Tag Archives: SVM

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

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We continue with our article series on the moons dataset as an entry point into “Machine Learning”:

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

The moons dataset and the related classification problem posed interesting challenges for us as beginners in ML:

Most people starting with ML have probably studied the topic of linear classification to separate distinct data sets by a linear decision surface (hyperplane) on data (y, X) with X=(x1,x2) defining points in a 2-dim space.

The SVM-approach studied in this article series follows a so called “soft-margin” classification: It tries to maximize the distances of the decision surface to the data points whilst it at the same time tries to reduce the number of points which violate the separation, i.e. points which end up on the wrong side of the decision hyperplane. This approach is controlled by so called hyper-parameters as a parameter “C” in the LinearSVC algorithm. If we just used LinearSVC on the original (x1,x2) data plane the hyperplane would be some linear line.

Unfortunately, for the moons dataset we must to overcome the fundamental problem that a linear classification approach to define a decision surface between the two data clusters is insufficient. (We have confirmed this in our last article by showing that even a quadratic approach does not give any reasonable decision surface.) We circumvented this problem by a trick – namely a polynomial extension of the parameter space (x1,x2) via a SciKit-Learn function “PolynomialFeatures()”.

We also had to tackle the technical problem of writing a simple Python class for creating plots of a decision surface in a 2 dim parameter space (x1,x2). After having solved this problem in the last articles it became easy to apply different or differently parameterized algorithms to the data set and display the results graphically. The functionalities of the SciKit and numpy libraries liberated us from dealing with complicated mathematical tasks. We also saw that using the “Pipeline” function helped to organize the sequential operations on the data sets – here transforming data, scaling data and training of a chosen algorithm on the data – in a very comfortable way.

In this article we want to visualize results of some other SVM approaches, which make use of the called “Kernel trick”. We shall explain this briefly and then use functions provided by SciKit to perform further experiments. On the Python side we shall learn, how to measure required computational times of the different algorithms.

Artificial polynomial feature extension

So far we moved
around the hurdle of non-linearity in combination with LinearSVC by a costly trick: We have extended the original 2-dimensional parameter space X(x1, x2) of our data points [y, X] artificially by more X-dimensions. We proclaimed that the distribution of data points is actually given in a multidimensional space constructed by a polynomial transformation: Each new and additional dimension is given by a term f*(x1**n)*(x2**m) with n+m = D, the degree of a polynomial function of x1 and x2.

This works as if the results “y” depended on further properties expressed by polynomial combinations of x1 and x2. Note that a 2-dim space (x1,x2) thus may be transformed into a 5-dimensional space with axes for x1, x2, x1**2, a*x1*x2, x**2. A data point (x1,x2) is transformed to a vector P(x1,x2) = [p_1=x1, p_2=x2, p_3=x1**2, p_4=a*x1*x2, p_5=x2**2]. Actually, for a broad class of problems it is enough to look at the 3-dim transformation space P([x1,x2])=[p_1=x1**2, p_2=f*x1*x2, p_3=x2**2].

In such a higher dimensional space we might actually find a “linear” hyperplane which allows for a suitable separation for the data clusters belonging to 2 different values of y=y(x1,x2) – here y=0 and y=1. The optimization algorithm then determines a suitable parameter vector (Theta= [theta_0, theta_1, … theta=n]), describing an optimal linear surface with respect to the distance of the data points to this surface. If we omit details then the separation surface is basically described by some scalar product of the form

theta_0*1 + theta1*p_1 + theta2*p_2 + … + theta_D * p_D = const.

Our algorithm calculates and optimizes the required theta-values.

Note that the projection of this hyperplane into the original (x1,x2)-feature-space becomes a non-linear hyperplane there. See the book of S. Raschka “Python machine Learning”, 20115, PACKT Publishing, chapter 3 for a nice example).

I cannot go into mathematical details in this article series. Nevertheless, this is a raw description of what we have done so far. But note, that there are other methods to extend the parameter space of the original data points to higher dimensions. The extension by the use of polynomials is just one of many approaches.

Why is a dimensional extension by polynomials computationally costly?

The soft margin optimization is a so called quadratic problem with linear constraints. To solve it you have both to transform all points into the higher dimensional space, i.e. to determine point coordinates there, and then determine distances in this space to a hyperplane with varying parameters.

This means we have at least to perform 2*D different calculations of the powers of the individual original coordinates of our input data points. As the power operation itself requires CPU-time depending on D the coordinate transformation operations vary with the

CPU-time ∝ number of points x the dimension of the original space x D**2

The “Kernel Trick”

In a certain formulation of the optimization problem the equation which determines the optimal linear separation hyperplane is governed by scalar products of the transformed vectors T(P(a1,a2)) * P(b1,b2) for all given data points a(x1=a1, x2=a2) and b(x1=b1,x2=b2), with T representing the transpose operation.

Now, instead of calculating the scalar product of the transformed vectors we would like to use a simpler “kernel” function

K(a, b = T(P(a)) * P(b)

It can indeed be shown that such a function K, which only operates on the lower dimensional space (!), really exists under fairly general conditions.

Kernel functions, which
are typically used in classification problems are:

  • Polynomial kernel : K(a, b) = [ f*P(a) * b + g]**D, with D = polynomial degree of a polynomial transformation
  • Gaussian RBF kernel: K(a, b) = exp(- g * || a – b ||**2 ), which also corresponds to an extension into a transformed parameter space of very high dimension (see below).

You can do the maths for the number and complexity operations for the polynomial kernel on your own. It is easy to see that it costs much less to perform a scalar product in the lower dimensional space and calculate the n-the power of the result just once – instead of transforming two points by around 2*D different power operations on the individual coordinates and then performing a scalar product in the higher dimensional space:

The difference in CPU costs between a non-kernel based polynomial extension and a kernel based grows quadratically, i.e. with D**2.

All good ?
Although it seems that the kernel trick saves us a lot of CPU-time, we also have to take into account convergence of the optimization process in the higher dimensional space. All in all the kernel trick works best on small complex datasets – but it may get slow on huge datasets. See for a discussion in the book “Hands on-On Machine Learning with Scikit-Learn and TensorFlow” of A.Geron,(2017, O’Reilly), chapter 5.

Gaussian RBF kernel

The Gaussian RBF kernel transforms the original feature space (x1,x2) into an extended multidimensional by a different approach: It looks at the similarity of a target point with selected other points – so called “landmarks”:

A new feature (=dimension) value is calculated by the Gaussian weight of the distance of a data point to one of the selected landmarks.

The number of landmarks can be chosen to be equal to the number N of all (other) data points in the training set. Thus we would add N-1 new dimensions – which would be a large number. The transformations operations for the coordinates of the data points in the original space (x1,x2) would, therefore, be numerous, too.

However, the Gaussian kernel enhances computational efficiency by large factors: it works on the lower dimensional parameter space, only, and determines the distance of data point pairs there! And still gives the correct results in the higher dimensional space for a linear separation interface there.

It is clear that the width of the Gaussian function is an important ingredient in this approach; this is controlled by the hyper-parameter “g” of the algorithm.

How to measure computational time?

This is relatively simple. We need to import the module “time”. It includes a suitable function “perf_counter()”. See: docs.python.org – perf_counter

We have to call it before a statement whose duration we want to measure and afterwards. The difference gives the CPU-time needed in fractions of a second. See below for the application.

Quadratic dependency of CPU time on the polynomial degree without the kernel trick

Let us measure a time series fro our standard polynomial approach. In our moons-notebook from the last session we add a cell and execute the following code:

And the plot looks like :

We recognize the expected quadratic behavior.

Polynomial kernel – almost constant low CPU-time independent of the polynomial degree

Let us now compare the output of the approach PolynomialsFeature + LinearSVC to an approach with the polynomial kernel. SciKit-learn provides us with an interface “SVC” to the kernel based algorithms. We have to specify the type of kernel besides other parameters. We execute the code in the following 2 cells to get a comparison for a polynomial of degree 3:

You saw that we specified a kernel-type “poly” in the second cell ?
The plots – first for LinearSVC and then for the polynomial kernel look like

We see a difference in the shape of separation line. And we notice already a slightly better performance for the kernel based algorithm.

Now let us prepare a similar time series for the kernel based approach:

The time series looks wiggled – but note that all numbers are below 1.3 msec ! What a huge difference!

Plot for the Gaussian RBF Kernel

Just to check how the separation surface looks like for the Gaussian kernel, we do the following experiment; note that we specify a kernel named “rbf“:

Oops, a very different surface in comparison to what we have seen before.
But: Even a minimum change of gamma gives us yet another surface:

Funny, isn’t it? We learn again that algorithms are sensitive!

Enough for today!

Conclusion

We have learned a bit about the so called kernel trick in the SVM business. Again, SciKit-Learn makes it very simple for us to make use of kernel based algorithms via an SVC-interface: different kernels and related feature space extension methods can be defined as a parameter.

We saw that a ”
poly”-kernel based approach in comparison to LinearSVC saves a lot of CPU-time when we use high order polynomials to extend the feature space to related higher dimensions.

The Gaussian RBF-kernel which extends the feature space by adding dimension based on weighted distances between data points proved to be interesting: It constructs a very different separation surface in comparison to polynomial approaches. We saw that RBF-kernel reacts sensitively to its configuration parameter “gamma” – i.e the width of the Gaussian weighing the similarity influence of other points.

Again we saw that in regions of the (x1,x2)-plane where no test data were provided the algorithms may predict very different memberships of new data points to either of the two moon clusters. Such extrapolations may depend on (small) parameter changes for the algorithms.

 

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

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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, 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[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” 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 …

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

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During this article series we use the moons dataset to acquire basic knowledge on Python based tools for machine learning [ML] – in this case for a classification task. 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 enabled 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, namely

  • to prepare the moons data set,
  • to analyze it
  • and to train a selected SVM-algorithm.

In this article we shall use a specific algorithm – 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 of 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 “X” to the input data points and a lowercase letter to the array with the classification information (per data point) – i.e. the target vector “y“.

The function “make_moons()” creates such an input array “X” of 2-dim data points and an associated target 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 – at its core it represents the 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 creating 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. The last value defines an alpha-opacity. Note that “green” in matplotlib is defined a bit strange in comparison to HTML.

Let us try it for a list (‘red’, ‘blue’, ‘green’, gray’, ‘yellow’, ‘#00ff00’):

The lower and upper limits of the the two axes must be given. Note that this sets the size of the region in our representation space which we want to analyze or get predictions for later on. We shall make the region big enough to willingly cover points outside the defined clusters. It will be interesting to see how an algorithm extrapolates its knowledge learned by training on the input data to regions beyond the
training area.

For the purpose of defining the length of the axes we can use the plot functions pyplot.xlim() and pyplot.ylim().

The central function, which we shall use for plotting data points in the defined area of the (x1,x2)-plane, 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 for-loop defined below:

In the code you 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 placed on 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 which 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()” (related to the predictor algorithm).

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.

The last statement

polynomial_svm_clf.fit(X, y)

starts the training based on our pipeline – with its algorithm.

PolynomialFatures

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. Still, we obviously want to use a linear classification algorithm …. Isn’t this a contradiction? What can be done about the problem of non-linearity?

In the first article of this series I briefly discussed an approach where data, which are apparently not linearly separable in their original representation space, can be placed into an extended feature space. For each data point we add new “features” by defining additional variables consisting of polynomial combinations of the points basic X-coordinates. We do this up to a maximum degree, namely the order of a polynomial function – e.g. T(x1,x2) = x1**3 + a* x1**2*x2 + b*x1*x2**2 + c*x1*x2 + x2**3.

Thereby, the dimensionality of the original X(x1,x2) set is extended by multiple further dimensions. Each data point is positioned in the extended feature space by a defined transformation T.

Our hope is that we can find a linear separation (“decision”) surface in the new extended multi-dimensional feature space.

The first step of our Pipeline enhances our X by additional and artificial 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 in a future article.

StandardScaler

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 predictor LinearSVC

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.

The basic parameters of LinearSVC, 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.

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 algorithm; see 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 throughout the selected region of our (x1,x2)-plane.

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. We just have to apply the function contourf() discussed in the second article of this series to our data:

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 in our contour plot. 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