## Elliptical representation of vector
error

Tieh-Yong Koh

School of Physical & Mathematical Sciences

Nanyang Technological University, Singapore

kohty@ntu.edu.sg
*Ellipse Representation of
Variance of
2D Vectors* (Koh
and Ng,
2009)

The variance of a 2D vector **A**,
e.g.,
horizontal wind, is a 2x2 matrix:

_{}

It can be represented schematically
by a
“variance ellipse”, with semi-major and semi-minor axes equal to the
square
root of the eigenvalues of the matrix, *a*^{2} and *b*^{2},
and oriented in the direction of the eigenvectors.

In
the figure
above the mean is shown as a bold line and the variance as an ellipse.

The variance ellipses of
modeled and
observed wind may be compared by computing for each ellipse:

·
standard deviation, _{};

·
eccentricity,_{};

·
orientation, *θ*
(in radians),
of the major axis.

The discrepancy between the
modeled
and
observed wind is a vector itself and its variance ellipse (dubbed
“error
ellipse”) can be similarly studied.

*Answers the questions: **How does the vector random fluctuations of the modeled
vector
compare with those of the observed vector? How does the vector error
between
the model and observation vary about the mean vector error (i.e.
bias)?*

*Range: *
*σ* ∈ [0,∞), *ε* ∈ [0,1],
*θ* ∈ [0,*π*)
*Perfect Score:* for a vector error
*σ* = 0, *ε* = 0

*Characteristics:* The representation using an ellipse is possible
because the
variance matrix is real, positive semi-definite and symmetric, so that
the
eigenvalues are always real and non-negative and the eigenvectors are
always
orthogonal. The set of diagnostics (*σ,**ε,**θ*) represents
the complete
set of information regarding the second-order moments of a
two-dimensional
vector. Note that higher order moments are not captured in this
representation.

For the variance ellipse of the
modeled or
observed wind, *σ *measures the extent
of
variability, *ε* measures the anisotropy
in the variations and *θ* measures
the preferred
direction of the variation.

For the error ellipse (i.e.
forecast minus
observation), *σ *indicates the
overall
magnitude of the random error.* θ*
is the preferred
direction of the vector random error and *ε* denotes
the degree of
preference for that direction.

__Reference__

Koh, T.
Y. and J. S. Ng
(2009), "**Improved
Diagnostics
for NWP Verification in the Tropics**", *J. Geophys. **Res.*,
114, D12102, doi:10.1029/2008JD011179.

http://www3.ntu.edu.sg/home/kohty/spms/publication.htm

(* In Koh
and Ng
(2009), *β*=(*b*-*a*)/(*b*+*a*) was used to
measure the
eccentricity of an ellipse, instead of the conventional mathematical
definition* ε*.
It is since realized that *ε* is
a more sensitive
diagnostic than* β* and it is recommended to
use *ε* instead
of *β*.)