# In the multi-linear regression, how is the value of weight $b_2$ calculated?

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 $$b_2$$?

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

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

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.

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).