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Y b , t = U b , t + U b e , t Y e , t = U e , t + U b e , t

where U i , t Pois ( λ i , t ) for i = e , b , b e . U b , t and U e , t represent the consumer's tendency to buy bacon or eggs independently while U b e , t represents consumer's tendency to buy the two products together. Note that Y b , t and Y e , t are marginally Poisson since sum of two Poisson variables is still Poisson.

Recall the Poisson log link function for GLM: log λ = Z T β . In the simulation, λ b , t and λ e , t are modeled using exogenous covariates (utility, price and product displays) as well as one lag of response, Y t - 1 i.e. the quantities of the product purchased last time period:

log λ b , t = β b , 0 + β b , 1 U t i l b , t + β b , 2 P r i c e b , t + β b , 3 D i s p b , t + β b , 4 D i s p e , t + β b , 5 Y b , t - 1 + β b , 6 Y e , t - 1 log λ e , t = β e , 0 + β e , 1 U t i l e , t + β e , 2 P r i c e e , t + β e , 3 D i s p b , t + β e , 4 D i s p e , t + β e , 5 Y b , t - 1 + β e , 6 Y e , t - 1

for simplicity, l o g λ b e , t = β b e , 0 .

The consumer's utility, U t i l , is assumed to follow a Gumbel distribution [link] with location = 0 and scale=0. After consulting local grocery stores, we let P r i c e b N ( 4 , 0 . 7 ) and P r i c e e N ( 3 , 0 . 3 ) . D i s p l a y indicates whether the product was advertized in store. This indicator variable is either on (1) or off (0) with probability p .

Simulated purchases of bacon and eggs for 100 weeks for one consumer with sensitivities: β b = ( - 2 . 4 , 1 , - 0 . 05 , 0 . 8 , 0 . 3 , - 0 . 5 , 0 . 2 ) , β e = ( - 0 . 6 , 1 , - 0 . 02 , 0 . 3 , 1 . 50 , 0 . 2 , - 0 . 5 ) and β b e = - 0 . 2 .

A realization of one consumer's purchase over time is plotted in Figure 3. We notice a few things in this plot that make it “realistic”: only small quantities are purchased; when higher quantity was purchased in a previous period, fewer units were purchase during the next period; the pruchases of the two products seem correlated as a number of peaks overlap.

Ongoing work

Modeling consumer purchases

Similar to data simulation, we model the consumer purchases of bacon and eggs using a trivariate reduction.

Y b , t = U b , t + U b e , t Y e , t = U e , t + U b e , t

Where U i , t Pois ( λ i , t ) for i = b , e , b e . However, we constrain the covariates to include only observable variables: price, display and past purchase. Thus in the log link function for Poisson GLM, we model λ like this:

log ( λ b , t ) = β b 0 + β b 1 P r i c e ( b , t ) + β b 2 D i s p ( b , t ) + β b 3 D i s p ( e , t ) + β b 4 Y ( b , t - 1 ) + β b 5 Y ( e , t - 1 ) log ( λ e , t ) = β e 0 + β e 1 P r i c e ( e , t ) + β e 2 D i s p ( b , t ) + β e 3 D i s p ( e , t ) + β e 4 Y ( b , t - 1 ) + β e 5 Y ( e , t - 1 ) log ( λ b e , t ) = β b e 0

Methods for estimating bivariate Poisson regression models are available in R package “bivpois” [link] . We can compare the λ ^ generated by the model against the “real” λ used in the simulation. Note that λ ^ i , t = λ i , t ^ + λ b e , t ^ for i = b , e .

Comparing fitted λ ^ (in solid line) with “real” λ (black dots) used in data simulation

We also tested the robustness of regression model to varying strengths of λ b e , t , the covariance term. A summary of the simulation studies is presented in Figure 5.

Median value of estimated regression coefficients compared to “real” betas used in simulation. (Top row: bacon; Bottom row: egg)

The regression is slightly more accurate with a lower λ b e , t

Next steps

We are currently working to extend the univariate MBC method to the bivariate case. The extension process consists of developing the bivariate model for the simulated consumer TSC data, deriving the bivariate KL metric, and improving the clustering algorithm to cluster bivariate models. So far, we have developed a working bivariate Poisson regression model using the bivpois package. The clustering algorithm is still under development.

Questions & Answers

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
Do somebody tell me a best nano engineering book for beginners?
s. Reply
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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?
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
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s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
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what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
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I'm interested in nanotube
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Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
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Prasenjit Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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