# Matplotlib, Jupyter and updating multiple interactive plots

For experiments in Machine Learning [ML] it is quite useful to see the development of some characteristic quantities during optimization processes for algorithms - e.g. the behaviour of the cost function during the training of Artificial Neural Networks. Beginners in Python the look for an option to continuously update plots by interactively changing or extending data from a running Python code.

Does Matplotlib offer an option for interactively updating plots? In a Jupyter notebook? Yes, it does. It is even possible to update multiple plot areas simultanously. The magic (meta) commands are "%matplotlib notebook" and "matplotlib.pyplot.ion()".

The following code for a Jupyter cell demonstrates the basic principles. I hope it is useful for other ML- and Python beginners as me.

```# Tests for dynamic plot updates
#-------------------------------
%matplotlib notebook
import numpy as np
import matplotlib.pyplot as plt
import time

x = np.linspace(0, 10*np.pi, 100)
y = np.sin(x)

# The really important command for interactive plot updating
plt.ion()

# sizing of the plots figure sizes
fig_size = plt.rcParams["figure.figsize"]
fig_size = 8
fig_size = 3

# Two figures
# -----------
fig1 = plt.figure(1)
fig2 = plt.figure(2)

# first figure with two plot-areas with axes
# --------------------------------------------
ax1_1 = fig1.add_subplot(121)
ax1_2 = fig1.add_subplot(122)

fig1.canvas.draw()

# second figure with just one plot area with axes
# -------------------------------------------------
ax2 = fig2.add_subplot(121)
line1, = ax2.plot(x, y, 'b-')
fig2.canvas.draw()

z= 32
b = np.zeros()
c = np.zeros()
c = 1000

for i in range(z):
# update data
phase = np.pi / z * i
line1.set_ydata(np.sin(0.5 * x + phase))
b = np.append(b, [i**2])
c = np.append(c, [1000.0 - i**2])

# re-plot area 1 of fig1
ax1_1.clear()
ax1_1.set_xlim (0, 100)
ax1_1.set_ylim (0, 1000)
ax1_1.plot(b)

# re-plot area 2 of fig1
ax1_2.clear()
ax1_2.set_xlim (0, 100)
ax1_2.set_ylim (0, 1000)
ax1_2.plot(c)

# redraw fig 1
fig1.canvas.draw()

# redraw fig 2 with updated data
fig2.canvas.draw()

time.sleep(0.1)
```

As you see clearly we defined two different "figures" to be plotted - fig1 and fig2. The first figure ist horizontally splitted into two plotting areas with axes "ax1_1" and "ax1_2". Such a plotting area is created via the "fig1.add_subplot()" function and suitable parameters. The second figure contains only one plotting area "ax2".

Then we update data for the plots within a loop witrh a timer of 0.1 secs. We clear the respective areas, redefine the axes and perform the plot for the updated data via the function "plt.figure.canvas.draw()".

In our case we see two parabolas develop in the upper figure; the lower figure shows a sinus-wave moving slowly from the right to the left.

The following plots show screenshots of the output in a Jupyter notebook in th emiddle of the loop and at its end: You see that we can deal with 3 plots at the same time. Try it yourself!

Hint:
There is small problem with the plot sizing when you have used the zoom-functionality of Chrome, Chromium or Firefox. You should work with interactive plots with the browser-zoom set to 100%.

# The moons dataset and decision surface graphics in a Jupyter environment – V – a class for plots and some experiments

We proceed with our exercises on the moons dataset. This series of articles is intended for readers which - as me - are relatively new both to Python and Machine Learning. By working with examples we try to extend our knowledge about the tools "Juypter notebooks" and "Eclipse/PyDev" for setting up experiments which require plots for an interpretation.

We have so far used a Jupyter notebook to perform some initial experiments for creating and displaying a decision surface between the moons dataset clusters with an algorithm called "LinearSVC". If you followed everything I described in the last articles

you may now have gathered around 20 different cells with code. Part of the cells' code was used to learn some basics about contour and scatter plots. This code is now irrelevant for further experiments. Time to consolidate our plotting knowledge.

In the last article I promised to put plot-related code into a Python class. The class itself can become a part of a Python module - which we in turn can import into the code of Jupyter notebook. By doing this we can reduce the number of cells in a notebook drastically. The importing of external classes is thus helpful for concentrating on "real" data analysis experiments with different learning and predicting algorithms and/or a variation of their parameters.

I assume that you have some basic knowledge on how classes are build in Python. If not please see an introductory book on Python 3.

# A class for plotting simple decision surfaces in a 2-dimensional space

In the articles

I had shown how to set up Eclipse PyDev to be used in the context of a Python virtual environment. In our special environment "ml1" used by our Jupyter notebook "moons1.ipynb" we have the following directory structure: "ml1" has a sub-directory "mynotebooks" which contains notebook files as our "moons1.ipynb". To provide a place for other general code there we open up a directory "mycode". In it we create a file "myplots.py" for a module "myplots", which shall comprise our own Python classes for plotting.

We distribute the code discussed in the last 2 articles of this series into methods of a class "MyDecisionPlot"; we put the following code into our file "myplots.py" with the Pydev editor.

```'''
Created on 15.07.2019
Module to gather classes for plotting
@author: rmo
'''
import numpy as np
import sys
from matplotlib import pyplot as plt
from matplotlib.colors import ListedColormap
import matplotlib.patches as mpat
#from matplotlib import ticker, cm
#from mpl_toolkits import mplot3d

class MyDecisionPlot():
'''
This class allows for
1) decision surfaces in 2D (x1,x2) planes
2) plotting scatter data of datapoints
'''

def __init__(self, X, y, predictor = None, ax_x_delta=1.0, ax_y_delta=1.0,
mesh_res=0.01, alpha=0.4, bcontour=1, bscatter=1,
figs_x1=12.0, figs_x2=8.0,
x1_lbl='x1', x2_lbl='x2',
legend_loc='upper right'
):
'''
Constructor of MyDecisionPlot
Input:
X: Input array (2D) for learning- and predictor-algorithm as VSM
y: result data for learning- and predictor-algorithm
ax_x_delta, ax_y_delta : delta for extension of both axis beyond the given X, y-data
mesh_res: resolution of the mesh spanned in the (x1,x2)-plane (x_max-x_min) * mesh_res
alpha:  transparency  of contours
bcontour: 0: Do not plot contour areas 1: plot contour areas
bscatter: 0: Do not plot scatter points of the input data sample 1: Plot scatter plot of the input data sample
figs_x1: plot size in x1 direction
figs_x2: plot size in x2 direction
x1_lbl, x2_lbl : axes lables
legend_loc : position of a legend
Ouptut:
Internal: self._mesh_points (mesh points created)
External: Plots - shoukd cone up automatically in Jupyter notebooks
'''

# initiate some internal variables
self._x1_min = 0.0
self._x1_max = 1.0
self._x2_min = 0.0
self._x2_max = 1.0

# Alternatives to resize plots
# 1: just resize figure  2: resize plus create subplots() [figure + axes]
self._plot_resize_alternative = 2

# X (x1,x2)-Input array
self.__X = X
self.__y = y
self._Z  = None

# predictor = algorithm to create y-values for new (x1,x2)-data points
self._predictor = predictor

# meshdata
self._resolution = mesh_res   # resolution of the mesh
self.__ax_x_delta = ax_x_delta
self.__ax_y_delta = ax_y_delta
self._alpha = alpha
self._bscatter = bscatter
self._bcontour = bcontour

self._xm1 = None
self._xm2 = None
self._mesh_points = None

# set marker array and define colormap
self._markers = ('s', 'x', 'o', '^', 'v')
self._colors = ('red', 'blue', 'lightgreen', 'gray', 'cyan')
self._cmap = ListedColormap(self._colors[:len(np.unique(y))])

self._x1_lbl = x1_lbl
self._x2_lbl = x2_lbl
self._legend_loc = legend_loc

# Plot-sizing
self._figs_x1 = figs_x1
self._figs_x2 = figs_x2
self._fig = None
self._ax  = None
# alternative 2 does resizing and (!) subplots()
self.initiate_and_resize_plot(self._plot_resize_alternative)

# create mesh in x1, x2 - direction with mesh_res resolution
# meshpoint-array will be creted with right dimension for plotting data
self.create_mesh()
# Array meshpoints should exist now

if(self._bcontour == 1):
try:
if self._predictor == None:
raise ValueError
except ValueError:
print ("You must provide an algorithm = 'predictor' as parameter")
#sys.exit(0)
sys.exit()

self.make_contourplot()
else:
if (self._bscatter == 1):
self.make_scatter_plot()

# method to create a dense mesh in the (x1,x2)-plane
def create_mesh(self):
'''
Method to create a dense mesh in an (x1,x2) plane
Input: x1, x2-data are constructed from array self.__X
Output: A suitable array of meshpoints is written to self._mesh_points()
'''
try:
self._x1_min = self.__X[:, 0].min()
self._x1_max = self.__X[:, 0].max()
except ValueError: # as e:
print ("cannot determine x1_min = X[:,0].min() or x1_max = X[:,0].max()")
try:
self._x2_min = self.__X[:, 1].min()
self._x2_max = self.__X[:, 1].max()
except ValueError: # as e:
print ("cannot determine x2_min =X[:,1].min()) or x2_max = X[:,1].max()")

self._x1_min, self._x1_max = self._x1_min - self.__ax_x_delta, self._x1_max + self.__ax_x_delta
self._x2_min, self._x2_max = self._x2_min - self.__ax_x_delta, self._x2_max + self.__ax_x_delta

#create mesh data (x1, x2)
self._xm1, self._xm2 = np.meshgrid( np.arange(self._x1_min, self._x1_max, self._resolution),
np.arange(self._x2_min, self._x2_max, self._resolution))

#print (self._xm1)
# for hasattr the variable cannot be provate !
#print ("xm1 is there: ", hasattr(self,'_xm1' ) )

# ordering and transposing of the mesh-matrix
# for understanding the structure and transpose requirements see linux-blog.anracom.con
self._mesh_points = np.array([self._xm1.ravel(), self._xm2.ravel()]).T

try:
if( hasattr(self, '_mesh_points') == False ):
raise ValueError
except ValueError:
print("The required array mesh_points has not been created!")
exit

# -------------
# Some helper functions to change valus on the fly if necessary

def set_mesh_res(self, new_mesh_res):
self._resolution = new_mesh_res

def change_predictor(self, new_predictor):
self._predictor = new_predictor

def change_alpha(self, new_alpha):
self._alpha = new_alpha

def change_figs(self, new_figs_x1, new_figs_x2):
self._figs_x1 = new_figs_x1
self._figs_x2 = new_figs_x2

# -------------
# method to get subplots and resize the figure
# -------------
def initiate_and_resize_plot(self, alternative=2 ):

# Alternative 1 to resize plots - works afte rimports to Jupyter notebooks, too
if alternative == 1:
self._fig_size = plt.rcParams["figure.figsize"]
self._fig_size = self._figs_x1
self._fig_size = self._figs_x2
plt.rcParams["figure.figsize"] = self._fig_size

# Not working for sizing if plain subplots() is used
#plt.figure(figsize=(self._figs_x1 , self._figs_x2))
#self._fig, self._ax = plt.subplots()
# instead you have to put the figsize-parameter into the subplots() function

# Alternative 2 for resizing plots and using subplots()
# we use this alternative as we may need the axes later for 3D plots
if alternative == 2:
self._fig, self._ax = plt.subplots(figsize=(self._figs_x1 , self._figs_x2))

# ***********************************************

# -------------
# method to create contour plots
# -------------
def make_contourplot(self):
'''
Method to create a contourplot based on a dense mesh of points in an (x1,x2) plane
and a predictor algorithm which allows for value calculations
'''

try:
if( not hasattr(self, '_mesh_points') ):
raise ValueError
except ValueError:
print("The required array mesh_points has not been created!")
exit

# Predict values for all meshpoints
try:
self._Z = self._predictor.predict(self._mesh_points)
except AttributeError:
print("method predictor.predict() does not exist")

#reshape
self._Z = self._Z.reshape(self._xm1.shape)
#print (self._Z)

# make the plot
plt.contourf(self._xm1, self._xm2, self._Z, alpha=self._alpha, cmap=self._cmap)

# create a scatter-plot of data sample as well
if (self._bscatter == 1):
self.make_scatter_plot()

self.make_plot_legend()

# -------------
# method to create a scatter plot of the data sample
# -------------
def make_scatter_plot(self):
alpha2 = self._alpha + 0.4
if (alpha2 > 1.0 ):
alpha2 = 1.0
for idx, yv in enumerate(np.unique(self.__y)):
plt.scatter(x=self.__X[self.__y==yv, 0], y=self.__X[self.__y==yv, 1],
alpha=alpha2, c=[self._cmap(idx)], marker=self._markers[idx], label=yv)

if self._bscatter == 0:
self._bscatter = 1

self.make_plot_legend()

# -------------
# method to add a legend
# -------------
def make_plot_legend(self):
plt.xlim(self._x1_min, self._x1_max)
plt.ylim(self._x2_min, self._x2_max)
plt.xlabel(self._x1_lbl)
plt.ylabel(self._x2_lbl)

# we have two cases
#     a) for a scatter plot we have array values where the legend is taken from automatically
#     b) For apure contourplot we need to prepare a legend with "patches" (kind og labels) used by pyplot.legend()
if (self._bscatter == 1):
plt.legend(loc=self._legend_loc)
else:
red_patch  = mpat.Patch(color='red',  label='0', alpha=0.4)
blue_patch = mpat.Patch(color='blue', label='1', alpha=0.4)
plt.legend(handles=[red_patch, blue_patch], loc=self._legend_loc)

```

This certainly is no masterpiece of superior code design; so you may change it. However, the code is good enough for our present purposes.

Note that we have to import basic Python modules into the namespace of this module. This is conventionally done at the top of the file.

Note also the 2 alternatives offered for resizing a plot! Both work also for "inline" plotting in a Jupyter environment; see the text below.

# Using the module "myplots" in a Jupyter notebook

In a terminal we move to our example directory "/projekte/GIT/ai/ml1" and start our virtual Python environment:

```myself@mytux: /projekte/GIT/ai/ml1> source bin/activate
(ml1) myself@mytux:/projekte/GIT/ai/ml1> jupyter notebook
[I 11:46:14.687 NotebookApp] Writing notebook server cookie secret to /run/user/1999/jupyter/notebook_cookie_secret
[I 11:46:15.942 NotebookApp] Serving notebooks from local directory: /projekte/GIT/ai/ml1
[I 11:46:15.942 NotebookApp] The Jupyter Notebook is running at:
....
....
```

We then open our old notebook "moons1" and save it under the name "moons2": We delete all old cells. Then we change the first cell of our new notebook to the following contents:

```import imp
%matplotlib inline
from mycode import myplots

from sklearn.datasets import make_moons
from sklearn.pipeline import Pipeline
from sklearn.preprocessing import StandardScaler
from sklearn.preprocessing import PolynomialFeatures
from sklearn.svm import LinearSVC
from sklearn.svm import SVC
```

You see that we imported the "myplots" module from the "package" directory "mycode". The Jupyter notebook will find the path to "mycode" as long as we have opened the notebook on a higher directory level. Which we did ... see above.

Note the statement with a so called "magic command" for our notebook:

%matplotlib inline

There are many other "magic commands" and parameters which you can study at
Ipython magic commands

The command "%matplotlib inline" informs the notebook to show plots created by any imported modules "inline", i.e. in the visual context of the affected cell. This specific magic directive should be issued before matplotlib/pyplot is imported into any of the external python modules which we in turn import into the cell code.

A call of plt.show() in our class's method "make_contourplot()" is no longer necessary afterwards.

If we, however, want to resize the plots in comparison to Jupyter standard values we have to control this by parameters of our plot class. Such parameters are offered already in the interface of the class; but they can be changed by a method "change_figs(new_figs_x1, new_figs_x2)), too.

In a second cell of our new notebook we now prepare the moon data set

```X, y = make_moons(noise=0.1, random_state=5)
```

Further cells will be used for quick individual experiments with the moons dataset. E.g.:

```imp.reload(myplots)
X, y = make_moons(noise=0.1, random_state=5)

# contour plot of the moons data - with scatter plot / training with LinearSVC
polynomial_degree = 3
max_iterations = 6000
polynomial_svm_clf = Pipeline([
("poly_features", PolynomialFeatures(degree=polynomial_degree)),
("scaler", StandardScaler()),
("svm_clf", LinearSVC(C=18, loss="hinge", max_iter=max_iterations))
])
#training
polynomial_svm_clf.fit(X, y)

#plotting
MyPlt = myplots.MyDecisionPlot(X, y, polynomial_svm_clf, bcontour = 1, bscatter=1 )

```

The last type of cell just handles the setup and training of our specific algorithm "LinearSVC" and delegates plotting to our class.

# Testing the new notebook

A test of the 3 cells in their order gives All well! This is exactly what we hoped to achieve.

# Three experiments with a varying polynomial degree

As we now have a simple and compact cell template for experiments we add three further cells where we vary the degree of the polynomials for LinearSVC. Below we show the results for degree 6, 7 and for comparison also for a degree of 2.   On a modern computer it should take almost no time to produce the results. (We shall learn how to measure CPU-time in the next article).

We understand that we at least need a polynomial of degree 3 to separate the data reasonably. However, polynomials with an even degree (>= 4) separate the 2 data regions very differently compared to polynomials with an uneven degree (>=3) in outer areas of the (x1,x2)-plane (where no training data were placed):

Depending on the polynomial degree our Linear SVC algorithm obviously extrapolates in different ways to regions without such data. And we have no clue which of the polynomials is the better choice ...

This poses a warning for the use of AI in general:

We should be extremely careful to trust predictions of any AI algorithm in parameter regions for which the algorithm must extrapolate - as long as we have no real data points available there to discriminate between multiple solutions that all work equally well in regions with given training data.

# Would general modules be imported twice in a Jupyter cell - via the import of an external module, which itself includes import statements, and a direct import statement in a Jupyter cell?

The question posed in the headline is an interesting one for me as a Python beginner. Coming from other programming languages I get a bit nervous when I see the possibility for import statements referring to a specific module both in another already imported module and by a direct import statement afterwards in a Jupyter cell. E.g. we import numpy indirectly via our "myplots" module, but we could and sometimes must import it in addition directly in our Jupyter cell.

Note that we must make the general modules as numpy, matplotlib, etc. available in the namespace environment of our private module "myplots". Otherwise the functions cannot be used there. The Jupyter cell, however, corresponds to an independent namespace environment. So, an import may indeed be required there, too, if we plan to handle numpy arrays via code in such a cell.

Reading a bit about the Python import logic on the Internet reveals that a double import or overwriting should not take place; instead an already imported piece of code only gets multiple references in the various specific namespaces of different code environments.

We can test this with the following code in a Jupyter cell:

Note that numpy is also imported by our "myplots". However, the length of the list produced by sys.modules.keys(), which enumerates all possible module reference points (including imports) does not change.

# Reloading modules in Jupyter cells

What if we in the course of or experiments need to change the code of our imported module? Then we need to reload the module in a Jupyter cell before we run it again. In our case (Python 3!) this can be done by the command

imp.reload(myplots)

As the code of our first cell reveals, the general package "imp" must have been imported before we can use its reload-function.

# Conclusion

We saw that it is easy to use our own modules with Python class code, which we created in an Eclipse/PyDev environment, in a Jupyter notebook. We just use Python's standard import mechanism in Jupyter cells to get access to our classes. We applied this to a module with a simple class for preparing decision surface plots based on contour and scatter plot routines of matplotlib. We imported the module in a new Jupyter notebook.

Plots created by our imported class-methods were displayed correctly within the cell environment as soon as we used the magic directive "%matplotlib inline" within our notebook.

In addition we used our new notebook with its compact cell structure for some quick experiments: We set different values for the polynomial degree parameter of our LinearSVC algorithm. We saw that the results of algorithms should be interpreted with caution if the predictions refer to data points in regions of the representation or feature space which were not at all covered by the data sets for training.

The prediction results (= extrapolations) of even one and the same algorithm with different parameters may deviate strongly in such regions - and we may not have any reliable indications to decide which of the predictions and their basic parameter setting are better.

In the next article of this series

The moons dataset and decision surface graphics in a Jupyter environment – VI – Kernel-based SVC algorithms

we shall have a look at what kind of decision surface some other SVM algorithms than LinearSVC create for the moons dataset. In addition shall briefly discuss kernel based algorithms.

# The moons dataset and decision surface graphics in a Jupyter environment – IV – plotting the decision surface

In this article series

we used the moons data set to build up some basic knowledge about using a Jupyter notebook for experiments, Pipelines and SVM-algorithms of SciKit-Sklearn and plotting functionality of matplotlib.

Our ultimate goal is to write some code for plotting a decision surface between the moon shaped data clusters. The ability to visualize data sets and related decision surfaces is a key to quickly testing the quality of different SVM-approaches. Otherwise, you would have to run some kind of analysis code to get an impression of what is going on and possible deficits of the determined separation surface.

In most cases, a visual impression of the separation surface for complexly shaped data sets will give you much clearer information. With just one look you get answers to the following questions:

• How well does the decision surface really separate the data points of the clusters? Are there data points which are placed on the wrong side of the decision surface?
• How reasonable does the decision surface look like? How does it extend into regions of the representation space not covered by the data points of the training set?
• Which parameters of our SVM-approach influences what regarding the shape of the surface?

In the second article of this series we saw how we would create contour-plots. The motivation behind this was that a decision surface is something as the border between different areas of data points in an (x1,x2)-plane for which we get different distinct Z(x1,x2)-values. I.e., a contour line separating contour areas is an example of a decision surface in a 2-dimensional plane.

During the third article we learned in addition how we could visualize the various distinct data points of a training set via a scatter-plot.

We then applied some analysis tools to analyze the moons data - namely the "LinearSVC" algorithm together with "PolynomialFeatures" to cover non-linearity by polynomial extensions of the input data.

We did this in form of a Sklearn Pipeline for a step-wise transformation of our data set plus the definition of a predictor algorithm. Our LinearSVC-algorithm was trained with 3000 iterations (for a polynomial degree of 3) - and we could predict values for new data points.

In this article we shall combine all previous insights to produce a visual impression of the decision interface determined by LinearSVC. We shall put part of our code into a wrapper function. This will help us to efficiently visualize the results of further classification experiments.

# Predicting Z-values for a contour plot in the (x1,x2) representation space of the moons dataset

To allow for the necessary interpolations done during contour plotting we need to cover the visible (x1,x2)-area relatively densely and systematically by data points. We then evaluate Z-values for all these points - in our case distinct values, namely 0 and 1. To achieve this we build a mesh of data points both in x1- and x2-direction. We saw already in the second article how numpy's meshgrid() function can help us with this objective:

```resolution = 0.02
x1_min, x1_max = X[:, 0].min()  - 1, X[:, 0].max() + 1
x2_min, x2_max = X[:, 1].min()  - 1, X[:, 1].max() + 1
xm1, xm2 = np.meshgrid( np.arange(x1_min, x1_max, resolution),
np.arange(x2_min, x2_max, resolution))
```

We extend our area quite a bit beyond the defined limits of (x1,x2) coordinates in our data set. Note that xm1 and xm2 are 2-dim arrays (!) of the same shape covering the surface with repeated values in either coordinate! We shall need this shape later on in our predicted Z-array.

To get a better understanding of the structure of the meshgrid data we start our Jupyter notebook (see the last article), and, of course, first run the cell with the import statements

```import numpy as np
import matplotlib
from matplotlib import pyplot as plt
from matplotlib import ticker, cm
from mpl_toolkits import mplot3d

from matplotlib.colors import ListedColormap
from sklearn.datasets import make_moons

from sklearn.pipeline import Pipeline
from sklearn.preprocessing import StandardScaler
from sklearn.preprocessing import PolynomialFeatures
from sklearn.svm import LinearSVC
from sklearn.svm import SVC
```

Then we run the cell that creates the moons data set to get the X-array of [x1,x2] coordinates plus the target values y:

```X, y = make_moons(noise=0.1, random_state=5)
#X, y = make_moons(noise=0.18, random_state=5)
print('X.shape: ' + str(X.shape))
print('y.shape: ' + str(y.shape))
print("\nX-array: ")
print(X)
print("\ny-array: ")
print(y)
```

Now we can apply the "meshgrid()"-function in a new cell: You see the 2-dim structure of the xm1- and xm2-arrays.

Rearranging the mesh data for predictions
How do we predict data values? In the last article we did this in the following form

```z_test = polynomial_svm_clf.predict([[x1_test_1, x2_test_1],
[x1_test_2, x2_test_2],
[x1_test_3, x2_test_3],
[x1_test_3, x2_test_3]
])
```

"polynomial_svm_clf" was the trained predictor we got by our pipeline with the LinearSVC algorithm and a subsequent training.

The "predict()"-function requires its input values as a 1-dim array, where each element provides a (x1, x2)-pair of coordinate values. But how do we get such pairs from our strange 2-dimensional xm1- and xm2-arrays?

We need a bit of array- or matrix-wizardry here:

Numpy gives us the function "ravel()" which transforms a 2d-array into a 1-d array AND numpy also gives us the possibility to transpose a matrix (invert the axes) via "array().T". (Have a look at the numpy-documentation e.g. at https://docs.scipy.org/doc/).

We can use these options in the following way - see the test example:

The involved logic should be clear by now. So, the next step should be something like

```Z = polynomial_svm_clf.predict( np.array([xm1.ravel(), xm2.ravel()] ).T)
```

However, in the second article we already learned that we need Z in the same shape as the 2-dim mesh coordinate-arrays to create a contour-plot with contourf(). We, therefore, need to reshape the Z-array; this is easy - numpy contains a method reshape() for numpy-array-objects : Z = Z.reshape(xm1.shape). It is sufficient to use xm1 - it has the same shape as xm2.

# Applying contourf()

To distinguish contour areas we need a color map for our y-target-values. Later on we will also need different optical markers for the data points. So, for the contour-plot we add some statements like

```markers = ('s', 'x', 'o', '^', 'v')
colors = ('red', 'blue', 'lightgreen', 'gray', 'cyan')
# fetch unique values of y into array and associate with colors
cmap = ListedColormap(colors[:len(np.unique(y))])

Z = Z.reshape(xm1.shape)

# see article 2 for the use of contourf()
plt.contourf(xm1, xm2, Z, alpha=0.4, cmap=cmap)
```

Let us put all this together; as the statements to create a plot obviously are many, we first define a function "plot_decision_surface()" in a notebook cell and run the cell contents: Now, let us test - with a new cell that repeats some of our code of the last article for training: Yeah - we eventually got our decision surface!

But this result still is not really satisfactory - we need the data set points in addition to see how good the 2 clusters are separated. But with the insights of the last article this is now a piece of cake; we extend our function and run the definition cell

```def plot_decision_surface(X, y, predictor, ax_delta=1.0, mesh_res = 0.01, alpha=0.4, bscatter=1,
figs_x1=12.0, figs_x2=8.0, x1_lbl='x1', x2_lbl='x2',
legend_loc='upper right'):

# some arrays and colormap
markers = ('s', 'x', 'o', '^', 'v')
colors = ('red', 'blue', 'lightgreen', 'gray', 'cyan')
cmap = ListedColormap(colors[:len(np.unique(y))])

# plot size
fig_size = plt.rcParams["figure.figsize"]
fig_size = figs_x1
fig_size = figs_x2
plt.rcParams["figure.figsize"] = fig_size

# mesh points
resolution = mesh_res
x1_min, x1_max = X[:, 0].min()  - 1, X[:, 0].max() + 1
x2_min, x2_max = X[:, 1].min()  - 1, X[:, 1].max() + 1
xm1, xm2 = np.meshgrid( np.arange(x1_min, x1_max, resolution),
np.arange(x2_min, x2_max, resolution))
mesh_points = np.array([xm1.ravel(), xm2.ravel()]).T

# predicted vals
Z = predictor.predict(mesh_points)
Z = Z.reshape(xm1.shape)

# plot contur areas
plt.contourf(xm1, xm2, Z, alpha=alpha, cmap=cmap)

# add a scatter plot of the data points
if (bscatter == 1):
alpha2 = alpha + 0.4
if (alpha2 > 1.0 ):
alpha2 = 1.0
for idx, yv in enumerate(np.unique(y)):
plt.scatter(x=X[y==yv, 0], y=X[y==yv, 1],
alpha=alpha2, c=[cmap(idx)], marker=markers[idx], label=yv)

plt.xlim(x1_min, x1_max)
plt.ylim(x2_min, x2_max)
plt.xlabel(x1_label)
plt.ylabel(x2_label)
if (bscatter == 1):
plt.legend(loc=legend_loc)
```

Now we get: So far, so good ! We see that our specific model of the moons data separates the (x1,x2)-plane into two areas - which has two wiggles near our data points, but otherwise asymptotically approaches almost a diagonal.

Hmmm, one could bet that this is model specific. Therefore, let us do a quick test for a polynomial_degree=4 and max_iterations=6000. We get Surprise, surprise ... We have already 2 different models fitting our data.

Which one do you believe to be "better" for extrapolations into the big (x1,x2)-plane? Even in the vicinity of the leftmost and rightmost points in x1-direction we would get different predictions of our models for some points. We see that our knowledge is insufficient - i.e. the test data do not provide enough information to really distinguish between different models.

# Conclusion

After some organization of our data we had success with our approach of using a contour plot to visualize a decision surface in the 2-dimensional space (x1,x2) of input data X for our moon clusters. A simple wrapper function for surface plotting equips us now for further fast experiments with other algorithms.

To become better organized, we should save this plot-function for decision surfaces as well as a simpler function for pure scatter plots in a Python class and import the functionality later on.

We shall create such a class within Eclipse PyDev as a first step in the next article:

The moons dataset and decision surface graphics in a Jupyter environment - V - a class for plots and some experiments

Afterward we shall look at other SVM algorithms - as the "polynomial kernel" and the "Gaussian kernel". We shall also have a look at the impact of some of the parameters of the algorithms. Stay tuned ...