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OCIAM**

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Oxford Centre for Industrial and Applied Mathematics
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Dr Leonard A Smith's Publications**

Disentangling Uncertainty and Error:
On the Predictability of Nonlinear Systems

Chapter 2 of Nonlinear Dynamics and Statistics, ed. Alistair I. Mees,
Boston: Birkhauser, 31--64, 2000.

ABSTRACT
Chaos places no a priori restrictions on predictability: any uncertainty in the
initial condition can be evolved and then quantified as a function of forecast time.
If a specified accuracy at a given future time is desired, a perfect model can
specify the initial accuracy required to obtain it, and accountable ensemble
forecasts can be obtained for each unknown initial condition. Statistics which
reflect the global properties of infinitesimals, such as Lyapunov exponents which
define ``chaos'', limit predictability only in the simplest mathematical examples.
Model error, on the other hand, makes forecasting a dubious endeavor. Forecasting
with uncertain initial conditions in the perfect model scenario is contrasted with
the case where a perfect model is unavailable, perhaps nonexistent. Applications to
both low (2 to 400) dimensional models and high (10^7) dimensional models are
discussed. For real physical systems no perfect model exists; the limitations of
near-perfect models are considered, as is the relevance of the recurrence time of the
system in terms of the likely duration of observations. It is argued that in the
absence of a perfect model, a perfect ensemble does not exist and hence no
accountable forecast scheme exists: accurate probabilistic forecasts cannot be made
even when the statistics of the observational uncertainty are known exactly.
Nevertheless, ensemble forecasts are required to maintain the uncertainty in initial
condition; returning to single best guess forecasts is not an option. Both the
relevance of these observations to operational forecasts and alternatives to aiming
for an accurate probabilistic forecasts are discussed.

## Contact Information

Tel: 01865-2-70517

E-mail: lenny@maths.ox.ac.uk
Last updated: 14 Feb 2001

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