# [R] "Special" LS estimation problem

(Ted Harding) Ted.Harding at manchester.ac.uk
Fri Aug 21 20:19:04 CEST 2009

```On 21-Aug-09 18:12:45, megh wrote:
>
> "Y" is 2 dimensional vector, X1 and X2 are 2x2 matrices.

Apologies.
Ted.

> Ted.Harding-2 wrote:
>>
>> On 21-Aug-09 16:28:26, megh wrote:
>>> Hi, I have following kind of model : Y = X1 * a1 + X2 * a2 + error
>>>
>>> Here sampled data for Y, X1, X2 are like that :
>>>
>>> Y <- replicate(10, matrix(rnorm(2),2), simplify = F)
>>> X1 <- replicate(10, matrix(rnorm(4),2), simplify = F)
>>> X2 <- replicate(10, matrix(rnorm(4),2), simplify = F)
>>>
>>> My goal is to calculate LS estimates of vectors "a1" and "a2".
>>> Can anyone please guide me how to do that in R? Is there any
>>> special function to handle this kind of problem?
>>>
>>> Thanks
>>
>> In your example, a1 and a2 cannot be vectors, since Y, X1 and X2
>> are all 2x2 matrices. On the other hand, either or both of a1, a2
>> could be either a scalar or a 2x2 matrix, any of which would make
>> X1*a1 + X2*a2 (provided you interpret "*" as "%*%" in R) into a
>> 2x2 matrix. But if (say) a1 is a vector it can only be a 2x1
>> vector (for multiplication conformity), and then X1*a1 (or X1%*%a1
>> in R) would be a 2x1 vector (in the linear algebra sense, i.e.
>> a 2x1 matrix in the R sense).
>>
>> So what is it that you want in the model?
>>
>> Next, if in fact a1 and a2 are scalars, then in effect you are
>> looking at
>>
>>   Y1[1,1] = a1*X1[1,1] + a2*X2[1,1] + error[1,1]
>>   Y1[1,2] = a1*X1[1,2] + a2*X2[1,2] + error[1,2]
>>   Y1[2,1] = a1*X1[2,1] + a2*X2[2,1] + error[2,1]
>>   Y1[2,2] = a1*X1[2,2] + a2*X2[2,2] + error[2,2]
>>
>> 10 times over. The relevant question here is: Are you wanting
>> to include possible covariance between the 4 elements of 'error'?
>>
>> One possibility in this case is to treat this as a multilevel
>> or longitudinal model, with 4 observations "per subject", and
>> correlated error within-subject.
>>
>> But you really need to spell out more detail of the kind of
>> model you are seeking to fit. What you described is not enough!
>>
>> Ted.
>>
>> --------------------------------------------------------------------
>> E-Mail: (Ted Harding) <Ted.Harding at manchester.ac.uk>
>> Fax-to-email: +44 (0)870 094 0861
>> Date: 21-Aug-09                                       Time: 18:48:31
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>>
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>> and provide commented, minimal, self-contained, reproducible code.
>>
>>
>
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--------------------------------------------------------------------
E-Mail: (Ted Harding) <Ted.Harding at manchester.ac.uk>
Fax-to-email: +44 (0)870 094 0861
Date: 21-Aug-09                                       Time: 18:59:54
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