In [1]:
import numpy as np
import matplotlib.pyplot as plt
from matplotlib import ticker
Circular Models¶
Table of Contents¶
Symmetric distributions¶
- Circular uniform distribution
- Triangular distribution
- Cardioid distribution
- Cartwright distribution
- wrapped Normal distribution
- wrapped Cauchy distribution
- von Mises distribution
- Jones-Pewsey distribution
- Flat-topped von Mises Distribution
Asymmetric distributions¶
Symmetric distributions¶
Circular uniform distribution¶
In [2]:
from pycircstat2.distributions import circularuniform
n = 100
x = np.linspace(0, 2 * np.pi, n)
fig, ax = plt.subplot_mosaic(
"abc",
figsize=(16, 5),
per_subplot_kw={"a": {"projection": "polar"}},
layout="constrained",
)
ax["a"].plot(
x,
circularuniform.pdf(x) + 1,
linestyle="-",
color="black",
)
rtick = [0, 1]
ax["a"].set_theta_zero_location("S")
ax["a"].spines["polar"].set_visible(False)
ax["a"].set_rgrids(rtick, ["" for _ in range(len(rtick))], fontsize=16)
gridlines = ax["a"].yaxis.get_gridlines()
gridlines[-1].set_color("k")
gridlines[-1].set_linewidth(1)
# position_major = np.arange(0, 2 * np.pi, 2 * np.pi / 4)
position_major = []
ax["a"].xaxis.set_major_locator(ticker.FixedLocator(position_major))
ax["a"].text(
x=0,
y=0,
s="+",
ha="center",
va="center",
color="black",
)
labels = ["0", "π/2", "π", "3π/2"]
for i, angle in enumerate([0, np.pi / 2, np.pi, 3 * np.pi / 2]):
ax["a"].text(x=angle, y=0.75, s=labels[i], ha="center", va="center", color="black")
ax["a"].legend(frameon=False)
ax["a"].set_title("Circular uniform distribution")
ax["b"].plot(x, circularuniform.cdf(x), linestyle="-", color="black")
ax["b"].set_xlabel("θ (rad)")
ax["b"].set_ylabel("CDF")
p = np.linspace(1e-16, 1 - 1e-16, 100)
ax["c"].plot(p, circularuniform.ppf(p), linestyle="-", color="black")
ax["c"].set_xlabel("Cumulative prob.")
ax["c"].set_ylabel("Quantile (rad)")
fig.savefig("../docs/docs/images/circ-mod-circularuniform.png")
/var/folders/2g/cw502gdd0hj9q05c6n34nkvr0000gn/T/ipykernel_93319/3993589730.py:44: UserWarning: No artists with labels found to put in legend. Note that artists whose label start with an underscore are ignored when legend() is called with no argument. ax["a"].legend(frameon=False)
Triangular distribution¶
In [3]:
from pycircstat2.distributions import triangular
n = 100
x = np.linspace(0, 2 * np.pi, n)
fig, ax = plt.subplot_mosaic(
"abc",
figsize=(16, 5),
per_subplot_kw={"a": {"projection": "polar"}},
layout="constrained",
)
ax["a"].plot(
x,
triangular.pdf(x, rho=0) + 1,
linestyle="-",
color="black",
label="ρ=0",
)
ax["a"].plot(
x,
triangular.pdf(x, rho=0.2) + 1,
linestyle="--",
color="black",
label="ρ=0.2",
)
ax["a"].plot(
x,
triangular.pdf(x, rho=0.4) + 1,
linestyle=":",
color="black",
label="ρ=0.4",
)
rtick = [0, 1]
ax["a"].set_theta_zero_location("S")
ax["a"].spines["polar"].set_visible(False)
ax["a"].set_rgrids(rtick, ["" for _ in range(len(rtick))], fontsize=16)
gridlines = ax["a"].yaxis.get_gridlines()
gridlines[-1].set_color("k")
gridlines[-1].set_linewidth(1)
# position_major = np.arange(0, 2 * np.pi, 2 * np.pi / 4)
position_major = []
ax["a"].xaxis.set_major_locator(ticker.FixedLocator(position_major))
ax["a"].text(
x=0,
y=0,
s="+",
ha="center",
va="center",
color="black",
)
labels = ["0", "π/2", "π", "3π/2"]
for i, angle in enumerate([0, np.pi / 2, np.pi, 3 * np.pi / 2]):
ax["a"].text(x=angle, y=0.75, s=labels[i], ha="center", va="center", color="black")
ax["a"].legend(frameon=False)
ax["a"].set_title("Triangular distribution")
ax["b"].plot(x, triangular.cdf(x, rho=0), linestyle="-", color="black")
ax["b"].plot(x, triangular.cdf(x, rho=0.2), linestyle="--", color="black")
ax["b"].plot(x, triangular.cdf(x, rho=0.4), linestyle=":", color="black")
ax["b"].set_xlabel("θ (rad)")
ax["b"].set_ylabel("CDF")
p = np.linspace(1e-16, 1 - 1e-16, 100)
ax["c"].plot(p, triangular.ppf(p, rho=0.0), linestyle="-", color="black")
ax["c"].plot(p, triangular.ppf(p, rho=0.2), linestyle="--", color="black")
ax["c"].plot(p, triangular.ppf(p, rho=0.4), linestyle=":", color="black")
ax["c"].set_xlabel("Cumulative prob.")
ax["c"].set_ylabel("Quantile (rad)")
fig.savefig("../docs/docs/images/circ-mod-triangular.png")
Cardioid Distribution¶
In [4]:
from pycircstat2.distributions import cardioid
n = 100
x = np.linspace(0, 2 * np.pi, n)
mu = np.pi
fig, ax = plt.subplot_mosaic(
"abc",
figsize=(16, 5),
per_subplot_kw={"a": {"projection": "polar"}},
layout="constrained",
)
ax["a"].plot(
x,
cardioid.pdf(x, rho=0.0, mu=mu) + 1,
linestyle=":",
color="black",
label="ρ=0",
)
ax["a"].plot(
x,
cardioid.pdf(x, rho=0.25, mu=mu) + 1,
linestyle="-",
color="black",
label="ρ=0.25",
)
ax["a"].plot(
x, cardioid.pdf(x, rho=0.5, mu=mu) + 1, linestyle="--", color="black", label="ρ=0.5"
)
rtick = [0, 1]
ax["a"].set_theta_zero_location("S")
ax["a"].spines["polar"].set_visible(False)
ax["a"].set_rgrids(rtick, ["" for _ in range(len(rtick))], fontsize=16)
gridlines = ax["a"].yaxis.get_gridlines()
gridlines[-1].set_color("k")
gridlines[-1].set_linewidth(1)
# position_major = np.arange(0, 2 * np.pi, 2 * np.pi / 4)
position_major = []
ax["a"].xaxis.set_major_locator(ticker.FixedLocator(position_major))
ax["a"].text(
x=0,
y=0,
s="+",
ha="center",
va="center",
color="black",
)
labels = ["0", "π/2", "π", "3π/2"]
for i, angle in enumerate([0, np.pi / 2, np.pi, 3 * np.pi / 2]):
ax["a"].text(x=angle, y=0.75, s=labels[i], ha="center", va="center", color="black")
ax["a"].legend(frameon=False)
ax["a"].set_title("Cardioid distr.")
ax["b"].plot(x, cardioid.cdf(x, rho=0.25, mu=mu), linestyle="-", color="black")
ax["b"].plot(x, cardioid.cdf(x, rho=0.5, mu=mu), linestyle="--", color="black")
ax["b"].plot(x, cardioid.cdf(x, rho=0.0, mu=mu), linestyle=":", color="black")
ax["b"].set_xlabel("θ (rad)")
ax["b"].set_ylabel("CDF")
p = np.linspace(1e-16, 1 - 1e-16, 100)
ax["c"].plot(p, cardioid.ppf(p, rho=0.25, mu=mu), linestyle="-", color="black")
ax["c"].plot(p, cardioid.ppf(p, rho=0.5, mu=mu), linestyle="--", color="black")
ax["c"].plot(p, cardioid.ppf(p, rho=0.0, mu=mu), linestyle=":", color="black")
ax["c"].set_xlabel("Cumulative prob.")
ax["c"].set_ylabel("Quantile (rad)")
fig.savefig("../docs/docs/images/circ-mod-cardioid.png")
Cartwright Distribution¶
In [5]:
from pycircstat2.distributions import cartwright
n = 100
x = np.linspace(0, 2 * np.pi, n)
mu = np.pi
fig, ax = plt.subplot_mosaic(
"abc",
figsize=(16, 5),
per_subplot_kw={"a": {"projection": "polar"}},
layout="constrained",
)
ax["a"].plot(
x,
cartwright.pdf(x, zeta=1, mu=mu) + 1,
linestyle=":",
color="black",
label="ζ=1",
)
ax["a"].plot(
x,
cartwright.pdf(x, zeta=10, mu=mu) + 1,
linestyle="-",
color="black",
label="ζ=10",
)
ax["a"].plot(
x,
cartwright.pdf(x, zeta=0.1, mu=mu) + 1,
linestyle="--",
color="black",
label="ζ=0.1",
)
rtick = [0, 1]
ax["a"].set_theta_zero_location("S")
ax["a"].spines["polar"].set_visible(False)
ax["a"].set_rgrids(rtick, ["" for _ in range(len(rtick))], fontsize=16)
gridlines = ax["a"].yaxis.get_gridlines()
gridlines[-1].set_color("k")
gridlines[-1].set_linewidth(1)
# position_major = np.arange(0, 2 * np.pi, 2 * np.pi / 4)
position_major = []
ax["a"].xaxis.set_major_locator(ticker.FixedLocator(position_major))
ax["a"].text(
x=0,
y=0,
s="+",
ha="center",
va="center",
color="black",
)
labels = ["0", "π/2", "π", "3π/2"]
for i, angle in enumerate([0, np.pi / 2, np.pi, 3 * np.pi / 2]):
ax["a"].text(x=angle, y=0.75, s=labels[i], ha="center", va="center", color="black")
ax["a"].legend(frameon=False)
ax["a"].set_title("Cartwright distr.")
ax["b"].plot(x, cartwright.cdf(x, zeta=1, mu=mu), linestyle="-", color="black")
ax["b"].plot(x, cartwright.cdf(x, zeta=10, mu=mu), linestyle="--", color="black")
ax["b"].plot(x, cartwright.cdf(x, zeta=0.1, mu=mu), linestyle=":", color="black")
ax["b"].set_xlabel("θ (rad)")
ax["b"].set_ylabel("CDF")
p = np.linspace(1e-16, 1 - 1e-16, 100)
ax["c"].plot(p, cartwright.ppf(p, zeta=1, mu=mu), linestyle="-", color="black")
ax["c"].plot(p, cartwright.ppf(p, zeta=10, mu=mu), linestyle="--", color="black")
ax["c"].plot(p, cartwright.ppf(p, zeta=0.1, mu=mu), linestyle=":", color="black")
ax["c"].set_xlabel("Cumulative prob.")
ax["c"].set_ylabel("Quantile (rad)")
fig.savefig("../docs/docs/images/circ-mod-cartwright.png")
Wrapped Normal Distribution¶
In [6]:
from pycircstat2.distributions import wrapnorm
n = 100
x = np.linspace(0, 2 * np.pi, n)
mu = np.pi
fig, ax = plt.subplot_mosaic(
"abc",
figsize=(16, 5),
per_subplot_kw={"a": {"projection": "polar"}},
layout="constrained",
)
ax["a"].plot(
x,
wrapnorm.pdf(x, rho=0.25, mu=mu) + 1,
linestyle="-",
color="black",
label="ρ=0.25",
)
ax["a"].plot(
x, wrapnorm.pdf(x, rho=0.5, mu=mu) + 1, linestyle="--", color="black", label="ρ=0.5"
)
ax["a"].plot(
x,
wrapnorm.pdf(x, rho=0.75, mu=mu) + 1,
linestyle=":",
color="black",
label="ρ=0.75",
)
rtick = [0, 1]
ax["a"].set_theta_zero_location("S")
ax["a"].spines["polar"].set_visible(False)
ax["a"].set_rgrids(rtick, ["" for _ in range(len(rtick))], fontsize=16)
gridlines = ax["a"].yaxis.get_gridlines()
gridlines[-1].set_color("k")
gridlines[-1].set_linewidth(1)
# position_major = np.arange(0, 2 * np.pi, 2 * np.pi / 4)
position_major = []
ax["a"].xaxis.set_major_locator(ticker.FixedLocator(position_major))
ax["a"].text(
x=0,
y=0,
s="+",
ha="center",
va="center",
color="black",
)
labels = ["0", "π/2", "π", "3π/2"]
for i, angle in enumerate([0, np.pi / 2, np.pi, 3 * np.pi / 2]):
ax["a"].text(x=angle, y=0.75, s=labels[i], ha="center", va="center", color="black")
ax["a"].legend(frameon=False)
ax["a"].set_title("Wrapped Normal distr.")
ax["b"].plot(x, wrapnorm.cdf(x, rho=0.25, mu=mu), linestyle="-", color="black")
ax["b"].plot(x, wrapnorm.cdf(x, rho=0.5, mu=mu), linestyle="--", color="black")
ax["b"].plot(x, wrapnorm.cdf(x, rho=0.75, mu=mu), linestyle=":", color="black")
ax["b"].set_xlabel("θ (rad)")
ax["b"].set_ylabel("CDF")
p = np.linspace(1e-16, 1 - 1e-16, 100)
ax["c"].plot(p, wrapnorm.ppf(p, rho=0.25, mu=mu), linestyle="-", color="black")
ax["c"].plot(p, wrapnorm.ppf(p, rho=0.5, mu=mu), linestyle="--", color="black")
ax["c"].plot(p, wrapnorm.ppf(p, rho=0.75, mu=mu), linestyle=":", color="black")
ax["c"].set_xlabel("Cumulative prob.")
ax["c"].set_ylabel("Quantile (rad)")
fig.savefig("../docs/docs/images/circ-mod-wrapnorm.png")
Wrapped Cauchy Distribution¶
In [7]:
from pycircstat2.distributions import wrapcauchy
n = 100
x = np.linspace(0, 2 * np.pi, n)
mu = np.pi
fig, ax = plt.subplot_mosaic(
"abc",
figsize=(16, 5),
per_subplot_kw={"a": {"projection": "polar"}},
layout="constrained",
)
ax["a"].plot(
x,
wrapcauchy.pdf(x, rho=0.25, mu=mu) + 1,
linestyle="-",
color="black",
label="ρ=0.25",
)
ax["a"].plot(
x,
wrapcauchy.pdf(x, rho=0.5, mu=mu) + 1,
linestyle="--",
color="black",
label="ρ=0.5",
)
ax["a"].plot(
x,
wrapcauchy.pdf(x, rho=0.75, mu=mu) + 1,
linestyle=":",
color="black",
label="ρ=0.75",
)
rtick = [0, 1]
ax["a"].set_theta_zero_location("S")
ax["a"].spines["polar"].set_visible(False)
ax["a"].set_rgrids(rtick, ["" for _ in range(len(rtick))], fontsize=16)
gridlines = ax["a"].yaxis.get_gridlines()
gridlines[-1].set_color("k")
gridlines[-1].set_linewidth(1)
position_major = []
ax["a"].xaxis.set_major_locator(ticker.FixedLocator(position_major))
ax["a"].text(
x=0,
y=0,
s="+",
ha="center",
va="center",
color="black",
)
labels = ["0", "π/2", "π", "3π/2"]
for i, angle in enumerate([0, np.pi / 2, np.pi, 3 * np.pi / 2]):
ax["a"].text(x=angle, y=0.75, s=labels[i], ha="center", va="center", color="black")
ax["a"].legend(frameon=False)
ax["a"].set_title("Wrapped Cauchy distr.")
ax["b"].plot(x, wrapcauchy.cdf(x, rho=0.25, mu=mu), linestyle="-", color="black")
ax["b"].plot(x, wrapcauchy.cdf(x, rho=0.5, mu=mu), linestyle="--", color="black")
ax["b"].plot(x, wrapcauchy.cdf(x, rho=0.75, mu=mu), linestyle=":", color="black")
ax["b"].set_xlabel("θ (rad)")
ax["b"].set_ylabel("CDF")
p = np.linspace(1e-16, 1 - 1e-16, 100)
ax["c"].plot(p, wrapcauchy.ppf(p, rho=0.25, mu=mu), linestyle="-", color="black")
ax["c"].plot(p, wrapcauchy.ppf(p, rho=0.5, mu=mu), linestyle="--", color="black")
ax["c"].plot(p, wrapcauchy.ppf(p, rho=0.75, mu=mu), linestyle=":", color="black")
ax["c"].set_xlabel("Cumulative prob.")
ax["c"].set_ylabel("Quantile (rad)")
fig.savefig("../docs/docs/images/circ-mod-wrapcauchy.png")
Von Mises Distribution¶
In [8]:
from pycircstat2.distributions import vonmises
n = 100
x = np.linspace(0, 2 * np.pi, n)
mu = np.pi
fig, ax = plt.subplot_mosaic(
"abc",
figsize=(16, 5),
per_subplot_kw={"a": {"projection": "polar"}},
layout="constrained",
)
ax["a"].plot(
x, vonmises.pdf(x, kappa=5, mu=mu) + 1, linestyle="-", color="black", label="κ=5"
)
ax["a"].plot(
x, vonmises.pdf(x, kappa=2, mu=mu) + 1, linestyle="-.", color="black", label="κ=2"
)
ax["a"].plot(
x, vonmises.pdf(x, kappa=1, mu=mu) + 1, linestyle="--", color="black", label="κ=1"
)
ax["a"].plot(
x,
vonmises.pdf(x, kappa=0.5, mu=mu) + 1,
linestyle=":",
color="black",
label="κ=0.5",
)
rtick = [0, 1]
ax["a"].set_theta_zero_location("S")
ax["a"].spines["polar"].set_visible(False)
ax["a"].set_rgrids(rtick, ["" for _ in range(len(rtick))], fontsize=16)
gridlines = ax["a"].yaxis.get_gridlines()
gridlines[-1].set_color("k")
gridlines[-1].set_linewidth(1)
position_major = []
ax["a"].xaxis.set_major_locator(ticker.FixedLocator(position_major))
ax["a"].text(
x=0,
y=0,
s="+",
ha="center",
va="center",
color="black",
)
labels = ["0", "π/2", "π", "3π/2"]
for i, angle in enumerate([0, np.pi / 2, np.pi, 3 * np.pi / 2]):
ax["a"].text(x=angle, y=0.75, s=labels[i], ha="center", va="center", color="black")
ax["a"].legend(frameon=False)
ax["a"].set_title("von Mises distr.")
ax["b"].plot(x, vonmises.cdf(x, kappa=5, mu=mu), linestyle="-", color="black")
ax["b"].plot(x, vonmises.cdf(x, kappa=2, mu=mu), linestyle="-.", color="black")
ax["b"].plot(x, vonmises.cdf(x, kappa=1, mu=mu), linestyle="--", color="black")
ax["b"].plot(x, vonmises.cdf(x, kappa=0.5, mu=mu), linestyle=":", color="black")
ax["b"].set_xlabel("θ (rad)")
ax["b"].set_ylabel("CDF")
p = np.linspace(1e-16, 1 - 1e-16, n)
ax["c"].plot(p, vonmises.ppf(p, kappa=5, mu=mu), linestyle="-", color="black")
ax["c"].plot(p, vonmises.ppf(p, kappa=2, mu=mu), linestyle="-.", color="black")
ax["c"].plot(p, vonmises.ppf(p, kappa=1, mu=mu), linestyle="--", color="black")
ax["c"].plot(p, vonmises.ppf(p, kappa=0.5, mu=mu), linestyle=":", color="black")
ax["c"].set_xlabel("Cumulative prob.")
ax["c"].set_ylabel("Quantile (rad)")
fig.savefig("../docs/docs/images/circ-mod-vonmises.png")
Jones-Pewsey Distribution¶
In [9]:
from pycircstat2.distributions import jonespewsey
n = 200
x = np.linspace(0, 2 * np.pi, n)
mu = np.pi / 2
fig, ax = plt.subplots(
1, 2, figsize=(10, 5), subplot_kw={"projection": "polar"}, layout="constrained"
)
ax[0].plot(
x,
jonespewsey.pdf(x, kappa=1, psi=-3 / 2, mu=mu) + 1,
linestyle="--",
color="black",
label="ψ=-1.5;-1;-2",
)
ax[0].plot(
x, jonespewsey.pdf(x, kappa=1, psi=-1, mu=mu) + 1, linestyle="--", color="black"
)
ax[0].plot(
x, jonespewsey.pdf(x, kappa=1, psi=-1 / 2, mu=mu) + 1, linestyle="--", color="black"
)
ax[0].plot(
x,
jonespewsey.pdf(x, kappa=1, psi=0, mu=mu) + 1,
linestyle="-",
color="black",
label="ψ=0",
)
ax[0].plot(
x,
jonespewsey.pdf(x, kappa=1, psi=1 / 2, mu=mu) + 1,
linestyle=":",
color="black",
label="ψ=0.5;1;1.5",
)
ax[0].plot(
x, jonespewsey.pdf(x, kappa=1, psi=1, mu=mu) + 1, linestyle=":", color="black"
)
ax[0].plot(
x, jonespewsey.pdf(x, kappa=1, psi=3 / 2, mu=mu) + 1, linestyle=":", color="black"
)
ax[1].plot(
x,
jonespewsey.pdf(x, kappa=2, psi=-3 / 2, mu=mu) + 1,
linestyle="--",
color="black",
label="ψ=-1.5;-1;-1.5",
)
ax[1].plot(
x, jonespewsey.pdf(x, kappa=2, psi=-1, mu=mu) + 1, linestyle="--", color="black"
)
ax[1].plot(
x, jonespewsey.pdf(x, kappa=2, psi=-1 / 2, mu=mu) + 1, linestyle="--", color="black"
)
ax[1].plot(
x,
jonespewsey.pdf(x, kappa=2, psi=0, mu=mu) + 1,
linestyle="-",
color="black",
label="ψ=0",
)
ax[1].plot(
x,
jonespewsey.pdf(x, kappa=2, psi=1 / 2, mu=mu) + 1,
linestyle=":",
color="black",
label="ψ=0.5;1;1.5",
)
ax[1].plot(
x, jonespewsey.pdf(x, kappa=2, psi=1, mu=mu) + 1, linestyle=":", color="black"
)
ax[1].plot(
x, jonespewsey.pdf(x, kappa=2, psi=3 / 2, mu=mu) + 1, linestyle=":", color="black"
)
ks = [1, 2]
for j in range(2):
rtick = [0, 1]
ax[j].spines["polar"].set_visible(False)
ax[j].set_rgrids(rtick, ["" for _ in range(len(rtick))], fontsize=16)
gridlines = ax[j].yaxis.get_gridlines()
gridlines[-1].set_color("k")
gridlines[-1].set_linewidth(1)
position_major = []
ax[j].xaxis.set_major_locator(ticker.FixedLocator(position_major))
ax[j].text(
x=0,
y=0,
s="+",
ha="center",
va="center",
color="black",
)
labels = ["0", "π/2", "π", "3π/2"]
for i, angle in enumerate([0, np.pi / 2, np.pi, 3 * np.pi / 2]):
ax[j].text(
x=angle,
y=0.75,
s=labels[i],
ha="center",
va="center",
color="black",
)
ax[j].set_ylim(0, 2.75)
ax[j].set_title(f"κ={ks[j]}", fontsize=14)
if j == 1:
ax[j].legend(frameon=False)
fig.savefig("../docs/docs/images/circ-mod-jonespewsey.png")
Flat-topped von Mises Distribution¶
In [10]:
from pycircstat2.distributions import vonmises_flattopped
n = 100
x = np.linspace(0, 2 * np.pi, n)
mu = np.pi
kappa = 3
fig, ax = plt.subplot_mosaic(
"abc",
figsize=(16, 5),
per_subplot_kw={"a": {"projection": "polar"}},
layout="constrained",
)
ax["a"].plot(
x,
vonmises_flattopped.pdf(x, nu=0, kappa=kappa, mu=mu) + 1,
linestyle="-",
color="black",
label="ν=0",
)
ax["a"].plot(
x,
vonmises_flattopped.pdf(x, nu=-0.999, kappa=kappa, mu=mu) + 1,
linestyle=":",
color="black",
label="ν=-0.999",
)
ax["a"].plot(
x,
vonmises_flattopped.pdf(x, nu=-0.75, kappa=kappa, mu=mu) + 1,
linestyle=":",
color="black",
label="ν=-0.75",
)
ax["a"].plot(
x,
vonmises_flattopped.pdf(x, nu=-0.5, kappa=kappa, mu=mu) + 1,
linestyle=":",
color="black",
label="ν=-0.5",
)
ax["a"].plot(
x,
vonmises_flattopped.pdf(x, nu=0.999, kappa=kappa, mu=mu) + 1,
linestyle="--",
color="black",
label="ν=0.999",
)
ax["a"].plot(
x,
vonmises_flattopped.pdf(x, nu=0.75, kappa=kappa, mu=mu) + 1,
linestyle="--",
color="black",
label="ν=0.75",
)
ax["a"].plot(
x,
vonmises_flattopped.pdf(x, nu=0.5, kappa=kappa, mu=mu) + 1,
linestyle="--",
color="black",
label="ν=0.5",
)
rtick = [0, 1]
ax["a"].set_theta_zero_location("S")
ax["a"].spines["polar"].set_visible(False)
ax["a"].set_rgrids(rtick, ["" for _ in range(len(rtick))], fontsize=16)
gridlines = ax["a"].yaxis.get_gridlines()
gridlines[-1].set_color("k")
gridlines[-1].set_linewidth(1)
position_major = []
ax["a"].xaxis.set_major_locator(ticker.FixedLocator(position_major))
ax["a"].text(
x=0,
y=0,
s="+",
ha="center",
va="center",
color="black",
)
labels = ["0", "π/2", "π", "3π/2"]
for i, angle in enumerate([0, np.pi / 2, np.pi, 3 * np.pi / 2]):
ax["a"].text(x=angle, y=0.75, s=labels[i], ha="center", va="center", color="black")
ax["a"].legend(frameon=False)
ax["a"].set_title("Flat-topped von Mises distr.")
ax["b"].plot(
x,
vonmises_flattopped.cdf(x, nu=0, kappa=kappa, mu=mu),
linestyle="-",
color="black",
)
ax["b"].plot(
x,
vonmises_flattopped.cdf(x, nu=-0.999, kappa=kappa, mu=mu),
linestyle=":",
color="black",
)
ax["b"].plot(
x,
vonmises_flattopped.cdf(x, nu=-0.75, kappa=kappa, mu=mu),
linestyle=":",
color="black",
)
ax["b"].plot(
x,
vonmises_flattopped.cdf(x, nu=-0.5, kappa=kappa, mu=mu),
linestyle=":",
color="black",
)
ax["b"].plot(
x,
vonmises_flattopped.cdf(x, nu=0.999, kappa=kappa, mu=mu),
linestyle="--",
color="black",
)
ax["b"].plot(
x,
vonmises_flattopped.cdf(x, nu=0.75, kappa=kappa, mu=mu),
linestyle="--",
color="black",
)
ax["b"].plot(
x,
vonmises_flattopped.cdf(x, nu=0.5, kappa=kappa, mu=mu),
linestyle="--",
color="black",
)
ax["b"].set_xlabel("θ (rad)")
ax["b"].set_ylabel("CDF")
p = np.linspace(1e-16, 1 - 1e-16, n)
ax["c"].plot(
p,
vonmises_flattopped.ppf(p, nu=0, kappa=kappa, mu=mu),
linestyle="-",
color="black",
)
ax["c"].plot(
p,
vonmises_flattopped.ppf(p, nu=-0.999, kappa=kappa, mu=mu),
linestyle=":",
color="black",
)
ax["c"].plot(
p,
vonmises_flattopped.ppf(p, nu=-0.75, kappa=kappa, mu=mu),
linestyle=":",
color="black",
)
ax["c"].plot(
p,
vonmises_flattopped.ppf(p, nu=-0.5, kappa=kappa, mu=mu),
linestyle=":",
color="black",
)
ax["c"].plot(
p,
vonmises_flattopped.ppf(p, nu=0.999, kappa=kappa, mu=mu),
linestyle="--",
color="black",
)
ax["c"].plot(
p,
vonmises_flattopped.ppf(p, nu=0.75, kappa=kappa, mu=mu),
linestyle="--",
color="black",
)
ax["c"].plot(
p,
vonmises_flattopped.ppf(p, nu=0.5, kappa=kappa, mu=mu),
linestyle="--",
color="black",
)
ax["c"].set_xlabel("Cumulative prob.")
ax["c"].set_ylabel("Quantile (rad)")
fig.savefig("../docs/docs/images/circ-mod-vonmises-flat-topped.png")
Asymmetric distributions¶
Sine-Skewed Jones-Pewsey Distribution¶
In [11]:
from pycircstat2.distributions import jonespewsey_sineskewed
n = 200
x = np.linspace(0, 2 * np.pi, n)
mu = np.pi / 2
fig, ax = plt.subplot_mosaic(
mosaic="abc",
figsize=(15, 5),
subplot_kw={"projection": "polar"},
layout="constrained",
)
ax["a"].plot(
x,
jonespewsey_sineskewed.pdf(x, kappa=2, psi=-1, lmbd=0, xi=mu) + 1,
linestyle="-",
color="black",
)
ax["a"].plot(
x,
jonespewsey_sineskewed.pdf(x, kappa=2, psi=-1, lmbd=0.25, xi=mu) + 1,
linestyle="--",
color="black",
)
ax["a"].plot(
x,
jonespewsey_sineskewed.pdf(x, kappa=2, psi=-1, lmbd=0.5, xi=mu) + 1,
linestyle="--",
color="black",
)
ax["a"].plot(
x,
jonespewsey_sineskewed.pdf(x, kappa=2, psi=-1, lmbd=0.75, xi=mu) + 1,
linestyle="--",
color="black",
)
ax["a"].plot(
x,
jonespewsey_sineskewed.pdf(x, kappa=2, psi=-1, lmbd=1, xi=mu) + 1,
linestyle=":",
color="black",
)
####
ax["b"].plot(
x,
jonespewsey_sineskewed.pdf(x, kappa=2, psi=0, lmbd=0, xi=mu) + 1,
linestyle="-",
color="black",
label="λ=0",
)
ax["b"].plot(
x,
jonespewsey_sineskewed.pdf(x, kappa=2, psi=0, lmbd=0.25, xi=mu) + 1,
linestyle="--",
color="black",
label="λ=0.25",
)
ax["b"].plot(
x,
jonespewsey_sineskewed.pdf(x, kappa=2, psi=0, lmbd=0.5, xi=mu) + 1,
linestyle="--",
color="black",
label="λ=0.5",
)
ax["b"].plot(
x,
jonespewsey_sineskewed.pdf(x, kappa=2, psi=0, lmbd=0.75, xi=mu) + 1,
linestyle="--",
color="black",
label="λ=0.75",
)
ax["b"].plot(
x,
jonespewsey_sineskewed.pdf(x, kappa=2, psi=0, lmbd=1, xi=mu) + 1,
linestyle=":",
color="black",
label="λ=1",
)
####
ax["c"].plot(
x,
jonespewsey_sineskewed.pdf(x, kappa=2, psi=1, lmbd=0, xi=mu) + 1,
linestyle="-",
color="black",
)
ax["c"].plot(
x,
jonespewsey_sineskewed.pdf(x, kappa=2, psi=1, lmbd=0.25, xi=mu) + 1,
linestyle="--",
color="black",
)
ax["c"].plot(
x,
jonespewsey_sineskewed.pdf(x, kappa=2, psi=1, lmbd=0.5, xi=mu) + 1,
linestyle="--",
color="black",
)
ax["c"].plot(
x,
jonespewsey_sineskewed.pdf(x, kappa=2, psi=1, lmbd=0.75, xi=mu) + 1,
linestyle="--",
color="black",
)
ax["c"].plot(
x,
jonespewsey_sineskewed.pdf(x, kappa=2, psi=1, lmbd=1, xi=mu) + 1,
linestyle=":",
color="black",
)
psis = [-1, 0, 1]
for j, s in enumerate(["a", "b", "c"]):
rtick = [0, 1]
ax[s].spines["polar"].set_visible(False)
ax[s].set_rgrids(rtick, ["" for _ in range(len(rtick))], fontsize=16)
gridlines = ax[s].yaxis.get_gridlines()
gridlines[-1].set_color("k")
gridlines[-1].set_linewidth(1)
position_major = []
ax[s].xaxis.set_major_locator(ticker.FixedLocator(position_major))
ax[s].text(
x=0,
y=0,
s="+",
ha="center",
va="center",
color="black",
)
labels = ["0", "π/2", "π", "3π/2"]
for i, angle in enumerate([0, np.pi / 2, np.pi, 3 * np.pi / 2]):
ax[s].text(
x=angle,
y=0.75,
s=labels[i],
ha="center",
va="center",
color="black",
)
ax[s].set_ylim(0, 2.2)
ax[s].set_title(f"ψ={psis[j]}", fontsize=14)
if j == 1:
ax[s].legend(frameon=False)
fig.savefig("../docs/docs/images/circ-mod-jonespewsey-sineskewed.png")
Asymmetric Extended Jones-Pewsey Distribution¶
In [12]:
from pycircstat2.distributions import jonespewsey_asym
n = 200
x = np.linspace(0, 2 * np.pi, n)
mu = np.pi / 2
fig, ax = plt.subplot_mosaic(
"abc", figsize=(15, 5), subplot_kw={"projection": "polar"}, layout="constrained"
)
### left
ax["a"].plot(
x,
jonespewsey_asym.pdf(x, kappa=2, psi=-1, nu=0, xi=mu) + 1,
linestyle="-",
color="black",
)
ax["a"].plot(
x,
jonespewsey_asym.pdf(x, kappa=2, psi=-1, nu=0.25, xi=mu) + 1,
linestyle="--",
color="black",
)
ax["a"].plot(
x,
jonespewsey_asym.pdf(x, kappa=2, psi=-1, nu=0.5, xi=mu) + 1,
linestyle="--",
color="black",
)
ax["a"].plot(
x,
jonespewsey_asym.pdf(x, kappa=2, psi=-1, nu=0.75, xi=mu) + 1,
linestyle="--",
color="black",
)
ax["a"].plot(
x,
jonespewsey_asym.pdf(x, kappa=2, psi=-1, nu=0.999, xi=mu) + 1,
linestyle=":",
color="black",
)
#### mid
ax["b"].plot(
x,
jonespewsey_asym.pdf(x, kappa=2, psi=0, nu=0, xi=mu) + 1,
linestyle="-",
color="black",
)
ax["b"].plot(
x,
jonespewsey_asym.pdf(x, kappa=2, psi=0, nu=0.25, xi=mu) + 1,
linestyle="--",
color="black",
)
ax["b"].plot(
x,
jonespewsey_asym.pdf(x, kappa=2, psi=0, nu=0.5, xi=mu) + 1,
linestyle="--",
color="black",
)
ax["b"].plot(
x,
jonespewsey_asym.pdf(x, kappa=2, psi=0, nu=0.75, xi=mu) + 1,
linestyle="--",
color="black",
)
ax["b"].plot(
x,
jonespewsey_asym.pdf(x, kappa=2, psi=0, nu=0.999, xi=mu) + 1,
linestyle=":",
color="black",
)
#### right
ax["c"].plot(
x,
jonespewsey_asym.pdf(x, kappa=2, psi=1, nu=0, xi=mu) + 1,
linestyle="-",
color="black",
label="ν=0",
)
ax["c"].plot(
x,
jonespewsey_asym.pdf(x, kappa=2, psi=1, nu=0.25, xi=mu) + 1,
linestyle="--",
color="black",
label="ν=0.25",
)
ax["c"].plot(
x,
jonespewsey_asym.pdf(x, kappa=2, psi=1, nu=0.5, xi=mu) + 1,
linestyle="--",
color="black",
label="ν=0.5",
)
ax["c"].plot(
x,
jonespewsey_asym.pdf(x, kappa=2, psi=1, nu=0.75, xi=mu) + 1,
linestyle="--",
color="black",
label="ν=0.75",
)
ax["c"].plot(
x,
jonespewsey_asym.pdf(x, kappa=2, psi=1, nu=0.999, xi=mu) + 1,
linestyle=":",
color="black",
label="ν=0.999",
)
psis = [-1, 0, 1]
for j, s in enumerate(["a", "b", "c"]):
rtick = [0, 1]
ax[s].spines["polar"].set_visible(False)
ax[s].set_rgrids(rtick, ["" for _ in range(len(rtick))], fontsize=16)
gridlines = ax[s].yaxis.get_gridlines()
gridlines[-1].set_color("k")
gridlines[-1].set_linewidth(1)
position_major = []
ax[s].xaxis.set_major_locator(ticker.FixedLocator(position_major))
ax[s].text(
x=0,
y=0,
s="+",
ha="center",
va="center",
color="black",
)
labels = ["0", "π/2", "π", "3π/2"]
for i, angle in enumerate([0, np.pi / 2, np.pi, 3 * np.pi / 2]):
ax[s].text(
x=angle,
y=0.75,
s=labels[i],
ha="center",
va="center",
color="black",
)
ax[s].set_ylim(0, 2.75)
ax[s].set_title(f"ψ={psis[j]}", fontsize=14)
if j == 2:
ax[s].legend(frameon=False)
fig.savefig("../docs/docs/images/circ-mod-jonespewsey-asym.png")
Inverse Batschelet Distribution¶
In [13]:
from pycircstat2.distributions import inverse_batschelet
n = 400
x = np.linspace(0, 2 * np.pi, n)
mu = np.pi / 2
fig, ax = plt.subplot_mosaic(
"abc", figsize=(15, 5), subplot_kw={"projection": "polar"}, layout="constrained"
)
ax["a"].plot(
x,
inverse_batschelet.pdf(x, kappa=2, nu=0, lmbd=-1.0, xi=mu) + 1,
linestyle="--",
color="black",
)
ax["a"].plot(
x,
inverse_batschelet.pdf(x, kappa=2, nu=0, lmbd=-0.75, xi=mu) + 1,
linestyle="--",
color="black",
)
ax["a"].plot(
x,
inverse_batschelet.pdf(x, kappa=2, nu=0, lmbd=-0.5, xi=mu) + 1,
linestyle="--",
color="black",
)
ax["a"].plot(
x,
inverse_batschelet.pdf(x, kappa=2, nu=0, lmbd=-0.25, xi=mu) + 1,
linestyle="--",
color="black",
)
ax["a"].plot(
x,
inverse_batschelet.pdf(x, kappa=2, nu=0, lmbd=0, xi=mu) + 1,
linestyle="-",
color="black",
)
ax["a"].plot(
x,
inverse_batschelet.pdf(x, kappa=2, nu=0, lmbd=0.25, xi=mu) + 1,
linestyle=":",
color="black",
)
ax["a"].plot(
x,
inverse_batschelet.pdf(x, kappa=2, nu=0, lmbd=0.5, xi=mu) + 1,
linestyle=":",
color="black",
)
ax["a"].plot(
x,
inverse_batschelet.pdf(x, kappa=2, nu=0, lmbd=0.75, xi=mu) + 1,
linestyle=":",
color="black",
)
ax["a"].plot(
x,
inverse_batschelet.pdf(x, kappa=2, nu=0, lmbd=1, xi=mu) + 1,
linestyle=":",
color="black",
)
ax["b"].plot(
x,
inverse_batschelet.pdf(x, kappa=2, nu=0.5, lmbd=-1, xi=mu) + 1,
linestyle="--",
color="black",
label="λ=-1;-0.75;-0.5;-0.25",
)
ax["b"].plot(
x,
inverse_batschelet.pdf(x, kappa=2, nu=0.5, lmbd=-0.75, xi=mu) + 1,
linestyle="--",
color="black",
)
ax["b"].plot(
x,
inverse_batschelet.pdf(x, kappa=2, nu=0.5, lmbd=-0.5, xi=mu) + 1,
linestyle="--",
color="black",
)
ax["b"].plot(
x,
inverse_batschelet.pdf(x, kappa=2, nu=0.5, lmbd=-0.25, xi=mu) + 1,
linestyle="--",
color="black",
)
ax["b"].plot(
x,
inverse_batschelet.pdf(x, kappa=2, nu=0.5, lmbd=0, xi=mu) + 1,
linestyle="-",
color="black",
label="λ=0",
)
ax["b"].plot(
x,
inverse_batschelet.pdf(x, kappa=2, nu=0.5, lmbd=0.25, xi=mu) + 1,
linestyle=":",
color="black",
label="λ=0.25;0.5;0.75;1",
)
ax["b"].plot(
x,
inverse_batschelet.pdf(x, kappa=2, nu=0.5, lmbd=0.5, xi=mu) + 1,
linestyle=":",
color="black",
)
ax["b"].plot(
x,
inverse_batschelet.pdf(x, kappa=2, nu=0.5, lmbd=0.75, xi=mu) + 1,
linestyle=":",
color="black",
)
ax["b"].plot(
x,
inverse_batschelet.pdf(x, kappa=2, nu=0.5, lmbd=1, xi=mu) + 1,
linestyle=":",
color="black",
)
ax["c"].plot(
x,
inverse_batschelet.pdf(x, kappa=2, nu=1, lmbd=-1, xi=mu) + 1,
linestyle="--",
color="black",
label="λ=-1;-0.75;-1;-0.25",
)
ax["c"].plot(
x,
inverse_batschelet.pdf(x, kappa=2, nu=1, lmbd=-0.75, xi=mu) + 1,
linestyle="--",
color="black",
)
ax["c"].plot(
x,
inverse_batschelet.pdf(x, kappa=2, nu=1, lmbd=-0.5, xi=mu) + 1,
linestyle="--",
color="black",
)
ax["c"].plot(
x,
inverse_batschelet.pdf(x, kappa=2, nu=1, lmbd=-0.25, xi=mu) + 1,
linestyle="--",
color="black",
)
ax["c"].plot(
x,
inverse_batschelet.pdf(x, kappa=2, nu=1, lmbd=0, xi=mu) + 1,
linestyle="-",
color="black",
label="λ=0",
)
ax["c"].plot(
x,
inverse_batschelet.pdf(x, kappa=2, nu=1, lmbd=0.25, xi=mu) + 1,
linestyle=":",
color="black",
label="λ=0.25;1;0.75;1",
)
ax["c"].plot(
x,
inverse_batschelet.pdf(x, kappa=2, nu=1, lmbd=0.5, xi=mu) + 1,
linestyle=":",
color="black",
)
ax["c"].plot(
x,
inverse_batschelet.pdf(x, kappa=2, nu=1, lmbd=0.75, xi=mu) + 1,
linestyle=":",
color="black",
)
ax["c"].plot(
x,
inverse_batschelet.pdf(x, kappa=2, nu=1, lmbd=1, xi=mu) + 1,
linestyle=":",
color="black",
)
nus = [0, 0.5, 1]
for j, s in enumerate(["a", "b", "c"]):
rtick = [0, 1]
ax[s].spines["polar"].set_visible(False)
ax[s].set_rgrids(rtick, ["" for _ in range(len(rtick))], fontsize=16)
gridlines = ax[s].yaxis.get_gridlines()
gridlines[-1].set_color("k")
gridlines[-1].set_linewidth(1)
position_major = []
ax[s].xaxis.set_major_locator(ticker.FixedLocator(position_major))
ax[s].text(
x=0,
y=0,
s="+",
ha="center",
va="center",
color="black",
)
labels = ["0", "π/2", "π", "3π/2"]
for i, angle in enumerate([0, np.pi / 2, np.pi, 3 * np.pi / 2]):
ax[s].text(
x=angle,
y=0.75,
s=labels[i],
ha="center",
va="center",
color="black",
)
ax[s].set_ylim(0, 2.75)
ax[s].set_title(f"ν={nus[j]}", fontsize=14)
if j == 1:
ax[s].legend(frameon=False)
fig.savefig("../docs/docs/images/circ-mod-inverse-batschelet.png")
Wrapped Stable Distribution¶
In [14]:
from pycircstat2.distributions import wrapstable
theta = np.linspace(-np.pi, np.pi, 1000)
fig_linear, ax_linear = plt.subplot_mosaic("abc", figsize=(15, 5), layout="constrained")
## Figure 2.5 (Jammalamadaka and SenGupta, 2001)
gamma = 1
delta = 0
alphas = [0.4, 0.8, 1.2]
beta = 0
for alpha in alphas:
pdf_values = wrapstable.pdf(theta, delta, alpha, beta, gamma)
ax_linear["a"].plot(theta, pdf_values, label=f"α={alpha}, γ=1")
alpha = 2
gammas = [0.5, 1, 1.2]
for gamma in gammas:
pdf_values = wrapstable.pdf(theta, delta, alpha, beta, gamma)
ax_linear["a"].plot(theta, pdf_values, label=f"α=2, γ={gamma}", linestyle="--")
## Figure 1 (Pewsey, 2008)
gamma = 1
delta = 0
alpha = 1
betas = np.arange(0, 1, 0.2)
for beta in betas:
pdf_values = wrapstable.pdf(theta, delta, alpha, beta, gamma)
ax_linear["b"].plot(theta, pdf_values, label=f"β={beta.round(2)}")
gamma = 1
delta = 0
alpha = 0.5
betas = np.arange(0, 1, 0.2)
for beta in betas:
pdf_values = wrapstable.pdf(theta, delta, alpha, beta, gamma)
ax_linear["c"].plot(theta, pdf_values, label=f"β={beta.round(2)}")
for j, s in enumerate(["a", "b", "c"]):
ax_linear[s].set_ylim(0, 1.1)
ax_linear[s].set_xlim(-np.pi, np.pi)
ax_linear[s].set_xlabel("θ")
ax_linear[s].set_ylabel("f(θ)")
ax_linear[s].spines[["top", "right"]].set_visible(False)
ax_linear[s].legend(frameon=False)
ax_linear["a"].set_title("δ=0, β=0", fontsize=14)
ax_linear["b"].set_title("δ=0, α=1, γ=1", fontsize=14)
ax_linear["c"].set_title("δ=0, α=0.5, γ=1", fontsize=14)
# Polar version
fig_polar, ax_polar = plt.subplot_mosaic(
"abc", figsize=(15, 5), subplot_kw={"projection": "polar"}, layout="constrained"
)
## Figure 2.5 (Jammalamadaka and SenGupta, 2001)
delta = np.pi / 2
gamma = 1
alphas = [0.4, 0.8, 1.2]
beta = 0
for alpha in alphas:
pdf_values = wrapstable.pdf(theta, delta, alpha, beta, gamma)
ax_polar["a"].plot(theta, pdf_values + 1, label=f"α={alpha}, γ=1")
alpha = 2
gammas = [0.5, 1, 1.2]
for gamma in gammas:
pdf_values = wrapstable.pdf(theta, delta, alpha, beta, gamma)
ax_polar["a"].plot(theta, pdf_values + 1, label=f"α=2, γ={gamma}", linestyle="--")
## Figure 1 (Pewsey, 2008)
delta = np.pi / 2
gamma = 1
alpha = 1
betas = np.arange(0, 1, 0.2)
for beta in betas:
pdf_values = wrapstable.pdf(theta, delta, alpha, beta, gamma)
ax_polar["b"].plot(theta, pdf_values + 1, label=f"β={beta.round(2)}")
gamma = 1
delta = np.pi / 2
alpha = 0.5
betas = np.arange(0, 1, 0.2)
for beta in betas:
pdf_values = wrapstable.pdf(theta, delta, alpha, beta, gamma)
ax_polar["c"].plot(theta, pdf_values + 1, label=f"β={beta.round(2)}")
for j, s in enumerate(["a", "b", "c"]):
rtick = [0, 1]
ax_polar[s].spines["polar"].set_visible(False)
ax_polar[s].set_rgrids(rtick, ["" for _ in range(len(rtick))], fontsize=16)
gridlines = ax_polar[s].yaxis.get_gridlines()
gridlines[-1].set_color("k")
gridlines[-1].set_linewidth(1)
position_major = []
ax_polar[s].xaxis.set_major_locator(ticker.FixedLocator(position_major))
ax_polar[s].text(
x=0,
y=0,
s="+",
ha="center",
va="center",
color="black",
)
labels = ["0", "π/2", "π", "3π/2"]
for i, angle in enumerate([0, np.pi / 2, np.pi, 3 * np.pi / 2]):
ax_polar[s].text(
x=angle,
y=0.75,
s=labels[i],
ha="center",
va="center",
color="black",
)
ax_polar[s].set_ylim(0, 2.5)
ax_polar[s].legend(frameon=False)
ax_polar["a"].set_title("δ=π/2, β=0", fontsize=14)
ax_polar["b"].set_title("δ=π/2, α=1, γ=1", fontsize=14)
ax_polar["c"].set_title("δ=π/2, α=0.5, γ=1", fontsize=14)
fig_polar.savefig("../docs/docs/images/circ-mod-wrapstable.png")
See also¶
Circular models in pycircstat2 mainly follows the implementations described in Pewsey, et al. (2014). For more examples can be found in notebook B3-Pewsey-2014).
In [15]:
%load_ext watermark
%watermark --time --date --timezone --updated --python --iversions --watermark -p pycircstat2
Last updated: 2025-01-31 14:49:02CET Python implementation: CPython Python version : 3.12.8 IPython version : 8.31.0 pycircstat2: 0.1.8 matplotlib: 3.10.0 numpy : 2.2.2 Watermark: 2.5.0