00001 """ Numerical python functions written for compatability with matlab commands with the same names. Matlab compatible functions: * cohere - Coherence (normalized cross spectral density) * corrcoef - The matrix of correlation coefficients * csd - Cross spectral density uing Welch's average periodogram * detrend -- Remove the mean or best fit line from an array * find - Return the indices where some condition is true * linspace -- Linear spaced array from min to max * hist -- Histogram * polyfit - least squares best polynomial fit of x to y * polyval - evaluate a vector for a vector of polynomial coeffs * prctile - find the percentiles of a sequence * prepca - Principal Component's Analysis * psd - Power spectral density uing Welch's average periodogram * rk4 - A 4th order runge kutta integrator for 1D or ND systems * vander - the Vandermonde matrix * trapz - trapeziodal integration Functions that don't exist in matlab, but are useful anyway: * cohere_pairs - Coherence over all pairs. This is not a matlab function, but we compute coherence a lot in my lab, and we compute it for alot of pairs. This function is optimized to do this efficiently by caching the direct FFTs. Credits: Unless otherwise noted, these functions were written by Author: John D. Hunter <jdhunter@ace.bsd.uchicago.edu> Some others are from the Numeric documentation, or imported from MLab or other Numeric packages """ from __future__ import division import MLab import sys from MLab import * from Numeric import * from LinearAlgebra import inverse, eigenvectors from FFT import fft from Matrix import Matrix from cbook import iterable def mean(x, dim=None): if len(x)==0: return None elif dim is None: return MLab.mean(x) else: return MLab.mean(x, dim) def linspace(xmin, xmax, N): if N==1: return xmax dx = (xmax-xmin)/(N-1) return xmin + dx*arange(N) def _norm(x): "return sqrt(x dot x)" return sqrt(dot(x,x)) def window_hanning(x): "return x times the hanning window of len(x)" return hanning(len(x))*x def window_none(x): "No window function; simply return x" return x def detrend(x, key=None): if key is None or key=='constant': return detrend_mean(x) elif key=='linear': return detrend_linear(x) def detrend_mean(x): "Return x minus the mean(x)" return x - mean(x) def detrend_none(x): "Return x: no detrending" return x def detrend_linear(x): "Return x minus best fit line; 'linear' detrending " # I'm going to regress x on xx=range(len(x)) and return x - # (b*xx+a). Now that I have polyfit working, I could convert the # code here, but if it ain't broke, don't fix it! xx = arange(len(x), typecode=x.typecode()) X = transpose(array([xx]+[x])) C = cov(X) b = C[0,1]/C[0,0] a = mean(x) - b*mean(xx) return x-(b*xx+a) 00118 def psd(x, NFFT=256, Fs=2, detrend=detrend_none, window=window_hanning, noverlap=0): """ The power spectral density by Welches average periodogram method. The vector x is divided into NFFT length segments. Each segment is detrended by function detrend and windowed by function window. noperlap gives the length of the overlap between segments. The absolute(fft(segment))**2 of each segment are averaged to compute Pxx, with a scaling to correct for power loss due to windowing. Fs is the sampling frequency. -- NFFT must be a power of 2 -- detrend and window are functions, unlike in matlab where they are vectors. -- if length x < NFFT, it will be zero padded to NFFT Returns the tuple Pxx, freqs Refs: Bendat & Piersol -- Random Data: Analysis and Measurement Procedures, John Wiley & Sons (1986) """ if NFFT % 2: raise ValueError, 'NFFT must be a power of 2' # zero pad x up to NFFT if it is shorter than NFFT if len(x)<NFFT: n = len(x) x = resize(x, (NFFT,)) x[n:] = 0 # for real x, ignore the negative frequencies if x.typecode()==Complex: numFreqs = NFFT else: numFreqs = NFFT//2+1 windowVals = window(ones((NFFT,),x.typecode())) step = NFFT-noverlap ind = range(0,len(x)-NFFT+1,step) n = len(ind) Pxx = zeros((numFreqs,n), Float) # do the ffts of the slices for i in range(n): thisX = x[ind[i]:ind[i]+NFFT] thisX = windowVals*detrend(thisX) fx = absolute(fft(thisX))**2 Pxx[:,i] = fx[:numFreqs] # Scale the spectrum by the norm of the window to compensate for # windowing loss; see Bendat & Piersol Sec 11.5.2 if n>1: Pxx = mean(Pxx,1) Pxx = divide(Pxx, norm(windowVals)**2) freqs = Fs/NFFT*arange(numFreqs) Pxx.shape = len(freqs), return Pxx, freqs 00178 def csd(x, y, NFFT=256, Fs=2, detrend=detrend_none, window=window_hanning, noverlap=0): """ The cross spectral density Pxy by Welches average periodogram method. The vectors x and y are divided into NFFT length segments. Each segment is detrended by function detrend and windowed by function window. noverlap gives the length of the overlap between segments. The product of the direct FFTs of x and y are averaged over each segment to compute Pxy, with a scaling to correct for power loss due to windowing. Fs is the sampling frequency. NFFT must be a power of 2 Returns the tuple Pxy, freqs Refs: Bendat & Piersol -- Random Data: Analysis and Measurement Procedures, John Wiley & Sons (1986) """ if NFFT % 2: raise ValueError, 'NFFT must be a power of 2' # zero pad x and y up to NFFT if they are shorter than NFFT if len(x)<NFFT: n = len(x) x = resize(x, (NFFT,)) x[n:] = 0 if len(y)<NFFT: n = len(y) y = resize(y, (NFFT,)) y[n:] = 0 # for real x, ignore the negative frequencies if x.typecode()==Complex: numFreqs = NFFT else: numFreqs = NFFT//2+1 windowVals = window(ones((NFFT,),x.typecode())) step = NFFT-noverlap ind = range(0,len(x)-NFFT+1,step) n = len(ind) Pxy = zeros((numFreqs,n), Complex) # do the ffts of the slices for i in range(n): thisX = x[ind[i]:ind[i]+NFFT] thisX = windowVals*detrend(thisX) thisY = y[ind[i]:ind[i]+NFFT] thisY = windowVals*detrend(thisY) fx = fft(thisX) fy = fft(thisY) Pxy[:,i] = conjugate(fx[:numFreqs])*fy[:numFreqs] # Scale the spectrum by the norm of the window to compensate for # windowing loss; see Bendat & Piersol Sec 11.5.2 if n>1: Pxy = mean(Pxy,1) Pxy = divide(Pxy, norm(windowVals)**2) freqs = Fs/NFFT*arange(numFreqs) Pxy.shape = len(freqs), return Pxy, freqs 00243 def cohere(x, y, NFFT=256, Fs=2, detrend=detrend_none, window=window_hanning, noverlap=0): """ cohere the coherence between x and y. Coherence is the normalized cross spectral density Cxy = |Pxy|^2/(Pxx*Pyy) The return value is (Cxy, f), where f are the frequencies of the coherence vector. See the docs for psd and csd for information about the function arguments NFFT, detrend, windowm noverlap, as well as the methods used to compute Pxy, Pxx and Pyy. Returns the tuple Cxy, freqs """ if len(x)<2*NFFT: print >>sys.stderr, 'Coherence is calculated by averaging over NFFT length segments. Your signal is too short for your choice of NFFT' Pxx,f = psd(x, NFFT, Fs, detrend, window, noverlap) Pyy,f = psd(y, NFFT, Fs, detrend, window, noverlap) Pxy,f = csd(x, y, NFFT, Fs, detrend, window, noverlap) Cxy = divide(absolute(Pxy)**2, Pxx*Pyy) Cxy.shape = len(f), return Cxy, f 00270 def corrcoef(*args): """ corrcoef(X) where X is a matrix returns a matrix of correlation coefficients for each row of X. corrcoef(x,y) where x and y are vectors returns the matrix or correlation coefficients for x and y. Numeric arrays can be real or complex The correlation matrix is defined from the covariance matrix C as r(i,j) = C[i,j] / (C[i,i]*C[j,j]) """ if len(args)==2: X = transpose(array([args[0]]+[args[1]])) elif len(args==1): X = args[0] else: raise RuntimeError, 'Only expecting 1 or 2 arguments' C = cov(X) d = resize(diagonal(C), (2,1)) r = divide(C,sqrt(matrixmultiply(d,transpose(d)))) try: return r.real except AttributeError: return r 00303 def polyfit(x,y,N): """ Do a best fit polynomial of order N of y to x. Return value is a vector of polynomial coefficients [pk ... p1 p0]. Eg, for N=2 p2*x0^2 + p1*x0 + p0 = y1 p2*x1^2 + p1*x1 + p0 = y1 p2*x2^2 + p1*x2 + p0 = y2 ..... p2*xk^2 + p1*xk + p0 = yk Method: if X is a the Vandermonde Matrix computed from x (see http://mathworld.wolfram.com/VandermondeMatrix.html), then the polynomial least squares solution is given by the 'p' in X*p = y where X is a len(x) x N+1 matrix, p is a N+1 length vector, and y is a len(x) x 1 vector This equation can be solved as p = (XT*X)^-1 * XT * y where XT is the transpose of X and -1 denotes the inverse. For more info, see http://mathworld.wolfram.com/LeastSquaresFittingPolynomial.html, but note that the k's and n's in the superscripts and subscripts on that page. The linear algebra is correct, however. See also polyval """ y = reshape(y, (len(y),1)) X = Matrix(vander(x, N+1)) Xt = Matrix(transpose(X)) c = array(inverse(Xt*X)*Xt*y) # convert back to array c.shape = (N+1,) return c 00350 def polyval(p,x): """ y = polyval(p,x) p is a vector of polynomial coeffients and y is the polynomial evaluated at x. Example code to remove a polynomial (quadratic) trend from y: p = polyfit(x, y, 2) trend = polyval(p, x) resid = y - trend See also polyfit """ p = reshape(p, (len(p),1)) X = vander(x,len(p)) y = matrixmultiply(X,p) return reshape(y, x.shape) 00373 def vander(x,N=None): """ X = vander(x,N=None) The Vandermonde matrix of vector x. The i-th column of X is the the i-th power of x. N is the maximum power to compute; if N is None it defaults to len(x). """ if N is None: N=len(x) X = ones( (len(x),N), x.typecode()) for i in range(N-1): X[:,i] = x**(N-i-1) return X def donothing_callback(*args): pass 00393 def cohere_pairs( X, ij, NFFT=256, Fs=2, detrend=detrend_none, window=window_hanning, noverlap=0, preferSpeedOverMemory=1, progressCallback=donothing_callback): """ Cxy, Phase, freqs = cohere_pairs( X, ij, ...) Compute the coherence for all pairs in ij. X is a numSamples,numCols Numeric array. ij is a list of tuples (i,j). Each tuple is a pair of indexes into the columns of X for which you want to compute coherence. For example, if X has 64 columns, and you want to compute all nonredundant pairs, define ij as ij = [] for i in range(64): for j in range(i+1,64): ij.append( (i,j) ) The other function arguments, except for 'preferSpeedOverMemory' (see below), are explained in the help string of 'psd'. Return value is a tuple (Cxy, Phase, freqs). Cxy -- a dictionary of (i,j) tuples -> coherence vector for that pair. Ie, Cxy[(i,j) = cohere(X[:,i], X[:,j]). Number of dictionary keys is len(ij) Phase -- a dictionary of phases of the cross spectral density at each frequency for each pair. keys are (i,j). freqs -- a vector of frequencies, equal in length to either the coherence or phase vectors for any i,j key. Eg, to make a coherence Bode plot: subplot(211) plot( freqs, Cxy[(12,19)]) subplot(212) plot( freqs, Phase[(12,19)]) For a large number of pairs, cohere_pairs can be much more efficient than just calling cohere for each pair, because it caches most of the intensive computations. If N is the number of pairs, this function is O(N) for most of the heavy lifting, whereas calling cohere for each pair is O(N^2). However, because of the caching, it is also more memory intensive, making 2 additional complex arrays with approximately the same number of elements as X. The parameter 'preferSpeedOverMemory', if false, limits the caching by only making one, rather than two, complex cache arrays. This is useful if memory becomes critical. Even when preferSpeedOverMemory is false, cohere_pairs will still give significant performace gains over calling cohere for each pair, and will use subtantially less memory than if preferSpeedOverMemory is true. In my tests with a 43000,64 array over all nonredundant pairs, preferSpeedOverMemory=1 delivered a 33% performace boost on a 1.7GHZ Athlon with 512MB RAM compared with preferSpeedOverMemory=0. But both solutions were more than 10x faster than naievly crunching all possible pairs through cohere. See test/cohere_pairs_test.py in the src tree for an example script that shows that this cohere_pairs and cohere give the same results for a given pair. """ if NFFT % 2: raise ValueError, 'NFFT must be a power of 2' numRows, numCols = X.shape # zero pad if X is too short if numRows < NFFT: tmp = X X = zeros( (NFFT, numCols), X.typecode()) X[:numRows,:] = tmp del tmp numRows, numCols = X.shape # get all the columns of X that we are interested in by checking # the ij tuples seen = {} for i,j in ij: seen[i]=1; seen[j] = 1 allColumns = seen.keys() Ncols = len(allColumns) del seen # for real X, ignore the negative frequencies if X.typecode()==Complex: numFreqs = NFFT else: numFreqs = NFFT//2+1 # cache the FFT of every windowed, detrended NFFT length segement # of every channel. If preferSpeedOverMemory, cache the conjugate # as well windowVals = window(ones((NFFT,), X.typecode())) ind = range(0, numRows-NFFT+1, NFFT-noverlap) numSlices = len(ind) FFTSlices = {} FFTConjSlices = {} Pxx = {} slices = range(numSlices) normVal = norm(windowVals)**2 for iCol in allColumns: progressCallback(i/Ncols, 'Cacheing FFTs') Slices = zeros( (numSlices,numFreqs), Complex) for iSlice in slices: thisSlice = X[ind[iSlice]:ind[iSlice]+NFFT, iCol] thisSlice = windowVals*detrend(thisSlice) Slices[iSlice,:] = fft(thisSlice)[:numFreqs] FFTSlices[iCol] = Slices if preferSpeedOverMemory: FFTConjSlices[iCol] = conjugate(Slices) Pxx[iCol] = divide(mean(absolute(Slices)**2), normVal) del slices, ind, windowVals # compute the coherences and phases for all pairs using the # cached FFTs Cxy = {} Phase = {} count = 0 N = len(ij) for i,j in ij: count +=1 if count%10==0: progressCallback(count/N, 'Computing coherences') if preferSpeedOverMemory: Pxy = FFTSlices[i] * FFTConjSlices[j] else: Pxy = FFTSlices[i] * conjugate(FFTSlices[j]) if numSlices>1: Pxy = mean(Pxy) Pxy = divide(Pxy, normVal) Cxy[(i,j)] = divide(absolute(Pxy)**2, Pxx[i]*Pxx[j]) Phase[(i,j)] = arctan2(Pxy.imag, Pxy.real) freqs = Fs/NFFT*arange(numFreqs) return Cxy, Phase, freqs 00538 def hist(y, bins=10, normed=0): """ Return the histogram of y with bins equally sized bins. If bins is an array, use the bins. Return value is (n,x) where n is the count for each bin in x If normed is False, return the counts in the first element of the return tuple. If normed is True, return the probability density n/(len(y)*dbin) Credits: the Numeric 22 documentation """ if not iterable(bins): ymin, ymax = min(y), max(y) if ymin==ymax: ymin -= 0.5 ymax += 0.5 bins = linspace(ymin, ymax, bins) n = searchsorted(sort(y), bins) n = diff(concatenate([n, [len(y)]])) if normed: db = bins[1]-bins[0] return 1/(len(y)*db)*n, bins else: return n, bins def normpdf(x, *args): "Return the normal pdf evaluated at x; args provides mu, sigma" mu, sigma = args return 1/(sqrt(2*pi)*sigma)*exp(-0.5 * (1/sigma*(x - mu))**2) def levypdf(x, gamma, alpha): "Returm the levy pdf evaluated at x for params gamma, alpha" N = len(x) if N%2 != 0: raise ValueError, 'x must be an event length array; try\n' + \ 'x = linspace(minx, maxx, N), where N is even' dx = x[1]-x[0] Nyq = 1/(2*dx) f = 1/(N*dx)*arange(-N/2, N/2, typecode=Float) ind = concatenate([arange(N/2, N, typecode=Int), arange(N/2,typecode=Int)]) df = f[1]-f[0] cfl = exp(-gamma*absolute(2*pi*f)**alpha) px = fft(take(cfl,ind)*df).astype(Float) return take(px, ind) def find(condition): "Return the indices where condition is true" return nonzero(condition) def trapz(x, y): if len(x)!=len(y): raise ValueError, 'x and y must have the same length' if len(x)<2: raise ValueError, 'x and y must have > 1 element' return sum(0.5*diff(x)*(y[1:]+y[:-1])) 00617 def longest_contiguous_ones(x): """ return the indicies of the longest stretch of contiguous ones in x, assuming x is a vector of zeros and ones. """ if len(x)==0: return array([]) ind = find(x==0) if len(ind)==0: return arange(len(x)) if len(ind)==len(x): return array([]) y = zeros( (len(x)+2,), x.typecode()) y[1:-1] = x dif = diff(y) up = find(d == 1); dn = find(d == -1); ind = find( dn-up == max(dn - up)) ind = arange(take(up, ind), take(dn, ind)) return ind 00639 def longest_ones(x): """ return the indicies of the longest stretch of contiguous ones in x, assuming x is a vector of zeros and ones. If there are two equally long stretches, pick the first """ x = asarray(x) if len(x)==0: return array([]) #print 'x', x ind = find(x==0) if len(ind)==0: return arange(len(x)) if len(ind)==len(x): return array([]) y = zeros( (len(x)+2,), Int) y[1:-1] = x d = diff(y) #print 'd', d up = find(d == 1); dn = find(d == -1); #print 'dn', dn, 'up', up, ind = find( dn-up == max(dn - up)) # pick the first if iterable(ind): ind = ind[0] ind = arange(up[ind], dn[ind]) return ind 00669 def prepca(P, frac=0): """ Compute the principal components of P. P is a numVars x numObservations numeric array. frac is the minimum fraction of variance that a component must contain to be included Return value are Pcomponents : a num components x num observations numeric array Trans : the weights matrix, ie, Pcomponents = Trans*P fracVar : the fraction of the variance accounted for by each component returned """ U,s,v = svd(P) varEach = s**2/P.shape[1] totVar = sum(varEach) fracVar = divide(varEach,totVar) ind = int(sum(fracVar>=frac)) # select the components that are greater Trans = transpose(U[:,:ind]) # The transformed data Pcomponents = matrixmultiply(Trans,P) return Pcomponents, Trans, fracVar[:ind] # From MLab2: http://pdilib.sourceforge.net/MLab2.py readme = \ """ MLab2.py, release 1 Created on February 2003 by Thomas Wendler as part of the Emotionis Project. This script is supposed to implement Matlab functions that were left out in MLab.py (part of Numeric Python). For further information on the Emotionis Project or on this script, please contact their authors: Rodrigo Benenson, rodrigob at elo dot utfsm dot cl Thomas Wendler, thomasw at elo dot utfsm dot cl Look at: http://pdilib.sf.net for new releases. """ import Numeric, LinearAlgebra, MLab _eps_approx = 1e-13 00712 def fix(x): """ Rounds towards zero. x_rounded = fix(x) rounds the elements of x to the nearest integers towards zero. For negative numbers is equivalent to ceil and for positive to floor. """ dim = Numeric.shape(x) if MLab.rank(x)==2: y = Numeric.reshape(x,(1,dim[0]*dim[1]))[0] y = y.tolist() elif MLab.rank(x)==1: y = x else: y = [x] for i in range(len(y)): if y[i]>0: y[i] = Numeric.floor(y[i]) else: y[i] = Numeric.ceil(y[i]) if MLab.rank(x)==2: x = Numeric.reshape(y,dim) elif MLab.rank(x)==0: x = y[0] return x 00740 def rem(x,y): """ Remainder after division. rem(x,y) is equivalent to x - y.*fix(x./y) in case y is not zero. By convention, rem(x,0) returns None. We keep the convention by Matlab: "The input x and y must be real arrays of the same size, or real scalars." """ x,y = Numeric.asarray(x),Numeric.asarray(y) if Numeric.shape(x) == Numeric.shape(y) or Numeric.shape(y) == (): try: return x - y * fix(x/y) except OverflowError: return None print >>sys.stderr, 'Dimension error' return None 00758 def norm(x,y=2): """ Norm of a matrix or a vector according to Matlab. The description is taken from Matlab: For matrices... NORM(X) is the largest singular value of X, max(svd(X)). NORM(X,2) is the same as NORM(X). NORM(X,1) is the 1-norm of X, the largest column sum, = max(sum(abs((X)))). NORM(X,inf) is the infinity norm of X, the largest row sum, = max(sum(abs((X')))). NORM(X,'fro') is the Frobenius norm, sqrt(sum(diag(X'*X))). NORM(X,P) is available for matrix X only if P is 1, 2, inf or 'fro'. For vectors... NORM(V,P) = sum(abs(V).^P)^(1/P). NORM(V) = norm(V,2). NORM(V,inf) = max(abs(V)). NORM(V,-inf) = min(abs(V)). """ x = Numeric.asarray(x) if MLab.rank(x)==2: if y==2: return MLab.max(MLab.svd(x)[1]) elif y==1: return MLab.max(MLab.sum(Numeric.absolute((x)))) elif y=='inf': return MLab.max(MLab.sum(Numeric.absolute((MLab.transpose(x))))) elif y=='fro': return MLab.sqrt(MLab.sum(MLab.diag(Numeric.matrixmultiply(MLab.transpose(x),x)))) else: print >>sys.stderr, 'Second argument not permitted for matrices' return None else: if y == 'inf': return MLab.max(Numeric.absolute(x)) elif y == '-inf': return MLab.min(Numeric.absolute(x)) else: return Numeric.power(MLab.sum(Numeric.power(Numeric.absolute(x),y)),1/float(y)) 00803 def orth(A): """ Orthogonalization procedure by Matlab. The description is taken from its help: Q = ORTH(A) is an orthonormal basis for the range of A. That is, Q'*Q = I, the columns of Q span the same space as the columns of A, and the number of columns of Q is the rank of A. """ A = Numeric.array(A) U,S,V = MLab.svd(A) m,n = Numeric.shape(A) if m > 1: s = S elif m == 1: s = S[0] else: s = 0 tol = MLab.max((m,n)) * MLab.max(s) * _eps_approx r = MLab.sum(s > tol) Q = Numeric.take(U,range(r),1) return Q 00831 def rank(x): """ Returns the rank of a matrix. The rank is understood here as the an estimation of the number of linearly independent rows or columns (depending on the size of the matrix). Note that MLab.rank() is not equivalent to Matlab's rank. This function is! """ x = Numeric.asarray(x) u,s,v = MLab.svd(x) # maxabs = MLab.max(Numeric.absolute(s)) is also possible. maxabs = norm(x) maxdim = MLab.max(Numeric.shape(x)) tol = maxabs*maxdim*_eps_approx r = s>tol return MLab.sum(r) 00850 def sqrtm(x): """ Returns the square root of a square matrix. This means that s=sqrtm(x) implies s*s = x. Note that s and x are matrices. """ return mfuncC(MLab.sqrt, x) 00858 def mfuncC(f, x): """ mfuncC(f, x) : matrix function with possibly complex eigenvalues. Note: Numeric defines (v,u) = eig(x) => x*u.T = u.T * Diag(v) This function is needed by sqrtm and allows further functions. """ x = Numeric.array(x) (v, u) = MLab.eig(x) uT = MLab.transpose(u) V = MLab.diag(f(v+0j)) y = Numeric.matrixmultiply( uT, Numeric.matrixmultiply( V, LinearAlgebra.inverse(uT))) return approx_real(y) 00874 def approx_real(x): """ approx_real(x) : returns x.real if |x.imag| < |x.real| * _eps_approx. This function is needed by sqrtm and allows further functions. """ if MLab.max(MLab.max(Numeric.absolute(x.imag))) <= MLab.max(MLab.max(Numeric.absolute(x.real))) * _eps_approx: return x.real else: return x 00888 def prctile(x, p = (0.0, 25.0, 50.0, 75.0, 100.0)): """ Return the percentiles of x. p can either be a sequence of percentil values or a scalar. If p is a sequence the i-th element of the return sequence is the p(i)-th percentile of x """ x = sort(x) Nx = len(x) if not iterable(p): return x[int(p*Nx/100.0)] p = multiply(array(p), Nx/100.0) ind = p.astype(Int) ind = where(ind>=Nx, Nx-1, ind) return take(x, ind) 00906 def center_matrix(M, dim=0): """ Return the matrix M with each row having zero mean and unit std if dim=1, center columns rather than rows """ # todo: implement this w/o loop. Allow optional arg to specify # dimension to remove the mean from if dim==1: M = transpose(M) M = array(M, Float) if len(M.shape)==1 or M.shape[0]==1 or M.shape[1]==1: M = M-mean(M) sigma = std(M) if sigma>0: M = divide(M, sigma) if dim==1: M=transpose(M) return M for i in range(M.shape[0]): M[i] -= mean(M[i]) sigma = std(M[i]) if sigma>0: M[i] = divide(M[i], sigma) if dim==1: M=transpose(M) return M 00932 def meshgrid(x,y): """ For vectors x, y with lengths Nx=len(x) and Ny=len(y), return X, Y where X and Y are (Ny, Nx) shaped arrays with the elements of x and y repeated to fill the matrix EG, [X, Y] = meshgrid([1,2,3], [4,5,6,7]) X = 1 2 3 1 2 3 1 2 3 1 2 3 Y = 4 4 4 5 5 5 6 6 6 7 7 7 """ x = asarray(x) y = asarray(y) numRows, numCols = len(y), len(x) # yes, reversed x.shape = 1, numCols X = repeat(x, numRows) y.shape = numRows,1 Y = repeat(y, numCols, 1) return X, Y 00967 def rk4(derivs, y0, t): """ Integrate 1D or ND system of ODEs from initial state y0 at sample times t. derivs returns the derivative of the system and has the signature dy = derivs(yi, ti) Example 1 : ## 2D system # Numeric solution def derivs6(x,t): d1 = x[0] + 2*x[1] d2 = -3*x[0] + 4*x[1] return (d1, d2) dt = 0.0005 t = arange(0.0, 2.0, dt) y0 = (1,2) yout = rk4(derivs6, y0, t) Example 2: ## 1D system alpha = 2 def derivs(x,t): return -alpha*x + exp(-t) y0 = 1 yout = rk4(derivs, y0, t) """ try: Ny = len(y0) except TypeError: yout = zeros( (len(t),), Float) else: yout = zeros( (len(t), Ny), Float) yout[0] = y0 i = 0 for i in arange(len(t)-1): thist = t[i] dt = t[i+1] - thist dt2 = dt/2.0 y0 = yout[i] k1 = asarray(derivs(y0, thist)) k2 = asarray(derivs(y0 + dt2*k1, thist+dt2)) k3 = asarray(derivs(y0 + dt2*k2, thist+dt2)) k4 = asarray(derivs(y0 + dt*k3, thist+dt)) yout[i+1] = y0 + dt/6.0*(k1 + 2*k2 + 2*k3 + k4) return yout

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