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Notes about distribution tables from Nistnet | |

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I. About the distribution tables | |

The table used for "synthesizing" the distribution is essentially a scaled, | |

translated, inverse to the cumulative distribution function. | |

Here's how to think about it: Let F() be the cumulative distribution | |

function for a probability distribution X. We'll assume we've scaled | |

things so that X has mean 0 and standard deviation 1, though that's not | |

so important here. Then: | |

F(x) = P(X <= x) = \int_{-inf}^x f | |

where f is the probability density function. | |

F is monotonically increasing, so has an inverse function G, with range | |

0 to 1. Here, G(t) = the x such that P(X <= x) = t. (In general, G may | |

have singularities if X has point masses, i.e., points x such that | |

P(X = x) > 0.) | |

Now we create a tabular representation of G as follows: Choose some table | |

size N, and for the ith entry, put in G(i/N). Let's call this table T. | |

The claim now is, I can create a (discrete) random variable Y whose | |

distribution has the same approximate "shape" as X, simply by letting | |

Y = T(U), where U is a discrete uniform random variable with range 1 to N. | |

To see this, it's enough to show that Y's cumulative distribution function, | |

(let's call it H), is a discrete approximation to F. But | |

H(x) = P(Y <= x) | |

= (# of entries in T <= x) / N -- as Y chosen uniformly from T | |

= i/N, where i is the largest integer such that G(i/N) <= x | |

= i/N, where i is the largest integer such that i/N <= F(x) | |

-- since G and F are inverse functions (and F is | |

increasing) | |

= floor(N*F(x))/N | |

as desired. | |

II. How to create distribution tables (in theory) | |

How can we create this table in practice? In some cases, F may have a | |

simple expression which allows evaluating its inverse directly. The | |

pareto distribution is one example of this. In other cases, and | |

especially for matching an experimentally observed distribution, it's | |

easiest simply to create a table for F and "invert" it. Here, we give | |

a concrete example, namely how the new "experimental" distribution was | |

created. | |

1. Collect enough data points to characterize the distribution. Here, I | |

collected 25,000 "ping" roundtrip times to a "distant" point (time.nist.gov). | |

That's far more data than is really necessary, but it was fairly painless to | |

collect it, so... | |

2. Normalize the data so that it has mean 0 and standard deviation 1. | |

3. Determine the cumulative distribution. The code I wrote creates a table | |

covering the range -10 to +10, with granularity .00005. Obviously, this | |

is absurdly over-precise, but since it's a one-time only computation, I | |

figured it hardly mattered. | |

4. Invert the table: for each table entry F(x) = y, make the y*TABLESIZE | |

(here, 4096) entry be x*TABLEFACTOR (here, 8192). This creates a table | |

for the ("normalized") inverse of size TABLESIZE, covering its domain 0 | |

to 1 with granularity 1/TABLESIZE. Note that even with the granularity | |

used in creating the table for F, it's possible not all the entries in | |

the table for G will be filled in. So, make a pass through the | |

inverse's table, filling in any missing entries by linear interpolation. | |

III. How to create distribution tables (in practice) | |

If you want to do all this yourself, I've provided several tools to help: | |

1. maketable does the steps 2-4 above, and then generates the appropriate | |

header file. So if you have your own time distribution, you can generate | |

the header simply by: | |

maketable < time.values > header.h | |

2. As explained in the other README file, the somewhat sleazy way I have | |

of generating correlated values needs correction. You can generate your | |

own correction tables by compiling makesigtable and makemutable with | |

your header file. Check the Makefile to see how this is done. | |

3. Warning: maketable, makesigtable and especially makemutable do | |

enormous amounts of floating point arithmetic. Don't try running | |

these on an old 486. (NIST Net itself will run fine on such a | |

system, since in operation, it just needs to do a few simple integral | |

calculations. But getting there takes some work.) | |

4. The tables produced are all normalized for mean 0 and standard | |

deviation 1. How do you know what values to use for real? Here, I've | |

provided a simple "stats" utility. Give it a series of floating point | |

values, and it will return their mean (mu), standard deviation (sigma), | |

and correlation coefficient (rho). You can then plug these values | |

directly into NIST Net. |