Data Analysis Tutorial : Two-dimensional vector: alternate description

2.3 Two-dimensional vector: alternate description

The discussion above suggests that rather than describing the velocity in terms of components alon the $x,u$ and $y,v$ axes, an alternative method would be to describe the characteristics of the ellipses. MORE HERE.

There is a confusion to be aware of: In geographical coordinates, the compass angle is measured positive clockwise from north, but in the complex plane, the angle ( $\theta$ ) is measured positive counterclockwise from the $x$ , or east direction. Now we begin with a complex series $W_ t=u_ t+iv_ t$ , where $u_ t, v_ t$ are the two components of the vector.

Mean

$\begin{equation} \label{eq:wbar} \overline{W}=\mu _ u+i\mu _ v \end{equation}$

(21)

Variance: Principal Axes

The description variance of a two-dimensional time series is complicated by the fact that the time series as measured can be correlated. If they are, then the variance is properly described as a four element symmetric matrix:

$\begin{equation} \label{eq:vcautocov} \begin{array}[t]{cc} C_{xx} & C_{xy}\\ C_{xy} & C_{yy} \end{array} \end{equation}$

(22)

The diagonal elements are equal to the variance in each component, as given by 7, and the off-diagonal element

$\begin{equation} \label{eq:tsvar} C_{xy}=\frac{\displaystyle {\sum _ t}{(u_ t-\mu _ u)(v_ t-\mu _ v)}}{N_ t} \end{equation}$

(23)

Matrices such as 22 can always be diagonalized by a suitable rotation. Physically, this means that if in the original coordinate system the two components are correlated, there exists another coordinate system, rotated by some angle $\theta$ in which the components are not correlated at zero time-lag. The axies of this rotated coordinate system are called Principal Axes. Since the covariance matrix is diagonal, the variance in this system has two parts, called the major and minor axis variance. To find the principal axes, find the eigenvalues and vectors of the matrix $C$ . The major and minor axis variance are the eigenvalues of $C$ , and the angle of the principal axis is $\arccos {e(1,1)}$ where $e$ is the eigenvector matrix.

Auto-covariance & auto-correlation

Even along the principal axes, the two components of the vector series may be correlated at a different lag from zero. An exaample of this might be a current that rotates in direction as a function of time (e.g. inertial currents). How to deal with this??

Spectrum

In general, in this case, $S_ p(\omega )\ne S_ p(-\omega )$ . If fft is called as $Z=fft(z)$ where z is a complex time series with an even number of discrete values (i.e. $N_ t$ is even), the the order of the output is as follows: $Z(1)$ is the mean, $Z(2)$ is the CCW fourier coefficient for for the lowest resolved frequency ( $df=1/T$ ) up to $Z(N_ t/2+1)$ corresponding to the frequency $N_ t df/2$ (this is the Nyquist frequency), the other elements of S correspond to negative frequencies in reverse order: $Z(N_ t)$ corresponds to the frequency $-df$ all the way down to $Z(N_ t/2+2)$ , corresponding to $(N_ t-1)*df/2$ . The first $N_ t/2+1$ elements correspond to positive trigonometric angles, or CCW spectra, whereas the higher elements are for CW spectra. Note that the Nyquist frequency is not quite resolved.

If we write for one element

$\begin{equation} \label{eq:z1} z=|Z_1|e^{i\phi _1}e^{i df t}+|Z_{-1}|e^{i\phi _{-1}}e^{-i df t} \end{equation}$

(24)

then

$\begin{equation} \label{eq:z2} z=e^{i(\phi _1+\phi _{-1})/2}\left[|Z_1|e^{i(df t+(\phi _1-\phi _{-1})/2)}+|Z_{-1}|e^{-i (df t+(\phi _1-\phi _{-1})/2)}\right] \end{equation}$

(25)

The real and imaginary parts of the term in the square bracket are:

$\begin{equation} \label{eq:realsqb} \Re {[]}=\left[|Z_1|+|Z_{-1}|\right]\cos {(df t+(\phi _1-\phi _{-1})/2)} \end{equation}$

(26)

$\begin{equation} \label{eq:imagsqb} \Im {[]}=\left[|Z_1|-|Z_{-1}|\right]\sin {(df t+(\phi _1-\phi _{-1})/2)} \end{equation}$

(27)

this shows that $z$ describes an ellipse of major axis $[|Z_1|+|Z_{-1}|]$ and minor axis $[|Z_1|-|Z_{-1}|]$