Add diagrams for rand_distr

requirements.txt

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+numpy==1.26.4
+matplotlib==3.8.4
+scipy==1.13.0
+python-ternary==1.0.8

src/distributions/beta.py

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+import numpy as np
+import matplotlib.pyplot as plt
+from scipy.stats import beta
+
+
+def save_to(directory: str, extension: str):
+    # Defining the Beta distribution PDF
+    def y(a, b, x):
+        y = beta.pdf(x, a, b)
+        y[y > 4] = np.nan
+        return y
+
+    inputs = [(0.5, 0.5), (5, 1), (1, 3), (2, 2), (2, 5)]
+    # Possible values for the distribution
+    x = np.linspace(0, 1, 1000)
+
+    # Creating the figure and the axis
+    fig, ax = plt.subplots()
+
+    # Plotting the PDF for each value of alpha and beta
+    for a, b in inputs:
+        ax.plot(x, y(a, b, x), label=f'α = {a}, β = {b}')
+
+    # Adding title and labels
+    ax.set_title('Beta distribution')
+    ax.set_xlabel('x')
+    ax.set_ylabel('Probability density')
+
+    # Adding a legend
+    ax.legend()
+    ax.grid()
+    ax.margins(x=0, y=0)
+
+    plt.savefig(f"{directory}/beta.{extension}")
+    plt.close()

src/distributions/binomial.py

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+import numpy as np
+import matplotlib.pyplot as plt
+from scipy.stats import binom
+
+
+def save_to(directory: str, extension: str):
+    inputs = [(10, 0.2), (10, 0.6)]
+    # Possible outcomes for a Binomial distributed variable
+    outcomes = np.arange(0, 11)
+    width = 0.5
+
+    # Creating the figure and the axis
+    fig, ax = plt.subplots()
+
+    # Plotting the PMF for each value of n and p
+    for i, (n, p) in enumerate(inputs):
+        ax.bar(outcomes + i * width - width / 2, binom.pmf(outcomes, n, p), width=width, label=f'n = {n}, p = {p}')
+
+    # Adding title and labels
+    ax.set_title('Binomial distribution')
+    ax.set_xlabel('k (number of successes)')
+    ax.set_ylabel('Probability')
+    ax.set_xticks(outcomes)  # set the ticks to be the outcome values
+
+    # Adding a legend
+    ax.legend()
+    ax.grid()
+    ax.margins(x=0, y=0)
+    ymin, ymax = ax.get_ylim()
+    ax.set_ylim(ymin, ymax * 1.05)
+
+    plt.savefig(f"{directory}/binomial.{extension}")
+    plt.close()

src/distributions/cauchy.py

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+import numpy as np
+import matplotlib.pyplot as plt
+
+
+def save_to(directory: str, extension: str):
+    # Possible values for the distribution
+    x = np.linspace(-7, 7, 1000)
+
+    # Creating the figure and the axis
+    fig, ax = plt.subplots()
+
+    inputs = [(0, 1), (0, 0.5), (0, 2), (-2, 1)]
+
+    # Plotting the PDF for the Cauchy distribution
+    for x0, gamma in inputs:
+        ax.plot(x, 1 / (np.pi * gamma * (1 + ((x - x0) / gamma)**2)), label=f'x₀ = {x0}, γ = {gamma}')
+
+    # Adding title and labels
+    ax.set_title('Cauchy distribution')
+    ax.set_xlabel('x')
+    ax.set_ylabel('Probability density')
+
+    # Adding a legend
+    ax.legend()
+    ax.grid()
+    ax.margins(x=0, y=0)
+    ymin, ymax = ax.get_ylim()
+    ax.set_ylim(ymin, ymax * 1.05)
+
+    plt.savefig(f"{directory}/cauchy.{extension}")
+    plt.close()

src/distributions/chi_squared.py

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+import numpy as np
+import matplotlib.pyplot as plt
+from scipy.stats import chi2
+
+
+def save_to(directory: str, extension: str):
+    def y(x, df):
+        y = chi2.pdf(x, df)
+        y[y > 1.05] = np.nan
+        return y
+    # Degrees of freedom for the distribution
+    df_values = [1, 2, 3, 5, 9]
+    # Possible values for the distribution
+    x = np.linspace(0, 10, 1000)
+
+    # Creating the figure and the axis
+    fig, ax = plt.subplots()
+
+    # Plotting the PDF for each value of the degrees of freedom
+    for df in df_values:
+        ax.plot(x, y(x, df), label=f'k = {df}')
+
+    # Adding title and labels
+    ax.set_title('Chi-squared distribution')
+    ax.set_xlabel('Chi-squared statistic')
+    ax.set_ylabel('Probability density')
+
+    # Adding a legend
+    ax.legend()
+    ax.grid()
+    ax.margins(x=0, y=0)
+
+    plt.savefig(f"{directory}/chi_squared.{extension}")
+    plt.close()

src/distributions/dirichlet.py

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+import numpy as np
+import matplotlib.pyplot as plt
+import matplotlib.tri as tri
+from math import gamma
+import ternary
+
+
+# Code source: https://blog.bogatron.net/blog/2014/02/02/visualizing-dirichlet-distributions/
+class Dirichlet(object):
+    def __init__(self, alpha):
+        self._alpha = np.array(alpha)
+        self._coef = gamma(np.sum(self._alpha)) / np.multiply.reduce([gamma(a) for a in self._alpha])
+
+    def pdf(self, x):
+        """Returns pdf value for `x`."""
+        return self._coef * np.multiply.reduce([xx ** (aa - 1) for (xx, aa) in zip(x, self._alpha)])
+
+
+def save_to(directory: str, _: str):
+    extension = "png"  # Hardcode png output format. SVG file size for this distribution is ~100x larger.
+    corners = np.array([[0, 0], [1, 0], [0.5, np.sqrt(0.75)]])
+    AREA = 0.5 * 1 * np.sqrt(0.75)
+    triangle = tri.Triangulation(corners[:, 0], corners[:, 1])
+
+    pairs = [corners[np.roll(range(3), -i)[1:]] for i in range(3)]
+    tri_area = lambda xy, pair: 0.5 * np.linalg.norm(np.cross(*(pair - xy)))
+
+    def xy2bc(xy, tol=1.e-4):
+        coords = np.array([tri_area(xy, p) for p in pairs]) / AREA
+        return np.clip(coords, tol, 1.0 - tol)
+
+    def draw_pdf_contours(ax, alphas, nlevels=200, subdiv=8):
+        refiner = tri.UniformTriRefiner(triangle)
+        trimesh = refiner.refine_triangulation(subdiv=subdiv)
+        pvals = [Dirichlet(alphas).pdf(xy2bc(xy)) for xy in zip(trimesh.x, trimesh.y)]
+
+        contour = ax.tricontourf(trimesh, pvals, nlevels, cmap='plasma')
+        ternary.plt.colorbar(contour, ax=ax, orientation='vertical', fraction=0.05, pad=0.05)
+
+        ax.axis('equal')
+        ax.spines['top'].set_visible(False)
+        ax.spines['right'].set_visible(False)
+        ax.spines['bottom'].set_visible(False)
+        ax.spines['left'].set_visible(False)
+
+        tax = ternary.TernaryAxesSubplot(ax=ax, scale=1.0)
+
+        tax.boundary(linewidth=1.0)
+
+        tax.ticks(axis='lbr', linewidth=1, multiple=0.2, tick_formats="%.1f", offset=0.02)
+
+        fontsize = 13
+        tax.set_title(f"α = {alphas}", fontsize=fontsize, pad=20)
+        tax.right_axis_label("$x_3$", fontsize=fontsize, offset=0.15)
+        tax.left_axis_label("$x_1$", fontsize=fontsize, offset=0.15)
+        tax.bottom_axis_label("$x_2$", fontsize=fontsize)
+        tax._redraw_labels()  # Won't do this automatically because of the way we are saving the plot
+
+        tax.clear_matplotlib_ticks()
+
+    inputs = [
+        [1.5, 1.5, 1.5],
+        [5.0, 5.0, 5.0],
+        [1.0, 2.0, 2.0],
+        [2.0, 4.0, 8.0]
+    ]
+
+    fig, axes = plt.subplots(2, 2, figsize=(12, 10))
+    for ax, alphas in zip(axes.flatten(), inputs):
+        draw_pdf_contours(ax, alphas)
+
+    # Adding the main title and colorbar
+    ternary.plt.suptitle('Dirichlet Distribution', fontsize=16)
+
+    ternary.plt.savefig(f"{directory}/dirichlet.{extension}")
+    plt.close()

src/distributions/exponential.py

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+import numpy as np
+import matplotlib.pyplot as plt
+from scipy.stats import expon
+
+
+def save_to(directory: str, extension: str):
+    # Possible values of lambda for the distribution
+    lambda_values = [1, 0.5, 2]
+    # Possible values for the distribution
+    x = np.linspace(0, 5, 1000)
+
+    # Creating the figure and the axis
+    fig, ax = plt.subplots()
+
+    # Plotting the PDF for each value of lambda
+    for lmbda in lambda_values:
+        ax.plot(x, expon.pdf(x, scale=1 / lmbda), label=f'λ = {lmbda}')
+
+    # Adding title and labels
+    ax.set_title('Exponential distribution')
+    ax.set_xlabel('x')
+    ax.set_ylabel('Probability density')
+
+    # Adding a legend
+    ax.legend()
+    ax.grid()
+    ax.margins(x=0, y=0)
+
+    plt.savefig(f"{directory}/exponential.{extension}")
+    plt.close()

src/distributions/exponential_exp1.py

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+import numpy as np
+import matplotlib.pyplot as plt
+from scipy.stats import expon
+
+
+def save_to(directory: str, extension: str):
+    # Fixed value for lambda
+    lmbda = 1
+    # Possible values for the distribution
+    x = np.linspace(0, 5, 1000)
+
+    # Creating the figure and the axis
+    fig, ax = plt.subplots()
+
+    # Plotting the PDF for each value of lambda
+    ax.plot(x, expon.pdf(x, scale=1 / lmbda), label=f'λ = {lmbda}')
+
+    # Adding title and labels
+    ax.set_title('Exponential distribution')
+    ax.set_xlabel('x')
+    ax.set_ylabel('Probability density')
+
+    # Adding a legend
+    ax.legend()
+    ax.grid()
+    ax.margins(x=0, y=0)
+
+    plt.savefig(f"{directory}/exponential_exp1.{extension}")
+    plt.close()

src/distributions/fisher_f.py

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+import numpy as np
+import matplotlib.pyplot as plt
+from scipy.stats import f
+
+
+def save_to(directory: str, extension: str):
+    def y(x, dfn, dfd):
+        y = f.pdf(x, dfn, dfd)
+        y[y > 2.3] = np.nan
+        return y
+
+    # Degrees of freedom for the distribution
+    d1_d2 = [(1, 1), (2, 1), (3, 1), (10, 1), (10, 10), (100, 100)]
+    # Possible values for the distribution
+    x = np.linspace(0, 5, 1000)
+
+    # Creating the figure and the axis
+    fig, ax = plt.subplots()
+
+    # Plotting the PDF for each value of the degrees of freedom
+    for m, n in d1_d2:
+        ax.plot(x, y(x, m, n), label=f'm = {m}, n = {n}')
+
+    # Adding title and labels
+    ax.set_title('F-distribution')
+    ax.set_xlabel('F-statistic')
+    ax.set_ylabel('Probability density')
+
+    # Adding a legend
+    ax.legend()
+    ax.grid()
+    ax.margins(x=0, y=0)
+
+    plt.savefig(f"{directory}/fisher_f.{extension}")
+    plt.close()

src/distributions/frechet.py

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+import numpy as np
+import matplotlib.pyplot as plt
+from scipy.stats import invweibull
+
+
+def save_to(directory: str, extension: str):
+    def y(x, shape, loc, scale):
+        return invweibull.pdf(x, shape, loc=loc, scale=scale)
+    # Inputs for the distribution
+    inputs = [(0, 1, 1), (1, 1, 1), (0, 0.5, 1), (0, 2, 1), (0, 1, 0.35), (0, 1, 2)]
+    # x values for the distribution
+    x = np.linspace(0, 5, 1000)
+
+    # Creating the figure and the axis
+    fig, ax = plt.subplots()
+
+    # Plotting the PDF for each value of a
+    for loc, scale, shape in inputs:
+        ax.plot(x, y(x, shape, loc, scale), label=f'μ = {loc}, σ = {scale}, α = {shape}')
+
+    # Adding title and labels
+    ax.set_title('Fréchet distribution')
+    ax.set_xlabel('x')
+    ax.set_ylabel('Probability density')
+
+    # Adding a legend
+    ax.legend()
+    ax.grid()
+    ax.margins(x=0, y=0)
+    ymin, ymax = ax.get_ylim()
+    ax.set_ylim(ymin, ymax * 1.05)
+
+    plt.savefig(f"{directory}/frechet.{extension}")
+    plt.close()

src/distributions/gamma.py

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+import numpy as np
+import matplotlib.pyplot as plt
+from scipy.stats import gamma
+
+
+def save_to(directory: str, extension: str):
+    inputs = [(1, 1), (2, 1), (3, 1), (1, 2), (2, 2), (3, 2)]
+    # Possible values for the distribution
+    x = np.linspace(0, 7, 1000)
+
+    # Creating the figure and the axis
+    fig, ax = plt.subplots()
+    colors = ["blue", "red", "green", "blue", "red", "green"]
+    alphas = [1.0, 1.0, 1.0, 0.6, 0.6, 0.6]
+
+    # Plotting the PDF for each value of alpha and beta
+    for i, (k, theta) in enumerate(inputs):
+        ax.plot(x, gamma.pdf(x, k, scale=theta), label=f'k = {k}, θ = {theta}', color=colors[i], alpha=alphas[i])
+
+    # Adding title and labels
+    ax.set_title('Gamma distribution')
+    ax.set_xlabel('x')
+    ax.set_ylabel('Probability density')
+
+    # Adding a legend
+    ax.legend()
+    ax.grid()
+    ax.margins(x=0, y=0)
+
+    plt.savefig(f"{directory}/gamma.{extension}")
+    plt.close()