# Linear regression - iterative approach

I have a single output variable $y$ and a number of inputs $x_1$, $x_2$, etc. These are time series. Each $x_i$ explains the changes in $y$ in specific circumstances, and the goal is to have a linear model that looks like $y=b_1x_1+b_2x_2+...$ The point is that each $x_i$ must be sampled on different clock $c_i$. Imagine that $y$ is continuous, but each $x_i$ is somewhat discrete, most of the time is 0, an occasionally it has an impulse, i.e. 'ticks'. Then, it decays quickly back to steady state of 0. We want to sample $x_i$ only then, when it "jumps", and decide the influence of these impulses of separate $x_i$s individually.

My approach is to take one sampling method $s_1$, sample $y$ and $x1$ on it, specific to $x_1$. Do the regression $y$ on $x_1$. Then, take the error $y-b_1x_1$, sample the error on another sampler $s2$, and regress this on $x_2$.

I can decide to sample $x$s and $y$ whenever I want, I can examine their value at any point in time, and decide whether to sample based on that. Most of time the $x$s will be 0, so I will choose to sample $x_i$ when it's actually trying to tell me something about $y$.

What's a good approach to solve this problem? Is there a regression form specific to this sort of situations?

Due to practical constraints, I want my model to include all $x$s at the same time, rather than switch between different models on the fly.

• Thanks for the suggestion. I have edited my question to make it more specific. – Baron Yugovich Dec 7 '15 at 3:27
• Cool, that was helpful! Thank you. So what information do you get to see? When you sample $x_i$, do you get to see the value of $y$ too? the value of all the other $x_j$'s? I'm not too clear how the sampler $s_1$ would work: how do you know at what times to sample the value of $x_1$ and $y$? Do you somehow have a way to learn when $x_1$ changes, or to learn when $x_1$ 'ticks'? Can $x_i$ 'tick' only when the clock happens (only every $c_i$ seconds), or can it 'tick' at arbitrary times? We're probably going to need some guarantee about the stochastic process governing each $x_i$... – D.W. Dec 7 '15 at 3:39
• I have made more edits. I can basically choose to sample whenever I want. But let's not focus on the logistics, instead, I'm interested in the sort of learning algorithm/regression format I can use. – Baron Yugovich Dec 7 '15 at 4:45