  1Notes about distribution tables from Nistnet
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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
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