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

In this article series

The moons dataset and decision surface graphics in a Jupyter environment – I
The moons dataset and decision surface graphics in a Jupyter environment – II – contourplots
The moons dataset and decision surface graphics in a Jupyter environment – III – Scatter-plots and LinearSVC

we used the moons data set to build up some basic knowledge about using a Jupyter notebook for experiments, SVM-tools of 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 analysis code to get an impression of what is going on and possible deficits. 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 ansers to the folowing 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 can 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 visualizing 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 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, fist 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. How do we get such pairs from our 2-dim xm1- and xm2-arrays?

We need a bit of array- or matrix-wizardry: 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 logic should be clear 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).

Applying contourf()

To distinguish contour areas we need a color map for our y-target-values. Later on we will also need 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 fro 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[0] = figs_x1 
    fig_size[1] = 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"? 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 distinguish between different models.

Conclusion

After some organization of our data we had success with out 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 becom 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. 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 ...

The moons dataset and decision surface graphics in a Jupyter environment – III – Scatter-plots and LinearSVC

During this article series we use the moons dataset to aquire basic knowledge on Python based tools for machine learning [ML]. The first article

The moons dataset and decision surface graphics in a Jupyter environment – I

provided us with some general information about the moons dataset. The second article

The moons dataset and decision surface graphics in a Jupyter environment – II – contourplots

explained how to use a Jupyter notebook for performing ML-experiments. We also had a look at some functions of "matplotlib" which enable us to create contour plots. We will need the latter to eventually visualize a decision surface between the two moon-like shaped clusters in the 2-dimensional representation space of the moons data points.

In this article we extend our plotting knowledge to the creation of a scatter-plot for visualizing data points of the moons data set. Then we will have a look at the "pipeline" feature of SciKit for a sequence of tasks to prepare the moons data set, analyze it and train various SVM-algorithms. We shall use one of these algorithms - namely LinearSVC - to predict the cluster association for some new data points.

Starting our Jupyter notebook, extending imports and loading the moons data set

At the end of the last session you certainly have found out, how to close the Jupyter notebook on a Linux system. Three steps were involved: 1) Logout via the button at the top-right corner of the web-page => 2) Ctrl-C in your terminal window => 3) Closing the tags in the browser.

For today#s session we start the notebook again from our dedicated Python "virtualenv" by

myself@mytux:/projekte/GIT/ai/ml1> source bin/activate
(ml1) myself@mytux:/projekte/GIT/ai/ml1> cd mynotebooks/
(ml1) myself@mytux:/projekte/GIT/ai/ml1/mynotebooks> jupyter notebook

We open "moons1.ipynb" from the list of available notebooks. (Note the move to the directory mynotebooks above; the Jupyter start page lists the notebooks in its present directory, which is used as a kind of "/"-directory for navigation. If you want the whole directory structure of the virtualenv accessible, you should choose a directory level higher as a starting point.)

For the work of today's session we need some more modules/classes from "sklearn" and "matplotlib". If you have not yet imported some of the most important ML-packages you should do so now. Probably, you need a second terminal - as the prompt of the first one is blocked by Jupyter:

myself@mytux:/projekte/GIT/ai/ml1> source bin/activate 
(ml1) myself@mytux:/projekte/GIT/ai/ml1> pip3 install --upgrade matplotlib numpy pandas scipy scikit-learn
Collecting matplotlib
  Downloading https://files.pythonhosted.org/packages/57/4f/dd381ecf6c6ab9bcdaa8ea912e866dedc6e696756156d8ecc087e20817e2/matplotlib-3.1.1-cp36-cp36m-manylinux1_x86_64.whl (13.1MB)
.....

The nice people from SciKit/SkLearn have already prepared data and functionality for the setup of the moons data set; We find the relevant function in sklearn.datasets. Later on we will also need some colormap functionality for scatter-plotting. And for doing the real work (training, SVM-analysis, ...) we need some special classes sklearn.

So, as a first step, we extend the import statements inside the first cell of our Jupyter notebook and run it:

Then we move to the end of our notebook to prepare new cells. (We can rerun already defined cell code at any time.)

We enter the following code that creates the moons data-points with some "noise", i.e. with a spread in the coordinates around a perfect moon-like line. You see the relevant function below; for a beginning it is wise to keep the spread limited - to avoid to many overlap points of the data clusters. I added some print-statements to get an impression of the data structure.

It is common use to assign an uppercase letter to the input data points and a lowercase letter to the array with the classification information (per data point). "make_moons()" creates an input array "X" of 2-dim data points and an associated array "y" with classification information for the data points. In our case the classification is binary, only; so we get an array with "0"s or "1"s for each point.

This basic (X,y)-structure of data is very common in classification tasks of ML - it represents information reduction: "multiple features" => "member of a class".

Scatter-plots: Plotting the raw data in 2D and 3D

We want to create a visual representation of the data points in their 2-dim feature space. We name the two elements of a data point array "x1" and "x2".

For a 2D-plot we need some symbols or "markers" to distinguish the different data points of our 2 classes. And we need at least 2 related colors to assign to the data points.

To work efficiently with colors, we create a list-like ColorMap-object from given color names (or RGB-values); see ListedColormap. We can access the RGBA-values from a ListedColormap by just calling it as a "list" with an integer index, i.e.:

colors= ('red', 'green', 'yellow')
cmap=ListedColormap(colors)
print(cmap(1)) // gives: (0.0, 0.5019607843137255, 0.0, 1.0)  
print(cmap(1)) // gives: (0.0, 0.5019607843137255, 0.0, 1.0)  

All RGBA-values are normalized between 0.0 and 1.0. Note that "green" in matplotlib is defined a bit strange in comparison to HTML. Let us try it :

The lower and upper limits of the the two axes must be given - we shall make them big enough. Here we shall use the plot functions pyplot.xlim() and pyplot.ylim().

The central function, which we shall use for plotting the data points, is "matplotlib.pyplot.scatter()"; see the documentation scatter() for parameters. Regarding the following code, please note that we plot all points of each of the two moon like cluster in one step. Therefore, we call scatter() exactly two times with the defined loop:

In the code you and me as beginners may stumble across the defined lists there with expressions included in the brackets. These are examples of so called Python "list comprehensions". You find an elementary introduction here.

As we are have come so far, lets try a 3D-scatter-plot, too. This is not required to achieve our objectives, but it is fun and it extends our knowledge base:

Of course all points of a class are placedon the same level (0 or 1) in z-direction. When we change the last statement to "ax.view_init(90, 0)" we get

As expected 🙂 .

Analyzing the data with the help of a "pipeline" and "LinearSVC" as an SVM classificator

Sklearn provides us with a very nice tool (actually a class) named "Pipeline":

Pipeline([]) allows us

  • to define a series of transformation operations which are successively applied to a data set
  • and to define exactly one predictor algorithm (e.g. a regression or classifier algorithm), which creates a model of the data and is optimized later on.

Transformers and predictors are also called "estimators". "Transformers" and "predictors" are defined by Python classes in Sklearn. All transformer classes must provide a method " fit_transform()" which operates on the (X,y)-data; the predictor class of a class provides a method "fit()".

The "Pipeline([])" is defined via rows of an array, each comprising a tuple with a chosen name for each step and the real class-names of the transformers/predictor. A pipeline of transformers and a predictor creates an object with a name, which also offers the method "fit()". Thus a pipeline prepares a data set(X,y) via a chain of operational steps for training.

This sounds complicated, but is actually pretty easy to use. How does such a pipeline look like for our moons dataset? One possible answer is:

polynomial_svm_clf = Pipeline([
  ("poly_features", PolynomialFeatures(degree=3)),
  ("scaler", StandardScaler()),
  ("svm_clf", LinearSVC(C=18, loss="hinge", max_iter=3000))
])
polynomial_svm_clf.fit(X, y)

The transformers obviously are "PolynomialFeatures" and "StandardScaler", the predictor is "LinearSVC" which is a special linear SVM method, trying to find a linear separation channel between the data in their representation space.

What is "PolynomialFeatures" in the first step of our Pipeline good for? Well, looking at the moons data plotted above, it becomes quite clear that in the conventional 2-dim space for the data points in the (x1, x2)-plane there is no linear decision surface. What can be done about this problem?

In the first article of this series I briefly discussed an approach where data, which are apparently not linearly separable in their representation space, can be extended by additional variables consisting of polynomial combinations of their basic X-data up to a maximum degree (i.e. the order of the polynomial).

Thereby, the dimensionality of the original X(x1,x2) set is extended by multiple further dimensions. The hope is that we can find a linear separation ("decision") surface in the new extended multi-dimensional representation space.

The first step of our Pipeline enhances our X by additional polynomial "features" (up to a degree of 3 in our example). We do not need to care for details - they are handled by the class "PolynomialFeatures". The choice of a polynomial of order 3 is a bit arbitrary at the moment; we shall play around with the polynomial degree a bit in a future article. Other parameters, as the number of iterations (3000) to find an optimal solution and the width "C" for the separation channel, will also be a subject of further experiments.

The second step in the Pipeline is a simple one: StandardScaler.fit_transform() scales all data such that they fit into standard ranges. This helps both for e.g. linear regression- and SVM-analysis.

The third step assigns a predictor - in our example a simple linear SVM-like algorithm. It is provided by the class LinearSVC (a linear soft margin classificator). See e.g
support-vector-machine-algorithm/,
LinearSVC vs SVC,
www.quora.com : What-is-the-difference-between-Linear-SVMs-and-Logistic-Regression.

Analyzing the moons data and fitting the LinearSVC algorithm

Let us apply our pipeline and predict for some data points outside the X-region whether they belong to the "red" or the "blue" cluster. But, how do we predict?

We are not surprised that we find a method predict() in the documentation for our classifier LinearSVC.

So:

We get for the different test points

[x1=1.50, x2=1.0] => 0  
[x1=1.92, x2=0.8] => 0
[x1=1.94, x2=0.8] => 1
[x1=2.20, x2=1.0] => 1               

Looking at our scatter plot above we can assume that the decision line predicted by LinearSVC moves through the right upper corner of the (x1,x2)-space.

However and of course, looking at some test data points is not enough to check the quality of our approach to find a decision surface. We absolutely need to plot the decision surface.

Conclusion

But enough for today's session. We have seen, how we can produce a scatter plot for our moons data. We have also learned a bit about Sklearn's "pipelines". And we have used the classes "PolynomialFeatures" and "LinearSVC" to try to separate our two data clusters.

By now, we have gathered so much knowledge that we should be able to use our predictor to create a contour plot - with just 2 contour areas in our representation space. If we cover the (x1,x2)-plane densely and associate the predicted values of 0 or 1 with colors we should clearly see the contour line, i.e. the decision surface, separating the two areas. And hopefully all data points of our original (X,y) set fall into the right region. This is the topic of the next article

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

Stay tuned.

Links

Understanding Support Vector Machine algorithm from examples (along with code) by Sunil Ray
Stackoverflow - What is exactly sklearn-pipeline?
LinearSVC