Matplotlib colours are wrong - python-3.x

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()

Related

How to do color code the 3D scatter plot according to time of occurrence in an animated graph?

I have developed a code to create an animated scatter graph.
About the dataset, I have the X,Y,Z coordinate of each point and each event point are assigned a value (M) and each happened at a specific time (t).
I have the size of each point to be proportional to their value (i.e., M), now I want to add the color to each point so that it also shows the time of occurrence. I know I have to use .set_color(c) but c value expects a tuple value. I tried to normalize the values of the time to map the color from this post. However, there is something that I miss because the code is not working to color the points with related time. I would appreciate it if someone could share their experiences?
import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
import matplotlib.animation as animation
from IPython.display import HTML # Animation on jupyter lab
from matplotlib.animation import PillowWriter # For GIF animation
#####Data Generation####
# Space Coordinate
X = np.random.random((100,)) * 255 * 2 - 255
Y = np.random.random((100,)) * 255 * 2 - 255
Z = np.random.random((100,)) * 255 * 2 - 255
# Magnitude of each point
# M = np.random.random((100,))*-1+0.5
M = np.random.randint(1,70, size=100)
# Time
t = np.sort(np.random.random((100,))*10)
#ID each point should be color coded. Moreover, each point belongs to a cluster `ID`
ID = np.sort(np.round([np.random.random((100,))*5]))
x = []
y = []
z = []
m = []
def update_lines(i):
# for i in range (df_IS["EASTING [m]"].size):
dx = X[i]
dy = Y[i]
dz = Z[i]
dm = M[i]
# text.set_text("{:d}: [{:.0f}] Mw[{:.2f}]".format(ID[i], t[i],ID[i])) # for debugging
x.append(dx)
y.append(dy)
z.append(dz)
m.append(dm)
graph._offsets3d = (x, y, z)
graph.set_sizes(m)
return graph,
fig = plt.figure(figsize=(5, 5))
ax = fig.add_subplot(111, projection="3d")
graph = ax.scatter(X, Y, Z, s=M, color='orange') # s argument here
text = fig.text(0, 1, "TEXT", va='top') # for debugging
ax.set_xlim3d(X.min(), X.max())
ax.set_ylim3d(Y.min(), Y.max())
ax.set_zlim3d(Z.min(), Z.max())
# Creating the Animation object
ani = animation.FuncAnimation(fig, update_lines, frames=100, interval=500, blit=False, repeat=False)
# plt.show()
ani.save('test3Dscatter.gif', writer='pillow')
plt.close()
HTML(ani.to_html5_video())
You need to change "Color" to "cmap" so that you are able to call set of colors, see below:
graph = ax.scatter(X, Y, Z, s=M, cmap='jet') #jet is similar to rainbow

Python: how to create a smoothed version of a 2D binned "color map"?

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')

Control marker properties in seaborn pairwise boxplot

I'm trying to plot a boxplot for two different datasets on the same plot. The x axis are the hours in a day, while the y axis goes from 0 to 1 (let's call it Efficiency). I would like to have different markers for the means of each dataset' boxes. I use the 'meanprops' for seaborn but that changes the marker style for both datasets at the same time. I've added 2000 lines of data in the excel that can be downloaded here. The values might not coincide with the ones in the picture but should be enough.
Basically I want the red squares to be blue on the orange boxplot, and red on the blue boxplot. Here is what I managed to do so far:
I tried changing the meanprops by using a dictionary with the labels as keys , but it seems to be entering a loop (in PyCharm is says Evaluating...)
import matplotlib.pyplot as plt
import seaborn as sns
import pandas as pd
#make sure you have your path sorted out
group1 = pd.read_excel('group1.xls')
ax,fig = plt.subplots(figsize = (20,10))
#does not work
#ax = sns.boxplot(data=group1, x='hour', y='M1_eff', hue='labels',showfliers=False, showmeans=True,\
# meanprops={"marker":{'7':"s",'8':'s'},"markerfacecolor":{'7':"white",'8':'white'},
#"markeredgecolor":{'7':"blue",'8':'red'})
#works but produces similar markers
ax = sns.boxplot(data=group1, x='hour', y='M1_eff', hue='labels',showfliers=False, showmeans=True,\
meanprops={"marker":"s","markerfacecolor":"white", "markeredgecolor":"blue"})
plt.legend(title='Groups', loc=2, bbox_to_anchor=(1, 1),borderaxespad=0.5)
# Add transparency to colors
for patch in ax.artists:
r, g, b, a = patch.get_facecolor()
patch.set_facecolor((r, g, b, .4))
ax.set_xlabel("Hours",fontsize=14)
ax.set_ylabel("M1 Efficiency",fontsize=14)
ax.tick_params(labelsize=10)
plt.show()
I also tried the FacetGrid but to no avail (Stops at 'Evaluating...'):
g = sns.FacetGrid(group1, col="M1_eff", hue="labels",hue_kws=dict(marker=["^", "v"]))
g = (g.map(plt.boxplot, "hour", "M1_eff")
.add_legend())
g.show()
Any help is appreciated!
I don't think you can do this using sns.boxplot() directly. I think you'll have to draw the means "by hand"
N=100
df = pd.DataFrame({'hour':np.random.randint(0,3,size=(N,)),
'M1_eff': np.random.random(size=(N,)),
'labels':np.random.choice([7,8],size=(N,))})
x_col = 'hour'
y_col = 'M1_eff'
hue_col = 'labels'
width = 0.8
hue_order=[7,8]
marker_colors = ['red','blue']
# get the offsets used by boxplot when hue-nesting is used
# https://github.com/mwaskom/seaborn/blob/c73055b2a9d9830c6fbbace07127c370389d04dd/seaborn/categorical.py#L367
n_levels = len(hue_order)
each_width = width / n_levels
offsets = np.linspace(0, width - each_width, n_levels)
offsets -= offsets.mean()
fig, ax = plt.subplots()
ax = sns.boxplot(data=df, x=x_col, y=y_col, hue=hue_col, hue_order=hue_order, showfliers=False, showmeans=False)
means = df.groupby([hue_col,x_col])[y_col].mean()
for (gr,temp),o,c in zip(means.groupby(level=0),offsets,marker_colors):
ax.plot(np.arange(temp.values.size)+o, temp.values, 's', c=c)

When plotting the Wigner function of a coherent state using QuTiP strange patterns appear

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.

Recreating decision-boundary plot in python with scikit-learn and matplotlib

I found this wonderful graph in post here Variation on "How to plot decision boundary of a k-nearest neighbor classifier from Elements of Statistical Learning?". In this example K-NN is used to clasify data into three classes. I especially enjoy that it features the probability of class membership as a indication of the "confidence".
r and ggplot seem to do a great job.I wonder, whether this can be re-created in python? My initial thought tends to scikit-learn and matplotlib. Here is the iris example from scikit:
print(__doc__)
import numpy as np
import matplotlib.pyplot as plt
from matplotlib.colors import ListedColormap
from sklearn import neighbors, datasets
n_neighbors = 15
# import some data to play with
iris = datasets.load_iris()
X = iris.data[:, :2] # we only take the first two features. We could
# avoid this ugly slicing by using a two-dim dataset
y = iris.target
h = .02 # step size in the mesh
# Create color maps
cmap_light = ListedColormap(['#FFAAAA', '#AAFFAA', '#AAAAFF'])
cmap_bold = ListedColormap(['#FF0000', '#00FF00', '#0000FF'])
for weights in ['uniform', 'distance']:
# we create an instance of Neighbours Classifier and fit the data.
clf = neighbors.KNeighborsClassifier(n_neighbors, weights=weights)
clf.fit(X, y)
# Plot the decision boundary. For that, we will assign a color to each
# point in the mesh [x_min, x_max]x[y_min, y_max].
x_min, x_max = X[:, 0].min() - 1, X[:, 0].max() + 1
y_min, y_max = X[:, 1].min() - 1, X[:, 1].max() + 1
xx, yy = np.meshgrid(np.arange(x_min, x_max, h),
np.arange(y_min, y_max, h))
Z = clf.predict(np.c_[xx.ravel(), yy.ravel()])
# Put the result into a color plot
Z = Z.reshape(xx.shape)
plt.figure()
plt.pcolormesh(xx, yy, Z, cmap=cmap_light)
# Plot also the training points
plt.scatter(X[:, 0], X[:, 1], c=y, cmap=cmap_bold)
plt.xlim(xx.min(), xx.max())
plt.ylim(yy.min(), yy.max())
plt.title("3-Class classification (k = %i, weights = '%s')"
% (n_neighbors, weights))
plt.show()
This produces a graph in a sense very similar:
I have three questions:
How can I introduce the confidence to the plot?
How can I plot the decision-boundaries with a connected line?
Let's say I have a new observation, how can I introduce it to the plot and plot if it is classified correctly?
I stumbled upon your question about a year ago, and loved the plot -- I just never got around to answering it, until now. Hopefully the code comments below are self-explanitory enough (I also blogged about, if you want more details). Maybe four years too late, haha.
import numpy as np
import matplotlib.pyplot as plt
from sklearn import datasets
from matplotlib.lines import Line2D
from matplotlib.ticker import MaxNLocator
from sklearn import neighbors
iris = datasets.load_iris()
x = iris.data[:,0:2]
y = iris.target
# create the x0, x1 feature
x0 = x[:,0]
x1 = x[:,1]
# set main parameters for KNN plot
N_NEIGHBORS = 15 # KNN number of neighbors
H = 0.1 # mesh stepsize
PROB_DOT_SCALE = 40 # modifier to scale the probability dots
PROB_DOT_SCALE_POWER = 3 # exponential used to increase/decrease size of prob dots
TRUE_DOT_SIZE = 50 # size of the true labels
PAD = 1.0 # how much to "pad" around the true labels
clf = neighbors.KNeighborsClassifier(N_NEIGHBORS, weights='uniform')
clf.fit(x, y)
# find the min/max points for both x0 and x1 features
# these min/max values will be used to set the bounds
# for the plot
x0_min, x0_max = np.round(x0.min())-PAD, np.round(x0.max()+PAD)
x1_min, x1_max = np.round(x1.min())-PAD, np.round(x1.max()+PAD)
# create 1D arrays representing the range of probability data points
# on both the x0 and x1 axes.
x0_axis_range = np.arange(x0_min,x0_max, H)
x1_axis_range = np.arange(x1_min,x1_max, H)
# create meshgrid between the two axis ranges
xx0, xx1 = np.meshgrid(x0_axis_range, x1_axis_range)
# put the xx in the same dimensional format as the original x
# because it's easier to work with that way (at least for me)
# * shape will be: [no_dots, no_dimensions]
# where no_dimensions = 2 (x0 and x1 axis)
xx = np.reshape(np.stack((xx0.ravel(),xx1.ravel()),axis=1),(-1,2))
yy_hat = clf.predict(xx) # prediction of all the little dots
yy_prob = clf.predict_proba(xx) # probability of each dot being
# the predicted color
yy_size = np.max(yy_prob, axis=1)
# make figure
plt.style.use('seaborn-whitegrid') # set style because it looks nice
fig, ax = plt.subplots(nrows=1, ncols=1, figsize=(8,6), dpi=150)
# establish colors and colormap
# * color blind colors, from
# https://towardsdatascience.com/two-simple-steps-to-create-colorblind-friendly-data-visualizations-2ed781a167ec
redish = '#d73027'
orangeish = '#fc8d59'
yellowish = '#fee090'
blueish = '#4575b4'
colormap = np.array([redish,blueish,orangeish])
# plot all the little dots, position defined by the xx values, color
# defined by the knn predictions (yy_hat), and size defined by the
# probability of that color (yy_prob)
# * because the yy_hat values are either 0, 1, 2, we can use
# these as values to index into the colormap array
# * size of dots (the probability) increases exponentially (^3), so that there is
# a nice difference between different probabilities. I'm sure there is a more
# elegant way to do this though...
# * linewidths=0 so that there are no "edges" around the dots
ax.scatter(xx[:,0], xx[:,1], c=colormap[yy_hat], alpha=0.4,
s=PROB_DOT_SCALE*yy_size**PROB_DOT_SCALE_POWER, linewidths=0,)
# plot the contours
# * we have to reshape the yy_hat to get it into a
# 2D dimensional format, representing both the x0
# and x1 axis
# * the number of levels and color scheme was manually tuned
# to make sense for this data. Would probably change, for
# instance, if there were 4, or 5 (etc.) classes
ax.contour(x0_axis_range, x1_axis_range,
np.reshape(yy_hat,(xx0.shape[0],-1)),
levels=3, linewidths=1,
colors=[redish,blueish, blueish,orangeish,])
# plot the original x values.
# * zorder is 3 so that the dots appear above all the other dots
ax.scatter(x[:,0], x[:,1], c=colormap[y], s=TRUE_DOT_SIZE, zorder=3,
linewidths=0.7, edgecolor='k')
# create legends
x_min, x_max = ax.get_xlim()
y_min, y_max = ax.get_ylim()
# set x-y labels
ax.set_ylabel(r"$x_1$")
ax.set_xlabel(r"$x_0$")
# create class legend
# Line2D properties: https://matplotlib.org/stable/api/_as_gen/matplotlib.lines.Line2D.html
# about size of scatter plot points: https://stackoverflow.com/a/47403507/9214620
legend_class = []
for flower_class, color in zip(['c', 's', 'v'], [blueish, redish, orangeish]):
legend_class.append(Line2D([0], [0], marker='o', label=flower_class,ls='None',
markerfacecolor=color, markersize=np.sqrt(TRUE_DOT_SIZE),
markeredgecolor='k', markeredgewidth=0.7))
# iterate over each of the probabilities to create prob legend
prob_values = [0.4, 0.6, 0.8, 1.0]
legend_prob = []
for prob in prob_values:
legend_prob.append(Line2D([0], [0], marker='o', label=prob, ls='None', alpha=0.8,
markerfacecolor='grey',
markersize=np.sqrt(PROB_DOT_SCALE*prob**PROB_DOT_SCALE_POWER),
markeredgecolor='k', markeredgewidth=0))
legend1 = ax.legend(handles=legend_class, loc='center',
bbox_to_anchor=(1.05, 0.35),
frameon=False, title='class')
legend2 = ax.legend(handles=legend_prob, loc='center',
bbox_to_anchor=(1.05, 0.65),
frameon=False, title='prob', )
ax.add_artist(legend1) # add legend back after it disappears
ax.set_yticks(np.arange(x1_min,x1_max, 1)) # I don't like the decimals
ax.grid(False) # remove gridlines (inherited from 'seaborn-whitegrid' style)
# only use integers for axis tick labels
# from: https://stackoverflow.com/a/34880501/9214620
ax.xaxis.set_major_locator(MaxNLocator(integer=True))
ax.yaxis.set_major_locator(MaxNLocator(integer=True))
# set the aspect ratio to 1, for looks
ax.set_aspect(1)
# remove first ticks from axis labels, for looks
# from: https://stackoverflow.com/a/19503828/9214620
ax.set_xticks(ax.get_xticks()[1:-1])
ax.set_yticks(np.arange(x1_min,x1_max, 1)[1:])
plt.show()

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