Best estimation of a fitting parameter to measured data My goal is to estimate a parameter $\alpha_1 = (\alpha_{11}, \alpha_{12})$ which provides the best fit of certain measured data (a readout of some currents in a set of positions for a set of loads) to a hypothesized model.
After some hypothesis on the underlying data, the model to be fitted is represented by:
$$
I_{Ndc_{(1,k)}l} = \frac{I_{dc_{(k)}} - \alpha_{(1,k)} i_{ac_{(k)}}} {I_{dc_{(1)}} + I_{dc_{(2)}} - \alpha_{(1,1)} i_{ac_{(1)}} - \alpha_{(1,2)} i_{ac_{(2)}}  }
$$
where $k=1,2$. Therefore there are two functions, $I_{Ndc_{(1,1)}l}$ and $I_{Ndc_{(1,2)}l}$.
The measured data are the $I_{dc_{(k)}}$ and $i_{ac_{(k)}}$ and are represented by a $n\times m$ matrix, where each column is a different position and each row is a different load.
The idea is that there is a certain $\alpha_1 = (\alpha_{11}, \alpha_{12})$ which can make both $I_{Ndc_{(1,k)}l}$ independent of load variations (or independent to a variation of the 0 axis) for each position (or 1 axis).
Since I don't know how to formalize it and find the best estimator, I tried by intuition by finding (with python) the $\alpha_1 = (\alpha_{11}, \alpha_{12})$ which minimizes the unbiased variance (ddof=1) of $I_{Ndc_{(1,k)}l}$ in each position (means computing the variance on axis 0)
def res(alpha):
    alpha_11, alpha_12 = alpha
    INdc11l_meas = ((Idc1_meas - alpha_11 * iac1_meas)/
                    (Idc1_meas + Idc2_meas - alpha_11 * iac1_meas - alpha_12 * iac2_meas))

    INdc12l_meas = ((Idc2_meas - alpha_12 * iac2_meas)/
                    (Idc1_meas + Idc2_meas - alpha_11 * iac1_meas - alpha_12 * iac2_meas))

    var_INdc11l = sc.var(INdc11l_meas, axis=0, ddof=1) #Unbiased estimator ddof=1
    var_INdc12l = sc.var(INdc12l_meas, axis=0, ddof=1)

    return var_INdc11l + var_INdc12l

alpha_opt = optimize.minimize(res, [0,0])['x']

$I_{Ndc_{(1,k)}l}$ with alpha_opt results in superimposed lines (each line a different load):

What I am asking at this point is:


*

*Can this method be formalized in some way or have a particular meaning?

*Is there a better way to estimate $\alpha_1$?

*Is it possible to prove that it is the best estimator?

*What is the difference with minimizing the pairwise distance squared since it provides the same result? (changing the return statement with: return sc.sum(distance.pdist(INdc11l_meas, 'sqeuclidean')))


As a second step, with the $\alpha_1$ already known, it was needed to find a relation that can be inverted to find the position by measuring the currents $I_{dc_{(k)}}$ and $i_{ac_{(k)}}$. At this step they are not any more a matrix $n \times m$, but a scalar value, which provides a scalar $I_{Ndc_{(1,k)}l}$.
Again, by intuition, I supposed that by minimizing the variance, it is equivalent to minimize the distance of each point to the expectation and therefore the expectation to be a good candidate for that function
E = sc.mean(INdc11l_meas, axis=0)
f = sc.interpolate.interp1d(position_meas.T[:,0], E.T, 'cubic')
f_inv11 = lambda I: sc.optimize.root(lambda p,I: f(p) - I, [10], (I))

# From here Idc1_meas, Idc2_meas, iac1_meas, iac2_meas are scalars and instantaneous measurements
def find_pos11(Idc1_meas, Idc2_meas, iac1_meas, iac2_meas, alpha_1 = alpha_opt, f_inv11 = f_inv11):
    return f_inv11(((Idc1_meas - alpha_11 * iac1_meas)/(Idc1_meas + Idc2_meas - alpha_1[0]* iac1_meas - alpha_1[1] * iac2_meas)))

The expectation is in yellow and the interpolated function in black:

What I am asking at this point is:


*

*Does it make sense?

*Is there a better way to estimate find_pos11?

*Is it possible to prove that it is the best available or to find which is the best?


Sorry in advance for my lack of rigour.
 A: Per the comments, here is an example Python surface fitter using
non-linear curve fitting with 3D scatter plot, 3D surface plot,
and contour plot. This example uses manually supplied initial
parameter estimates, but the scipy differential_evolution module
has a genetic algorithm to help determine these initial
estimates and I can give an example of that also.
import numpy, scipy, scipy.optimize
import matplotlib
from mpl_toolkits.mplot3d import  Axes3D
from matplotlib import cm # to colormap 3D surfaces from blue to red
import matplotlib.pyplot as plt

graphWidth = 800 # units are pixels
graphHeight = 600 # units are pixels

# 3D contour plot lines
numberOfContourLines = 16


def SurfacePlot(func, data, fittedParameters):
    f = plt.figure(figsize=(graphWidth/100.0, graphHeight/100.0), dpi=100)

    matplotlib.pyplot.grid(True)
    axes = Axes3D(f)

    x_data = data[0]
    y_data = data[1]
    z_data = data[2]

    xModel = numpy.linspace(min(x_data), max(x_data), 20)
    yModel = numpy.linspace(min(y_data), max(y_data), 20)
    X, Y = numpy.meshgrid(xModel, yModel)

    Z = func(numpy.array([X, Y]), *fittedParameters)

    axes.plot_surface(X, Y, Z, rstride=1, cstride=1, cmap=cm.coolwarm, linewidth=1, antialiased=True)

    axes.scatter(x_data, y_data, z_data) # show data along with plotted surface

    axes.set_title('Surface Plot (click-drag with mouse)') # add a title for surface plot
    axes.set_xlabel('X Data') # X axis data label
    axes.set_ylabel('Y Data') # Y axis data label
    axes.set_zlabel('Z Data') # Z axis data label

    plt.show()
    plt.close('all') # clean up after using pyplot or else thaere can be memory and process problems


def ContourPlot(func, data, fittedParameters):
    f = plt.figure(figsize=(graphWidth/100.0, graphHeight/100.0), dpi=100)
    axes = f.add_subplot(111)

    x_data = data[0]
    y_data = data[1]
    z_data = data[2]

    xModel = numpy.linspace(min(x_data), max(x_data), 20)
    yModel = numpy.linspace(min(y_data), max(y_data), 20)
    X, Y = numpy.meshgrid(xModel, yModel)

    Z = func(numpy.array([X, Y]), *fittedParameters)

    axes.plot(x_data, y_data, 'o')

    axes.set_title('Contour Plot') # add a title for contour plot
    axes.set_xlabel('X Data') # X axis data label
    axes.set_ylabel('Y Data') # Y axis data label

    CS = matplotlib.pyplot.contour(X, Y, Z, numberOfContourLines, colors='k')
    matplotlib.pyplot.clabel(CS, inline=1, fontsize=10) # labels for contours

    plt.show()
    plt.close('all') # clean up after using pyplot or else thaere can be memory and process problems


def ScatterPlot(data):
    f = plt.figure(figsize=(graphWidth/100.0, graphHeight/100.0), dpi=100)

    matplotlib.pyplot.grid(True)
    axes = Axes3D(f)
    x_data = data[0]
    y_data = data[1]
    z_data = data[2]

    axes.scatter(x_data, y_data, z_data)

    axes.set_title('Scatter Plot (click-drag with mouse)')
    axes.set_xlabel('X Data')
    axes.set_ylabel('Y Data')
    axes.set_zlabel('Z Data')

    plt.show()
    plt.close('all') # clean up after using pyplot or else thaere can be memory and process problems


def func(data, a, alpha, beta):
    x = data[0]
    y = data[1]
    return a * (x**alpha) * (y**beta)


if __name__ == "__main__":
    xData = numpy.array([1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0])
    yData = numpy.array([11.0, 12.1, 13.0, 14.1, 15.0, 16.1, 17.0, 18.1, 90.0])
    zData = numpy.array([1.1, 2.2, 3.3, 4.4, 5.5, 6.6, 7.7, 8.0, 9.9])

    data = [xData, yData, zData]

    # these are the same as scipy default values in this example
    # if you do not have initial parameter estimates, they can
    # be estimated using scipy's differential_evolution genetic
    # algorithm module, I can supply an example if needed
    initialParameters = [1.0, 1.0, 1.0]

    # here a non-linear surface fit is made with scipy's curve_fit()
    fittedParameters, pcov = scipy.optimize.curve_fit(func, [xData, yData], zData, p0 = initialParameters)

    ScatterPlot(data)
    SurfacePlot(func, data, fittedParameters)
    ContourPlot(func, data, fittedParameters)

    print('fitted prameters', fittedParameters)

    modelPredictions = func(data, *fittedParameters) 

    absError = modelPredictions - zData

    SE = numpy.square(absError) # squared errors
    MSE = numpy.mean(SE) # mean squared errors
    RMSE = numpy.sqrt(MSE) # Root Mean Squared Error, RMSE
    Rsquared = 1.0 - (numpy.var(absError) / numpy.var(zData))
    print('RMSE:', RMSE)
    print('R-squared:', Rsquared)

A: It seems that minimizing the ensemble variance (as an objective function) is equivalent to minimizing the pair-wise distances between the variables
$$\text{FullSimplify}\left(\sum _{i=1}^N \left(y_i-\frac{\sum _{j=1}^N y_j}{N}\right)^2-\frac{\sum _{i=1}^N \sum _{j=i}^N \left(y_i-y_j\right){}^2}{N}\text{/.}\, \{N\to 20\}\right)=0$$
