변수가 여러개인 함수 예를 들어 f(d,t; c_0,c_1...) 인 경우에는
어떻게 fitting을 할까? 여기서, d,t는 변수이고, c_0,c_1..등은
fitting parameter 이다.
Suppose we have data which corresponds to
y=f[d,t]
Something like (d,t)=(0,0) gives y=0, (0,1) gives y=1....
We want to fit these data with certain fitting function
with parameters c1,c2,c3...
fit[d,t; c1,c2,c3]
Even in this case, we can use curve_fit
(Following example copied from http://stackoverflow.com/a/27096056/2775514
from scipy.optimize import curve_fit
import scipy
def fn(x, a, b, c):
return a + b*x[0] + c*x[1]
# y(x0,x1) data:
# x0=0 1 2
# ___________
# x1=0 |0 1 2
# x1=1 |1 2 3
# x1=2 |2 3 4
x = scipy.array([[0,1,2,0,1,2,0,1,2,],[0,0,0,1,1,1,2,2,2]])
y = scipy.array([0,1,2,1,2,3,2,3,4])
popt, pcov = curve_fit(fn, x, y)
print popt
In this example,
x[:, i-th], y[i-th] corresponds to i-th data set.
result returns popt -> [0,1,1].
One can use different form :
x = np.array([ [1.0, 1.0 ],[1.0,1.5],[2.0,0.5],[2.1,1.6],[0.5,0.5],[0.8,1.2] ] )
y = 0.5*x[:,0] + 2.0 *x[:,1]**2
def fitf(x,*paras):
return paras[0]*x[:,0]+paras[1]*x[:,1]**2
xx = np.array( [x[:,0],x[:,1]])
def testf(x,*para):
return para[0]*x[0]+x[1]**2*para[1]
print( curve_fit(fitf,x,y,p0=[1,1]) )
print( curve_fit(testf,xx,y,p0=[1,1]) )
Note that curve_fit calls the function as
ydata = f(xdata, *params) +eps
In other words, when xdata is an (k,M) array with M-data,
the function should return array ydata with M-data.
if function only works work for a scalar f(x), it cannot be used directly with curve_fit.
On the other hand, leastsq(cost, p0) assumes the
cost function returns a number or array of length of p0
#--------------------------------------------------------------------------------------------
scipy.optimize.curve_fit(f, xdata, ydata, p0=None, sigma=None, absolute_sigma=False, check_finite=True, **kw)[source]
Use non-linear least squares to fit a function, f, to data.
Assumes ydata = f(xdata, *params) + eps
| Parameters: |
f : callable
The model function, f(x, ...). It must take the independent variable as the first argument and the parameters to fit as separate remaining arguments.
xdata : An M-length sequence or an (k,M)-shaped array
for functions with k predictors. The independent variable where the data is measured.
ydata : M-length sequence
The dependent data — nominally f(xdata, ...)
p0 : None, scalar, or N-length sequence
Initial guess for the parameters. If None, then the initial values will all be 1 (if the number of parameters for the function can be determined using introspection, otherwise a ValueError is raised).
sigma : None or M-length sequence, optional
If not None, the uncertainties in the ydata array. These are used as weights in the least-squares problem i.e. minimising np.sum( ((f(xdata, *popt) -ydata) / sigma)**2 ) If None, the uncertainties are assumed to be 1.
absolute_sigma : bool, optional
If False, sigma denotes relative weights of the data points. The returned covariance matrix pcov is based on estimated errors in the data, and is not affected by the overall magnitude of the values in sigma. Only the relative magnitudes of the sigma values matter.
If True, sigma describes one standard deviation errors of the input data points. The estimated covariance in pcov is based on these values.
check_finite : bool, optional
If True, check that the input arrays do not contain nans of infs, and raise a ValueError if they do. Setting this parameter to False may silently produce nonsensical results if the input arrays do contain nans. Default is True.
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| Returns: |
popt : array
Optimal values for the parameters so that the sum of the squared error of f(xdata, *popt) - ydata is minimized
pcov : 2d array
The estimated covariance of popt. The diagonals provide the variance of the parameter estimate. To compute one standard deviation errors on the parameters use perr = np.sqrt(np.diag(pcov)).
How the sigma parameter affects the estimated covariance depends on absolute_sigma argument, as described above.
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| Raises: |
OptimizeWarning
if covariance of the parameters can not be estimated.
ValueError
if ydata and xdata contain NaNs.
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