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Re: st: 2 dimensional graph for joint distribution
From
David Hoaglin <[email protected]>
To
[email protected]
Subject
Re: st: 2 dimensional graph for joint distribution
Date
Fri, 1 Nov 2013 08:01:48 -0400
Dear Lulu,
I have not had contact with the literature of choice models, but I
looked at the paper by Train. (Thank you for including the link.)
I do not recall seeing an explanation of why odo and bill are independent draws.
In your initial message you said that you had estimated those
coefficients "from a nonparametric choice model using fixed mass point
method following Kenneth Train's approach in his 2008 paper."
According to the text of that paper, Figure 5 and Figure 6 show the
joint distribution of two pairs of coefficients (out of the seven
coefficients in his model). That is, the height of each bar is the
sum of the values of share for the combinations of values of the two
coefficients that fall in that particular bin. The pattern of heights
of the bars in the two figures does not seem compatible with taking
the product of independent marginal distributions. Indeed, I would
not expect the distributions of a pair of coefficients to be
independent.
I don't know whether your data will produce a reasonably smooth
surface, but Train's Figure 5 and Figure 6 are definitely based on
(estimated) joint probabilities. In your last paragraph, your data
will determine whether you have the joint probability for odo = 0.26
and bill = 0.30 and also the joint probability for odo = 0.26 and bill
= 0.56. You have displayed share as a column vector. For a figure
like those in Train's paper, however, you should think of it as the
height of the bar for the particular combination of values of odo and
bill. The combinations of values of odo and bill in your data will
determine the "grid" (which may be reduced to the bins of a
histogram).
As I recall, you have 1,000 combinations of odo and bill (with the
share for each). It may be useful to look at the marginal
distribution of odo and the marginal distribution of bill, to choose a
reasonable set of bins for each (i.e., not leave the choice to a
histogram command), and then produce (as panels of the same display) a
separate histogram of bill for each category of odo. This approach is
not elegant, but it avoids the shortcomings of 3D in Train's Figure 5
and Figure 6.
David Hoaglin
On Fri, Nov 1, 2013 at 12:04 AM, Lulu Zeng <[email protected]> wrote:
> Dear David, Alfonso and others,
>
> Thank you for your comments.
>
> My understanding of the fixed mass point choice model (using EM
> algorithm) is that -- share here is the individual probability of odo
> and bill (share is the same for odo and bill), not the joint
> probability.
>
> odo and bill are independent draws (using the mdraws command),
> therefore the joint probability is the product of their individual
> probability.
>
> Please let me know if I have misunderstood the model. But if share is
> the joint probability, then it would be just a colum vector, not a
> grid (e.g., I can only have the joint probability for odo of 0.26 and
> bill of 0.30, can't have the joint probability for odo of 0.26 and
> bill of 0.56?) how can I produce a surface graph version of Train's
> graph on page 65 of http://elsa.berkeley.edu/~train/EMtrain.pdf?
>
> It would be really appreciated if I could have your advice on this.
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