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Variation in many of the stochastic rate parameters is described by a broken-stick probability distribution function (pdf). This is a simple function based on a piecewise linear distribution. The probability density function and the cumulative density function are illustrated in Fig. 65. It is described using the 0, 50 and 100% cumulative probability levels.
Random deviates for this broken stick density distribution are obtained from the following transformation formula
(163)
where
- x = unit uniform random deviate range 0 < x < 1
- yl = lower limit of the distribution range
- ym = distribution of the median value
- yu = upper limit of the distribution range.
Although the distribution uses the median, the broken-stick input windows in CRiSP.1 use the mean value since most data reports include a mean in addition to the minimum and maximum values. The median is estimated from these three measures as
(164)
assuming the mean of the distribution is equal to the average of the mean of the lowest 50% of the distribution and the highest 50%. These are simply the average of the minimum and median, and maximum and median, respectively.
Note that in a skewed distribution the mean and median are different. The result is that the mean specified by the user must fall in the middle two quartiles of the distribution, i.e. if the user specifies a minimum of 0 and a maximum of 100 for some distribution, the mean must lie between 25 and 75, inclusive. If the user specifies a distribution outside this range, CRiSP.1 will post a message to that effect in the message window and will direct the user to choose a mean that lies in the acceptable range.
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Columbia River Salmon Passage Model CRiSP.1.5 Theory, Calibration & Validation Manual
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