1Notes about distribution tables from Nistnet 2------------------------------------------------------------------------------- 3I. About the distribution tables 4 5The table used for "synthesizing" the distribution is essentially a scaled, 6translated, inverse to the cumulative distribution function. 7 8Here's how to think about it: Let F() be the cumulative distribution 9function for a probability distribution X. We'll assume we've scaled 10things so that X has mean 0 and standard deviation 1, though that's not 11so important here. Then: 12 13 F(x) = P(X <= x) = \int_{-inf}^x f 14 15where f is the probability density function. 16 17F is monotonically increasing, so has an inverse function G, with range 180 to 1. Here, G(t) = the x such that P(X <= x) = t. (In general, G may 19have singularities if X has point masses, i.e., points x such that 20P(X = x) > 0.) 21 22Now we create a tabular representation of G as follows: Choose some table 23size N, and for the ith entry, put in G(i/N). Let's call this table T. 24 25The claim now is, I can create a (discrete) random variable Y whose 26distribution has the same approximate "shape" as X, simply by letting 27Y = T(U), where U is a discrete uniform random variable with range 1 to N. 28To see this, it's enough to show that Y's cumulative distribution function, 29(let's call it H), is a discrete approximation to F. But 30 31 H(x) = P(Y <= x) 32 = (# of entries in T <= x) / N -- as Y chosen uniformly from T 33 = i/N, where i is the largest integer such that G(i/N) <= x 34 = i/N, where i is the largest integer such that i/N <= F(x) 35 -- since G and F are inverse functions (and F is 36 increasing) 37 = floor(N*F(x))/N 38 39as desired. 40 41II. How to create distribution tables (in theory) 42 43How can we create this table in practice? In some cases, F may have a 44simple expression which allows evaluating its inverse directly. The 45pareto distribution is one example of this. In other cases, and 46especially for matching an experimentally observed distribution, it's 47easiest simply to create a table for F and "invert" it. Here, we give 48a concrete example, namely how the new "experimental" distribution was 49created. 50 511. Collect enough data points to characterize the distribution. Here, I 52collected 25,000 "ping" roundtrip times to a "distant" point (time.nist.gov). 53That's far more data than is really necessary, but it was fairly painless to 54collect it, so... 55 562. Normalize the data so that it has mean 0 and standard deviation 1. 57 583. Determine the cumulative distribution. The code I wrote creates a table 59covering the range -10 to +10, with granularity .00005. Obviously, this 60is absurdly over-precise, but since it's a one-time only computation, I 61figured it hardly mattered. 62 634. Invert the table: for each table entry F(x) = y, make the y*TABLESIZE 64(here, 4096) entry be x*TABLEFACTOR (here, 8192). This creates a table 65for the ("normalized") inverse of size TABLESIZE, covering its domain 0 66to 1 with granularity 1/TABLESIZE. Note that even with the granularity 67used in creating the table for F, it's possible not all the entries in 68the table for G will be filled in. So, make a pass through the 69inverse's table, filling in any missing entries by linear interpolation. 70 71III. How to create distribution tables (in practice) 72 73If you want to do all this yourself, I've provided several tools to help: 74 751. maketable does the steps 2-4 above, and then generates the appropriate 76header file. So if you have your own time distribution, you can generate 77the header simply by: 78 79 maketable < time.values > header.h 80 812. As explained in the other README file, the somewhat sleazy way I have 82of generating correlated values needs correction. You can generate your 83own correction tables by compiling makesigtable and makemutable with 84your header file. Check the Makefile to see how this is done. 85 863. Warning: maketable, makesigtable and especially makemutable do 87enormous amounts of floating point arithmetic. Don't try running 88these on an old 486. (NIST Net itself will run fine on such a 89system, since in operation, it just needs to do a few simple integral 90calculations. But getting there takes some work.) 91 924. The tables produced are all normalized for mean 0 and standard 93deviation 1. How do you know what values to use for real? Here, I've 94provided a simple "stats" utility. Give it a series of floating point 95values, and it will return their mean (mu), standard deviation (sigma), 96and correlation coefficient (rho). You can then plug these values 97directly into NIST Net. 98