Annotation of OpenXM_contrib/gnuplot/demo/stat.inc, Revision 1.1.1.1
1.1 maekawa 1: #
2: # $Id: stat.inc,v 1.3 1998/04/14 00:16:54 drd Exp $
3: #
4: # Library of Statistical Functions version 3.0
5: #
6: # Permission granted to distribute freely for non-commercial purposes only
7: #
8: # Copyright (c) 1991, 1992 Jos van der Woude, jvdwoude@hut.nl
9:
10: # If you don't have the gamma() and/or lgamma() functions in your library,
11: # you can use the following recursive definitions. They are correct for all
12: # values i / 2 with i = 1, 2, 3, ... This is sufficient for most statistical
13: # needs.
14: #logsqrtpi = log(sqrt(pi))
15: #lgamma(x) = (x<=0.5)?logsqrtpi:((x==1)?0:log(x-1)+lgamma(x-1))
16: #gamma(x) = exp(lgamma(x))
17:
18: # If you have the lgamma() function compiled into gnuplot, you can use
19: # alternate definitions for some PDFs. For larger arguments this will result
20: # in more efficient evalution. Just uncomment the definitions containing the
21: # string `lgamma', while at the same time commenting out the originals.
22: # NOTE: In these cases the recursive definition for lgamma() is NOT sufficient!
23:
24: # Some PDFs have alternate definitions of a recursive nature. I suppose these
25: # are not really very efficient, but I find them aesthetically pleasing to the
26: # brain.
27:
28: # Define useful constants
29: fourinvsqrtpi=4.0/sqrt(pi)
30: invpi=1.0/pi
31: invsqrt2pi=1.0/sqrt(2.0*pi)
32: log2=log(2.0)
33: sqrt2=sqrt(2.0)
34: sqrt2invpi=sqrt(2.0/pi)
35: twopi=2.0*pi
36:
37: # define variables plus default values used as parameters in PDFs
38: # some are integers, others MUST be reals
39: a=1.0
40: alpha=0.5
41: b=2.0
42: df1=1
43: df2=1
44: g=1.0
45: lambda=1.0
46: m=0.0
47: mm=1
48: mu=0.0
49: nn=2
50: n=2
51: p=0.5
52: q=0.5
53: r=1
54: rho=1.0
55: sigma=1.0
56: u=1.0
57:
58: #
59: #define 1.0/Beta function
60: #
61: Binv(p,q)=exp(lgamma(p+q)-lgamma(p)-lgamma(q))
62:
63: #
64: #define Probability Density Functions (PDFs)
65: #
66:
67: # NOTE:
68: # The discrete PDFs are calulated for all real values, using the int()
69: # function to truncate to integers. This is a monumental waste of processing
70: # power, but I see no other easy solution. If anyone has any smart ideas
71: # about this, I would like to know. Setting the sample size to a larger value
72: # makes the discrete PDFs look better, but takes even more time.
73:
74: # Arcsin PDF
75: arcsin(x)=invpi/sqrt(r*r-x*x)
76:
77: # Beta PDF
78: beta(x)=Binv(p,q)*x**(p-1.0)*(1.0-x)**(q-1.0)
79:
80: # Binomial PDF
81: #binom(x)=n!/(n-int(x))!/int(x)!*p**int(x)*(1.0-p)**(n-int(x))
82:
83: bin_s(x)=n!/(n-int(x))!/int(x)!*p**int(x)*(1.0-p)**(n-int(x))
84: bin_l(x)=exp(lgamma(n+1)-lgamma(n-int(x)+1)-lgamma(int(x)+1)\
85: +int(x)*log(p)+(n-int(x))*log(1.0-p))
86: binom(x)=(n<20)?bin_s(x):bin_l(x)
87:
88: # Cauchy PDF
89: cauchy(x)=b/(pi*(b*b+(x-a)**2))
90:
91: # Chi-square PDF
92: #chi(x)=x**(0.5*df1-1.0)*exp(-0.5*x)/gamma(0.5*df1)/2**(0.5*df1)
93: chi(x)=exp((0.5*df1-1.0)*log(x)-0.5*x-lgamma(0.5*df1)-df1*0.5*log2)
94:
95: # Erlang PDF
96: erlang(x)=lambda**n/(n-1)!*x**(n-1)*exp(-lambda*x)
97:
98: # Extreme (Gumbel extreme value) PDF
99: extreme(x)=alpha*(exp(-alpha*(x-u)-exp(-alpha*(x-u))))
100:
101: # F PDF
102: f(x)=Binv(0.5*df1,0.5*df2)*(df1/df2)**(0.5*df1)*x**(0.5*df1-1.0)/\
103: (1.0+df1/df2*x)**(0.5*(df1+df2))
104:
105: # Gamma PDF
106: #g(x)=lambda**rho*x**(rho-1.0)*exp(-lambda*x)/gamma(rho)
107: g(x)=exp(rho*log(lambda)+(rho-1.0)*log(x)-lgamma(rho)-lambda*x)
108:
109: # Geometric PDF
110: #geometric(x)=p*(1.0-p)**int(x)
111: geometric(x)=exp(log(p)+int(x)*log(1.0-p))
112:
113: # Half normal PDF
114: halfnormal(x)=sqrt2invpi/sigma*exp(-0.5*(x/sigma)**2)
115:
116: # Hypergeometric PDF
117: hypgeo(x)=(int(x)>mm||int(x)<mm+n-nn)?0:\
118: mm!/(mm-int(x))!/int(x)!*(nn-mm)!/(n-int(x))!/(nn-mm-n+int(x))!*(nn-n)!*n!/nn!
119:
120: # Laplace PDF
121: laplace(x)=0.5/b*exp(-abs(x-a)/b)
122:
123: # Logistic PDF
124: logistic(x)=lambda*exp(-lambda*(x-a))/(1.0+exp(-lambda*(x-a)))**2
125:
126: # Lognormal PDF
127: lognormal(x)=invsqrt2pi/sigma/x*exp(-0.5*((log(x)-mu)/sigma)**2)
128:
129: # Maxwell PDF
130: maxwell(x)=fourinvsqrtpi*a**3*x*x*exp(-a*a*x*x)
131:
132: # Negative binomial PDF
133: #negbin(x)=(r+int(x)-1)!/int(x)!/(r-1)!*p**r*(1.0-p)**int(x)
134: negbin(x)=exp(lgamma(r+int(x))-lgamma(r)-lgamma(int(x)+1)+\
135: r*log(p)+int(x)*log(1.0-p))
136:
137: # Negative exponential PDF
138: nexp(x)=lambda*exp(-lambda*x)
139:
140: # Normal PDF
141: normal(x)=invsqrt2pi/sigma*exp(-0.5*((x-mu)/sigma)**2)
142:
143: # Pareto PDF
144: pareto(x)=x<a?0:b/x*(a/x)**b
145:
146: # Poisson PDF
147: poisson(x)=mu**int(x)/int(x)!*exp(-mu)
148: #poisson(x)=exp(int(x)*log(mu)-lgamma(int(x)+1)-mu)
149: #poisson(x)=(x<1)?exp(-mu):mu/int(x)*poisson(x-1)
150: #lpoisson(x)=(x<1)?-mu:log(mu)-log(int(x))+lpoisson(x-1)
151:
152: # Rayleigh PDF
153: rayleigh(x)=lambda*2.0*x*exp(-lambda*x*x)
154:
155: # Sine PDF
156: sine(x)=2.0/a*sin(n*pi*x/a)**2
157:
158: # t (Student's t) PDF
159: t(x)=Binv(0.5*df1,0.5)/sqrt(df1)*(1.0+(x*x)/df1)**(-0.5*(df1+1.0))
160:
161: # Triangular PDF
162: triangular(x)=1.0/g-abs(x-m)/(g*g)
163:
164: # Uniform PDF
165: uniform(x)=1.0/(b-a)
166:
167: # Weibull PDF
168: weibull(x)=lambda*n*x**(n-1)*exp(-lambda*x**n)
169:
170: #
171: #define Cumulative Distribution Functions (CDFs)
172: #
173:
174: # Arcsin CDF
175: carcsin(x)=0.5+invpi*asin(x/r)
176:
177: # incomplete Beta CDF
178: cbeta(x)=ibeta(p,q,x)
179:
180: # Binomial CDF
181: #cbinom(x)=(x<1)?binom(0):binom(x)+cbinom(x-1)
182: cbinom(x)=ibeta(n-x,x+1.0,1.0-p)
183:
184: # Cauchy CDF
185: ccauchy(x)=0.5+invpi*atan((x-a)/b)
186:
187: # Chi-square CDF
188: cchi(x)=igamma(0.5*df1,0.5*x)
189:
190: # Erlang CDF
191: # approximation, using first three terms of expansion
192: cerlang(x)=1.0-exp(-lambda*x)*(1.0+lambda*x+0.5*(lambda*x)**2)
193:
194: # Extreme (Gumbel extreme value) CDF
195: cextreme(x)=exp(-exp(-alpha*(x-u)))
196:
197: # F CDF
198: cf(x)=1.0-ibeta(0.5*df2,0.5*df1,df2/(df2+df1*x))
199:
200: # incomplete Gamma CDF
201: cgamma(x)=igamma(rho,x)
202:
203: # Geometric CDF
204: cgeometric(x)=(x<1)?geometric(0):geometric(x)+cgeometric(x-1)
205:
206: # Half normal CDF
207: chalfnormal(x)=erf(x/sigma/sqrt2)
208:
209: # Hypergeometric CDF
210: chypgeo(x)=(x<1)?hypgeo(0):hypgeo(x)+chypgeo(x-1)
211:
212: # Laplace CDF
213: claplace(x)=(x<a)?0.5*exp((x-a)/b):1.0-0.5*exp(-(x-a)/b)
214:
215: # Logistic CDF
216: clogistic(x)=1.0/(1.0+exp(-lambda*(x-a)))
217:
218: # Lognormal CDF
219: clognormal(x)=cnormal(log(x))
220:
221: # Maxwell CDF
222: cmaxwell(x)=igamma(1.5,a*a*x*x)
223:
224: # Negative binomial CDF
225: cnegbin(x)=(x<1)?negbin(0):negbin(x)+cnegbin(x-1)
226:
227: # Negative exponential CDF
228: cnexp(x)=1.0-exp(-lambda*x)
229:
230: # Normal CDF
231: cnormal(x)=0.5+0.5*erf((x-mu)/sigma/sqrt2)
232: #cnormal(x)=0.5+((x>mu)?0.5:-0.5)*igamma(0.5,0.5*((x-mu)/sigma)**2)
233:
234: # Pareto CDF
235: cpareto(x)=x<a?0:1.0-(a/x)**b
236:
237: # Poisson CDF
238: #cpoisson(x)=(x<1)?poisson(0):poisson(x)+cpoisson(x-1)
239: cpoisson(x)=1.0-igamma(x+1.0,mu)
240:
241: # Rayleigh CDF
242: crayleigh(x)=1.0-exp(-lambda*x*x)
243:
244: # Sine CDF
245: csine(x)=x/a-sin(n*twopi*x/a)/(n*twopi)
246:
247: # t (Student's t) CDF
248: ct(x)=(x<0.0)?0.5*ibeta(0.5*df1,0.5,df1/(df1+x*x)):\
249: 1.0-0.5*ibeta(0.5*df1,0.5,df1/(df1+x*x))
250:
251: # Triangular PDF
252: ctriangular(x)=0.5+(x-m)/g-(x-m)*abs(x-m)/(2.0*g*g)
253:
254: # Uniform CDF
255: cuniform(x)=(x-a)/(b-a)
256:
257: # Weibull CDF
258: cweibull(x)=1.0-exp(-lambda*x**n)
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