# An application of model-based clustering in market segmentation  (Page 4/5)

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$\begin{array}{cc}\hfill {Y}_{b,t}& ={U}_{b,t}+{U}_{be,t}\hfill \\ \hfill {Y}_{e,t}& ={U}_{e,t}+{U}_{be,t}\hfill \end{array}$

where ${U}_{i,t}\sim Pois\left({\lambda }_{i,t}\right)$ for $i=e,b,be$ . ${U}_{b,t}$ and ${U}_{e,t}$ represent the consumer's tendency to buy bacon or eggs independently while ${U}_{be,t}$ 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: $log\lambda ={Z}^{T}\beta$ . In the simulation, ${\lambda }_{b,t}$ and ${\lambda }_{e,t}$ 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:

$\begin{array}{c}\hfill log{\lambda }_{b,t}={\beta }_{b,0}+{\beta }_{b,1}Uti{l}_{b,t}+{\beta }_{b,2}Pric{e}_{b,t}+{\beta }_{b,3}Dis{p}_{b,t}+{\beta }_{b,4}Dis{p}_{e,t}+{\beta }_{b,5}{Y}_{b,t-1}+{\beta }_{b,6}{Y}_{e,t-1}\\ \hfill log{\lambda }_{e,t}={\beta }_{e,0}+{\beta }_{e,1}Uti{l}_{e,t}+{\beta }_{e,2}Pric{e}_{e,t}+{\beta }_{e,3}Dis{p}_{b,t}+{\beta }_{e,4}Dis{p}_{e,t}+{\beta }_{e,5}{Y}_{b,t-1}+{\beta }_{e,6}{Y}_{e,t-1}\end{array}$

for simplicity, $log{\lambda }_{be,t}={\beta }_{be,0}$ .

The consumer's utility, $Util$ , is assumed to follow a Gumbel distribution [link] with location $=0$ and scale=0. After consulting local grocery stores, we let $Pric{e}_{b}\sim N\left(4,0.7\right)$ and $Pric{e}_{e}\sim N\left(3,0.3\right)$ . $Display$ indicates whether the product was advertized in store. This indicator variable is either on (1) or off (0) with probability p .

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.

## Modeling consumer purchases

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

$\begin{array}{cc}\hfill {Y}_{b,t}& ={U}_{b,t}+{U}_{be,t}\hfill \\ \hfill {Y}_{e,t}& ={U}_{e,t}+{U}_{be,t}\hfill \end{array}$

Where ${U}_{i,t}\sim Pois\left({\lambda }_{i,t}\right)$ for $i=b,e,be$ . 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:

$\begin{array}{cc}\hfill log\left({\lambda }_{b,t}\right)& ={\beta }_{b}^{0}+{\beta }_{b}^{1}Pric{e}_{\left(b,t\right)}+{\beta }_{b}^{2}Dis{p}_{\left(b,t\right)}+{\beta }_{b}^{3}Dis{p}_{\left(e,t\right)}+{\beta }_{b}^{4}{Y}_{\left(b,t-1\right)}+{\beta }_{b}^{5}{Y}_{\left(e,t-1\right)}\hfill \\ \hfill log\left({\lambda }_{e,t}\right)& ={\beta }_{e}^{0}+{\beta }_{e}^{1}Pric{e}_{\left(e,t\right)}+{\beta }_{e}^{2}Dis{p}_{\left(b,t\right)}+{\beta }_{e}^{3}Dis{p}_{\left(e,t\right)}+{\beta }_{e}^{4}{Y}_{\left(b,t-1\right)}+{\beta }_{e}^{5}{Y}_{\left(e,t-1\right)}\hfill \\ \hfill log\left({\lambda }_{be,t}\right)& ={\beta }_{be}^{0}\hfill \end{array}$

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

We also tested the robustness of regression model to varying strengths of ${\lambda }_{be,t}$ , the covariance term. A summary of the simulation studies is presented in Figure 5.

The regression is slightly more accurate with a lower ${\lambda }_{be,t}$

## 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.

can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
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Sherica
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Sherica
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Tamia
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a perfect square v²+2v+_
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algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
is it 3×y ?
J, combine like terms 7x-4y
im not good at math so would this help me
yes
Asali
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Samantha
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Asali
how do you translate this in Algebraic Expressions
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
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Yasmin
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Cesar
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Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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Stotaw
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
Azam
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Prasenjit
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
Azam
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Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
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