Robust Analysis, Inc. is a consulting and software company that
specializes in analyzing problems that involve heavy tailed
distributions. Our initial focus has been on stable
distributions, a class of probability distributions that generalize
normal distributions and allow heavy tails and skewness.
This rich class of probability distributions can model data
in a wide range of applications. In engineering, noise in some
communications, radar and sonar systems is heavy tailed.
In economics and finance, stable laws are
used to model fluctuations of stocks prices and foreign
exchange rates, compute Value At Risk, and model large insurance claims.
They are used in hydrology to model super-diffusion
of a fluid through an aquifer. Also some network
traffic quantities, e.g. file size of downloads, are heavy tailed.
Stable densities computed with the STABLE program.
The values of α are indicated on the plots, skewness is indicated by color:
β=0 (black), β=0.25 (red), β=0.5 (green), β=0.75 (yellow), β=1 (blue).
Basic information on stable
Basic facts about stable distributions can be found in an
introduction to stable distributions (PDF format, 360 kb). This
is a draft version of Chapter 1 in a forthcoming book on stable
distributions. It describes one dimensional stable distributions in
a non-technical way, explaining the two main parameterizations of
John Nolan's academic web page has
more information about stable distributions and a free Windows
programs for doing basic calculations with stable distributions.
STABLE library and interfaces
We have developed a library of STABLE functions for programmers to call directly (from C,
Fortran, or Visual Basic) and interfaces for matlab, Excel,
R, and Mathematica. All the capabilities of the free STABLE program are available, plus
more features, at increased speed. The library has functions to compute stable
densities, cumulative distribution functions, quantiles, hazard
function, etc. at an arbitrary vector of values. It will estimate
stable parameters by six methods: quantile method, empirical
characteristic function method, numerical maximum likelihood
estimation, fractional moments, log absolute moment, and modified
quantile. Linear regression with stable error terms is now included.
New features include support for bivariate stable distributions:
density calculations, cdf (probability of being in a rectangle),
simulation and estimation of spectral measure. For higher dimensional data,
the STABLE program supports isotropic and elliptically contoured stable
distributions for up to 100 dimensions. For an
isotropic/radially symmetric distribution X, the amplitude is
defined by R=|X| (a generalized Rayleigh distribution).
Functions are included to calculate the density
and cumulative distribution of the amplitude distribution, as well
as simulating samples from the amplitude distribution. For isotropic
or elliptically contoured distributions, the program can fit data,
compute densities, and simulate for dimension up to 100.
The user manual contains a description of all the functions, see online versions below.
Here is a one page Power Point information sheet on STABLE.
This library and/or interfaces for matlab, Excel,
R, and Mathematica can be purchased below.
STABLE version 5.1 was released in January 2009. It is completely rewritten
and now runs under both Windows and linux.
In July 2009 we released the α-stable signal filtering toolbox for matlab.
This toolbox implements nonlinear stable filters for filtering out
spikey/impulsive/heavy tailed noise. The filters can be unweighted or
weighted. In addition, signed weighted and matched filters are defined
for the detection of waveforms, e.g. a chirp, in a noisy signal. In 2010,
we added support for 64-bit linux and Excel.
Purchase STABLE online
The STABLE package can be ordered online using a credit card.
All versions include the STABLE software, getting started
instructions, a user manual, and scripts demonstrating how to use each function.
To order, look down the rows of the table below for the version you want.
Note that for matlab, you can purchase just the STABLE core toolbox (first row),
or you can purchase the combined STABLE core and the signal filtering (second row).
Then go over to the ``Select platform'' column. On the next page, select the version
of the software that you want: Windows, linux, or Mac. We now support 64 bit platforms.
This will connect you to a secure server (esellerate.net) that will process your order and verify your
credit card. After that is done, they will send us your order and we will ship the software
to you by e-mail.
Note: these prices are for single user licenses. For other licensing arrangements,
information about other environments, volume pricing (more than 2 copies),
to order by mail, or other issues, use the
NOTE: Orders are not available immediately, so please be patient.
When you place an order, it goes through a process to check your credit card.
When that is approved, we get an e-mail informing us of your order,
we build the software you requested, and we e-mail it to you. This usually occurs within one business day.
* Prices are in US dollars. The company that handles our credit card sales will convert to other currencies.
Tax will be added at checkout: U.S. customers will be charged the rate for their state;
international customers will be charged appropriate VAT for their country.
Note: this grant program is officially over,
but a limited number of grants may still be made.
Academic and non-profit organizations may apply for a grant program that
awards a limited number of free copies of STABLE. Decisions on grant
applications will be based on concrete plans for use of the STABLE program and for disseminating results
in publications, talks, etc.
Click STABLE Software Grant Application
for more information.
More information, other platforms and consulting
If you have any questions about the above products, are interested in a custom
interface for STABLE to a different program, on a different
platform, multi-user licenses, or other consulting work, please contact us using our
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