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POPython parameter transformation according to MINUIT
 

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  1. POPython parameter transformation according to MINUIT
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    <p>I am writing an automated curve fitting routine for 2D data based on scipy's optimize.leastsq, and it works. However when adding many curves with starting values slighly off I get non-physical results (negative amplitude, for example).</p> <p>I found this post <a href="https://stackoverflow.com/questions/7409694/scipy-bounds-for-fitting-parameters-when-using-optimize-leastsq">Scipy: bounds for fitting parameter(s) when using optimize.leastsq</a> and was trying to use the parameter transformation according to Minuit from Cern. In the above mentioned question somebody provided a link to some python code.</p> <pre><code>code.google.com/p/nmrglue/source/browse/trunk/nmrglue/analysis/leastsqbound.py </code></pre> <p>I wrote this minimal working example (extending the code)</p> <pre><code>""" http://code.google.com/p/nmrglue/source/browse/trunk/nmrglue/analysis/leastsqbound.py Constrained multivariate Levenberg-Marquardt optimization """ from scipy.optimize import leastsq import numpy as np import matplotlib.pyplot as plt #new def internal2external_grad(xi, bounds): """ Calculate the internal to external gradiant Calculates the partial of external over internal """ ge = np.empty_like(xi) for i, (v, bound) in enumerate(zip(xi, bounds)): a = bound[0] # minimum b = bound[1] # maximum if a == None and b == None: # No constraints ge[i] = 1.0 elif b == None: # only min ge[i] = v / np.sqrt(v ** 2 + 1) elif a == None: # only max ge[i] = -v / np.sqrt(v ** 2 + 1) else: # both min and max ge[i] = (b - a) * np.cos(v) / 2. return ge def i2e_cov_x(xi, bounds, cov_x): grad = internal2external_grad(xi, bounds) grad = grad = np.atleast_2d(grad) return np.dot(grad.T, grad) * cov_x def internal2external(xi, bounds): """ Convert a series of internal variables to external variables""" xe = np.empty_like(xi) for i, (v, bound) in enumerate(zip(xi, bounds)): a = bound[0] # minimum b = bound[1] # maximum if a == None and b == None: # No constraints xe[i] = v elif b == None: # only min xe[i] = a - 1. + np.sqrt(v ** 2. + 1.) elif a == None: # only max xe[i] = b + 1. - np.sqrt(v ** 2. + 1.) else: # both min and max xe[i] = a + ((b - a) / 2.) * (np.sin(v) + 1.) return xe def external2internal(xe, bounds): """ Convert a series of external variables to internal variables""" xi = np.empty_like(xe) for i, (v, bound) in enumerate(zip(xe, bounds)): a = bound[0] # minimum b = bound[1] # maximum if a == None and b == None: # No constraints xi[i] = v elif b == None: # only min xi[i] = np.sqrt((v - a + 1.) ** 2. - 1) elif a == None: # only max xi[i] = np.sqrt((b - v + 1.) ** 2. - 1) else: # both min and max xi[i] = np.arcsin((2.*(v - a) / (b - a)) - 1.) return xi def err(p, bounds, efunc, args): pe = internal2external(p, bounds) # convert to external variables return efunc(pe, *args) def calc_cov_x(infodic, p): """ Calculate cov_x from fjac, ipvt and p as is done in leastsq """ fjac = infodic['fjac'] ipvt = infodic['ipvt'] n = len(p) # adapted from leastsq function in scipy/optimize/minpack.py perm = np.take(np.eye(n), ipvt - 1, 0) r = np.triu(np.transpose(fjac)[:n, :]) R = np.dot(r, perm) try: cov_x = np.linalg.inv(np.dot(np.transpose(R), R)) except LinAlgError: cov_x = None return cov_x def leastsqbound(func, x0, bounds, args = (), **kw): """ Constrained multivariant Levenberg-Marquard optimization Minimize the sum of squares of a given function using the Levenberg-Marquard algorithm. Contraints on parameters are inforced using variable transformations as described in the MINUIT User's Guide by Fred James and Matthias Winkler. Parameters: * func functions to call for optimization. * x0 Starting estimate for the minimization. * bounds (min,max) pair for each element of x, defining the bounds on that parameter. Use None for one of min or max when there is no bound in that direction. * args Any extra arguments to func are places in this tuple. Returns: (x,{cov_x,infodict,mesg},ier) Return is described in the scipy.optimize.leastsq function. x and con_v are corrected to take into account the parameter transformation, infodic is not corrected. Additional keyword arguments are passed directly to the scipy.optimize.leastsq algorithm. """ # check for full output if "full_output" in kw and kw["full_output"]: full = True else: full = False # convert x0 to internal variables i0 = external2internal(x0, bounds) # perfrom unconstrained optimization using internal variables r = leastsq(err, i0, args = (bounds, func, args), **kw) # unpack return convert to external variables and return if full: xi, cov_xi, infodic, mesg, ier = r xe = internal2external(xi, bounds) cov_xe = i2e_cov_x(xi, bounds, cov_xi) # XXX correct infodic 'fjac','ipvt', and 'qtf' return xe, cov_xe, infodic, mesg, ier else: xi, ier = r xe = internal2external(xi, bounds) return xe, ier # new below def _evaluate(x, p): ''' Linear plus Lorentzian curve p = list with three parameters ([a, b, I, Pos, FWHM]) ''' return p[0] + p[1] * x + p[2] / (1 + np.power((x - p[3]) / (p[4] / 2), 2)) def residuals(p, y, x): err = _evaluate(x, p) - y return err if __name__ == '__main__': data = np.loadtxt('constraint.dat') # read data p0 = [5000., 0., 500., 2450., 3] #Start values for a, b, I, Pos, FWHM constraints = [(4000., None), (-50., 20.), (0., 2000.), (2400., 2451.), (None, None)] p, res = leastsqbound(residuals, p0, constraints, args = (data[:, 1], data[:, 0]), maxfev = 20000) print p, res plt.plot(data[:, 0], data[:, 1]) # plot data plt.plot(data[:, 0], _evaluate(data[:, 0], p0)) # plot start values plt.plot(data[:, 0], _evaluate(data[:, 0], p)) # plot fit values plt.show() </code></pre> <p>Thats the plot output, where green is the starting conditions and red the fit result: <img src="https://i.stack.imgur.com/VisvI.png" alt="plot results"></p> <p>Is this the correct usage? External2internal conversion just throws a nan if outside the bounds. leastsq seems to be able to handle this?</p> <p>I uploaded the fitting data <a href="http://pastebin.com/cn6kR6Lf" rel="nofollow noreferrer">here</a>. Just paste into a text file named constraint.dat.</p>
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