I would like to know how to use Euclidean distance to find similarity between two multivariate time series.
Suppose, I have two $N$-variate time series $u$ and $v$, with $u_i(t)$ denoting the $i$-th component of time series $u$ at time $t$.
Would it make sense to calculate distance between values of same parameters and then sum that distances for all parameters to get the final distance?
$$ d_i = \sqrt{(u_i(0)-v_i(0))^2 + (u_i(1)-v_i(1))^2 + … + (u_i(T)-v_i(T))^2}$$ $$ d = d_0 + d_1 + … + d_N$$
Or would it make sense to treat all the values equally, no matter which parameter is considered?
$$\check{d} = \sqrt{ \hphantom{+\,}(u_1(0)-v_1(0))^2 + (u_1(1)-v_1(1))^2 + … + (u_1(T)-v_1(T))^2 \\ + (u_2(0)-v_2(0))^2 + (u_2(1)-v_2(1))^2 + … + (u_2(T)-v_2(T))^2 \\ + … \\ + (u_N(0)-v_N(0))^2 + (u_N(1)-v_N(1))^2 + … + (u_N(T)-v_N(T))^2 }$$
Or to consider distance between each record (values of each parameter at certain point in time):
$$ \hat{d}_0 = \sqrt{(u_1(0)-v_1(0))^2 + (u_2(0)-v_2(0))^2 + … + (u_m(0)-v_m(0))^2}$$
$$ \hat{d}_1 = \sqrt{(u_1(1)-v_1(1))^2 + (u_2(1)-v_2(1))^2 + … + (u_m(1)-v_m(1))^2}$$
$$ \hat{d}_T = \sqrt{(u_1(T)-v_1(T))^2 + (u_2(T)-v_2(T))^2 + … + (u_m(T)-v_m(T))^2}$$
$$ \hat{d} = \hat{d}_0 + \hat{d}_1 + … + \hat{d}_T$$
Or something else?
Example
Multivariate series 1:
power, current, voltage
100, 10, 10
400, 20, 20
900, 30, 30
Multivariate series 2
power, current, voltage
600, 20, 30
1000, 50, 20
450, 15, 30
What would the Euclidean distance look like?