2. Visualization

Visualizatin is vital in data analysis and scientific computing, to

• get intuitive understanding

• come up with a new hypothesis

• detect a bug or data anormaly

That is why we’ll cover this topic at the beginning of this course.

Matplotlib

Matplotlib is the standard graphics package for Python. It mimics many graphics functions of MATLAB.
The Matplotlib gallery (http://matplotlib.org/stable/gallery) illustrates variety of plots.

Usually matplotlib opens a window for a new plot.

A nice feature of Jupyter notebook is that you can embed the figures produced by your program within the notebook by the following magic command
%matplotlib inline
(It may be a default setting in recent jupyter notebook).

%matplotlib inline

import numpy as np
import matplotlib.pyplot as plt

Plotting functions

The standard way is to prepare an array for x values, compute y values, and call plot( ) function.

# make an array from 0 to 10, the default is 50 points
x = np.linspace(0, 10)

# comupute a function for each point
y = x*np.sin(x)
# plot the points
plt.plot(y)  # x is the index of y

[<matplotlib.lines.Line2D at 0x115a54680>]

There are multiple ways to pass variables to plot():

• plot(y): x is assumed as the indices of y

• plot(x, y): specify both x and y values

• plot(x1, y1, x2, y2,...): multiple lines

• plot(x, Y): lines for columns of matrix Y

# specify both x and y values
plt.plot(x, y)

[<matplotlib.lines.Line2D at 0x115aa1010>]

# take another function of x
y2 = x*np.cos(x)

# plot two lines
plt.plot(x, y, x, y2)

[<matplotlib.lines.Line2D at 0x115b7c8f0>,
 <matplotlib.lines.Line2D at 0x115b7c920>]

# phase plot
plt.plot(y, y2);
# you can supress <matplotlib...> output by ;

# plot multiple lines by a matrix
Y = np.array([y, y2])  # stack data in two rows
plt.plot(x, Y.T); # transpose to give data in two columns

# plot multiple lines by a matrix
Y = np.array([np.sin(k*x) for k in range(4)])
plt.plot(x, Y.T);

Options for plotting

Line styles can be specified by

• color= (or c= ) for color by code, name, RGB or RGBA

  • code: ‘r’, ‘g’, ‘b’, ‘y’, ‘c’, ‘m’, ‘k’, ‘w’

• marker= for marker style

  • code: ‘.’, ‘o’, ‘+’, ‘*’, ‘^’, …

• linestyle= (or ls= ) for line style

  • code: ‘-’, ‘–’, ‘:’, ‘-.’, …

• linewidth= (or lw= ) for line width

• a string of color, marker and line sytel codes, e.g. ‘ro:’

# using code string
plt.plot(x, y, 'm:', x, y2, 'c*');  # magenta dash-dot, cyan circle

# using keyword=value
plt.plot(x, y, c=[0.2,0.5,0.8,0.5], marker='o', markersize=10, ls='-.', lw=2);

It’s a good practice to add axis lables and plot title.

plt.plot(x, y)
plt.title('oscillation')
plt.xlabel('time ($\mu s$)')
plt.ylabel('amplitude')

<>:3: SyntaxWarning: invalid escape sequence '\m'
<>:3: SyntaxWarning: invalid escape sequence '\m'
/var/folders/0z/3cttnw852b959kp4wjh_z_wc0000gn/T/ipykernel_51581/2066067687.py:3: SyntaxWarning: invalid escape sequence '\m'
  plt.xlabel('time ($\mu s$)')

Text(0, 0.5, 'amplitude')

It is also nice to add a legend box.

ax = plt.plot(x, y, x, y2)
plt.legend(('x sin x','x cos x'))

<matplotlib.legend.Legend at 0x115c77e00>

You can control axis ranges and scaling.

plt.plot(x, y)
plt.xlim(-1, 5)
plt.ylim(-4, 4)

(-4.0, 4.0)

plt.plot(y, y2)
plt.axis('equal')  # equal scaling for x and y

(-6.107980677841252, 8.582949839004911, -10.247801600732576, 7.121229063313089)

plt.plot(y, y2)
plt.axis('square')  # in square plot area

(-6.107980677841252,
 11.261049986204412,
 -10.247801600732576,
 7.121229063313088)

You can create a fiure of your preferred size by plt.figure() function

fig = plt.figure(figsize=(6, 6))
plt.plot(y, y2)

[<matplotlib.lines.Line2D at 0x115f8dd30>]

Bar plot and histogram

i = np.arange(10)
j = i**2
plt.bar(i, j)

<BarContainer object of 10 artists>

np.random.randn() gives random numbers from the normal distribution

z = np.random.randn(100)
plt.hist(z)

(array([ 1.,  2., 14., 13., 20., 26., 11.,  8.,  4.,  1.]),
 array([-2.53293677, -1.99641202, -1.45988728, -0.92336253, -0.38683778,
         0.14968697,  0.68621172,  1.22273647,  1.75926122,  2.29578597,
         2.83231072]),
 <BarContainer object of 10 artists>)

Subplot and axes

You can create multiple axes in a figure by subplot(rows, columns, index).
It uses a MATLAB legacy for index starting from 1.

plt.subplot(2, 2, 1)
plt.plot(x, y)
plt.subplot(2, 2, 2)
plt.plot(y, x)
plt.subplot(2, 2, 3)
plt.plot(x, y2)
plt.subplot(2, 2, 4)
plt.plot(y, y2)

[<matplotlib.lines.Line2D at 0x115d1d070>]

Figure and axes

When you make a plot, matplotlib creates a figure object with an axes object.
You can use gcf() and gca() to identify them and getp() and setp() to access their parameters.

plt.plot(x, y)
fig = plt.gcf()  # get current figure
plt.getp(fig)  # show all parameters

    agg_filter = None
    alpha = None
    animated = False
    axes = [<Axes: >]
    children = [<matplotlib.patches.Rectangle object at 0x115f265...
    clip_box = None
    clip_on = True
    clip_path = None
    constrained_layout = False
    constrained_layout_pads = (None, None, None, None)
    default_bbox_extra_artists = [<Axes: >, <matplotlib.spines.Spine object at 0x11...
    dpi = 100.0
    edgecolor = (1.0, 1.0, 1.0, 1.0)
    facecolor = (1.0, 1.0, 1.0, 1.0)
    figheight = 4.8
    figure = Figure(640x480)
    figwidth = 6.4
    frameon = True
    gid = None
    in_layout = True
    label = 
    layout_engine = None
    linewidth = 0.0
    mouseover = False
    path_effects = []
    picker = None
    rasterized = False
    size_inches = [6.4 4.8]
    sketch_params = None
    snap = None
    suptitle = 
    supxlabel = 
    supylabel = 
    tight_layout = False
    tightbbox = TransformedBbox(     Bbox(x0=49.77777777777777, y0...
    transform = IdentityTransform()
    transformed_clip_path_and_affine = (None, None)
    url = None
    visible = True
    window_extent = TransformedBbox(     Bbox(x0=0.0, y0=0.0, x1=6.4, ...
    zorder = 0

plt.plot(x, y)
fig = plt.gcf()
plt.setp(fig, size_inches=(8,4), facecolor=[0.5,0.5,0.5])

[None, None]

plt.plot(x, y)
ax = plt.gca()  # get current axes
plt.getp(ax)
plt.setp(ax, facecolor='y')  # change parameters

    adjustable = box
    agg_filter = None
    alpha = None
    anchor = C
    animated = False
    aspect = auto
    autoscale_on = True
    autoscalex_on = True
    autoscaley_on = True
    axes_locator = None
    axisbelow = line
    box_aspect = None
    children = [<matplotlib.lines.Line2D object at 0x115c0b500>, ...
    clip_box = None
    clip_on = True
    clip_path = None
    data_ratio = 1.3355391378951056
    default_bbox_extra_artists = [<matplotlib.spines.Spine object at 0x11611c0b0>, ...
    facecolor or fc = (1.0, 1.0, 1.0, 1.0)
    figure = Figure(640x480)
    frame_on = True
    gid = None
    gridspec = GridSpec(1, 1)
    images = <a list of 0 AxesImage objects>
    in_layout = True
    label = 
    legend = None
    legend_handles_labels = ([], [])
    lines = <a list of 1 Line2D objects>
    mouseover = False
    navigate = True
    navigate_mode = None
    path_effects = []
    picker = None
    position = Bbox(x0=0.125, y0=0.10999999999999999, x1=0.9, y1=...
    rasterization_zorder = None
    rasterized = False
    shared_x_axes = <matplotlib.cbook.GrouperView object at 0x115f6958...
    shared_y_axes = <matplotlib.cbook.GrouperView object at 0x115f683e...
    sketch_params = None
    snap = None
    subplotspec = GridSpec(1, 1)[0:1, 0:1]
    tightbbox = Bbox(x0=49.77777777777777, y0=29.077777777777776, ...
    title = 
    transform = IdentityTransform()
    transformed_clip_path_and_affine = (None, None)
    url = None
    visible = True
    window_extent = TransformedBbox(     Bbox(x0=0.125, y0=0.109999999...
    xaxis = XAxis(80.0,52.8)
    xaxis_transform = BlendedGenericTransform(     CompositeGenericTrans...
    xbound = (-0.5, 10.5)
    xgridlines = <a list of 8 Line2D gridline objects>
    xlabel = 
    xlim = (-0.5, 10.5)
    xmajorticklabels = [Text(-2.0, 0, '−2'), Text(0.0, 0, '0'), Text(2.0,...
    xminorticklabels = []
    xscale = linear
    xticklabels = [Text(-2.0, 0, '−2'), Text(0.0, 0, '0'), Text(2.0,...
    xticklines = <a list of 16 Line2D ticklines objects>
    xticks = [-2.  0.  2.  4.  6.  8.]...
    yaxis = YAxis(80.0,52.8)
    yaxis_transform = BlendedGenericTransform(     BboxTransformTo(     ...
    ybound = (-6.107980677841252, 8.582949839004911)
    ygridlines = <a list of 10 Line2D gridline objects>
    ylabel = 
    ylim = (-6.107980677841252, 8.582949839004911)
    ymajorticklabels = [Text(0, -8.0, '−8'), Text(0, -6.0, '−6'), Text(0,...
    yminorticklabels = []
    yscale = linear
    yticklabels = [Text(0, -8.0, '−8'), Text(0, -6.0, '−6'), Text(0,...
    yticklines = <a list of 20 Line2D ticklines objects>
    yticks = [-8. -6. -4. -2.  0.  2.]...
    zorder = 0

[None]

Visualizing data in 2D

A standard way for visualizing pariwise data is a scatter plot.

n = 100
x = np.random.uniform(-1, 1, n)  # n points in [-1,1]
y = 2*x + np.random.randn(n)  # scale and add noise
plt.plot(x, y, 'o')

[<matplotlib.lines.Line2D at 0x1160496d0>]

By scatterplot( ) you can specify the size and the color of each point to visualize higher dimension information.

z = x**2 + y**2
c = y - 2*x
# z for size, c for color
plt.scatter(x, y, z, c)
plt.colorbar()

<matplotlib.colorbar.Colorbar at 0x116189130>

Visualizing a matrix or a function in 2D space

meshgrid() is for preparing x and y values in a grid.

x = np.linspace(-4, 4, 9)
y = np.linspace(-3, 3, 7)
print(x, y)
X, Y = np.meshgrid(x, y)
print(X, Y)

[-4. -3. -2. -1.  0.  1.  2.  3.  4.] [-3. -2. -1.  0.  1.  2.  3.]
[[-4. -3. -2. -1.  0.  1.  2.  3.  4.]
 [-4. -3. -2. -1.  0.  1.  2.  3.  4.]
 [-4. -3. -2. -1.  0.  1.  2.  3.  4.]
 [-4. -3. -2. -1.  0.  1.  2.  3.  4.]
 [-4. -3. -2. -1.  0.  1.  2.  3.  4.]
 [-4. -3. -2. -1.  0.  1.  2.  3.  4.]
 [-4. -3. -2. -1.  0.  1.  2.  3.  4.]] [[-3. -3. -3. -3. -3. -3. -3. -3. -3.]
 [-2. -2. -2. -2. -2. -2. -2. -2. -2.]
 [-1. -1. -1. -1. -1. -1. -1. -1. -1.]
 [ 0.  0.  0.  0.  0.  0.  0.  0.  0.]
 [ 1.  1.  1.  1.  1.  1.  1.  1.  1.]
 [ 2.  2.  2.  2.  2.  2.  2.  2.  2.]
 [ 3.  3.  3.  3.  3.  3.  3.  3.  3.]]

We can use imshow() to visualize a matrix as an image.

Z = X**2 * Y
print(Z)
plt.imshow(Z)

[[-48. -27. -12.  -3.  -0.  -3. -12. -27. -48.]
 [-32. -18.  -8.  -2.  -0.  -2.  -8. -18. -32.]
 [-16.  -9.  -4.  -1.  -0.  -1.  -4.  -9. -16.]
 [  0.   0.   0.   0.   0.   0.   0.   0.   0.]
 [ 16.   9.   4.   1.   0.   1.   4.   9.  16.]
 [ 32.  18.   8.   2.   0.   2.   8.  18.  32.]
 [ 48.  27.  12.   3.   0.   3.  12.  27.  48.]]

<matplotlib.image.AxesImage at 0x11629d070>

# some more options
plt.imshow(Z, origin='lower', extent=(-4.5, 4.5, -3.5, 3.5))
plt.colorbar()

<matplotlib.colorbar.Colorbar at 0x116344a10>

color maps

imshow( ) maps a scalar Z value to color by a colormap. The standard color map viridis is friedly to color blindness and monochrome printing. You can also choose other color maps.

plt.imshow(Z, cmap='jet')

<matplotlib.image.AxesImage at 0x116419fa0>

contour plot

x = np.linspace(-4, 4, 25)
y = np.linspace(-4, 4, 25)
X, Y = np.meshgrid(x, y)
Z = X**2 + 2*Y**2
plt.contour(X, Y, Z)
plt.axis('square')

(-4.0, 4.0, -4.0, 4.0)

vector field by quiver( )

x = np.linspace(-3, 3, 15)
y = np.linspace(-3, 3, 15)
X, Y = np.meshgrid(x, y)
# Van del Pol model
k = 1  # paramter
U = Y  # dx/dt
V = k*(1 - X**2)*Y - X  # dy/dt
plt.quiver(X, Y, U, V)

<matplotlib.quiver.Quiver at 0x116419d30>