How does one use LinearModelFit with multiple y variables?

Questions
,
1

Using a list as the y variable would seem most natural but doesn’t work.

2

Could you please post the example of code that you tried and couldn’t get to work? Also you probably use some data that you are fitting; - it’d be very useful to point us to them or give a code that generates a similar test data set. Without this info it is hard to proceed.

3

For example let’s say we want to simultaneously predict GDP and Population as a linear function of the Area and Coastline Length for each country having sufficient data:

data = Cases[  Table[CountryData[country, property], {country,     CountryData[]}, {property, {"Area", "CoastlineLength", "GDP",      "Population"}}], {_Real ..}];

There is no problem predicting either GDP or Population separately. Each of the following works:

LinearModelFit[data[[All, {1, 2, 3}]], {area, coast}, {area, coast}]

LinearModelFit[data[[All, {1, 2, 4}]], {area, coast}, {area, coast}]

And one would imagine this would generalize to multivariate regressions with multiple outcome variables naturally:

LinearModelFit[{#1, #2, {#3, #4}} & @@@ data, {area, coast}, {area,   coast}]

But this comes back with an error:

LinearModelFit::notdata: The first argument is not a vector, matrix, or a list containing a design matrix and response vector. >>
4

Is there some reason to do this other than one at a time? Offhand I do not see a drawback to that approach.

5

You’d lose the covarianece between the two variables. In some applications, that’s fine, and people just do several independent univariate regressions. But sometimes you really want to do a single multivariate one. There’s a whole literature on it, the general linear model, MANOVA, etc.

http://en.wikipedia.org/wiki/General_linear_model

http://en.wikipedia.org/wiki/Multivariate_analysis_of_variance

6

In Wolfram Library there is an upload:

Multiple-Response Fitting by Bruce Miller

Daniel sent a few links - perhaps they will be useful:

7

Wonderful! Thank you Vitaliy and Daniel.