Python's curve_fit calculates the best-fit parameters for a function with a single independent variable, but is there a way, using curve_fit or something else, to fit for a function with multiple independent variables? For example:
def func(x, y, a, b, c): return log(a) + b*log(x) + c*log(y) where x and y are the independent variable and we would like to fit for a, b, and c.
You can pass curve_fit a multi-dimensional array for the independent variables, but then your func must accept the same thing. For example, calling this array X and unpacking it to x, y for clarity:
import numpy as np from scipy.optimize import curve_fit def func(X, a, b, c): x,y = X return np.log(a) + b*np.log(x) + c*np.log(y) # some artificially noisy data to fit x = np.linspace(0.1,1.1,101) y = np.linspace(1.,2., 101) a, b, c = 10., 4., 6. z = func((x,y), a, b, c) * 1 + np.random.random(101) / 100 # initial guesses for a,b,c: p0 = 8., 2., 7. print(curve_fit(func, (x,y), z, p0)) Gives the fit:
(array([ 9.99933937, 3.99710083, 6.00875164]), array([[ 1.75295644e-03, 9.34724308e-05, -2.90150983e-04], [ 9.34724308e-05, 5.09079478e-06, -1.53939905e-05], [ -2.90150983e-04, -1.53939905e-05, 4.84935731e-05]])) 
func represents a bivariate function (taking two independent variables), so for the fit it should be defined as giving a result f(x_i,y_i) for any provided input values x_i and y_i. If x and y don't have the same size then you're trying to evaluate it e.g. at some x but with y undefined which surely can't be done.x,y = np.mesgrid(x,y) and then np.stack((x,y), axis=2).reshape(-1, 2) to get a (600,2) array which will be xdata, with all 600 x and y combinations. Then I flattened by data from the 600 samples to a 1d (600,) array which will be ydata instead of a 2d (20, 30) array. And then you can just unpack your your data in func with x, y = np.hsplit(X, 2). (X is xdata)
X contains various types of variables. Here, I have x0, x4, y0, mylist. The curve_fit seems to unpack these under the hood into numpy arrays but runs into an error with mylist. I get: "ValueError: setting an array element with a sequence. The requested array has an inhomogeneous shape after 1 dimensions. The detected shape was (4,) + inhomogeneous part.". Any idea on how this can be treated? Surprisingly, also, leastsq seems to work & I thought curve_fit is simply a wrapper around this?This example shows how to fit a polynomial with a two dimensional input (R^2 -> R) by an increasing number of coefficients. The design is very flexible so that the callable f from curve_fit is defined once for any number of non-keyword arguments.
minimal reproducible example
import numpy as np import matplotlib.pyplot as plt from scipy.optimize import curve_fit def poly2d(xy, *coefficients): x = xy[:, 0] y = xy[:, 1] proj = x + y res = 0 for order, coef in enumerate(coefficients): res += coef * proj ** order return res nx = 31 ny = 21 range_x = [-1.5, 1.5] range_y = [-1, 1] target_coefficients = (3, 0, -19, 7) xs = np.linspace(*range_x, nx) ys = np.linspace(*range_y, ny) im_x, im_y = np.meshgrid(xs, ys) xdata = np.c_[im_x.flatten(), im_y.flatten()] im_target = poly2d(xdata, *target_coefficients).reshape(ny, nx) fig, axs = plt.subplots(2, 3, figsize=(29.7, 21)) axs = axs.flatten() ax = axs[0] ax.set_title('Unknown polynomial P(x+y)\n[secret coefficients: ' + str(target_coefficients) + ']') sm = ax.imshow( im_target, cmap = plt.get_cmap('coolwarm'), origin='lower' ) fig.colorbar(sm, ax=ax) for order in range(5): ydata=im_target.flatten() popt, pcov = curve_fit(poly2d, xdata=xdata, ydata=ydata, p0=[0]*(order+1) ) im_fit = poly2d(xdata, *popt).reshape(ny, nx) ax = axs[1+order] title = 'Fit O({:d}):'.format(order) for o, p in enumerate(popt): if o%2 == 0: title += '\n' if o == 0: title += ' {:=-{w}.1f} (x+y)^{:d}'.format(p, o, w=int(np.log10(max(abs(p), 1))) + 5) else: title += ' {:=+{w}.1f} (x+y)^{:d}'.format(p, o, w=int(np.log10(max(abs(p), 1))) + 5) title += '\nrms: {:.1f}'.format( np.mean((im_fit-im_target)**2)**.5 ) ax.set_title(title) sm = ax.imshow( im_fit, cmap = plt.get_cmap('coolwarm'), origin='lower' ) fig.colorbar(sm, ax=ax) for ax in axs.flatten(): ax.set_xlabel('x') ax.set_ylabel('y') plt.show() P.S. The concept of this answer is identical to my other answer here, but the code example is way more clear. At the time given, I will delete the other answer.
Fitting to an unknown numer of parameters
In this example, we try to reproduce some measured data measData. In this example measData is generated by the function measuredData(x, a=.2, b=-2, c=-.8, d=.1). I practice, we might have measured measData in a way - so we have no idea, how it is described mathematically. Hence the fit.
We fit by a polynomial, which is described by the function polynomFit(inp, *args). As we want to try out different orders of polynomials, it is important to be flexible in the number of input parameters. The independent variables (x and y in your case) are encoded in the 'columns'/second dimension of inp.
import numpy as np import matplotlib import matplotlib.pyplot as plt from scipy.optimize import curve_fit def measuredData(inp, a=.2, b=-2, c=-.8, d=.1): x=inp[:,0] y=inp[:,1] return a+b*x+c*x**2+d*x**3 +y def polynomFit(inp, *args): x=inp[:,0] y=inp[:,1] res=0 for order in range(len(args)): print(14,order,args[order],x) res+=args[order] * x**order return res +y inpData=np.linspace(0,10,20).reshape(-1,2) inpDataStr=['({:.1f},{:.1f})'.format(a,b) for a,b in inpData] measData=measuredData(inpData) fig, ax = plt.subplots() ax.plot(np.arange(inpData.shape[0]), measData, label='measuered', marker='o', linestyle='none' ) for order in range(5): print(27,inpData) print(28,measData) popt, pcov = curve_fit(polynomFit, xdata=inpData, ydata=measData, p0=[0]*(order+1) ) fitData=polynomFit(inpData,*popt) ax.plot(np.arange(inpData.shape[0]), fitData, label='polyn. fit, order '+str(order), linestyle='--' ) ax.legend( loc='upper left', bbox_to_anchor=(1.05, 1)) print(order, popt) ax.set_xticklabels(inpDataStr, rotation=90) Result:
Yes. We can pass multiple variables for curve_fit. I have written a piece of code:
import numpy as np x = np.random.randn(2,100) w = np.array([1.5,0.5]).reshape(1,2) esp = np.random.randn(1,100) y = np.dot(w,x)+esp y = y.reshape(100,) In the above code I have generated x a 2D data set in shape of (2,100) i.e, there are two variables with 100 data points. I have fit the dependent variable y with independent variables x with some noise.
def model_func(x,w1,w2,b): w = np.array([w1,w2]).reshape(1,2) b = np.array([b]).reshape(1,1) y_p = np.dot(w,x)+b return y_p.reshape(100,) We have defined a model function that establishes relation between y & x.
Note: The shape of output of the model function or predicted y should be (length of x,)
popt, pcov = curve_fit(model_func,x,y) The popt is an 1D numpy array containing predicted parameters. In our case there are 3 parameters.
Yes, there is: simply give curve_fit a multi-dimensional array for xData.
z=[x,y], so that x=z[0] and y=z[1]. But curve_fit doesn't seem to like that and gives me an error: TypeError: unsupported operand type(s) for /: 'list' and 'float'
curve_fit