I need to calculate the time average of a measured quantity during some period $T$. During that time the quantity was recorded as a series of equidistant data points $\{s_j\}_{j=1,...,n}$.
To reduce the amount of data, the signal has already been averaged, i.e. the original measured data set was reduced to a smaller set $\{S_i\}_{i=1,...,k}$ with $S_i=\frac{1}{N}\sum_{j=1}^N s_{(i-1)\cdot N+j}$ and $n=N \cdot k$.
Example for $n=6$, $N=2$, $k=3$:
- Original series$=\{s_1,s_2,s_3,s_4,s_5,s_6\}$
- Reduced series$=\{S_1,S_2,S_3\}=\{\frac{s_1+s_2}{2},\frac{s_3+s_4}{2},\frac{s_5+s_6}{2}\}$
The new time series $\{S_i\}_{i=1,...,k}$ is the one I am interested in and the one that I have to average.
In a perfect world we would not just have samples $s_j$ of the measured signal but the complete function $s(t)$. Then the average is $\frac{1}{T} \int_0^Ts(t) dt$. Taking the sum as above is just an approximation of this integral. But one can do better (I guess?) by approximating the integral by methods like the trapezoidal rule or the Simpson method (approximating s(t) with polynomials of higher degree).
My question is the following. Can I apply the Simpson method on the already averaged time series $\{S_i\}_{i=1,...,k}$ to get a more accurate average of the measured quantity during period $T$ than averaging again by the sum? It is not possible for me to work directly with the series $\{s_j\}_{i=1,...,n}$.
Thanks for any help.
There is nothing you can do to regain the data lost in the transition from $\{s_i\} \to \{S_i\}$. However, there is also no reason you can't apply the same techniques to $\{S_i\}$.
Note that the average of $\{s_i\}$ and $\{S_i\}$ will be the same. To clean up the math a little bit, I am shifting your indexing to being from $0$ to $n-1$ or $k-1$ instead of $1$ to $n$ or $k$.
$$S := \dfrac 1k \sum_{i=0}^{k-1} S_i = \dfrac 1k\dfrac 1N\sum_{i=0}^{k-1} \sum_{j = 0}^{N-1} s_{iN + j} = \dfrac 1n \sum_{i=0}^{n-1}s_i$$
And the $S_i$ can also be considered approximate values of the function - though more widely separated than the $s_i$. Don't let the "approximate" worry you. All measurements are by their very nature just approximations. The only physical values we know exactly are because we defined the units of measure to make them exact.
If the function you are measuring is well-behaved, then it is likely that Simpson's rule (when $k$ is even) will give a somewhat better predictor of the function's average than the trapezoidal rule, which in turn will be a slightly better predictor than $S$. However, you may find that the improved accuracy is so slight as to not make it worthwhile.
If the function is not well-behaved - turning sharply back and forth, then it is a crapshoot which will be better. The overall error in any of the estimates will be much bigger than the difference between them.