0
$\begingroup$

In multivariate linear regression (linear regression with more than one variable) the model is $yi = b_0 + b_1x_{1i} + b_2x_{2i} + ...$ , and so on. But how is the $b_n$ value calculated iteratively? Can it be calculated non-iteratively? What is the intuition behind using that method to calculate $b_2$?

$\endgroup$
2
$\begingroup$

It is calculated the same way $b_1$ is calculated.

Nearly following your notation, say your multiple linear regression function is

$H(X_i) = b_0 + b_1x_{1,i} + ...+ b_nx_{n, i}$

for data instance $X_i=x_{1,i},...,x_{n, i}$ and weights $b_0,...,b_n$.

And say your error function is $E(X,Y) = \sum_i(H(X_i)-Y_i)^2$

where $X$ is the collection of all data points $X_i, Y_i$.

From your error function $E$, for whatever weights you have (with a gradient based method), calculate the partial derivative $\partial E /\partial b_i$ and use this to update all of your weights at once in each iteration of your optimization routine.

$\endgroup$
-1
$\begingroup$

put your b_1, ... ,b_n coefficients into a vector b

put all your x_{ij} into a matrix X

then all components of b are calculated at the same time with this equation

b = (H(X) - Xb)^T (H(x) - Xb)

But this calculation (estimation) is only consistent (search consistent estimator) when certain assumptions are present (read here https://en.wikipedia.org/wiki/Ordinary_least_squares).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.