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About Sun's f95 with Interval Support

Philippe Nivlet -- Interval statistics?

Philippe Nivlet

See: Walster, G. W.,
`` Philosophy and Practicalities of Interval Arithmetic'',
in *Reliability in Computing,* R. Moore (ed.).
Academic Press: San Diego, California (1988), pp 309-323.

There are a great many opportunities in the intersection of intervals and statistics. In fact, my interest in intervals was originally motivated by the problem of computing bounds on the noncentral cumulative distribution function of various test statistics.

Bill Walster

Dear Dr. Nivlet,

There exist plenty of literature about finding (interval) values for a and b
whenever intervals (from nonstatistical origin) for x_i and y_i are given.
Is this your problem?

Philippe Nivlet replies:

You pointed out the right
question. In fact, the problem could be formulated this way: The
statistical theory argues that a = c(x,y) /c(x,x) and
b = E(y) - a.E(x), where C is the covariance, and E() is the mean. I
have already tried to find an optimal interval for C(x,x) and C(x,y),
which is equivalent to finding the minimum and the maximum of a
n-variate quadratic function on a convex subset of R^{n}. Yet,
I haven't tried for the regression a and b values, which seems a
little bit more complex.

If you have the references about that subject, I would greatly appreciate reading these.

Svetoslav Markov:

Here are some related references. Interval values (only) for y_i are
considered. You may find more references in the cited literature.

- Markov, S.: Least square approximations under interval input data, Contributions to Computer Arithmetic and Self-Validating Numerical Methods (Ed. C. P. Ullrich), IMACS Annals on computing and applied mathematics, vol. 7, J. C. Baltzer Sci. Publ., Basel, 1990, 133--147.
- Markov, S.; Popova, E.: Estimation and Identification using Interval Arithmetic. Preprints, v. 1--2. Identification and System Parameter Estimation, vol 1, IFAC, 1991, 769--772.
- Markov, S., E. D. Popova, U. Schneider, J. Schulze: On Linear Interpolation under Interval Data. Mathematics and Computers in Simulation, 42, 1 (1996), 35--45.
- Markov, S., Y. Akyildiz: Curve Fitting and Interpolation of Biological Data under Uncertainties. J. UCS. 2, 2 (1996), 59--69. www.iicm.edu/jucs_2_2
- S. M. Markov, E. D. Popova: Linear Interpolation and Estimation using Interval Analysis. In: Milanese, M.; Norton, J. P.; P.-Lahanier H.; Water, E. (Eds.): Bounding Approaches to System Identification. Plenum Press, London, N. Y., 1996, 139--157.

Svetoslav Markov

Interval FAQ: [ Entry page | Contents | Search ]

About Sun's f95 with Interval Support

If *you* have a question related to validated computing, interval
analysis, or related matters, I recommend

- Vladik Kreinovich's Interval Computation web site, or
- Submit it to the interval computations listserv: reliable_computing@interval.louisiana.edu

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