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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258
|
#
# $Id: stat.inc,v 1.3 1998/04/14 00:16:54 drd Exp $
#
# Library of Statistical Functions version 3.0
#
# Permission granted to distribute freely for non-commercial purposes only
#
# Copyright (c) 1991, 1992 Jos van der Woude, jvdwoude@hut.nl
# If you don't have the gamma() and/or lgamma() functions in your library,
# you can use the following recursive definitions. They are correct for all
# values i / 2 with i = 1, 2, 3, ... This is sufficient for most statistical
# needs.
#logsqrtpi = log(sqrt(pi))
#lgamma(x) = (x<=0.5)?logsqrtpi:((x==1)?0:log(x-1)+lgamma(x-1))
#gamma(x) = exp(lgamma(x))
# If you have the lgamma() function compiled into gnuplot, you can use
# alternate definitions for some PDFs. For larger arguments this will result
# in more efficient evalution. Just uncomment the definitions containing the
# string `lgamma', while at the same time commenting out the originals.
# NOTE: In these cases the recursive definition for lgamma() is NOT sufficient!
# Some PDFs have alternate definitions of a recursive nature. I suppose these
# are not really very efficient, but I find them aesthetically pleasing to the
# brain.
# Define useful constants
fourinvsqrtpi=4.0/sqrt(pi)
invpi=1.0/pi
invsqrt2pi=1.0/sqrt(2.0*pi)
log2=log(2.0)
sqrt2=sqrt(2.0)
sqrt2invpi=sqrt(2.0/pi)
twopi=2.0*pi
# define variables plus default values used as parameters in PDFs
# some are integers, others MUST be reals
a=1.0
alpha=0.5
b=2.0
df1=1
df2=1
g=1.0
lambda=1.0
m=0.0
mm=1
mu=0.0
nn=2
n=2
p=0.5
q=0.5
r=1
rho=1.0
sigma=1.0
u=1.0
#
#define 1.0/Beta function
#
Binv(p,q)=exp(lgamma(p+q)-lgamma(p)-lgamma(q))
#
#define Probability Density Functions (PDFs)
#
# NOTE:
# The discrete PDFs are calulated for all real values, using the int()
# function to truncate to integers. This is a monumental waste of processing
# power, but I see no other easy solution. If anyone has any smart ideas
# about this, I would like to know. Setting the sample size to a larger value
# makes the discrete PDFs look better, but takes even more time.
# Arcsin PDF
arcsin(x)=invpi/sqrt(r*r-x*x)
# Beta PDF
beta(x)=Binv(p,q)*x**(p-1.0)*(1.0-x)**(q-1.0)
# Binomial PDF
#binom(x)=n!/(n-int(x))!/int(x)!*p**int(x)*(1.0-p)**(n-int(x))
bin_s(x)=n!/(n-int(x))!/int(x)!*p**int(x)*(1.0-p)**(n-int(x))
bin_l(x)=exp(lgamma(n+1)-lgamma(n-int(x)+1)-lgamma(int(x)+1)\
+int(x)*log(p)+(n-int(x))*log(1.0-p))
binom(x)=(n<20)?bin_s(x):bin_l(x)
# Cauchy PDF
cauchy(x)=b/(pi*(b*b+(x-a)**2))
# Chi-square PDF
#chi(x)=x**(0.5*df1-1.0)*exp(-0.5*x)/gamma(0.5*df1)/2**(0.5*df1)
chi(x)=exp((0.5*df1-1.0)*log(x)-0.5*x-lgamma(0.5*df1)-df1*0.5*log2)
# Erlang PDF
erlang(x)=lambda**n/(n-1)!*x**(n-1)*exp(-lambda*x)
# Extreme (Gumbel extreme value) PDF
extreme(x)=alpha*(exp(-alpha*(x-u)-exp(-alpha*(x-u))))
# F PDF
f(x)=Binv(0.5*df1,0.5*df2)*(df1/df2)**(0.5*df1)*x**(0.5*df1-1.0)/\
(1.0+df1/df2*x)**(0.5*(df1+df2))
# Gamma PDF
#g(x)=lambda**rho*x**(rho-1.0)*exp(-lambda*x)/gamma(rho)
g(x)=exp(rho*log(lambda)+(rho-1.0)*log(x)-lgamma(rho)-lambda*x)
# Geometric PDF
#geometric(x)=p*(1.0-p)**int(x)
geometric(x)=exp(log(p)+int(x)*log(1.0-p))
# Half normal PDF
halfnormal(x)=sqrt2invpi/sigma*exp(-0.5*(x/sigma)**2)
# Hypergeometric PDF
hypgeo(x)=(int(x)>mm||int(x)<mm+n-nn)?0:\
mm!/(mm-int(x))!/int(x)!*(nn-mm)!/(n-int(x))!/(nn-mm-n+int(x))!*(nn-n)!*n!/nn!
# Laplace PDF
laplace(x)=0.5/b*exp(-abs(x-a)/b)
# Logistic PDF
logistic(x)=lambda*exp(-lambda*(x-a))/(1.0+exp(-lambda*(x-a)))**2
# Lognormal PDF
lognormal(x)=invsqrt2pi/sigma/x*exp(-0.5*((log(x)-mu)/sigma)**2)
# Maxwell PDF
maxwell(x)=fourinvsqrtpi*a**3*x*x*exp(-a*a*x*x)
# Negative binomial PDF
#negbin(x)=(r+int(x)-1)!/int(x)!/(r-1)!*p**r*(1.0-p)**int(x)
negbin(x)=exp(lgamma(r+int(x))-lgamma(r)-lgamma(int(x)+1)+\
r*log(p)+int(x)*log(1.0-p))
# Negative exponential PDF
nexp(x)=lambda*exp(-lambda*x)
# Normal PDF
normal(x)=invsqrt2pi/sigma*exp(-0.5*((x-mu)/sigma)**2)
# Pareto PDF
pareto(x)=x<a?0:b/x*(a/x)**b
# Poisson PDF
poisson(x)=mu**int(x)/int(x)!*exp(-mu)
#poisson(x)=exp(int(x)*log(mu)-lgamma(int(x)+1)-mu)
#poisson(x)=(x<1)?exp(-mu):mu/int(x)*poisson(x-1)
#lpoisson(x)=(x<1)?-mu:log(mu)-log(int(x))+lpoisson(x-1)
# Rayleigh PDF
rayleigh(x)=lambda*2.0*x*exp(-lambda*x*x)
# Sine PDF
sine(x)=2.0/a*sin(n*pi*x/a)**2
# t (Student's t) PDF
t(x)=Binv(0.5*df1,0.5)/sqrt(df1)*(1.0+(x*x)/df1)**(-0.5*(df1+1.0))
# Triangular PDF
triangular(x)=1.0/g-abs(x-m)/(g*g)
# Uniform PDF
uniform(x)=1.0/(b-a)
# Weibull PDF
weibull(x)=lambda*n*x**(n-1)*exp(-lambda*x**n)
#
#define Cumulative Distribution Functions (CDFs)
#
# Arcsin CDF
carcsin(x)=0.5+invpi*asin(x/r)
# incomplete Beta CDF
cbeta(x)=ibeta(p,q,x)
# Binomial CDF
#cbinom(x)=(x<1)?binom(0):binom(x)+cbinom(x-1)
cbinom(x)=ibeta(n-x,x+1.0,1.0-p)
# Cauchy CDF
ccauchy(x)=0.5+invpi*atan((x-a)/b)
# Chi-square CDF
cchi(x)=igamma(0.5*df1,0.5*x)
# Erlang CDF
# approximation, using first three terms of expansion
cerlang(x)=1.0-exp(-lambda*x)*(1.0+lambda*x+0.5*(lambda*x)**2)
# Extreme (Gumbel extreme value) CDF
cextreme(x)=exp(-exp(-alpha*(x-u)))
# F CDF
cf(x)=1.0-ibeta(0.5*df2,0.5*df1,df2/(df2+df1*x))
# incomplete Gamma CDF
cgamma(x)=igamma(rho,x)
# Geometric CDF
cgeometric(x)=(x<1)?geometric(0):geometric(x)+cgeometric(x-1)
# Half normal CDF
chalfnormal(x)=erf(x/sigma/sqrt2)
# Hypergeometric CDF
chypgeo(x)=(x<1)?hypgeo(0):hypgeo(x)+chypgeo(x-1)
# Laplace CDF
claplace(x)=(x<a)?0.5*exp((x-a)/b):1.0-0.5*exp(-(x-a)/b)
# Logistic CDF
clogistic(x)=1.0/(1.0+exp(-lambda*(x-a)))
# Lognormal CDF
clognormal(x)=cnormal(log(x))
# Maxwell CDF
cmaxwell(x)=igamma(1.5,a*a*x*x)
# Negative binomial CDF
cnegbin(x)=(x<1)?negbin(0):negbin(x)+cnegbin(x-1)
# Negative exponential CDF
cnexp(x)=1.0-exp(-lambda*x)
# Normal CDF
cnormal(x)=0.5+0.5*erf((x-mu)/sigma/sqrt2)
#cnormal(x)=0.5+((x>mu)?0.5:-0.5)*igamma(0.5,0.5*((x-mu)/sigma)**2)
# Pareto CDF
cpareto(x)=x<a?0:1.0-(a/x)**b
# Poisson CDF
#cpoisson(x)=(x<1)?poisson(0):poisson(x)+cpoisson(x-1)
cpoisson(x)=1.0-igamma(x+1.0,mu)
# Rayleigh CDF
crayleigh(x)=1.0-exp(-lambda*x*x)
# Sine CDF
csine(x)=x/a-sin(n*twopi*x/a)/(n*twopi)
# t (Student's t) CDF
ct(x)=(x<0.0)?0.5*ibeta(0.5*df1,0.5,df1/(df1+x*x)):\
1.0-0.5*ibeta(0.5*df1,0.5,df1/(df1+x*x))
# Triangular PDF
ctriangular(x)=0.5+(x-m)/g-(x-m)*abs(x-m)/(2.0*g*g)
# Uniform CDF
cuniform(x)=(x-a)/(b-a)
# Weibull CDF
cweibull(x)=1.0-exp(-lambda*x**n)
|