Generating Random Numbers/de

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Zufallszahlen sind wichtige Quellen für wissenschaftliche Anwendungen, Bildung, Spiele-Entwicklung und Visualisierung.

Die Standard Funktion random der RTL erzeugt Zufallszahlen, die eine gleichmäßige Verteilung zu erfüllen. Gleichmäßig verteilte Zufallszahlen sind nicht für jede Anwendung sinnvoll. Um Zufallszahlen mit einer anderen Verteilung zu erstellen sind spezielle Algorithmen notwendig.


Normale (gaußsche) Verteilung

Einer der häufigsten Algorithmen normalverteilte Zufallszahlen aus gleichmäßig verteilten Zufallszahlen erzeugen ist der Box-Müller Ansatz. Die folgende Funktion berechnet die nach Gauß verteilten Zufallszahlen:

 function rnorm (mean, sd: real): real;
 {Berechnet Gaußsche Zufallszahlen nach dem Box-Müller-Ansatz}
  var
   u1, u2: real;
 begin
   u1 := random;
   u2 := random;
   rnorm := mean * abs(1 + sqrt(-2 * (ln(u1))) * cos(2 * pi * u2) * sd);
  end;

Der gleiche Algorithmus wird durch die Funktion randg aus der RTL unit math verwendet:

function randg(mean,stddev: float): float;

Exponentielle Verteilung

An exponential distribution occurs frequently in real-world problems. A classical example is the distribution of waiting times between independent Poisson-random events, e.g. the radioactive decay of nuclei [Press et al. 1989].

The following function delivers a single real random number out of an exponential distribution. Rate is the inverse of the mean and the constant RESOLUTION determines the granularity of generated random numbers.

function randomExp(a, rate: real): real;
const
  RESOLUTION = 1000;
var
  unif: real;
begin
  if rate = 0 then
    randomExp := NaN
  else
  begin
    repeat
      unif := random(RESOLUTION) / RESOLUTION;
    until unif <> 0;
    randomExp := a - rate * ln(unif);
  end;
end;

Gamma Verteilung

The gamma distribution is a two-parameter family of continuous random distributions. It is a generalization of both the exponential distribution and the Erlang distribution. Possible applications of the gamma distribution include modelling and simulation of waiting lines, or queues, and actuarial science.

The following function delivers a single real random number out of a gamma distribution. The shape of the distribution is defined by the parameters a, b and c. The function makes use of the function randomExp as defined above.

function randomGamma(a, b, c: real): real;
const
  RESOLUTION = 1000;
  T = 4.5;
  D = 1 + ln(T);
var
  unif: real;
  A2, B2, C2, Q, p, y: real;
  p1, p2, v, w, z: real;
  found: boolean;
begin
  A2 := 1 / sqrt(2 * c - 1);
  B2 := c - ln(4);
  Q := c + 1 / A2;
  C2 := 1 + c / exp(1);
  found := False;
  if c < 1 then
  begin
    repeat
      repeat
        unif := random(RESOLUTION) / RESOLUTION;
      until unif > 0;
      p := C2 * unif;
      if p > 1 then
      begin
        repeat
          unif := random(RESOLUTION) / RESOLUTION;
        until unif > 0;
        y := -ln((C2 - p) / c);
        if unif <= power(y, c - 1) then
        begin
          randomGamma := a + b * y;
          found := True;
        end;
      end
      else
      begin
        y := power(p, 1 / c);
        if unif <= exp(-y) then
        begin
          randomGamma := a + b * y;
          found := True;
        end;
      end;
    until found;
  end
  else if c = 1 then
    { Gamma distribution becomes exponential distribution, if c = 1 }
  begin
    randomGamma := randomExp(a, b);
  end
  else
  begin
    repeat
      repeat
        p1 := random(RESOLUTION) / RESOLUTION;
      until p1 > 0;
      repeat
        p2 := random(RESOLUTION) / RESOLUTION;
      until p2 > 0;
      v := A2 * ln(p1 / (1 - p1));
      y := c * exp(v);
      z := p1 * p1 * p2;
      w := B2 + Q * v - y;
      if (w + D - T * z >= 0) or (w >= ln(z)) then
      begin
        randomGamma := a + b * y;
        found := True;
      end;
    until found;
  end;
end;

Erlang Verteilung

The Erlang distribution is a two parameter family of continuous probability distributions. It is a generalization of the exponential distribution and a special case of the gamma distribution, where c is an integer. The Erlang distribution has been first described by Agner Krarup Erlang in order to model the time interval between telephone calls. It is used for queuing theory and for simulating waiting lines.

  function randomErlang(mean: real; k: integer): real;
  const
    RESOLUTION = 1000;
  var
    i: integer;
    unif, prod: real;
  begin
    if (mean <= 0) or (k < 1) then
      randomErlang := NaN
    else
    begin
      prod := 1;
      for i := 1 to k do
      begin
        repeat
          unif := random(RESOLUTION) / RESOLUTION;
        until unif <> 0;
        prod := prod * unif;
      end;
      randomErlang := -mean * ln(prod);
    end;
  end;

Poisson Verteilung

Studentsche t-Verteilung

The t distribution (also referred to a Student's t distribution, since it was published by William Sealy Gosset in 1908 under the pseudonym Student) is a continuous probability distribution. Its shape is defined by one parameter, the degrees of freedom (df). In statistics, many estimators are t distributed. Therefore, Student's t-distribution plays a major role in a number of widely used statistical analyses, including Student's t-test for assessing the statistical significance of the difference between two sample means, the construction of confidence intervals for the difference between two population means, and in linear regression analysis. The t-distribution also arises in Bayesian analysis of data from a normal family.

The following algorithm depends on the [RTL]] function random and on the randomChisq function

function randomT(df: integer): real;
{ Generator for Student's t distribution }
begin
  if df < 1 then randomT := NaN
  else
  begin
    randomT := randg(0, 1) / sqrt(randomChisq(df) / df);
  end;
end;

Chi Squared Distribution

The chi squared distribution is a continuous distribution of random numbers with df degrees of freedom. It is the distribution of a sum of the squares of df independent standard normal random variables. The chi squared distribution has numerous applications in inferential statistics, e.g. in estimating variances and for chi-squared tests. It is a special gamma distribution with c = df/ 2 and b = 2. Therefore the following function depends on the function randomGamma.

function randomChisq(df: integer): real;
begin
  if df < 1 then randomChisq := NaN
  else
  randomChisq := randomGamma(0, 2, 0.5 * df);
end;

F Distribution

The F distribution, also referred to as Fisher-Snedecor distribution, is a continuous probability distribution. It is used for F Test and ANOVA. It has two degrees of freedom that serve as shape parameters v and w and that are positive integers. The following function randomF makes use of randomChisq.

function randomF(v, w: integer): real;
begin
  if (v < 1) or (w < 1) then
    randomF := NaN
  else
  randomF := randomChisq(v) / v / (randomChisq(w) / w);
end;

See also

References

  1. G. E. P. Box and Mervin E. Muller, A Note on the Generation of Random Normal Deviates, The Annals of Mathematical Statistics (1958), Vol. 29, No. 2 pp. 610–611
  2. Dietrich, J. W. (2002). Der Hypophysen-Schilddrüsen-Regelkreis. Berlin, Germany: Logos-Verlag Berlin. ISBN 978-3-89722-850-4. OCLC 50451543.
  3. Press, W. H., B. P. Flannery, S. A. Teukolsky, W. T. Vetterling (1989). Numerical Recipes in Pascal. The Art of Scientific Computing, Cambridge University Press, ISBN 0-521-37516-9.
  4. Richard Saucier, Computer Generation of Statistical Distributions, ARL-TR-2168, US Army Research Laboratory, Aberdeen Proving Ground, MD, 21005-5068, March 2000.
  5. R.U. Seydel, Generating Random Numbers with Specified Distributions. In: Tools for Computational Finance, Universitext, DOI 10.1007/978-1-4471-2993-6_2, © Springer-Verlag London Limited 2012
  6. Christian Walck, Hand-book on STATISTICAL DISTRIBUTIONS for experimentalists, Internal Report SUF–PFY/96–01, University of Stockholm 2007