Now that we can start doing serious numerical analysis with Numpy arrays, we also reach the stage where we can no longer print out hundreds or thousands of values, so we need to be able to make plots to show the results.
The Matplotlib package can be used to make scientific-grade plots. You can import it with:
import matplotlib.pyplot as plt
If you are using IPython and you want to make interactive plots, you can start up IPython with:
ipython --matplotlib
If you now type a plotting command, an interactive plot will pop up.
If you use the IPython notebook, add a cell containing:
%matplotlib inline
and the plots will appear inside the notebook.
The main plotting function is called plot:
plt.plot([1,2,3,6,4,2,3,4])
In the above example, we only gave a single list, so it will assume the x values are the indices of the list/array.
However, we can instead specify the x values:
plt.plot([3.3, 4.4, 4.5, 6.5], [3., 5., 6., 7.])
Matplotlib can take Numpy arrays, so we can do for example:
import numpy as np
x = np.linspace(0., 10., 50)
y = np.sin(x)
plt.plot(x, y)
The plot function is actually quite complex, and for example can take arguments specifying the type of point, the color of the line, and the width of the line:
plt.plot(x, y, marker='o', color='green', linewidth=2)
The line can be hidden with:
plt.plot(x, y, marker='o', color='green', linewidth=0)
If you are interested, you can specify some of these attributes with a special syntax, which you can read up more about in the Matplotlib documentation:
plt.plot(x, y, 'go') # means green and circles
We start off by loading the data/munich_temperatures_average_with_bad_data.txt file which we encountered in the introduction to Numpy (section 10):
# The following code reads in the file and removes bad values
import numpy as np
date, temperature = np.loadtxt('data/munich_temperatures_average_with_bad_data.txt', unpack=True)
keep = np.abs(temperature) < 90
date = date[keep]
temperature = temperature[keep]
Now that the data has been read in, plot the temperature against time:
# your solution here
plt.plot(date, temperature, 'r.')
Next, try plotting the data against the fraction of the year (all years on top of each other). Note that you can use the % (modulo) operator to find the fractional part of the dates:
# your solution here
plt.plot(date%1, temperature, 'r,')
While the plot function can be used to show scatter plots, it is mainly used for line plots, and the scatter function is more often used for scatter plots, because it allows more fine control of the markers:
x = np.random.random(1000)
y = np.random.random(1000)
plt.scatter(x, y)
Histograms are easy to plot using the hist function:
v = np.random.uniform(0., 10., 100)
h = plt.hist(v) # we do h= to capture the output of the function, but we don't use it
h = plt.hist(v, range=[-5., 15.], bins=100)
You can also show two-dimensional arrays with the imshow function:
array = np.random.random((64, 64))
plt.imshow(array)
And the colormap can be changed:
plt.imshow(array, cmap=plt.cm.gist_heat)
For a vector field, you can plot a depiction of the field lines with plt.streamplot
Here this is done in the x-y plane for the field $$\vec{K}(\vec{r})=\vec{K}(x,y,z) = (-y,x,0) = \vec{e}_z \times \vec{r}$$
We use np.meshgrid that returns N-D arrays according to the number of input arrays.
# Set up a figure
fig, ax = plt.subplots()
ax.set_aspect('equal')
x = np.arange(-10,10,0.1)
y = np.arange(-10,10,0.1)
# The purpose of the meshgrid function is to create a rectangular
# grid out of two input arrays (x,y)
# meshgrid will use all possible combinations of points from the
# two input arrays to generate this grid
xx,yy = np.meshgrid(x,y) # define meshgrid of all coordinates
# (can also work out radii etc)
U = -yy # Here we define the 'strength' of
V = xx # the vectors in the x and y directions
plt.streamplot(x,y,U,V,color='k')
ax.set_title("Field Lines")
# For reference:
x=[1,2,3]
y=[1,2]
(a,b)=np.meshgrid(x,y)
print(np.shape(a),np.shape(b))
print(a)
print(b)
You can easily customize plots. For example, the following code adds axis labels, and sets the x and y ranges explicitly:
x = np.random.random(100)
y = np.random.random(100)
plt.scatter(x, y)
plt.xlabel('x values')
plt.ylabel(r'$\Delta\chi$ values')
plt.xlim(0., 1.)
plt.ylim(0., 1.)
To save a plot to a file, you can do for example:
plt.savefig('my_plot.png')
and you can then view the resulting file like you would view a normal image. On Linux, you can also do:
$ eog my_plot.png
in the terminal (or whichever image viewer is available).
One of the nice features of Matplotlib is the ability to make interactive plots. When using IPython, you can do:
%matplotlib qt
to change the backend to be interactive, after which plots that you make will be interactive.
The easiest way to find out more about a function and available options is to use the ? help in IPython:
In [11]: plt.hist?
Definition: plt.hist(x, bins=10, range=None, normed=False, weights=None, cumulative=False, bottom=None, histtype='bar', align='mid', orientation='vertical', rwidth=None, log=False, color=None, label=None, stacked=False, hold=None, **kwargs)
Docstring:
Plot a histogram.
Call signature::
hist(x, bins=10, range=None, normed=False, weights=None,
cumulative=False, bottom=None, histtype='bar', align='mid',
orientation='vertical', rwidth=None, log=False,
color=None, label=None, stacked=False,
**kwargs)
Compute and draw the histogram of *x*. The return value is a
tuple (*n*, *bins*, *patches*) or ([*n0*, *n1*, ...], *bins*,
[*patches0*, *patches1*,...]) if the input contains multiple
data.
etc.
But sometimes you don't even know how to make a specific type of plot, in which case you can look at the Matplotlib Gallery for example plots and scripts.
Use Numpy to generate 10000 random values following a Gaussian/Normal distribution, and make a histogram. Try changing the number of bins to properly see the Gaussian. Try overplotting a Gaussian function on top of it using a colored line, and adjust the normalization so that the histogram and the line are aligned.
Do the same for a Poisson distribution. Compare the Poisson distribution for expectation values $\lambda <15$ with the appropriate Gaussian.
# your solution here
Work out the magnetic field lines of two equal but infinite line currents along the z-axis that are separated by a distance $a$ along the y-axis.
Since the stream lines are scaled, one can drop constants.
The $\vec{B}$-Field of a single line current $I$ is $$\vec{B} (\vec{r})= \frac{\mu_0 I}{2 \pi} \frac{1}{|\vec{r}|} \vec{e}_{\varphi}$$
Only with time left: Can you plot the fieldlines in the x-y plane for a circular coil with radius a in the x-z plane? The Biot-Savart law for a wire element is
$$d\vec{B} (\vec{r})= \frac{\mu_0 I}{4 \pi} \frac{d \vec{l} \times \vec{r}}{|\vec{r}|^3}$$where $\vec{r}$ is the vector from the current element to the place of measurement.
You can even code the problem in a way that you can add an aribitrary number of coils, e.g. for a long coil or a ring coil (torus).
# your solution here