Data Analysis Tutorial : Scalar

3.1 Scalar

Examples will be worked out using two time series, from an ADCP moored for a two-week period, recording one second samples of pressure and velocity. The variable $p$ is the pressure measured at the bottom in 15 m water, and $w$ is the vertical velocity measured $xx~ m$ beneath the surface, or $yy~ m$ above the bottom. A 20 second time series of both variables is ilustrated in Fig. xx

Semantics and definitions

There is a distinction between Covariance and Correlation. The covariance is

$\begin{equation} \label{eq:defcov} Cov(p,w)=Cov(w,p)=\frac{1}{n-1}\sum _ n(p_ n-\overline{p}))(w_ n-\overline{w}) \end{equation}$

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whereas the correlation is the normalized covariance:

$\begin{equation} \label{eq:defcorr} r_{w\: p}=\frac{Cov(p,w)}{\sqrt {Cov(p,p)Cov(w,w)}} \end{equation}$

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This guarantees that the correlation is between $-1$ and $1$ .

It is also critical to distinguish between Correlation and Cross-correlation, also known as Lagged correlation. In matlab xcov(w,p,maxlag,’coeff’) returns an array sized $2*maxlag+1$ long that has the values of the lagged correlations, defined as:

$\begin{equation} \label{eq:crosscorr} r_{w\: p}(m)=\frac{\sum _ n(p_{(n-m)}-\overline{p}))(w_{(n+m)}-\overline{w})}{(n-1)\sqrt {Cov(p,p)Cov(w,w)}} \end{equation}$

3.1 Scalar

Semantics and definitions

The correlation matrix

Regression

Empirical Orthogonal Functions

Lagged correlations: the Hilbert transform

The cross-spectrum: Coherence, Phase and Gain