I have a whole set of data on
[0,T] with an observation variable
y(t), and a feature
x(t), the two being univariates with no missing data.
For a given period
[t, t+h], I am applying a dynamic linear regression:
y(t) = a(t) + b(t) * x(t) + w(t) a(t) = a(t-1) + w_a(t) b(t) = b(t-1) + w_b(t)
w, w_a, w_b are the variance of the last term on these
3 lines (following a centered normal distribution).
What matters is that for window
[0, h], I normalize my data (x, y) - aka
zero mean and
1 std - and then will have a series of estimated parameters
b after maximum likehood estimation:
time a b 0 0 0 1 0.41 0.72 ... h-1 0.432 0.55 h 0.435 0.567
I am interested in the last couple
a(h), b(h) giving the intercept and the slope.
Then I take as training set
x(1), x(2) ... x(h+1) and
y(1), y(2), .., y(h+1), normalize thse two series, and have as well some best estimations for the parameters
I take parameters
And so on. By rolling the window, I thus have a set of slopes
b(h), b(h+1), ..., b(T) and intercept
a(h), a(h+1), ..., a(T)
However the normalization differs between each rolling windows. So that I am not sure at all it is a good way to deduce some behaviour about the time series
b of the "last estimation".
What would be the best method:
still perform some analysis on
(x,y)normalisation is different on each window?
apply the above by normalizing the data on each window
[t, t+h]with the means/std of
xon [0, T]?
- applying the above without normalizing the data?