1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
|
[vset VERSION 1]
[manpage_begin math::quasirandom n [vset VERSION]]
[keywords {quasi-random}]
[keywords mathematics]
[moddesc {Tcl Math Library}]
[titledesc {Quasi-random points for integration and Monte Carlo type methods}]
[category Mathematics]
[require Tcl 8.5]
[require TclOO]
[require math::quasirandom [vset VERSION]]
[description]
[para]
In many applications pseudo-random numbers and pseudo-random points in a (limited)
sample space play an important role. For instance in any type of Monte Carlo simulation.
Pseudo-random numbers, however, may be too random and as a consequence a large
number of data points is required to reduce the error or fluctuation in the results
to the desired value.
[para]
Quasi-random numbers can be used as an alternative: instead of "completely" arbitrary
points, points are generated that are diverse enough to cover the entire sample space
in a more or less uniform way. As a consequence convergence to the limit can be
much faster, when such quasi-random numbers are well-chosen.
[para]
The package defines a [term class] "qrpoint" that creates a command to generate
quasi-random points in 1, 2 or more dimensions. The command can either generate
separate points, so that they can be used in a user-defined algorithm or use these
points to calculate integrals of functions defined over 1, 2 or more dimensions.
It also holds several other common algorithms. (NOTE: these are not implemented yet)
[para]
One particular characteristic of the generators is that there are no tuning parameters
involved, which makes the use particularly simple.
[section "COMMANDS"]
A quasi-random point generator is created using the [term qrpoint] class:
[list_begin definitions]
[call [cmd "::math::quasirandom::qrpoint create"] [arg NAME] [arg DIM] [opt ARGS]]
This command takes the following arguments:
[list_begin arguments]
[arg_def string NAME] The name of the command to be created (alternatively: the [term new] subcommand
will generate a unique name)
[arg_def integer/string DIM] The number of dimensions or one of: "circle", "disk", "sphere" or "ball"
[arg_def strings ARGS] Zero or more key-value pairs. The supported options are:
[list_begin itemized]
[item] [term {-start index}]: The index for the next point to be generated (default: 1)
[item] [term {-evaluations number}]: The number of evaluations to be used by default (default: 100)
[list_end]
[list_end]
[list_end]
The points that are returned lie in the hyperblock [lb]0,1[lb]^n (n the number of dimensions)
or on the unit circle, within the unit disk, on the unit sphere or within the unit ball.
[para]
Each generator supports the following subcommands:
[list_begin definitions]
[call [cmd "gen next"]]
Return the coordinates of the next quasi-random point
[nl]
[call [cmd "gen set-start"] [arg index]]
Reset the index for the next quasi-random point. This is useful to control which list of points is returned.
Returns the new or the current value, if no value is given.
[nl]
[call [cmd "gen set-evaluations"] [arg number]]
Reset the default number of evaluations in compound algorithms. Note that the actual number is the
smallest 4-fold larger or equal to the given number. (The 4-fold plays a role in the detailed integration
routine.)
[nl]
[call [cmd "gen integral"] [arg func] [arg minmax] [arg args]]
Calculate the integral of the given function over the block (or the circle, sphere etc.)
[list_begin arguments]
[arg_def string func] The name of the function to be integrated
[arg_def list minmax] List of pairs of minimum and maximum coordinates. This can be used to
map the quasi-random coordinates to the desired hyper-block.
[nl]
If the space is a circle, disk etc. then this argument should be a single value, the radius.
The circle, disk, etc. is centred at the origin. If this is not what is required, then a coordinate
transformation should be made within the function.
[arg_def strings args] Zero or more key-value pairs. The following options are supported:
[list_begin itemized]
[item] [term {-evaluations number}]: The number of evaluations to be used. If not specified use the
default of the generator object.
[list_end]
[list_end]
[list_end]
[section TODO]
Implement other algorithms and variants
[para]
Implement more unit tests.
[para]
Comparison to pseudo-random numbers for integration.
[section References]
Various algorithms exist for generating quasi-random numbers. The generators created in this package are based on:
[uri http://extremelearning.com.au/unreasonable-effectiveness-of-quasirandom-sequences/]
[manpage_end]
|
