Alternative Simulation Core for Material Reliability Assessments Speculation how to heighten random character of probability calculations (concerning the basic uniform density)

There are two problems thought over in this contemplation: 1. It is perpetual interest to look into the matter of the best fidelity of random of Monte Carlo simulation, which is the fundament for reliability assessments. Here is presented an engineering approach to the improvement of random, similar to dice, with very simple logic, based on the usage of random digits table. The efficiency of such approach is tested and interpreted. 2. The well-known imperative is: the more the extent of the random digits table is, the more random is the result of random sampling. Let us investigate, if also smaller segments of random digits table will do for very random sampling, under the usage of patent sampling method advertised in the previous count.

Patent method evolution: Many random digits/numbers generators are available. However, let us emulate to pan out from well-known random digits table. It was sure attentively coined to secure the highest possible random. To gain random numbers from it, there are used various procedures of selection of digits. Trying to heighten the random of selected numbers, let be exploited the fundamental random of digits implicated in the table. That means, the selection of wanted next digit let be managed by last chosen digit alone. Then the sampling proceeds as leaping along the table segment. F rom various formulae probed for such leaping, as most suitable were found relations plot on the next slide. Theirs structure ensures chaotic motion (both in interval and in direction) along the table segment. Chaos is governed by random digits values.

Using following (completely empirical) formulae  i = (D i,j + I)(-1) D i,j  j = (D i,j+1 + J)(-1) D i,j+1 there may be got new indices of hitting point: i new = i +  i j new = j +  j where D = last found random digit, i,j = indices of D in the matrix, I,J = indices i,j used in previous cycle and stored,  i,  j = increases of indices.

Shown theoretical basement of leaping generation might be taken as sufficient simple and transparent, because the random digits table segment is in computer memory stored like 2D matrix, enlarged in all directions with its own copy – then the motion to the new digit proceeds continuously. The starting point could be chosen at will (however, built into the computer program). The process might be called Frog-leaping Method because of mode of hitting of digits that is visible from the following slide 

Random numbers formation Coming now to the task to form a random number then on the hitting point may be hung a mask shaped as a table, e.g. M(3  3), containing powers of 10 or zeros (ride on wanted decimal places). During the running leaping process, each new location of this mask covers a group of random digits in the matrix. Those covered digits together with cells content of the mask, multiplied by each other and added up, give the wanted random number. The mathematical receipt for this operation is given by relation:  = m = i – 1 + k n = j – 1 + l where i,j denote the position of the hitting point within random digits matrix and k,l precise the turn of the mask cell.

Random numbers formation Example   = =

Disscussion and testig However, now it becomes proper to search into the question if there are any advantages of such random digits/numbers choice. To be observed are two features of Frog-leaping Method: It may be rightly supposed that due to the leaping method, managed by random digits alone, the random character of simulations increases. It is proposed to make certain that the application of the Frog-leaping Method enables the use of random digits tables of smaller extent for Monte Carlo simulations as well. (But in fact, there is any reason against usage of this algorithm for choice from large random digits tables, too.)

The testing could be started from the Chi-square tests comparison. As it is depicted below, the Chi-square results using picked table segments of extent 100/100, 45/67, 20/20 are sufficiently close to results published by Snedecor, thus the random should be saved (more details see in presented paper).

From this point of view, the usage of tables of small extent seems to be competent. However, if a very large sample is drawn from small table, it shows itself exhausted, thus by non-uniform digits density, that remains practically the same independent of cycles of drawing. See the lower graph of starting distribution of random digits in table 100/100 and 20/20 in comparison with distribution in the sample of size 1000 drawn from table 20/20 by frog-leaping. The density should be 10% overall. Naturally, closer is the segment 100/100.

To reach more successful tendency to the uniform probability density of drawn samples, let us start with the idea (that should be constitutional) to add to all of the random digits in the random digits table segment an arbitrary, but constant digit, shifting cyclic the results back to the range 0; 1;... 8; 9. The developed table may be supposed to be the random digits table as well. If the small table-segment is modified/regenerated in such a way, then it is possible to draw large sample from it. The most suitable period of such modification/regeneration (here adding of 1 is cycled) may be approximated with the minor dimension of the 2D matrix of random digits used. In the graph it is played out through marks pointed out, accompanied with numerical data, in dependence on period magnitude ADD.

Conclusions Computer program for Frog-leaping Method for choice of random digits/numbers was construct. Virtual-generator analogy might be hold up. Method seems to be useful also for small random digits table segments. Creating of random numbers (brought together from random digits) is understood as an impulsive choice of a group of digits chosen by leaping, too. It may be supposed that the random character of simulations increases. Mathematic-statistical test of uniform distribution of chosen random numbers is consistent with this null hypothesis at the 5% significance level.

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