How to do multiple regression where all dependent and independent variables have uncertainties?

I have some data that I would like to run multiple regression on. The important thing here is that I would like the multiple regression to take into account the uncertainties in each of the data points -- something that mvregress doesn't do, as far as I can tell. What's more, each measurement in each of the variables has an uncertainty associated with it. For example, each measurement y(i) in the dependent variable y has its own uncertainty sy(i).

y = [y(1) y(2) ... y(n)]; % measured values of y

sy = [sy(1) sy(2) ... sy(n)] % uncertainties in measured values of y

The same goes for each of the independent variables x, w, z, etc.

x = [x(1) x(2) ... x(n)]

sx = [sx(1) sx(2) ... sx(n)]

When doing a single linear regression analysis, the function york_fit is able to handle data that have uncertainties in both the measured x- and y-values. I'd like to use something analogous to that -- just for multiple regression.

Anybody know how to do that? Is there a variant on mvregress that does that? Are there any canned functions that do that?

-Ken

Answer 1

[TED MOSBY]

Hi,

A convenient implementation for this case can be the File‑Exchange package “Total Least Squares with mixed and/or weighted disturbances” (gtls.m / gmtls.m)

https://www.mathworks.com/matlabcentral/fileexchange/50332-total-least-squares-with-mixed-and-or-weighted-disturbances

Have a look at the examples in the link above.

It lets you give:

• A – the predictor matrix [x  w  z  … ]
• b – the response vector y
• W – a block‑diagonal weight or full covariance matrix built from your per‑point uncertainties

A basic implementation can be as below:

n  = numel(y);
A  = [x(:)  w(:)  z(:)  ones(n,1)];       
b  = y(:);

Sig = diag([sx(:);  sw(:);  sz(:);  sy(:)].^2);

beta = gtls(A,b,Sig);

Modify this according to your use-case.

Hope this helps!