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  1. Random Component: Distribution of { Y t | Y t - 1 , } belongs to the exponential family of distributions.
    f ( y t ; θ t , φ | F ) = e x p y t θ t - b ( θ t ) a t ( θ t ) + c ( y t , φ ) ,
    where θ t is the parameter of the distribution and φ is a dispersion parameter. In the Poisson case, we have: a t ( φ ) = 1 ; θ t = log ( μ t ) ; b ( θ t ) = μ t ; c ( y t ; φ ) = - log y .
  2. Systematic Component: μ t , the mean of Y t , is modeled by a monotone link function g ( · ) such that
    g ( μ t ) = X t T β ,
    where X is the set of covariates and β is a vector of coefficients. In the Poisson case μ = λ , g ( λ ) = log ( λ ) .

In the case of time series data, we can augment the exogenous covariates in the model with lagged values of the response variable, i.e. the observed counts at previous time points. Thus the model is “observation-driven.” Lags of exogenous covariates can also be included. For instance let the new covariate matrix be represented by Z , where

Z t T = ( X t , X t - 1 , , X t - p , Y t - 1 , , Y t - n ) .

For more details see [link] .

Clustering of tsc models

Armed with the GLM model for Poisson regression, we can begin clustering the TSC. In order to determine the similarity or dissimilarity between two TSC, a metric is needed to measure the “distance.” The classic Euclidean metric is not adequate for data with time dependence. We will use the empirical Kullback-Leibler (KL) likelihood metric [link] , which calculates the distance between two TSC by evaluating the relative fit of their respective models.

Let λ j be a given “model structure” for the data, i.e. an observation-driven Poisson model with specified covariates. The KL metric has the following expression.

D K ( λ k , λ j ) = 1 | Y K | y Y K ( log p ( y | λ k ) - log p ( y | λ j ) )

where Y K is the set of data objects which belong to cluster k . Note that log p ( y | λ k ) is an expression for the likelihood of the model. See [link] for discussion on the likelihood of observation-driven Poisson models. The measure is made symmetric by,

D S K = D K ( λ k , λ j ) + D K ( λ j , λ k ) 2

With the KL metric, we apply a hierarchical bottom-up clustering algorithm. A flowchart of the algorithm is displayed in Figure 1.

The MBC algorithm

The algorithm produces a cluster tree similar to the figure below. The bottom-up clustering method is easy to visualize and break down objects into groups and eliminates the need for any stopping criterion.

Sample hierarchical cluster tree


Finding relevant data

Though count data are prevalent in consumer behavior, obtaining commercial commercial data for MBC is expensive. Thus, for this project, we use results from previous studies on marketing data to creat a data set that realistically mimics consumer behavior.

Data simulation

Niraj et al. [link] proposed an economic model for consumer purchases of bacon and eggs. Based on store scanner data, the authors studied the consumer sensitivities to various variables such as personal utility, product prices, product displays, and purchase history. For the purpose of data simulation, key elements from this economic model were borrowed to create our own consumer bacon and eggs purchase data.

We let Y b , t and Y e , t be a bivariate Poisson random variable which represent a consumer's purchase of bacon and eggs during time window t respectively, then Y b , t and Y e , t can be modeled using a trivariate reduction [link] :

Questions & Answers

How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
How can I make nanorobot?
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
how can I make nanorobot?
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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