I noticed something strange this day when I plotted the Wigner function of a coherent state using the open source quantum toolbox QuTiP in python.
When I do the plot I noticed these strange patterns just around the edge of the plot that are not supposed to be there. I believe it's just some sort of numerical error but I don't know how I can get rid or minimize them or most impartant: what's causing them.
Here is the code
# import packages
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.colors as colors
import matplotlib as mpl
from matplotlib import cm
from qutip import *
N = 60 # number of levels in Hilbert space
# density matrix of a coherent state
rho_coherent = coherent_dm(N, 1-1j)
X = np.linspace(-3, 3, 300)
Y = np.linspace(-3, 3, 300)
# Wigner function
W = wigner(rho_coherent, X, Y, 'iterative', 2)
X, Y = np.meshgrid(X, Y)
# Color Normalization
class MidpointNormalize(colors.Normalize):
def __init__(self, vmin=None, vmax=None, midpoint=None, clip=False):
self.midpoint = midpoint
colors.Normalize.__init__(self, vmin, vmax, clip)
def __call__(self, value, clip=None):
x, y = [self.vmin, self.midpoint, self.vmax], [0, 0.5, 1]
return np.ma.masked_array(np.interp(value, x, y))
# contour plot
plt.subplot(111, aspect='equal')
plt.contourf(X, Y, W, 100, cmap = cm.RdBu_r, norm = MidpointNormalize(midpoint=0.))
plt.show()
and here is the plot
The blue spots as you can clearly see that's around the edges are not supposed to be there! The blue spots indicate that the Wigner function is negative at that point, but a coherent state should have a Wigner function thats positive everywhere!
I also noticed that when I reduce the linspace steps from 300 to 100 the blue parts disappear.
Would appreciate very much if someone can explain what's causing this problem to appear.
This is simply due to truncation. When using a finite number of modes (in your case N=60), the Wigner function will go negative at some point.
Reducing the linspace steps brings the negative regions you see on the plot into the zero value increment and displays these regions as zero. Reducing the linspace steps is probably the best solution to your problem. Your plot will only be as accurate as the errors introduced by truncation, so simply reduce the resolution until those errors disappear.
Related
I would like to create a version of this 2D binned "color map" with smoothed colors.
I am not even sure this would be the correct nomenclature for the plot, but, essentially, I want my figure to be color coded by the median values of a third variable for points that reside in each defined bin of my (X, Y) space.
Even though I am able to accomplish that to a certain degree (see example), I would like to find a way to create a version of the same plot with a smoothed color gradient. That would allow me to visualize the overall behavior of my distribution.
I tried ideas described here: Smoothing 2D map in python
and here: Python: binned_statistic_2d mean calculation ignoring NaNs in data
as well as links therein, but could not find a clear solution to the problem.
This is what I have so far:
import numpy as np
import matplotlib.pyplot as plt
from matplotlib import cm
from scipy.stats import binned_statistic_2d
import random
random.seed(999)
x = np.random.normal (0,10,5000)
y = np.random.normal (0,10,5000)
z = np.random.uniform(0,10,5000)
fig = plt.figure(figsize=(20, 20))
plt.rcParams.update({'font.size': 10})
ax = fig.add_subplot(3,3,1)
ax.set_axisbelow(True)
plt.grid(b=True, lw=0.5, zorder=-1)
x_bins = np.arange(-50., 50.5, 1.)
y_bins = np.arange(-50., 50.5, 1.)
cmap = plt.cm.get_cmap('jet_r',1000) #just a colormap
ret = binned_statistic_2d(x, y, z, statistic=np.median, bins=[x_bins, y_bins]) # Bin (X, Y) and create a map of the medians of "Colors"
plt.imshow(ret.statistic.T, origin='bottom', extent=(-50, 50, -50, 50), cmap=cmap)
plt.xlim(-40,40)
plt.ylim(-40,40)
plt.xlabel("X", fontsize=15)
plt.ylabel("Y", fontsize=15)
ax.set_yticks([-40,-30,-20,-10,0,10,20,30,40])
bounds = np.arange(2.0, 20.0, 1.0)
plt.colorbar(ticks=bounds, label="Color", fraction=0.046, pad=0.04)
# save plots
plt.savefig("Whatever_name.png", bbox_inches='tight')
Which produces the following image (from random data):
Therefore, the simple question would be: how to smooth these colors?
Thanks in advance!
PS: sorry for excessive coding, but I believe a clear visualization is crucial for this particular problem.
Thanks to everyone who viewed this issue and tried to help!
I ended up being able to solve my own problem. In the end, it was all about image smoothing with Gaussian Kernel.
This link: Gaussian filtering a image with Nan in Python gave me the insight for the solution.
I, basically, implemented the exactly same code, but, in the end, mapped the previously known NaN pixels from the original 2D array to the resulting smoothed version. Unlike the solution from the link, my version does NOT fill NaN pixels with some value derived from the pixels around. Or, it does, but then I erase those again.
Here is the final figure produced for the example I provided:
Final code, for reference, for those who might need in the future:
import numpy as np
import matplotlib.pyplot as plt
from matplotlib import cm
from scipy.stats import binned_statistic_2d
import scipy.stats as st
import scipy.ndimage
import scipy as sp
import random
random.seed(999)
x = np.random.normal (0,10,5000)
y = np.random.normal (0,10,5000)
z = np.random.uniform(0,10,5000)
fig = plt.figure(figsize=(20, 20))
plt.rcParams.update({'font.size': 10})
ax = fig.add_subplot(3,3,1)
ax.set_axisbelow(True)
plt.grid(b=True, lw=0.5, zorder=-1)
x_bins = np.arange(-50., 50.5, 1.)
y_bins = np.arange(-50., 50.5, 1.)
cmap = plt.cm.get_cmap('jet_r',1000) #just a colormap
ret = binned_statistic_2d(x, y, z, statistic=np.median, bins=[x_bins, y_bins]) # Bin (X, Y) and create a map of the medians of "Colors"
sigma=1 # standard deviation for Gaussian kernel
truncate=5.0 # truncate filter at this many sigmas
U = ret.statistic.T.copy()
V=U.copy()
V[np.isnan(U)]=0
VV=sp.ndimage.gaussian_filter(V,sigma=sigma)
W=0*U.copy()+1
W[np.isnan(U)]=0
WW=sp.ndimage.gaussian_filter(W,sigma=sigma)
np.seterr(divide='ignore', invalid='ignore')
Z=VV/WW
for i in range(len(Z)):
for j in range(len(Z[0])):
if np.isnan(U[i][j]):
Z[i][j] = np.nan
plt.imshow(Z, origin='bottom', extent=(-50, 50, -50, 50), cmap=cmap)
plt.xlim(-40,40)
plt.ylim(-40,40)
plt.xlabel("X", fontsize=15)
plt.ylabel("Y", fontsize=15)
ax.set_yticks([-40,-30,-20,-10,0,10,20,30,40])
bounds = np.arange(2.0, 20.0, 1.0)
plt.colorbar(ticks=bounds, label="Color", fraction=0.046, pad=0.04)
# save plots
plt.savefig("Whatever_name.png", bbox_inches='tight')
I have a set of data which correspond to ages (in steps of 0.1) along the x axis, and probabilities along the y axis. I'm trying to interpolate the data so I can find the maximum and a range of ages which covers 95% of the probability.
I've tried a simple interpolation using the code below, taken from the SciPy help pages, and it produces good results (I change the x and y variables to read my data), except for one feature.
from scipy.interpolate import interp1d
x = np.linspace(72, 100, num=29, endpoint=True)
y = df.iloc[:,0].values
f = interp1d(x, y)
f2 = interp1d(x, y, kind='cubic')
xnew = np.linspace(0, 10, num=41, endpoint=True)
import matplotlib.pyplot as plt
plt.plot(x, y, 'o', xnew, f(xnew), '-', xnew, f2(xnew), '--')
plt.legend(['data', 'linear', 'cubic'], loc='best')
plt.show()
The problem is, the cubic function works best, with the smoothest fit. However, it gives negative values for some parts of the probability curve, which is obviously not acceptable. Is there some way of setting a floor at y=0? I thought maybe switching to a quadratic kind would fix it, but it doesn't seem to. The linear fit does, but it's not smoothed, so is not a very good match.
I'm also not sure how to perform the second part of what I'm trying to do. It's probably very simple, but I don't know how to find the mean when I don't have a frequency table, but a grid of interpolated points which form a function. If I knew the function, I could integrate it, but I'm not sure how to do that in Python.
EDIT to include some data:
This is what my y data looks like:
array([3.41528917e-08, 7.81041275e-05, 9.60711716e-04, 5.75868934e-05,
6.50260297e-05, 2.95556411e-05, 2.37331370e-05, 9.11990619e-05,
1.08003254e-04, 4.16800419e-05, 6.63673113e-05, 2.57934035e-04,
3.42235937e-03, 5.07534495e-03, 1.76603165e-02, 1.69535370e-01,
2.67624254e-01, 4.29420872e-01, 8.25165926e-02, 2.08367339e-02,
2.01227453e-03, 1.15405995e-04, 5.40163098e-07, 1.66905537e-10,
8.31862858e-18, 4.14093219e-23, 8.32103362e-29, 5.65637769e-34,
7.93547444e-40])
NOTE: This is now resolved, although I made no changes to my code, but the images are now mysteriously coming out with perfect colour.
I am making plots using Python3 with a Spyder interface. I have looked at the plots both in the Spyder terminal, and in the saved PNGs of the images. Both show that the colours are coming out very badly.
A simple line in the script
plt.plot(x1, beta.transpose()[x1], color = 'r', linewidth = 0.2)
The red colour is showing as a light pink. When I change 'r' for 'k' to get a black line, it comes as a very light grey. Does anyone know of issues affecting Matplotlib, or Matplotlib used through Spyder, that might account for this?
Example:
Code:
Data-generating script logit_data.py
#Logistic regression data and link function.
import numpy as np
#Define link function here
def g(z):
g=1/(1+np.exp(-z))
return g
#For producing y data values given true paramters theta and number of covariates
def logit_data(n,p, theta):
#Define parameters
#1)Number of covariates
p_i = p+1 #with intercept
p_i=np.int(p_i)
#2) m as correct data type
n=np.int(n)
#4)Specify parameter valueas to be estimated
theta=np.reshape(theta, (p_i,1))
#5)Define distribution from which covariate values are drawn i.i.d., and initiate data values
X=np.zeros((n,p_i))
X[:,0]=1 #intercept
mean=0
sigma=1.5
X[:,1:]=np.random.normal(mean,sigma,(n,p))
#6)Produce y values treating y as a Bernoulli variable with p=g(X*theta)
r=np.random.uniform(0,1,n)
r=np.reshape(r, (len(r),1))
htrue=g(X.dot(theta))
y=htrue-r
y[y>=0]=1
y[y<0]=0
return X, y
Plotting script:
#Script for producing y data from p covariates from a specified distribution, specified beta paraemters,
#and n data samples for logit link function.
import numpy as np
import matplotlib.pyplot as plt
from logit_data import logit_data
import pylab
import statsmodels.stats as sms
import statsmodels.api as sma
import csv
def figure2():
#def MLE_logistic_function():
#1)Sample and observation numbers
samples = 30
observations = 40000
#2)Number of independent covariates
p=299
#3)True beta to be estimated (parameter values)
nonzerosN=30
beta1=np.append(np.full((1, nonzerosN),10),np.full((1,nonzerosN),-10), axis=1)
print(np.shape(beta1))
beta=np.append(beta1,np.zeros((1,p+1-2*nonzerosN)), axis=1)
print(np.shape(beta))
#4)#Initiate arrays to store estimates of beta (and errors) computed at specified sample numbers N
#Betas=np.zeros((len(npowers),p+1))
#Errors=np.zeros((len(npowers),p+1))
#5)Obtain random covariate values from specified distribution, and corresponding y values using true beta
#plus gaussian noise term.
X,y = logit_data(observations,p,beta)
logit = sma.Logit(y,X)
result = logit.fit()
print(result.summary())
MLEcoefficients = result.params
x1 = np.arange(0, nonzerosN,1)
x2 = np.arange(nonzerosN, 2*nonzerosN,1)
x3 = np.arange(2*nonzerosN, p+1,1)
plt.scatter(index, MLEcoefficients, 0.2)
plt.plot(x1, beta.transpose()[x1], color = 'black', linewidth = 0.2, alpha=1)
plt.plot(x2, beta.transpose()[x2], color = 'black', linewidth = 0.2)
plt.plot(x3, beta.transpose()[x3], color = 'black', linewidth = 0.2)
plt.xlabel('Index')
plt.ylabel('Coefficient values (true and fitted)')
plt.savefig('MMLTfig2_p%s_o%d.png' %(p,observations))
plt.show()
return
figure2()
I'm struggling to draw a power law graph for Facebook Data that I found online. I'm using Networkx and I've found how to draw a Degree Histogram and a degree rank. The problem that I'm having is I want the y axis to be a probability so I'm assuming I need to sum up each y value and divide by the total number of nodes? Can anyone please help me do this? Once I've got this I'd like to draw a log-log graph to see if I can obtain a straight line. I'd really appreciate it if anyone could help! Here's my code:
import collections
import networkx as nx
import matplotlib.pyplot as plt
from networkx.algorithms import community
import math
import pylab as plt
g = nx.read_edgelist("/Users/Michael/Desktop/anaconda3/facebook_combined.txt","r")
nx.info(g)
degree_sequence = sorted([d for n, d in g.degree()], reverse=True)
degreeCount = collections.Counter(degree_sequence)
deg, cnt = zip(*degreeCount.items())
fig, ax = plt.subplots()
plt.bar(deg, cnt, width=0.80, color='b')
plt.title("Degree Histogram for Facebook Data")
plt.ylabel("Count")
plt.xlabel("Degree")
ax.set_xticks([d + 0.4 for d in deg])
ax.set_xticklabels(deg)
plt.show()
plt.loglog(degree_sequence, 'b-', marker='o')
plt.title("Degree rank plot")
plt.ylabel("Degree")
plt.xlabel("Rank")
plt.show()
You seem to be on the right tracks, but some simplifications will likely help you. The code below uses only 2 libraries.
Without access your graph, we can use some graph generators instead. I've chosen 2 qualitatively different types here, and deliberately chosen different sizes so that the normalization of the histogram is needed.
import networkx as nx
import matplotlib.pyplot as plt
g1 = nx.scale_free_graph(1000, )
g2 = nx.watts_strogatz_graph(2000, 6, p=0.8)
# we don't need to sort the values since the histogram will handle it for us
deg_g1 = nx.degree(g1).values()
deg_g2 = nx.degree(g2).values()
# there are smarter ways to choose bin locations, but since
# degrees must be discrete, we can be lazy...
max_degree = max(deg_g1 + deg_g2)
# plot different styles to see both
fig = plt.figure()
ax = fig.add_subplot(111)
ax.hist(deg_g1, bins=xrange(0, max_degree), density=True, histtype='bar', rwidth=0.8)
ax.hist(deg_g2, bins=xrange(0, max_degree), density=True, histtype='step', lw=3)
# setup the axes to be log/log scaled
ax.set_yscale('log')
ax.set_xscale('log')
ax.set_xlabel('degree')
ax.set_ylabel('relative density')
ax.legend()
plt.show()
This produces an output plot like this (both g1,g2 are randomised so won't be identical):
Here we can see that g1 has an approximately straight line decay in the degree distribution -- as expected for scale-free distributions on log-log axes. Conversely, g2 does not have a scale-free degree distribution.
To say anything more formal, you could look at the toolboxes from Aaron Clauset: http://tuvalu.santafe.edu/~aaronc/powerlaws/ which implement model fitting and statistical testing of power-law distributions.
I have the results of a (H,ranges) = numpy.histogram2d() computation and I'm trying to plot it.
Given H I can easily put it into plt.imshow(H) to get the corresponding image. (see http://matplotlib.org/api/pyplot_api.html#matplotlib.pyplot.imshow )
My problem is that the axis of the produced image are the "cell counting" of H and are completely unrelated to the values of ranges.
I know I can use the keyword extent (as pointed in: Change values on matplotlib imshow() graph axis ). But this solution does not work for me: my values on range are not growing linearly (actually they are going exponentially)
My question is: How can I put the value of range in plt.imshow()? Or at least, or can I manually set the label values of the plt.imshow resulting object?
Editing the extent is not a good solution.
You can just change the tick labels to something more appropriate for your data.
For example, here we'll set every 5th pixel to an exponential function:
import numpy as np
import matplotlib.pyplot as plt
im = np.random.rand(21,21)
fig,(ax1,ax2) = plt.subplots(1,2)
ax1.imshow(im)
ax2.imshow(im)
# Where we want the ticks, in pixel locations
ticks = np.linspace(0,20,5)
# What those pixel locations correspond to in data coordinates.
# Also set the float format here
ticklabels = ["{:6.2f}".format(i) for i in np.exp(ticks/5)]
ax2.set_xticks(ticks)
ax2.set_xticklabels(ticklabels)
ax2.set_yticks(ticks)
ax2.set_yticklabels(ticklabels)
plt.show()
Expanding a bit on #thomas answer
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.image as mi
im = np.random.rand(20, 20)
ticks = np.exp(np.linspace(0, 10, 20))
fig, ax = plt.subplots()
ax.pcolor(ticks, ticks, im, cmap='viridis')
ax.set_yscale('log')
ax.set_xscale('log')
ax.set_xlim([1, np.exp(10)])
ax.set_ylim([1, np.exp(10)])
By letting mpl take care of the non-linear mapping you can now accurately over-plot other artists. There is a performance hit for this (as pcolor is more expensive to draw than AxesImage), but getting accurate ticks is worth it.
imshow is for displaying images, so it does not support x and y bins.
You could either use pcolor instead,
H,xedges,yedges = np.histogram2d()
plt.pcolor(xedges,yedges,H)
or use plt.hist2d which directly plots your histogram.