def circstd(samples, high=2*pi, low=0):
"""Compute the circular standard deviation for samples assumed to be in the range [low to high]
"""
ang = (samples - low)*2*pi / (high-low)
res = stats.mean(exp(1j*ang))
V = 1-abs(res)
return ((high-low)/2.0/pi) * sqrt(V)
def rootfunc(ab,xj,N):
a,b = ab
tmp = (xj-a)/b
tmp2 = exp(tmp)
val = [sum(1.0/(1+tmp2))-0.5*N,
sum(tmp*(1.0-tmp2)/(1+tmp2))+N]
return array(val)
def circvar(samples, high=2*pi, low=0):
"""Compute the circular variance for samples assumed to be in the range [low to high]
"""
ang = (samples - low)*2*pi / (high-low)
res = stats.mean(exp(1j*ang))
V = 1-abs(res)
return ((high-low)/2.0/pi)**2 * V
def comb(N,k,exact=0):
"""Combinations of N things taken k at a time.
If exact==0, then floating point precision is used, otherwise
exact long integer is computed.
Notes:
- Array arguments accepted only for exact=0 case.
- If k > N, N < 0, or k < 0, then a 0 is returned.
"""
if exact:
if (k > N) or (N < 0) or (k < 0):
return 0L
N,k = map(long,(N,k))
top = N
val = 1L
while (top > (N-k)):
val *= top
top -= 1
n = 1L
while (n < k+1L):
val /= n
n += 1
return val
else:
k,N = asarray(k), asarray(N)
lgam = special.gammaln
cond = (k <= N) & (N >= 0) & (k >= 0)
sv = special.errprint(0)
vals = exp(lgam(N+1) - lgam(N-k+1) - lgam(k+1))
sv = special.errprint(sv)
return where(cond, vals, 0.0)
def circmean(samples, high=2*pi, low=0):
"""Compute the circular mean for samples assumed to be in the range [low to high]
"""
ang = (samples - low)*2*pi / (high-low)
res = angle(stats.mean(exp(1j*ang)))
if (res < 0):
res = res + 2*pi
return res*(high-low)/2.0/pi + low
def thisfunc(x):
xn = (x-mu)/sig
return totp(xn)*exp(-xn*xn/2.0)/sqrt(2*pi)/sig
def fixedsolve(th,xj,N):
val = stats.sum(xj)*1.0/N
tmp = exp(-xj/th)
term = sum(xj*tmp)
term /= sum(tmp)
return val - term
def anderson(x,dist='norm'):
"""Anderson and Darling test for normal, exponential, or Gumbel
(Extreme Value Type I) distribution.
Given samples x, return A2, the Anderson-Darling statistic,
the significance levels in percentages, and the corresponding
critical values.
Critical values provided are for the following significance levels
norm/expon: 15%, 10%, 5%, 2.5%, 1%
Gumbel: 25%, 10%, 5%, 2.5%, 1%
logistic: 25%, 10%, 5%, 2.5%, 1%, 0.5%
If A2 is larger than these critical values then for that significance
level, the hypothesis that the data come from a normal (exponential)
can be rejected.
"""
if not dist in ['norm','expon','gumbel','extreme1','logistic']:
raise ValueError, "Invalid distribution."
y = sort(x)
xbar = stats.mean(x)
N = len(y)
if dist == 'norm':
s = stats.std(x)
w = (y-xbar)/s
z = distributions.norm.cdf(w)
sig = array([15,10,5,2.5,1])
critical = around(_Avals_norm / (1.0 + 4.0/N - 25.0/N/N),3)
elif dist == 'expon':
w = y / xbar
z = distributions.expon.cdf(w)
sig = array([15,10,5,2.5,1])
critical = around(_Avals_expon / (1.0 + 0.6/N),3)
elif dist == 'logistic':
def rootfunc(ab,xj,N):
a,b = ab
tmp = (xj-a)/b
tmp2 = exp(tmp)
val = [sum(1.0/(1+tmp2))-0.5*N,
sum(tmp*(1.0-tmp2)/(1+tmp2))+N]
return array(val)
sol0=array([xbar,stats.std(x)])
sol = optimize.fsolve(rootfunc,sol0,args=(x,N),xtol=1e-5)
w = (y-sol[0])/sol[1]
z = distributions.logistic.cdf(w)
sig = array([25,10,5,2.5,1,0.5])
critical = around(_Avals_logistic / (1.0+0.25/N),3)
else:
def fixedsolve(th,xj,N):
val = stats.sum(xj)*1.0/N
tmp = exp(-xj/th)
term = sum(xj*tmp)
term /= sum(tmp)
return val - term
s = optimize.fixed_point(fixedsolve, 1.0, args=(x,N),xtol=1e-5)
xbar = -s*log(sum(exp(-x/s))*1.0/N)
w = (y-xbar)/s
z = distributions.gumbel_l.cdf(w)
sig = array([25,10,5,2.5,1])
critical = around(_Avals_gumbel / (1.0 + 0.2/sqrt(N)),3)
i = arange(1,N+1)
S = sum((2*i-1.0)/N*(log(z)+log(1-z[::-1])))
A2 = -N-S
return A2, critical, sig
def thefunc(x):
xn = (x-mu)/sig
return totp(xn)*exp(-xn*xn/2.0)
