I have two time series. The two series look like they follow the same function, but one appears to lag slightly behind the other. I would like to determine if there are phase differences between the points in these time series, but each time series is only 4 points long. Is this enough data to get accurate phase differences using the Hilbert transform?
Forgive me for not having a numerical answer to the question. But here are a few things to think about. The answer depends on how the Hilbert transform (HT) is implemented. If the HT is implemented using an $N$-point tapped-delay line filter-like block diagram the first and last $N$-1 output samples are incorrect. So in this situation four input times samples is definitely not enough samples to produce useful results.
If HT is implemented by way of a forward FFT, zero the neg-frequency spectral components, and inverse FFT (as is done in Matlab) you have a better chance of computing meaning results. However, in this second implementation the input time sequence must be long enough to contain at least one cycle of all the spectral components of your input signal in order to compute somewhat meaningful results. It seems to me that four times samples is too short to reliably estimate any signal parameters of most real-world (information-carrying) signals.
The answer depends on the signal to noise ratio and whether the signal is a single sinusoid that is integer periodic in a window that number of samples in length. Or perhaps a number of pure integer periodic sinusoids that is low enough compared to N/2.
The a-priori bandwidth of the signal would also need to be small enough with respect to the number of samples, otherwise aliasing ambiguity leads to phase ambiguity.
A Hilbert transform doesn't magically add information to frequency estimation methods. And frequency estimation errors lead to phase estimation errors.
I would not recommend that, because, 4 points may not describe your signal really well. These 4 points can be coming from anywhere. To experiment with that I wrote the following extremely simple code:
% generate a signal x = cos(pi/4*(0:99)); y = hilbert(x); sigphase = (unwrap(angle(y)))'; X = ones(length(sigphase),2); X(:,2) = (1:length(sigphase))'; beta = X\sigphase; beta(2)
beta(2) will be very close to the frequency. If you generate x using:
x = cos(pi/4*(0:3));
you will immediately see that the frequencies do not match. This is of course a tough one due to the cosine signal, but there might be ambiguous cases, similar to that one.
Why don't you use a running correlation analysis instead?