Correlation Regression Tests¶

In [1]:

import numpy as np
from matplotlib import pyplot as plt
import yaml

%matplotlib inline
%config InlineBackend.figure_format = 'retina'

import numpy as np from matplotlib import pyplot as plt import yaml %matplotlib inline %config InlineBackend.figure_format = 'retina'

Test Correlations in the Hammersley Generator¶

---

The Hammerlsey sequence should produce random numbers on (0,1) that are less correlated than the basic random generator:

In [2]:

from distgen.dist import random_generator

rands = random_generator((4, 1000), sequence="hammersley")
cov = np.cov(rands)

print("Covariance of Hammersley samples:")
print(cov)
plt.plot(rands[0, :], rands[1, :], ".");

from distgen.dist import random_generator rands = random_generator((4, 1000), sequence="hammersley") cov = np.cov(rands) print("Covariance of Hammersley samples:") print(cov) plt.plot(rands[0, :], rands[1, :], ".");

Covariance of Hammersley samples:
[[ 8.33224805e-02 -3.68343309e-04 -6.68187387e-05  4.25010191e-04]
 [-3.68343309e-04  8.33878181e-02  7.56326104e-05  2.25221076e-04]
 [-6.68187387e-05  7.56326104e-05  8.34060346e-02  4.12948253e-04]
 [ 4.25010191e-04  2.25221076e-04  4.12948253e-04  8.32500833e-02]]

In [3]:

rands = random_generator((4, 1000), sequence="pseudo")
cov = np.cov(rands)
print("Covariance of Rand samples:")
print(cov)
plt.plot(rands[0, :], rands[1, :], ".")

rands = random_generator((4, 1000), sequence="pseudo") cov = np.cov(rands) print("Covariance of Rand samples:") print(cov) plt.plot(rands[0, :], rands[1, :], ".")

Covariance of Rand samples:
[[ 0.08391168 -0.00307985  0.00329383 -0.00347822]
 [-0.00307985  0.08696711 -0.00257096  0.00138202]
 [ 0.00329383 -0.00257096  0.08786434  0.00127328]
 [-0.00347822  0.00138202  0.00127328  0.08356148]]

Out[3]:

[<matplotlib.lines.Line2D at 0x7f3e9fe8ed20>]

Radial Distributions¶

---

Test correlation in sin(theta)cos(theta)¶

In [4]:

import math

rands = random_generator((2, 10000), sequence="hammersley")
np.sum(np.cos(2 * math.pi) * rands[1, :] * np.sin(2 * math.pi) * rands[1, :])

import math rands = random_generator((2, 10000), sequence="hammersley") np.sum(np.cos(2 * math.pi) * rands[1, :] * np.sin(2 * math.pi) * rands[1, :])

Out[4]:

np.float64(-8.163903819533452e-13)

In [5]:

rands = random_generator((2, 10000), sequence="pseudo")
np.sum(np.cos(2 * math.pi) * rands[1, :] * np.sin(2 * math.pi) * rands[1, :])

rands = random_generator((2, 10000), sequence="pseudo") np.sum(np.cos(2 * math.pi) * rands[1, :] * np.sin(2 * math.pi) * rands[1, :])

Out[5]:

np.float64(-8.179767467773059e-13)

Generate x,y for uniform dist and check correlation¶

In [6]:

R = 1
N = 100000

rands1 = random_generator((2, N), sequence="hammersley")
rands2 = random_generator((2, N), sequence="pseudo")
rands3 = np.linspace(0, 1, N)

rs = R * np.sqrt(rands1[0, :])

xs1 = rs * np.cos(2 * math.pi * rands1[1, :])
ys1 = rs * np.sin(2 * math.pi * rands1[1, :])

xs2 = rs * np.cos(2 * math.pi * rands2[1, :])
ys2 = rs * np.sin(2 * math.pi * rands2[1, :])

xs3 = rs * np.cos(2 * math.pi * rands3)
ys3 = rs * np.sin(2 * math.pi * rands3)

plt.plot(xs1, ys1, ".");

R = 1 N = 100000 rands1 = random_generator((2, N), sequence="hammersley") rands2 = random_generator((2, N), sequence="pseudo") rands3 = np.linspace(0, 1, N) rs = R * np.sqrt(rands1[0, :]) xs1 = rs * np.cos(2 * math.pi * rands1[1, :]) ys1 = rs * np.sin(2 * math.pi * rands1[1, :]) xs2 = rs * np.cos(2 * math.pi * rands2[1, :]) ys2 = rs * np.sin(2 * math.pi * rands2[1, :]) xs3 = rs * np.cos(2 * math.pi * rands3) ys3 = rs * np.sin(2 * math.pi * rands3) plt.plot(xs1, ys1, ".");

In [7]:

print(np.mean((xs1 - xs1.mean()) * (ys1 - ys1.mean())))
print(np.mean((xs2 - xs2.mean()) * (ys2 - ys2.mean())))
print(np.mean((xs3 - xs3.mean()) * (ys3 - ys3.mean())))

print(np.mean((xs1 - xs1.mean()) * (ys1 - ys1.mean()))) print(np.mean((xs2 - xs2.mean()) * (ys2 - ys2.mean()))) print(np.mean((xs3 - xs3.mean()) * (ys3 - ys3.mean())))

-9.407807239222166e-07
5.846480065920893e-05
-9.400468250214544e-07

In [8]:

sigma = np.cov(rands1)
v, V = np.linalg.eig(sigma)

# np.matmul(np.matmul(V.T, sigma), V)

randsp = np.matmul(V.T, rands1)

np.cov(randsp)
rs = R * np.sqrt(randsp[0, :])

xsp = rs * np.cos(2 * math.pi * randsp[1, :])
ysp = rs * np.sin(2 * math.pi * randsp[1, :])
print(np.mean((xsp - xsp.mean()) * (ysp - ysp.mean())))

sigma = np.cov(rands1) v, V = np.linalg.eig(sigma) # np.matmul(np.matmul(V.T, sigma), V) randsp = np.matmul(V.T, rands1) np.cov(randsp) rs = R * np.sqrt(randsp[0, :]) xsp = rs * np.cos(2 * math.pi * randsp[1, :]) ysp = rs * np.sin(2 * math.pi * randsp[1, :]) print(np.mean((xsp - xsp.mean()) * (ysp - ysp.mean())))

0.008920000391575414

In [9]:

input_str = """
n_particle: 100000
species: electron
random: 
  type: hammersley
total_charge:
  value: 1
  units: pC
start:
  type: time
r_dist:
  max_r:
    units: m
    value: 2
  min_r:
    units: m
    value: 0
  type: radial_uniform
"""

yaml.safe_load(input_str)

input_str = """ n_particle: 100000 species: electron random: type: hammersley total_charge: value: 1 units: pC start: type: time r_dist: max_r: units: m value: 2 min_r: units: m value: 0 type: radial_uniform """ yaml.safe_load(input_str)

Out[9]:

{'n_particle': 100000,
 'species': 'electron',
 'random': {'type': 'hammersley'},
 'total_charge': {'value': 1, 'units': 'pC'},
 'start': {'type': 'time'},
 'r_dist': {'max_r': {'units': 'm', 'value': 2},
  'min_r': {'units': 'm', 'value': 0},
  'type': 'radial_uniform'}}

In [10]:

from distgen import Generator

gen = Generator(input_str)
gen.run()
xs = gen.particles["x"]
ys = gen.particles["y"]

gen.particles.cov("x", "y")

from distgen import Generator gen = Generator(input_str) gen.run() xs = gen.particles["x"] ys = gen.particles["y"] gen.particles.cov("x", "y")

Out[10]:

array([[ 1.00001000e+00, -3.76331398e-06],
       [-3.76331398e-06,  1.00001000e+00]])

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