Multiple Operands Calculation

Objective

Original P-bounds software supports two operands to calculation. But in the real application, there are over 2 operands to calculation, for example, x+y+z, max(x,y,z).

Association is directly way to solute this question. E.g. x+y+z, we can first calculate x+y, and save the result to temporary variable w=x+y, then we calculate w+z. through this way, we can get the result of x+y+z. but there is a constraint, that is, this kind operation supports the association and commutation. In another word, it doesn’t care the order to be calculated. It is the same that you first calculate x+y or y+z for x+y+z.

For P-bounds software, we would get the CDFs for result of two variables’ operation under unknown dependence, not IDF file (not histogram bar, IDF is used in this software. It is similar to PDF, but it can be overlapped). So we can solute this question if we can transfer CDFs to IDF format.

Solution

In interval mathematics, association and commutation work (reference to report about interval analysis). So we can apply this method to transfer CDF to IDF bar. Based on upper and lower bound, we can express them using overlap bars. Every bar would include the upper and lower bound, left bound point is in upper bound, right bound point is in lower bound. The probability of this bar is the probability difference of its top and bottom level in CDF bound. For example, four bars will be gotten from the following CDF figure.

Result

This method is implemented in P-bounds software using VB. Now both CDF and IDF data format can be saved or displayed. From the following figures, data before transference is the same as that after transference.

This figure shows the CDF of two variables’ operation

Part 3 is the result of operation. Part 2 is the IDF transferred from CDF. Part1 is retransferred from IDF to CDF using the IDF transferred from original CDF.

The following figures show the procedure to calculate multiple operands (x+y+z).

First, we get the result of x+y.

Then load the result of x+y as a new operand. Let it operate with z. So we get result of x+y+z.