Tag Archives: decision surface

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

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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

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

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

Eclipse, PyDev, virtualenv and graphical output of matplotlib on KDE – I
Eclipse, PyDev, virtualenv and graphical output of matplotlib on KDE – II
Eclipse, PyDev, virtualenv and graphical output of matplotlib on KDE – III

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) 
               
        
r
        # 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[0] = self._figs_x1
            self._fig_size[1] = 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

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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