Add sums of distributions

src/sdr/_probability.py

@@ -8,7 +8,7 @@
 import numpy.typing as npt
 import scipy.special
 
-from ._helper import convert_output, export, verify_arraylike
+from ._helper import convert_output, export, verify_arraylike, verify_scalar
 
 
 @export
@@ -75,3 +75,256 @@ def Qinv(p: npt.ArrayLike) -> npt.NDArray[np.float64]:
  x = np.sqrt(2) * scipy.special.erfcinv(2 * p)
 
  return convert_output(x)
+
+
+@export
+def sum_distribution(
+ X: scipy.stats.rv_continuous | scipy.stats.rv_histogram,
+ n_terms: int,
+ p: float = 1e-16,
+) -> scipy.stats.rv_histogram:
+ r"""
+ Numerically calculates the distribution of the sum of $n$ i.i.d. random variables $X_i$.
+
+ Arguments:
+ X: The distribution of the i.i.d. random variables $X_i$.
+ n_terms: The number $n$ of random variables to sum.
+ p: The probability of exceeding the x axis, on either side, for each distribution. This is used to determine
+ the bounds on the x axis for the numerical convolution. Smaller values of $p$ will result in more accurate
+ analysis, but will require more computation.
+
+ Returns:
+ The distribution of the sum $Z = X_1 + X_2 + \ldots + X_n$.
+
+ Notes:
+ The PDF of the sum of $n$ independent random variables is the convolution of the PDF of the base distribution.
+
+ $$f_{X_1 + X_2 + \ldots + X_n}(t) = (f_X * f_X * \ldots * f_X)(t)$$
+
+ Examples:
+ Compute the distribution of the sum of two normal distributions.
+
+ .. ipython:: python
+
+ X = scipy.stats.norm(loc=-1, scale=0.5)
+ n_terms = 2
+ x = np.linspace(-6, 2, 1_001)
+
+ @savefig sdr_sum_distribution_1.png
+ plt.figure(); \
+ plt.plot(x, X.pdf(x), label="X"); \
+ plt.plot(x, sdr.sum_distribution(X, n_terms).pdf(x), label="X + X"); \
+ plt.hist(X.rvs((100_000, n_terms)).sum(axis=1), bins=101, density=True, histtype="step", label="X + X empirical"); \
+ plt.legend(); \
+ plt.xlabel("Random variable"); \
+ plt.ylabel("Probability density"); \
+ plt.title("Sum of two Normal distributions");
+
+ Compute the distribution of the sum of three Rayleigh distributions.
+
+ .. ipython:: python
+
+ X = scipy.stats.rayleigh(scale=1)
+ n_terms = 3
+ x = np.linspace(0, 10, 1_001)
+
+ @savefig sdr_sum_distribution_2.png
+ plt.figure(); \
+ plt.plot(x, X.pdf(x), label="X"); \
+ plt.plot(x, sdr.sum_distribution(X, n_terms).pdf(x), label="X + X + X"); \
+ plt.hist(X.rvs((100_000, n_terms)).sum(axis=1), bins=101, density=True, histtype="step", label="X + X + X empirical"); \
+ plt.legend(); \
+ plt.xlabel("Random variable"); \
+ plt.ylabel("Probability density"); \
+ plt.title("Sum of three Rayleigh distributions");
+
+ Compute the distribution of the sum of four Rician distributions.
+
+ .. ipython:: python
+
+ X = scipy.stats.rice(2)
+ n_terms = 4
+ x = np.linspace(0, 18, 1_001)
+
+ @savefig sdr_sum_distribution_3.png
+ plt.figure(); \
+ plt.plot(x, X.pdf(x), label="X"); \
+ plt.plot(x, sdr.sum_distribution(X, n_terms).pdf(x), label="X + X + X + X"); \
+ plt.hist(X.rvs((100_000, n_terms)).sum(axis=1), bins=101, density=True, histtype="step", label="X + X + X + X empirical"); \
+ plt.legend(); \
+ plt.xlabel("Random variable"); \
+ plt.ylabel("Probability density"); \
+ plt.title("Sum of four Rician distributions");
+
+ Group:
+ probability
+ """
+ verify_scalar(n_terms, int=True, positive=True)
+ verify_scalar(p, float=True, exclusive_min=0, inclusive_max=0.1)
+
+ if n_terms == 1:
+ return X
+
+ # Determine the x axis of each distribution such that the probability of exceeding the x axis, on either side,
+ # is p.
+ x1_min, x1_max = _x_range(X, p)
+ x = np.linspace(x1_min, x1_max, 1_001)
+ dx = np.mean(np.diff(x))
+
+ # Compute the PDF of the base distribution
+ f_X = X.pdf(x)
+
+ # The PDF of the sum of n_terms independent random variables is the convolution of the PDF of the base distribution.
+ # This is efficiently computed in the frequency domain by exponentiating the FFT. The FFT must be zero-padded
+ # enough that the circular convolutions do not wrap around.
+ n_fft = scipy.fft.next_fast_len(f_X.size * n_terms)
+ f_X_fft = np.fft.fft(f_X, n_fft)
+ f_X_fft = f_X_fft**n_terms
+ f_Y = np.fft.ifft(f_X_fft).real
+ f_Y /= f_Y.sum() * dx
+ x = np.arange(f_Y.size) * dx + x[0] * (n_terms)
+
+ # Adjust the histograms bins to be on either side of each point. So there is one extra point added.
+ x = np.append(x, x[-1] + dx)
+ x -= dx / 2
+
+ return scipy.stats.rv_histogram((f_Y, x))
+
+
+@export
+def sum_distributions(
+ X: scipy.stats.rv_continuous | scipy.stats.rv_histogram,
+ Y: scipy.stats.rv_continuous | scipy.stats.rv_histogram,
+ p: float = 1e-16,
+) -> scipy.stats.rv_histogram:
+ r"""
+ Numerically calculates the distribution of the sum of two independent random variables $X$ and $Y$.
+
+ Arguments:
+ X: The distribution of the first random variable $X$.
+ Y: The distribution of the second random variable $Y$.
+ p: The probability of exceeding the x axis, on either side, for each distribution. This is used to determine
+ the bounds on the x axis for the numerical convolution. Smaller values of $p$ will result in more accurate
+ analysis, but will require more computation.
+
+ Returns:
+ The distribution of the sum $Z = X + Y$.
+
+ Notes:
+ The PDF of the sum of two independent random variables is the convolution of the PDF of the two distributions.
+
+ $$f_{X+Y}(t) = (f_X * f_Y)(t)$$
+
+ Examples:
+ Compute the distribution of the sum of two normal distributions.
+
+ .. ipython:: python
+
+ X = scipy.stats.norm(loc=-1, scale=0.5)
+ Y = scipy.stats.norm(loc=2, scale=1.5)
+ x = np.linspace(-5, 10, 1_001)
+
+ @savefig sdr_sum_distributions_1.png
+ plt.figure(); \
+ plt.plot(x, X.pdf(x), label="X"); \
+ plt.plot(x, Y.pdf(x), label="Y"); \
+ plt.plot(x, sdr.sum_distributions(X, Y).pdf(x), label="X + Y"); \
+ plt.hist(X.rvs(100_000) + Y.rvs(100_000), bins=101, density=True, histtype="step", label="X + Y empirical"); \
+ plt.legend(); \
+ plt.xlabel("Random variable"); \
+ plt.ylabel("Probability density"); \
+ plt.title("Sum of two Normal distributions");
+
+ Compute the distribution of the sum of two Rayleigh distributions.
+
+ .. ipython:: python
+
+ X = scipy.stats.rayleigh(scale=1)
+ Y = scipy.stats.rayleigh(loc=1, scale=2)
+ x = np.linspace(0, 12, 1_001)
+
+ @savefig sdr_sum_distributions_2.png
+ plt.figure(); \
+ plt.plot(x, X.pdf(x), label="X"); \
+ plt.plot(x, Y.pdf(x), label="Y"); \
+ plt.plot(x, sdr.sum_distributions(X, Y).pdf(x), label="X + Y"); \
+ plt.hist(X.rvs(100_000) + Y.rvs(100_000), bins=101, density=True, histtype="step", label="X + Y empirical"); \
+ plt.legend(); \
+ plt.xlabel("Random variable"); \
+ plt.ylabel("Probability density"); \
+ plt.title("Sum of two Rayleigh distributions");
+
+ Compute the distribution of the sum of two Rician distributions.
+
+ .. ipython:: python
+
+ X = scipy.stats.rice(2)
+ Y = scipy.stats.rice(3)
+ x = np.linspace(0, 12, 1_001)
+
+ @savefig sdr_sum_distributions_3.png
+ plt.figure(); \
+ plt.plot(x, X.pdf(x), label="X"); \
+ plt.plot(x, Y.pdf(x), label="Y"); \
+ plt.plot(x, sdr.sum_distributions(X, Y).pdf(x), label="X + Y"); \
+ plt.hist(X.rvs(100_000) + Y.rvs(100_000), bins=101, density=True, histtype="step", label="X + Y empirical"); \
+ plt.legend(); \
+ plt.xlabel("Random variable"); \
+ plt.ylabel("Probability density"); \
+ plt.title("Sum of two Rician distributions");
+
+ Group:
+ probability
+ """
+ verify_scalar(p, float=True, exclusive_min=0, inclusive_max=0.1)
+
+ # Determine the x axis of each distribution such that the probability of exceeding the x axis, on either side,
+ # is p.
+ x1_min, x1_max = _x_range(X, p)
+ x2_min, x2_max = _x_range(Y, p)
+ dx1 = (x1_max - x1_min) / 1_000
+ dx2 = (x2_max - x2_min) / 1_000
+ dx = np.min([dx1, dx2]) # Use the smaller delta x -- must use the same dx for both distributions
+ x1 = np.arange(x1_min, x1_max, dx)
+ x2 = np.arange(x2_min, x2_max, dx)
+
+ # Compute the PDF of each distribution
+ f_X = X.pdf(x1)
+ f_Y = Y.pdf(x2)
+
+ # The PDF of the sum of two independent random variables is the convolution of the PDF of the two distributions
+ f_Z = np.convolve(f_X, f_Y, mode="full") * dx
+
+ # Determine the x axis for the output convolution
+ x = np.arange(f_Z.size) * dx + x1[0] + x2[0]
+
+ # Adjust the histograms bins to be on either side of each point. So there is one extra point added.
+ x = np.append(x, x[-1] + dx)
+ x -= dx / 2
+
+ return scipy.stats.rv_histogram((f_Z, x))
+
+
+def _x_range(X: scipy.stats.rv_continuous, p: float) -> tuple[float, float]:
+ r"""
+ Determines the range of x values for a given distribution such that the probability of exceeding the x axis, on
+ either side, is p.
+ """
+ # Need to have these loops because for very small p, sometimes SciPy will return NaN instead of a valid value.
+ # The while loops will increase the p value until a valid value is returned.
+
+ pp = p
+ while True:
+ x_min = X.ppf(pp)
+ if not np.isnan(x_min):
+ break
+ pp *= 10
+
+ pp = p
+ while True:
+ x_max = X.isf(pp)
+ if not np.isnan(x_max):
+ break
+ pp *= 10
+
+ return x_min, x_max

tests/probability/test_sum_distribution.py

@@ -0,0 +1,50 @@
+import numpy as np
+import scipy.stats
+
+import sdr
+
+
+def test_normal():
+ X = scipy.stats.norm(loc=-1, scale=0.5)
+ _verify(X, 2)
+ _verify(X, 3)
+ _verify(X, 4)
+
+
+def test_rayleigh():
+ X = scipy.stats.rayleigh(scale=1)
+ _verify(X, 2)
+ _verify(X, 3)
+ _verify(X, 4)
+
+
+def test_rician():
+ X = scipy.stats.rice(2)
+ _verify(X, 2)
+ _verify(X, 3)
+ _verify(X, 4)
+
+
+def _verify(X, n):
+ # Numerically compute the distribution
+ Y = sdr.sum_distribution(X, n)
+
+ # Empirically compute the distribution
+ y = X.rvs((100_000, n)).sum(axis=1)
+ hist, bins = np.histogram(y, bins=101, density=True)
+ x = bins[1:] - np.diff(bins) / 2
+
+ if False:
+ import matplotlib.pyplot as plt
+
+ plt.figure()
+ plt.plot(x, X.pdf(x), label="X")
+ plt.plot(x, Y.pdf(x), label="X + ... + X")
+ plt.hist(y, bins=101, cumulative=False, density=True, histtype="step", label="X + .. + X empirical")
+ plt.legend()
+ plt.xlabel("Random variable")
+ plt.ylabel("Probability density")
+ plt.title("Sum of distribution")
+ plt.show()
+
+ assert np.allclose(Y.pdf(x), hist, atol=np.max(hist) * 0.1)

tests/probability/test_sum_distributions.py

@@ -0,0 +1,60 @@
+import numpy as np
+import scipy.stats
+
+import sdr
+
+
+def test_normal_normal():
+ X = scipy.stats.norm(loc=-1, scale=0.5)
+ Y = scipy.stats.norm(loc=2, scale=1.5)
+ _verify(X, Y)
+
+
+def test_rayleigh_rayleigh():
+ X = scipy.stats.rayleigh(scale=1)
+ Y = scipy.stats.rayleigh(loc=1, scale=2)
+ _verify(X, Y)
+
+
+def test_rician_rician():
+ X = scipy.stats.rice(2)
+ Y = scipy.stats.rice(3)
+ _verify(X, Y)
+
+
+def test_normal_rayleigh():
+ X = scipy.stats.norm(loc=-1, scale=0.5)
+ Y = scipy.stats.rayleigh(loc=2, scale=1.5)
+ _verify(X, Y)
+
+
+def test_rayleigh_rician():
+ X = scipy.stats.rayleigh(scale=1)
+ Y = scipy.stats.rice(3)
+ _verify(X, Y)
+
+
+def _verify(X, Y):
+ # Numerically compute the distribution
+ Z = sdr.sum_distributions(X, Y)
+
+ # Empirically compute the distribution
+ z = X.rvs(100_000) + Y.rvs(100_000)
+ hist, bins = np.histogram(z, bins=101, density=True)
+ x = bins[1:] - np.diff(bins) / 2
+
+ if False:
+ import matplotlib.pyplot as plt
+
+ plt.figure()
+ plt.plot(x, X.pdf(x), label="X")
+ plt.plot(x, Y.pdf(x), label="Y")
+ plt.plot(x, Z.pdf(x), label="X + Y")
+ plt.hist(z, bins=101, cumulative=False, density=True, histtype="step", label="X + Y empirical")
+ plt.legend()
+ plt.xlabel("Random variable")
+ plt.ylabel("Probability density")
+ plt.title("Sum of two distributions")
+ plt.show()
+
+ assert np.allclose(Z.pdf(x), hist, atol=np.max(hist) * 0.1)