Struggling with Python extrapolation

Question

I have a recorded data from lab equipment. In several cases I would like to interpolate and extrapolate from recorded data.

I will be using I_id and I_iq as my main control variables.

I have tried many different variations to get this working but I cannot.

My data looks like this for the first piece of measurement equipment:

I_id = [0,-25,-50,-75,-100,-125,-150,-175,0,-25,-50,-75,-100,-125,-150,-175,0,-25,-50,-75,-100,-125,-150,-175,0,-25,-50,-75,-100,-125,-150,-175,0,-25,-50,-75,-100,-125,-150,0,-25,-50,-75,-100,-125,-150,0,-25,-50,-75,-100,-125,0,-25,-50,-75,0,0,-25,-50,-75,-100,-125,-150,-175]
I_iq = [0,0,0,0,0,0,0,0,25,25,25,25,25,25,25,25,50,50,50,50,50,50,50,50,75,75,75,75,75,75,75,75,100,100,100,100,100,100,100,125,125,125,125,125,125,125,150,150,150,150,150,150,175,175,175,175,200,0,0,0,0,0,0,0,0]
var = [-0.040032,0.011188,0.030851,0.183906,0.258842,0.355956,0.560895,0.753436,3.325974,11.611581,12.113206,12.804795,13.11953,13.423358,13.689702,13.899162,17.267299,23.553225,24.495611,25.086743,25.559352,25.953261,26.248565,26.534781,34.935503,35.761774,36.52968,37.227405,37.834295,38.310515,38.715564,38.944562,46.322635,47.382142,48.31467,49.163737,49.897316,50.510074,50.936424,57.367325,58.686137,59.86712,60.871714,61.727998,62.407764,62.902043,68.254704,69.745637,71.075856,72.232987,73.282945,74.110145,78.724496,80.425047,81.965227,83.270788,88.79109,69.950271,1.538601,0.005484,0.160758,0.336944,0.44188,0.568149,0.825262]

I_id	I_iq	var
0	0	0
1	-25	0
2	-50	0
3	-75	0
4	-100	0
5	-125	0
6	-150	0
7	-175	0
8	0	25
9	-25	25
10	-50	25
11	-75	25
12	-100	25
13	-125	25
14	-150	25
15	-175	25
16	0	50
17	-25	50
18	-50	50
19	-75	50
20	-100	50
21	-125	50
22	-150	50
23	-175	50
24	0	75
25	-25	75
26	-50	75
27	-75	75
28	-100	75
29	-125	75
30	-150	75
31	-175	75
32	0	100
33	-25	100
34	-50	100
35	-75	100
36	-100	100
37	-125	100
38	-150	100
39	0	125
40	-25	125
41	-50	125
42	-75	125
43	-100	125
44	-125	125
45	-150	125
46	0	150
47	-25	150
48	-50	150
49	-75	150
50	-100	150
51	-125	150
52	0	175
53	-25	175
54	-50	175
55	-75	175
56	0	200
57	0	0
58	-25	0
59	-50	0
60	-75	0
61	-100	0
62	-125	0
63	-150	0
64	-175	0

I have tried creating a meshgrid:

for var in target_variables:

    x = idq_data["Id"]
    y = idq_data["Iq"]
    z = idq_data[var]
    
    # Create a grid of data for the target variables
    xi = np.arange(np.min(x), np.max(x), 1)
    yi = np.arange(np.min(y), np.max(y), 1)
    xx, yy = np.meshgrid(xi, yi)

    # Interpolation function using gridata
    zi = griddata((x, y), z, (xx, yy), method='cubic')

However, when using griddata when I plot the extrapolated points do not exist.

    # plot the interpolated data on a contour plot
    fig = go.Figure()
    
    fig.add_trace(go.Contour(
        x=xi,
        y=yi,
        z=zi,
        colorscale='Jet',
    ))
    
    fig.show()

I understand they have not been plotted as they are NaN's but why has the values not been extrapolated?

Answer 1

You can tinker with the parameters of the Radial Basis method:

xi = np.arange(I_id.min(), 1+I_id.max())
yi = np.arange(I_iq.min(), 1+I_iq.max())
xgrid, ygrid = np.meshgrid(xi, yi)

interpolator = scipy.interpolate.RBFInterpolator(
    y=np.stack((I_id, I_iq), axis=1),
    d=var,
    smoothing=0.1,
)
zi = interpolator(np.stack((xgrid, ygrid)).reshape((-1, 2))).reshape(xgrid.shape)

but it doesn't work very well. Instead I will recommend a linear regression over (most of) your data:

'''
[ Id Iq 1 ] [ a ] = [ var ]
[ Id Iq 1 ] [ b ]   [ var ]
[ ...   1 ] [ c ]   [ ... ]
'''
linear_region = I_id <= -25
affine = np.stack((I_id, I_iq, np.ones_like(I_id)), axis=1)[linear_region, :]
params, *_ = np.linalg.lstsq(affine, var[linear_region])
print(params)

di = np.arange(I_id.min(), 1+I_id.max())[np.newaxis, :]
qi = np.arange(I_iq.min(), 1+I_iq.max())[:, np.newaxis]
vi = params[0]*di + params[1]*qi + params[2]

fig, ax = plt.subplots()
graph: matplotlib.contour.QuadContourSet = ax.contourf(di.ravel(), qi.ravel(), vi)
bar: matplotlib.colorbar.Colorbar = fig.colorbar(graph)
bar.set_label('var')
ax.set_xlabel('I_id')
ax.set_ylabel('I_iq')
plt.show()

producing planar parameters

[-0.01984498  0.47951953 -1.0313259 ]

Don't use the jet colormap.