Wolf's paper Determining Lyapunov Exponents from a Time Series states that:
Experimental data typically consist of discrete measurements of a single observable. The well-known technique of phase space reconstruction with delay coordinates [2, 33, 34] makes it possible to obtain from such a time series an attractor whose Lyapunov spectrum is identical to that of the original attractor.
One of the cited papers, Geometry from a Time Series, elaborates:
The heuristic idea behind the reconstruction method is that to specify the state of a three-dimensional system at any given time, the measurement of any three independent quantities should be sufficient [...]. The three quantities typically used are the values of each state-space coordinate, $x(t)$, $y(t)$, and $z(t)$. We have found that [...] one can obtain a variety of three independent quantities which appear to yield a faithful phase-space representation of the dynamics of the original $x$, $y$, $z$ space. One possible set of three such quantities is the value of the coordinate with its values at two previous times, e.g. $x(t)$, $x(t - \tau)$, and $x(t - 2\tau)$.
Finally, Rosenstein's paper A practical method for calculating largest Lyapunov exponents from small data sets states that:
The first step of our approach involves reconstructing the attractor dynamics from a single time series. We use the method of delays [27, 37] since one goal of our work is to develop a fast and easily implemented algorithm. The reconstructed trajectory, X, can be expressed as a matrix where each row is a phase-space vector. That is, $$ X = [X_1\;X_2\; ...\,X_M]^T$$ where $X_i$ is the state of the system at discrete time $i$.
All three papers seem to implicitly assume that the system under study has a multi-dimensional phase space, but that only one dimension can be measured experimentally, and therefore that the full phase space data must be reconstructed from a one-dimensional time series.
However, what if the time series is multi-dimensional, indeed of the same dimension as the phase space, to begin with? For instance, consider the problem of showing experimentally that a simple pendulum is not chaotic. The phase space is 4-dimensional ($r$, $\dot r$, $\phi$, $\dot \phi$) and it is straight-forward to design an experiment which generates a 4-dimensional time series of the values of these variables at each time step.
Is it possible to skip the reconstruction in this case, and use $X = [r\; \dot r\; \phi\; \dot \phi]^T$ in place of the reconstructed trajectory in Rosenstein's paper, with no additional modifications? Is there a simpler way to calculate Lyapunov exponents when the full phase-space state of the system is known?