseclassifications, we can easily characterize each topic. Topic 1 contains “phone,” “n95” (Nokia
cellphone), “nokia,” “good,” etc., which are used regarding reviews. Topic 2 includes thewords “ur,” “me,” “install,” “tell,” “help,” “thanks,” “plz,” and other words used in the context
of discussions. Topic 3 contains “apple,” “will,” “market,” “Europe,”“us,” “released,” “uk,” and other words in the context of marketing. Thus, we call Topic 1“Reviews,” Topic 2 “Discussion,” and Topic 3 “Marketing.”
Figure 4.3 shows time-series plots for the number of words in each topic and global
sales data.
----------------------------------------------------------------------------Figure 4.3: Number of Words in each Topic----------------------------------------------------------------------------The figure shows that the number of topics, especially Reviews and Marketing (Topics 1
and 3) leads to sales with a one-period lag and suggests that they could be leading indicators
for accurate sales forecasting.

In contrast, Discussion (Topic 2) is synchronized with sales.

4.4 Model Comparison
The models were estimated by generated sample of θt ( k ) , k = 1,..., M , and we used M =5,000 samples for constructing the posterior density after discarding the previous 1,000
samples as the burn-in period. This required many iterations and almost 20 hours for the
MCMC sequence to converge for Model 4.

Other models did not need such a high numberof iterations. We confirmed their convergence using the Geweke’s test (Geweke, 1992), witha significance level of 95%. In the above, the non-informative diffuse prior was set forparameters. The specification of prior distribution and necessary conditional posterior
densities are provided in the Appendix.
11

Five models defined in previous section are compared based on three measures: log of
marginal likelihood (LMD), deviance information criteria (DIC), and root mean squared
errors of forecasts for holdout samples (RMSE). The results are provided in Table 4.3.
----------------------------------------------------------------------------Table 4.3: Model Comparison----------------------------------------------------------------------------First, the models with time-varying parameters perform significantly better than the static
model (Model 1) in terms of respective criterion. This means that BBS contains useful
information to describe the new product diffusion process. Among the dynamic models, themodel with sentiment analysis (Model 2) is supported slightly better than the model with
topics (Model 3).

However, their combined model (Model 4) performs best.

4.5 Parameter Estimates
Table 4.4 shows the estimates of coefficient matrix G in (5) for the respective models. The
first column of each table show the estimates of constant term of time evolution model fortransformed parameters.

This means the estimation of transformed parameters for theoriginal Bass model.

Other columns show the time-varying factors of transformed

parameters induced by several variables constructed by sentiment and topic analysis and using
BBS information.
----------------------------------------------------------------------------Table 4.4: Parameter Estimates----------------------------------------------------------------------------The estimates are defined as posterior mean and 95% CI (credible interval) with theboundary created by upper and lower 2.5 percentiles of posterior density given in parentheses
below the estimate. First, the estimates of the intercept term are shown in the column()

denoted “0” in the tables, and the time-invariant part of key parameters qt = mt* , pt* , qt* ' issignificantly estimated in the sense that 95% CI does not include zero. This means that theoriginal Bass model by itself (Model 1) and the parts of original Bass model for other models
are well estimated if we interpret them when they are inversely transformed. They drive a12smooth orbit of sales and the mechanism of the original Bass model works for all models as
an intrinsic part of the diffusion process. Second, three topics extracted by topic model
affect all parameters not only when used solely, i.e., in the case of Model 3, but also together
with sentiment variables, i.e., Model 4. Third, the numbers of total, positive and negative,comments from the sentiment analysis almost explain the changes of mt* , pt* , but it does nothold for qt* from the results of Models 2 and 4, and we could explain that imitators rely noton emotional but subjective factors such as review, discussion, and marketing by other people.
Next, we consider the result of the most supported model (Model 4) in more detail. The
time transition equation of mt* has significant positive coefficients on all covariates and they
induce a positive increase of mt* when they are increased.

According to the magnitude ofestimates due to the fact that the measurement scale is common in each category, the order ofeffectiveness is as follows: (i) Topic 2 (discussion) > Topic 3 (marketing) > Topic 1 (review)for topic models and (ii) positive comments > negative comments > number of comments.
As for pt* , Topic 3 (marketing) has a positive effect, implying that the recognition of anew product through the firm’s marketing activity would be creating new innovators at eachperiod.

On the other hand, Topic 1 (review) and Topic 2 (discussion) have negative

coefficients, and we interpret that active review and discussion are reflecting the circumstance
where there is too much product information, which would discourage innovative offerings on
the part of consumers.
Finally, the qt* equation has positive significant estimates of coefficient on three topic
variables; however, there is no effective variable in sentiment analysis. This means that thechange of imitator would be induced not by the subjective emotional factors in sentiment
analysis, but rather by objective product evaluation through review and discussion in topic
analysis.

4.6 Temporal Change of Key Parameters
Figure 4.4 shows the posterior density of key parameter estimates for Model 4, where the
13

(

)

estimates of qt = mt* , pt* , qt* ' , t = 1…, 4 (estimates), and 5, 6 (forecasting) are inverselytransformed to their original scales( m , p , q ) for the model interpretations. Mostttt

posterior densities are skewed by the form of log and logistic transformations. We share thisskewness throughout the models and then define the estimates of the original key parametersby the median, which provides a more reasonable point estimate in the case of a skewed
distribution.
----------------------------------------------------------------------------Figure 4.4: Posterior Density of Key Parameters----------------------------------------------------------------------------The temporal change of key parameter estimates is depicted in Figure 4.5.
----------------------------------------------------------------------------Figure 4.5: Temporal Change of Key Parameters----------------------------------------------------------------------------First, as for the market potential parameter m, the dynamic models have larger values than theoriginal Bass model (Model 1) and show the same pattern, growing with the peak at the third
period and declining after that, although the levels are different with the highest potential
numbers for Model 4. The variables constructed by sentiment analysis and topic models
have similar effects on determining the orbit of mt .
Second, the innovator p’s estimates take similar low values and Model 3 produces rather
fluctuating innovator estimates with the highest at the third period. The estimates of imitator

qt are heterogeneous among models. In particular, Models 2 and 3 have relatively lower
values, around 0.6–0.8 and, thus, share higher values with Model 4.

4.7 Forecasting
Bayesian inference in this model constitutes unconditional predictive density. The predictive
density for s-step ahead forecast yT + s can be written by the model structure asp ( yT + s | Data ) =∫ p ( yT + s | θT + s , G, Σ ) p (θT + s | G, Data ) p ( G | Σ, Data ) p ( Σ | Data ) dθ dGΣ
14(9)where yT + s is the s-step ahead forecast and θT + s is the corresponding time-varying
parameter vector. The integration in (9) can be numerically evaluated by efficient Monte
Carlo methods, i.e., by sequentially generating samples in addition to MCMC iterations forposterior density. That is, starting from some initial values ofsteps: (i)Σ( k )

is generated from p ( Σ | Data ) ; (ii)( G , Σ ) , we take the
(0)

(0)

G ( k ) is generated from ;p ( G | Σ ( k ) , Data ) , (iii) θT + s ( k ) is generated from p (θT + s ( k ) | G ( k ) , Data ) using equation((6); and (iv) yT + s ( k ) is generated from p yT + s ( k ) | θT + s ( k ) , G ( k ) , Σ ( k )) using equation (5).

We note that when the diffusion model contains an explanatory variable of “time,” thestructural equation is easily updated by shifting T to T+s, without assuming scenarios for
future explanatory variables, as is done by Takada et al. (2014).
----------------------------------------------------------------------------Figure 4.6: In-Sample and Out of Sample Fit--------------------------------------------------------------------------------------------------------------------------------------------------------Figure 4.7: Predictive Density for Model 4----------------------------------------------------------------------------Figure 4.6 shows the generated forecasts of respective models from the fifth and sixth
periods, where in-sample fits from the first to fourth periods are also depicted and where theforecasts are defined as the mean of predictive density. The predictive densities for Model 4
are shown with observation by the x-mark in Figure 4.7.

They are well-defined andaccommodate holdout observations in the center of density, implying that the forecast using
predictive density has high precision. In addition, we can evaluate the predictive interval
easily by evaluating percentiles of predictive density.

5. Concluding Remarks
The fusion of numeric structured data and unstructured text data is a challenging issue in big
data analysis and it is also demanded in marketing research.

15

In this article, we proposed time-varying diffusion models to accommodate social media
information. These models belong to the class of systematic variation models and provide
useful insights on parameter variations, where we enlarge the information set regarding the
diffusion process using product-related BBS text data from before and after the launch of anew product. We use this information based on the recognition that communications in BBS
reflect changes in consumer expectations before launch as well as changes in productevaluations of not only the product itself but also the marketing activity and its competitive
products. In particular, the communications among potential customers waiting to launchinnovative IT products used in our study contain a sort of a proxy variable for consumers’
expectations before launch and changes in perception and evaluation after launch.
Our proposed models contain additional variables constituted from BBS text data by
applying two approaches for analyzing text data, i.e., sentiment analysis and topic analysis.
These variables are used as covariates to explain parameter temporal transitions. These
analytical techniques are expected to extract subjective emotional variables andevaluation-based objective variables in BBS, respectively. The empirical study showed thatthese additional variables lead to an improvement in the model fit and precision of forecastingby filling a gap between smooth transitions of sales generated by a static diffusion model andrealized sales, and they provide the roles of constructed variables in text analysis for thechange in model parameters.

For example, both of the emotional sentiment variables, rather

than objective topic variables, have positive effects on market potential; on the other hand,
topic variables affect the innovator and imitator with reasonable interpretation while
sentiment variables affect the change of innovator transition, but not for that of the imitator.
We also showed that the proposed model with the augmented information set produces a great
improvement in the precision of forecasting.
IT products, such as the iPhone, continue to evolve and, together with growing social
media networks, we can consider the extension of our model to successive productgenerations, including second- and third- generation products, by using the models of Norton
and Bass (1987), Mahajan and Muller (1996), Kim et al. (2000) and others. Future research
can investigate into this problem.
16

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18

Appendix: MCMC Algorithm
I.

LDA Topic Model

Under the prior distributions of Dirichllet distributions on hyperparameter α, β𝜃𝜃𝑑𝑑 ~ 𝐷𝐷𝐷𝐷𝐷𝐷(𝜶𝜶) (𝑑𝑑 = 1, … , 𝑀𝑀), and 𝜙𝜙𝑘𝑘 ~ 𝐷𝐷𝐷𝐷𝐷𝐷( 𝛃𝛃) (𝑘𝑘 = 1, … , 𝐾𝐾), where M is the number of

documents, and K means the number of topics. Following Griffiths and Steyvers (2004), we
set the vectors with element of 50/K for α and 0.1 for 𝜷𝜷 respectively. Then, the collapsed

Gibbs Sampler provides the posterior density in the closed form,| wd ,i v, w
p=
( zd ,i k=\ d ,i,z\ d ,i, α, β )=nk\ d,v,i + β v

nd\ d,k,i + α k

nk\ d,⋅ ,i + ∑ β v ' nd\ d ,i + ∑ α k '
v'

,

k'

where wd ,i means word i in document d, zd ,i is latent topic of word i in document d, θ dmeans topic distribution of document d, and φk is vocabulary distribution of topic k.

w \ d ,iis all words from the text data except word 𝑤𝑤𝑑𝑑,𝑖𝑖 , z \ d ,i is all topics except 𝑧𝑧𝑑𝑑,𝑖𝑖 , 𝑛𝑛𝑑𝑑 is the\ d ,inumber of words in document d, nk ,v means frequency of word v in topic k except word i in

document d, and nk\ d,⋅ ,i =∑n

\ d ,ik ,v .

v

II.

Proposed Model

1. Prior Distributions
Parameter

Setting

𝑚𝑚𝑘𝑘 ~ 𝑁𝑁(𝜇𝜇𝑚𝑚0 , 𝜏𝜏𝑚𝑚0 −1 )𝜇𝜇𝑚𝑚0 = 0, 𝜏𝜏𝑚𝑚0 = 0.3

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

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

τ~ 𝐼𝐼𝐼𝐼(𝛼𝛼, 𝛽𝛽)

α = 3 , β = 10

τ𝑗𝑗 ~ 𝐼𝐼𝐼𝐼(𝛼𝛼, 𝛽𝛽)

α = 3 , β = 10γ~ 𝐼𝐼𝐼𝐼(𝛼𝛼, 𝛽𝛽)

α = 3 , β = 10

2. Conditional Posterior Distributions
19

(1)

𝐺𝐺𝑗𝑗𝑗𝑗 |{𝜃𝜃𝑡𝑡 }, {𝑧𝑧𝑡𝑡 }, Σ(2)

τ𝑗𝑗 | {𝜃𝜃𝑡𝑡 }, {𝑧𝑧𝑡𝑡 }, 𝐺𝐺−1

𝑁𝑁 ��𝑛𝑛𝜏𝜏𝑗𝑗 + 𝜏𝜏𝑗𝑗0 �

𝑛𝑛

𝐾𝐾≠𝑘𝑘
−1
∗
−1
�𝜏𝜏𝑗𝑗 � �𝜃𝜃𝑗𝑗𝑗𝑗 − � �𝐺𝐺𝑗𝑗𝑗𝑗 𝑧𝑧𝑖𝑖 � � 𝐺𝐺𝑗𝑗𝑗𝑗+ 𝜇𝜇𝑗𝑗0 𝜏𝜏𝑗𝑗0 � , �𝑛𝑛𝜏𝜏𝑗𝑗 + 𝜏𝜏𝑗𝑗0 � �
𝑖𝑖=1
𝑧𝑧=1
𝑛𝑛

𝐾𝐾

𝑖𝑖=1

𝑧𝑧=1

2

𝐼𝐼𝐼𝐼 �α + n/2, β + � �𝜃𝜃𝑗𝑗𝑗𝑗∗ − ��𝐺𝐺𝑗𝑗𝑗𝑗 𝑧𝑧𝑖𝑖 � � /2�(3) {θt } | { yt , t} ,t , G , { zt } , Σ

The conditional posterior density of θt is generated by Metropolis–Hastings samplingby the proposed density on the right hand side ofp ({θt } | { yt , t} ,t , G, { zt } , Σ ) ∝ p ({ yt , t} | {θt } ,t ) p ({θt } | G, { zt } , Σ ) .
For iter(=1,…,R) of MCMC iterations, we use Metropolis–Hastings with a random walkalgorithm,(𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖)

𝜃𝜃𝑗𝑗𝑗𝑗α = min �1,(𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖−1)= 𝜃𝜃𝑗𝑗𝑗𝑗(𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖)+ 𝜆𝜆𝜃𝜃 ; 𝜆𝜆𝜃𝜃 ~ 𝑁𝑁(0,0.05) , where the acceptance probability is| {𝑦𝑦𝑡𝑡 ,𝑡𝑡},𝜏𝜏,𝐺𝐺,{𝑧𝑧𝑡𝑡 },τ𝑗𝑗 �(𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖−1)𝑝𝑝�𝜃𝜃𝑗𝑗𝑗𝑗| {𝑦𝑦𝑡𝑡 ,𝑡𝑡},𝜏𝜏,𝐺𝐺,{𝑧𝑧𝑡𝑡 },τ𝑗𝑗 �𝑝𝑝�𝜃𝜃𝑗𝑗𝑗𝑗(4) τ|{𝑦𝑦𝑡𝑡 , 𝑡𝑡}, {𝜃𝜃𝑡𝑡 }�, where t = 1, …, N and j = 1,2,3..

𝑛𝑛

2

𝐼𝐼𝐼𝐼 �α + n/2, β + � �y𝑡𝑡 − 𝑚𝑚𝑡𝑡 �𝐹𝐹(𝑡𝑡|𝑝𝑝𝑡𝑡 , 𝑞𝑞𝑡𝑡 ) − 𝐹𝐹(𝑡𝑡 − 1|𝑝𝑝𝑡𝑡−1 , 𝑞𝑞𝑡𝑡−1 )�� /2�
𝑖𝑖=1

20

Table 4.1: Summary of Naïve Bayes ClassifierClass
Positive
Negative
No relation

Prior(Trainning data)
0.390
0.331
0.278

Accuracy(Test data)
0.942
0.900
0.847

Table 4.2: Top Words for each Topic

Topic1 (0.554)
phone
0.0093n95
0.0072nokia
0.0071
good
0.0070but
0.0069better
0.0069people0.0054
camera0.0053
think
0.0041like
0.0041
really0.0039because0.0038Topic2 (0.224)
ur
0.0117me
0.0081can
0.0080fone
0.0080install
0.0080tell
0.0061
help
0.0061
thanks
0.0060
bluetooth0.0049plz
0.0048
installer
0.0041
files
0.0040Topic3 (0.222)
apple
0.0063
mobile0.0059will
0.0058market0.0058contract
0.0054
network
0.0051
europe0.0049sim
0.0048us
0.0047
released
0.0041uk
0.0032june
0.0032

Table 4.3: Model Comparison
Model 1Model 2Model 3Model 4

RMSE
0.376
0.101
0.125
0.016

Log(ML)
1.619
3.059
2.472
5.017

21

DIC
2.522
2.365
2.403
2.217

Table 4.4: Parameter Estimates
Model 1m*p*q*

02.035 *
[2.026,2.044]-3.019 *[-3.036,-3.002]
24.473 *[24.511,24.438]

num.comment-

pos.comment-

neg.comment-

Topic1-

Topic2-

Topic3-0
0.145[-1.177,0.468]-3.899 *[-4.627,-3.171]
0.526[-0.128,1.181]

num.comment
1.173 *
[0.834,1.512]
4.067 *
[3.364,4.771]
0.445[-0.245,1.137]

pos.comment
0.338[-0.145,0.823]-4.539 *[-6.016,-3.063]
0.667[-0.218,1.553]

neg.comment
0.902 *
[0.336,1.469]-3.839 *[-5.277,-2.401]
1.160 *
[0.274,2.045]

Topic1-

Topic2-

Topic3-00.624 *
[0.530,0.691]-1.139 *[-1.215,-1.097]-0.289 *[-0.368,-0.210]

num.comment-

pos.comment-

neg.comment-

Topic10.320 *
[0.233,0.417]-3.144 *[-3.699,-2.589]
0.931 *
[0.548,1.314]

Topic20.886 *
[0.552,1.120]
1.305[-0.104,2.716]
1.674 *
[0.930,2.418]

Topic3-0.431 *[-0.531,-0.367]
4.513 *
[4.218,4.807]
1.907 *
[1.627,2.187]

0-0.473 *[-0.498,-0.449]-1.826 *[-1.862,-1.774]-0.260 *[-0.302,-0.218]

num.comment
0.090 *
[0.063,0.117]
0.609 *
[0.551,0.666]
0.011[-0.036,0.059]

pos.comment
0.805 *
[0.037,0.123]-0.398 *[-0.299,-0.497]-0.023[-0.099,0.052]

neg.comment
0.198 *
[0.151,0.245]
0.019[-0.087,0.126]-0.013[-0.094,0.068]

Topic10.330 *
[0.308,0.352]-0.612 *[-0.676,-0.548]
0.248 *
[0.208,0.288]

Topic20.831*[0.796,0.866]-0.209 *[-0.284,-0.134]
0.503*[0.429,0.577]

Topic30.442 *
[0.402,0.482]
0.131 *

Model 2m*p*q*Model 3m*p*q*Model 4
m*
p*
q*

22

[0.017,0.244]
0.964 *
[0.906,1.023]

Figure 4.1: iPhone Sales (June 2007–September 2008)

Figure 4.2 Time Series Plots of Positive and Negative Comments

Figure 4.3: Number of Topics

23

Figure 4.4: Posterior Density of Key Parameters: Model 424

Figure 4.5: Key Parameter Estimates

25

Figure 4.6: In-sample and Out of Sample Fit

Figure 4.7: Predictive Density for Model 426