from math import * # function [alpha, xmin, L]=plfit(x, varargin) # PLFIT fits a power-law distributional model to data. # Source: http://www.santafe.edu/~aaronc/powerlaws/ # # PLFIT(x) estimates x_min and alpha according to the goodness-of-fit # based method described in Clauset, Shalizi, Newman (2007). x is a # vector of observations of some quantity to which we wish to fit the # power-law distribution p(x) ~ x^-alpha for x >= xmin. # PLFIT automatically detects whether x is composed of real or integer # values, and applies the appropriate method. For discrete data, if # min(x) > 1000, PLFIT uses the continuous approximation, which is # a reliable in this regime. # # The fitting procedure works as follows: # 1) For each possible choice of x_min, we estimate alpha via the # method of maximum likelihood, and calculate the Kolmogorov-Smirnov # goodness-of-fit statistic D. # 2) We then select as our estimate of x_min, the value that gives the # minimum value D over all values of x_min. # # Note that this procedure gives no estimate of the uncertainty of the # fitted parameters, nor of the validity of the fit. # # Example: # x = [500,150,90,81,75,75,70,65,60,58,49,47,40] # [alpha, xmin, L] = plfit(x) # or a = plfit(x) # # The output 'alpha' is the maximum likelihood estimate of the scaling # exponent, 'xmin' is the estimate of the lower bound of the power-law # behavior, and L is the log-likelihood of the data x>=xmin under the # fitted power law. # # For more information, try 'type plfit' # # See also PLVAR, PLPVA # Version 1.0.10 (2010 January) # Copyright (C) 2008-2011 Aaron Clauset (Santa Fe Institute) # Ported to Python by Joel Ornstein (2011 July) # (joel_ornstein@hmc.edu) # Distributed under GPL 2.0 # http://www.gnu.org/copyleft/gpl.html # PLFIT comes with ABSOLUTELY NO WARRANTY # # # The 'zeta' helper function is modified from the open-source library 'mpmath' # mpmath: a Python library for arbitrary-precision floating-point arithmetic # http://code.google.com/p/mpmath/ # version 0.17 (February 2011) by Fredrik Johansson and others # # Notes: # # 1. In order to implement the integer-based methods in Matlab, the numeric # maximization of the log-likelihood function was used. This requires # that we specify the range of scaling parameters considered. We set # this range to be 1.50 to 3.50 at 0.01 intervals by default. # This range can be set by the user like so, # # a = plfit(x,'range',[1.50,3.50,0.01]) # # 2. PLFIT can be told to limit the range of values considered as estimates # for xmin in three ways. First, it can be instructed to sample these # possible values like so, # # a = plfit(x,'sample',100) # # which uses 100 uniformly distributed values on the sorted list of # unique values in the data set. Second, it can simply omit all # candidates above a hard limit, like so # # a = plfit(x,'limit',3.4) # # Finally, it can be forced to use a fixed value, like so # # a = plfit(x,'xmin',3.4) # # In the case of discrete data, it rounds the limit to the nearest # integer. # # 3. When the input sample size is small (e.g., < 100), the continuous # estimator is slightly biased (toward larger values of alpha). To # explicitly use an experimental finite-size correction, call PLFIT like # so # # a = plfit(x,'finite') # # which does a small-size correction to alpha. # # 4. For continuous data, PLFIT can return erroneously large estimates of # alpha when xmin is so large that the number of obs x >= xmin is very # small. To prevent this, we can truncate the search over xmin values # before the finite-size bias becomes significant by calling PLFIT as # # a = plfit(x,'nosmall') # # which skips values xmin with finite size bias > 0.1. def plfit(x, *varargin): vec = [] sample = [] xminx = [] limit = [] finite = False nosmall = False nowarn = False # parse command-line parameters trap for bad input i=0 while iRange[0]: argok=0 vec=[] try: vec=map(lambda X:X*float(Range[2])+Range[0],\ range(int((Range[1]-Range[0])/Range[2]))) except: argok=0 vec=[] if Range[0]>=Range[1]: argok=0 vec=[] i-=1 i+=1 elif varargin[i]== 'sample': sample = varargin[i+1] i = i + 1 elif varargin[i]== 'limit': limit = varargin[i+1] i = i + 1 elif varargin[i]== 'xmin': xminx = varargin[i+1] i = i + 1 elif varargin[i]== 'finite': finite = True elif varargin[i]== 'nowarn': nowarn = True elif varargin[i]== 'nosmall': nosmall = True else: argok=0 if not argok: print '(PLFIT) Ignoring invalid argument #',i+1 i = i+1 if vec!=[] and (type(vec)!=list or min(vec)<=1): print '(PLFIT) Error: ''range'' argument must contain a vector or minimum <= 1. using default.\n' vec = [] if sample!=[] and sample<2: print'(PLFIT) Error: ''sample'' argument must be a positive integer > 1. using default.\n' sample = [] if limit!=[] and limit= 1. using default.\n' limit = [] if xminx!=[] and xminx>=max(x): print'(PLFIT) Error: ''xmin'' argument must be a positive value < max(x). using default behavior.\n' xminx = [] # select method (discrete or continuous) for fitting if reduce(lambda X,Y:X==True and floor(Y)==float(Y),x,True): f_dattype = 'INTS' elif reduce(lambda X,Y:X==True and (type(Y)==int or type(Y)==float or type(Y)==long),x,True): f_dattype = 'REAL' else: f_dattype = 'UNKN' if f_dattype=='INTS' and min(x) > 1000 and len(x)>100: f_dattype = 'REAL' # estimate xmin and alpha, accordingly if f_dattype== 'REAL': xmins = unique(x) xmins.sort() xmins = xmins[0:-1] if xminx!=[]: xmins = [min(filter(lambda X: X>=xminx,xmins))] if limit!=[]: xmins=filter(lambda X: X<=limit,xmins) if xmins==[]: xmins = [min(x)] if sample!=[]: step = float(len(xmins))/(sample-1) index_curr=0 new_xmins=[] for i in range (0,sample): if round(index_curr)==len(xmins): index_curr-=1 new_xmins.append(xmins[int(round(index_curr))]) index_curr+=step xmins = unique(new_xmins) xmins.sort() dat = [] z = sorted(x) for xm in range(0,len(xmins)): xmin = xmins[xm] z = filter(lambda X:X>=xmin,z) n = len(z) # estimate alpha using direct MLE a = float(n) / sum(map(lambda X: log(float(X)/xmin),z)) if nosmall: if (a-1)/sqrt(n) > 0.1 and dat!=[]: xm = len(xmins)+1 break # compute KS statistic #cx = map(lambda X:float(X)/n,range(0,n)) cf = map(lambda X:1-pow((float(xmin)/X),a),z) dat.append( max( map(lambda X: abs(cf[X]-float(X)/n),range(0,n)))) D = min(dat) xmin = xmins[dat.index(D)] z = filter(lambda X:X>=xmin,x) z.sort() n = len(z) alpha = 1 + n / sum(map(lambda X: log(float(X)/xmin),z)) if finite: alpha = alpha*float(n-1)/n+1./n # finite-size correction if n < 50 and not finite and not nowarn: print '(PLFIT) Warning: finite-size bias may be present.\n' L = n*log((alpha-1)/xmin) - alpha*sum(map(lambda X: log(float(X)/xmin),z)) elif f_dattype== 'INTS': x=map(int,x) if vec==[]: for X in range(150,351): vec.append(X/100.) # covers range of most practical # scaling parameters zvec = map(zeta, vec) xmins = unique(x) xmins.sort() xmins = xmins[0:-1] if xminx!=[]: xmins = [min(filter(lambda X: X>=xminx,xmins))] if limit!=[]: limit = round(limit) xmins=filter(lambda X: X<=limit,xmins) if xmins==[]: xmins = [min(x)] if sample!=[]: step = float(len(xmins))/(sample-1) index_curr=0 new_xmins=[] for i in range (0,sample): if round(index_curr)==len(xmins): index_curr-=1 new_xmins.append(xmins[int(round(index_curr))]) index_curr+=step xmins = unique(new_xmins) xmins.sort() if xmins==[]: print '(PLFIT) Error: x must contain at least two unique values.\n' alpha = 'Not a Number' xmin = x[0] D = 'Not a Number' return [alpha,xmin,D] xmax = max(x) z = x z.sort() datA=[] datB=[] for xm in range(0,len(xmins)): xmin = xmins[xm] z = filter(lambda X:X>=xmin,z) n = len(z) # estimate alpha via direct maximization of likelihood function # force iterative calculation L = [] slogz = sum(map(log,z)) xminvec = map(float,range(1,xmin)) for k in range(0,len(vec)): L.append(-vec[k]*float(slogz) - float(n)*log(float(zvec[k]) - sum(map(lambda X:pow(float(X),-vec[k]),xminvec)))) I = L.index(max(L)) # compute KS statistic fit = reduce(lambda X,Y: X+[Y+X[-1]],\ (map(lambda X: pow(X,-vec[I])/(float(zvec[I])-sum(map(lambda X: pow(X,-vec[I]),map(float,range(1,xmin))))),range(xmin,xmax+1))),[0])[1:] cdi=[] for XM in range(xmin,xmax+1): cdi.append(len(filter(lambda X: floor(X)<=XM,z))/float(n)) datA.append(max( map(lambda X: abs(fit[X] - cdi[X]),range(0,xmax-xmin+1)))) datB.append(vec[I]) # select the index for the minimum value of D I = datA.index(min(datA)) xmin = xmins[I] z = filter(lambda X:X>=xmin,x) n = len(z) alpha = datB[I] if finite: alpha = alpha*(n-1.)/n+1./n # finite-size correction if n < 50 and not finite and not nowarn: print '(PLFIT) Warning: finite-size bias may be present.\n' L = -alpha*sum(map(log,z)) - n*log(zvec[vec.index(max(filter(lambda X:X<=alpha,vec)))] - \ sum(map(lambda X: pow(X,-alpha),range(1,xmin)))) else: print '(PLFIT) Error: x must contain only reals or only integers.\n' alpha = [] xmin = [] L = [] return [alpha,xmin,L] # helper functions (unique and zeta) def unique(seq): # not order preserving set = {} map(set.__setitem__, seq, []) return set.keys() def _polyval(coeffs, x): p = coeffs[0] for c in coeffs[1:]: p = c + x*p return p _zeta_int = [\ -0.5, 0.0, 1.6449340668482264365,1.2020569031595942854,1.0823232337111381915, 1.0369277551433699263,1.0173430619844491397,1.0083492773819228268, 1.0040773561979443394,1.0020083928260822144,1.0009945751278180853, 1.0004941886041194646,1.0002460865533080483,1.0001227133475784891, 1.0000612481350587048,1.0000305882363070205,1.0000152822594086519, 1.0000076371976378998,1.0000038172932649998,1.0000019082127165539, 1.0000009539620338728,1.0000004769329867878,1.0000002384505027277, 1.0000001192199259653,1.0000000596081890513,1.0000000298035035147, 1.0000000149015548284] _zeta_P = [-3.50000000087575873, -0.701274355654678147, -0.0672313458590012612, -0.00398731457954257841, -0.000160948723019303141, -4.67633010038383371e-6, -1.02078104417700585e-7, -1.68030037095896287e-9, -1.85231868742346722e-11][::-1] _zeta_Q = [1.00000000000000000, -0.936552848762465319, -0.0588835413263763741, -0.00441498861482948666, -0.000143416758067432622, -5.10691659585090782e-6, -9.58813053268913799e-8, -1.72963791443181972e-9, -1.83527919681474132e-11][::-1] _zeta_1 = [3.03768838606128127e-10, -1.21924525236601262e-8, 2.01201845887608893e-7, -1.53917240683468381e-6, -5.09890411005967954e-7, 0.000122464707271619326, -0.000905721539353130232, -0.00239315326074843037, 0.084239750013159168, 0.418938517907442414, 0.500000001921884009] _zeta_0 = [-3.46092485016748794e-10, -6.42610089468292485e-9, 1.76409071536679773e-7, -1.47141263991560698e-6, -6.38880222546167613e-7, 0.000122641099800668209, -0.000905894913516772796, -0.00239303348507992713, 0.0842396947501199816, 0.418938533204660256, 0.500000000000000052] def zeta(s): """ Riemann zeta function, real argument """ if not isinstance(s, (float, int)): try: s = float(s) except (ValueError, TypeError): try: s = complex(s) if not s.imag: return complex(zeta(s.real)) except (ValueError, TypeError): pass raise NotImplementedError if s == 1: raise ValueError("zeta(1) pole") if s >= 27: return 1.0 + 2.0**(-s) + 3.0**(-s) n = int(s) if n == s: if n >= 0: return _zeta_int[n] if not (n % 2): return 0.0 if s <= 0.0: return 0 if s <= 2.0: if s <= 1.0: return _polyval(_zeta_0,s)/(s-1) return _polyval(_zeta_1,s)/(s-1) z = _polyval(_zeta_P,s) / _polyval(_zeta_Q,s) return 1.0 + 2.0**(-s) + 3.0**(-s) + 4.0**(-s)*z