# truncated distribution

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## truncated distribution

[¦trəŋ‚kād·əd ‚dis·trə′byü·shən]
(statistics)
A distribution fashioned from another distribution by deleting that part of the distribution to the right or left of a random variable value.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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According to the results of statistical hypotheses testing it is determined that the translog function with the truncated distribution of a random component best describes the investigated functional dependences.
Truncated distribution enables the travel time to be restricted within a certain limited range.
In spite of that, the residual symmetries characterizing the truncated distribution help simplify the problem in the following respects: (i) the reflection invariance of the whole setup still yields E[[X.sub.k] | X [member of] [B.sub.v]([rho])] = 0 [for all]k and (ii) the rotational invariance of [B.sub.v]([rho]) preserves the possibility of defining the principal components of the distribution just like in the unconstrained case.
On the other hand, when we are completely sure about the priori interval constraint of 9, a suitable restriction on the parameter space [THETA] such as using a truncated distribution is expected.
Let [phi](x) and [PHI](x) denote the PDF and CDF of the standard Normal distribution which is supported in (-[infinity], +[infinity]), and the new support interval is [[b.sub.l], [b.sub.u]]; the PDF and CDF of the truncated distribution with mean value [mu] and standard deviation [sigma] are given by the following equations:
Because vehicles with speeds v < [v.sub.min] and v > [v.sub.max] ([v.sub.min] and [v.sub.max] denote minimum speed and maximum speed, resp.) are rarely observed in the actual world, which is confirmed by field data as showninthe data acquisitionand analysis section given below, therefore, the assumed speed following truncated distribution ranging from [v.sub.min] to [v.sub.max] is more suitable.
Removing unrelated factors from Equation 6 shows that each of the full conditionals is now a tractable truncated distribution:
Boyle, Welsh, and Bishop (1988) present a refinement of that approach where the truncated distribution is normalized, so that all of the distribution's density lies below the truncation limit, and the distribution is everywhere continuous.
We assumed that the following modifications of Horris algorithm could possibly help to eliminate some of the negative effects of outliers on the transformation: (a) truncate the distribution to eliminate possible outliers by temporarily excluding, e.g., 5% of the extreme values at each tail; (b) then estimate the transformation parameter on the truncated distribution; and (c) finally transform all data, including the outliers, with this parameter and continue with steps 2-4 of Horris algorithm (see the Materials and Methods).
Visually, the truncated distributions are similar to the nontruncated distributions for the degrees of truncation used here.
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