最終的に、目標は、モデルと観測値の差の二乗和の絶対値を減らすことですZ
。
abs(((model(w, *params) - Z)**2).sum())
私の最初の答えは、実数と虚数の差の二乗和を表すスカラーを返す関数に
適用leastsq
することを提案しました。residuals
def residuals(params, w, Z):
R, C, L = params
diff = model(w, R, C, L) - Z
return diff.real**2 + diff.imag**2
Mike Sulzer は、float のベクトルを返すresidual 関数を使用することを提案しました。
これらの残差関数を使用した結果の比較は次のとおりです。
from __future__ import print_function
import random
import numpy as np
import scipy.optimize as optimize
j = 1j
def model1(w, R, C, L):
Z = 1.0/(1.0/R + j*w*C) + j*w*L
return Z
def model2(w, R, C, L):
Z = 1.0/(1.0/R + j*w*C) + j*w*L
# make Z non-contiguous and of a different complex dtype
Z = np.repeat(Z, 2)
Z = Z[::2]
Z = Z.astype(np.complex64)
return Z
def make_data(R, C, L):
N = 10000
w = np.linspace(0.1, 2, N)
Z = model(w, R, C, L) + 0.1*(np.random.random(N) + j*np.random.random(N))
return w, Z
def residuals(params, w, Z):
R, C, L = params
diff = model(w, R, C, L) - Z
return diff.real**2 + diff.imag**2
def MS_residuals(params, w, Z):
"""
https://stackoverflow.com/a/20104454/190597 (Mike Sulzer)
"""
R, C, L = params
diff = model(w, R, C, L) - Z
z1d = np.zeros(Z.size*2, dtype=np.float64)
z1d[0:z1d.size:2] = diff.real
z1d[1:z1d.size:2] = diff.imag
return z1d
def alt_residuals(params, w, Z):
R, C, L = params
diff = model(w, R, C, L) - Z
return diff.astype(np.complex128).view(np.float64)
def compare(*funcs):
fmt = '{:15} | {:37} | {:17} | {:6}'
header = fmt.format('name', 'params', 'sum(residuals**2)', 'ncalls')
print('{}\n{}'.format(header, '-'*len(header)))
fmt = '{:15} | {:37} | {:17.2f} | {:6}'
for resfunc in funcs:
# params, cov = optimize.leastsq(resfunc, p_guess, args=(w, Z))
params, cov, infodict, mesg, ier = optimize.leastsq(
resfunc, p_guess, args=(w, Z),
full_output=True)
ssr = abs(((model(w, *params) - Z)**2).sum())
print(fmt.format(resfunc.__name__, params, ssr, infodict['nfev']))
print(end='\n')
R, C, L = 3, 2, 4
p_guess = 1, 1, 1
seed = 2013
model = model1
np.random.seed(seed)
w, Z = make_data(R, C, L)
assert np.allclose(model1(w, R, C, L), model2(w, R, C, L))
print('Using model1')
compare(residuals, MS_residuals, alt_residuals)
model = model2
print('Using model2')
compare(residuals, MS_residuals, alt_residuals)
収量
Using model1
name | params | sum(residuals**2) | ncalls
------------------------------------------------------------------------------------
residuals | [ 2.86950167 1.94245378 4.04362841] | 9.41 | 89
MS_residuals | [ 2.85311972 1.94525477 4.04363883] | 9.26 | 29
alt_residuals | [ 2.85311972 1.94525477 4.04363883] | 9.26 | 29
Using model2
name | params | sum(residuals**2) | ncalls
------------------------------------------------------------------------------------
residuals | [ 2.86590332 1.9326829 4.0450271 ] | 7.81 | 483
MS_residuals | [ 2.85422448 1.94853383 4.04333851] | 9.78 | 754
alt_residuals | [ 2.85422448 1.94853383 4.04333851] | 9.78 | 754
したがって、どの残差関数を使用するかは、モデル関数に依存するようです。model1
と の類似性を
考えると、結果の違いを説明するのに途方に暮れていmodel2
ます。