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Interval FAQ
Philippe Nivlet -- Interval statistics?

From Philippe Nivlet (philippe.nivlet@ifp.fr), Institut Francais du Petrole:

Interval statistics?

Could anybody recommend to me references or software available on the web, concerning the application of interval arithmetics to statistics? More particularly, I would like to calculate the interval slope [a] and the interval intercept [b] estimated with the regression analysis of a sample ([x_i], [y_i]).

Philippe Nivlet

Response by Bill Walster:

(bill.walster@eng.sun.com), Sun Microsystems:

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

Response by Svetoslav Markov:

(smarkov@iph.bio.bas.bg), Bulgarian Academy of Sciences:

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 Rn. 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.

  1. 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.
  2. 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.
  3. 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.
  4. 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
  5. 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

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