Estimate correlation between two time series

Hello, I’m a new Turing user and I’m not very familiar with probabilistic modeling, but the package is so cool that I couldn’t help trying to implement some of it

Basically I have 2 Bernoulli/Poisson processes (binned spike trains) and I’d like to know if and how much one influences the other.

So my data has this form:

``````s₁ = [0, 0, 1, 0, 1]
s₂ = [1, 0, 0, 0, 1]
``````

and I’ve computed a feature which is the delay of the closest spike of s₂

``````x = [0, 1, 2, 1, 0]
``````

Then I’ve built the model:

``````@model my_naive_model(x, y, n, σ²) = begin
b0 ~ Normal(0, σ²)
b1 ~ Normal(0, σ²)
for i = 1:n
theta = b0 + b1*x[i, 1]
y[i] ~ Bernoulli(sigmoid(theta))
end
end;
``````

Trained it:

``````chain = sample(my_naive_model(x, s₁, length(x), 10), NUTS(200, 0.65), 1500, discard_adapt=false)
``````

And took the predictions with:

``````function prediction(x::Vector, chain)
p0 = mean(chain, :b0)
p1 = mean(chain, :b1)
r = zeros(size(x, 1))
for i = 1:n
theta = p0 + p1*x[i]
r[i] = sigmoid(theta)
end
r
end;
``````

And that was awesomely easy, but I’d like to do something a bit more complicated, so these are my questions:

1.

What could be a sensible transformation of x, or how could I figure this out?
For example, I guess I could fit a polynomial, but what about embedding a smoothing spline or another non parametric regression method?

2.

How do I understand the magnitude of the effect that `s₂` has on `s₁`? In particular if I’m using a polynomial fit, how can I say that `s₂` is actually having an effect on `s₁`?

3.

This follows directly from the second question: Let’s say that I have a third spike train, and I want to `s₃`, and I want to prove that it has no effect at all on `s₁`. How do I do that?

Thank you very much in advance, and thanks to all Turing developers for the marvelous work!

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