Complex to real

Complex-valued linear problems can be reformulated as real-valued problems by splitting out the real and imaginary parts of the equations.

\[a + ib = C_1 (x + iy) .\]

Remember that we are solving for \(C_1 = (c_{1r} + c_{1i})\), which is also complex-valued, therefore,

\[a + ib = (c_{1r} + c_{1i}) (x + iy) .\]

This can be expanded out

\[\begin{split}a + ib &= (c_{1r} + ic_{1i}) (x + iy) \\ &= c_{1r} x - c_{1i} y + i c_{1r} y + i c_{1i} x \\ &= (c_{1r} x - c_{1i} y) + i (c_{1r} y + i c_{1i} x) ,\end{split}\]

which gives,

\[\begin{split}a &= c_{1r} x - c_{1i} y \\ b &= c_{1r} y + i c_{1i} x .\end{split}\]

For the complex-valued problem, the aim is to solve for \(C_1\). Making this real-valued means we are solving for \(c_{1r}\) and \(c_{1i}\).

Moving from complex-valued to real-valued results in the following

• Doubling the number of observations as the real and imaginary parts of the observations are split up

• Doubling the number of regressors as we are now solving for the real and imaginary component of each regressor explicitly

import numpy as np
import matplotlib.pyplot as plt
from regressioninc.linear import add_intercept, LeastSquares, M_estimate
from regressioninc.testing.complex import ComplexGrid, generate_linear_grid
from regressioninc.testing.complex import add_gaussian_noise
from regressioninc.testing.complex import add_outliers, plot_complex
from regressioninc.linear import complex_to_glr, glr_coef_to_complex

np.random.seed(42)

Let’s setup another linear regression problem with complex values

coef = np.array([0.5 + 2j, -3 - 1j])
grid_r1 = ComplexGrid(r1=0, r2=10, nr=11, i1=-5, i2=5, ni=11)
grid_r2 = ComplexGrid(r1=-25, r2=-5, nr=11, i1=-5, i2=5, ni=11)
X, y = generate_linear_grid(coef, [grid_r1, grid_r2], intercept=20 + 20j)

fig = plot_complex(X, y, {})
fig.set_size_inches(7, 6)
plt.tight_layout()
plt.show()

Add high leverage points to our regressors

seeds = [22, 36]
for ireg in range(X.shape[1]):
    np.random.seed(seeds[ireg])
    X[:, ireg] = add_outliers(
        X[:, ireg],
        outlier_percent=20,
        mult_min=7,
        mult_max=10,
        random_signs_real=True,
        random_signs_imag=True,
    )
np.random.seed(42)

intercept = 20 + 20j
y = np.matmul(X, coef) + intercept

fig = plot_complex(X, y, {})
fig.set_size_inches(7, 6)
plt.tight_layout()
plt.show()

Solve

X = add_intercept(X)
model = LeastSquares()
model.fit(X, y)
for idx, coef in enumerate(model.coef):
    print(f"Coefficient {idx}: {coef:.6f}")

Coefficient 0: 0.500000+2.000000j
Coefficient 1: -3.000000-1.000000j
Coefficient 2: 20.000000+20.000000j

Add some outliers

y_noise = add_gaussian_noise(y, loc=(0, 0), scale=(21, 21))
model_ls = LeastSquares()
model_ls.fit(X, y_noise)
for idx, coef in enumerate(model_ls.coef):
    print(f"Coefficient {idx}: {coef:.6f}")

fig = plot_complex(X, y_noise, {"least squares": model_ls}, y_orig=y)
fig.set_size_inches(7, 9)
plt.tight_layout()
plt.show()

Coefficient 0: 0.504175+1.992051j
Coefficient 1: -3.002991-1.001990j
Coefficient 2: 19.634988+20.233521j

Add some outliers

y_noise = add_gaussian_noise(y, loc=(0, 0), scale=(21, 21))
model_mest = M_estimate()
model_mest.fit(X, y_noise)
for idx, coef in enumerate(model_mest.coef):
    print(f"Coefficient {idx}: {coef:.6f}")

fig = plot_complex(
    X, y_noise, {"least squares": model_ls, "M_estimate": model_mest}, y_orig=y
)
fig.set_size_inches(7, 9)
plt.tight_layout()
plt.show()

Coefficient 0: 0.503769+1.990233j
Coefficient 1: -3.005947-1.000431j
Coefficient 2: 20.021505+19.804239j

Try running as a real-valued problem

X_real, y_real = complex_to_glr(X, y_noise)
model_ls = LeastSquares()
model_ls.fit(X_real, y_real)
coef = glr_coef_to_complex(model_ls.coef)
for idx, coef in enumerate(coef):
    print(f"Coefficient {idx}: {coef:.6f}")

Coefficient 0: 0.509492+1.998838j
Coefficient 1: -2.999218-0.998313j
Coefficient 2: 20.273556+19.907992j

Try running using real-valued M_estimates

model_mest = M_estimate()
model_mest.fit(X_real, y_real)
coef = glr_coef_to_complex(model_mest.coef)
for idx, coef in enumerate(coef):
    print(f"Coefficient {idx}: {coef:.6f}")

Coefficient 0: 0.505285+1.993242j
Coefficient 1: -3.005300-0.999968j
Coefficient 2: 20.040712+19.764842j