How to plot support vectors for support vector regression

0 votes

I am trying to solve hard margin support vector regression and plot hyperplane and support vectors for a dataset.

As you know, hard margin is solved with the below assumption:

Formulation of “hard-margin” e-SVR

I solved the problem but when I want to plot decision boundaries and support vectors, I face the below problem. All of point should be located between two decision boundaries and support vectors should be drawn on the decision boundaries. Can you help me to find the problem?

Hyperplane, decision boundaries and support vectors

Here is the full code:

import pandas as pd
import numpy as np
from pandas import DataFrame
from sklearn import metrics

Data = pd.read_csv("Data.txt",delimiter="\t")

X=Data['waterlevel(x)'].values
y=Data['Area(y)'].values

# Plot Data
import matplotlib.pyplot as plt
fig,ax = plt.subplots(1, 1,constrained_layout=True,figsize=(8, 4))
ax.plot(X, y,'k.')
ax.set_title('Urmia lake Area versus Level')
ax.set_xlabel('Water level (M)',fontsize=15)
ax.set_ylabel('Area (km^2)',fontsize=15)
#plt.axis([0, 25, 0, 25])
plt.grid(True)
plt.show()

# find max and min values of predictor variables (here X) to use it to specify initial values of w and b 

max_feature_value=np.amax(X)
min_feature_value=np.amin(X)
w_optimum = max_feature_value*0.5

w = [w_optimum for i in range(1)]   # w shoulb be a vector with dimension of the independent features (here:1)
wt_b=w

b_sum=0
for i in range(X.shape[0]):
    b_sum+=y[i]-np.dot(wt_b,X[i])

b_ini=b_sum/len(X)
b_step_size_lower = 0.9
b_step_size_upper = 0.2
b_multiple = 500   # step size for b
b_range = np.arange((b_ini*b_step_size_lower), -b_ini*b_step_size_upper, b_multiple)
print(len(b_range))

# Estimate w and b using stochastic gradient descent and trial and error
l_rate=0.1
n_epoch = 250
epsilon=150 # acceptable error
length_Wvector_list=[]

for i in range (len(b_range)):
    correctly_regressed = True
    for j in range(X.shape[0]):
        print(i,j,wt_b,b_range[i])
        if (y[j]-(np.dot(wt_b,X[j])+b_range[i]) > epsilon) or (y[j]-(np.dot(wt_b,X[j])+b_range[i]) < -epsilon)==True:
            correctly_regressed = False 
            wt_b = np.asarray(wt_b) - l_rate
        if correctly_regressed==True:
            length_Wvector_list.append([wt_b[0],wt_b,b_range[i]])
        if wt_b[0] < 0:
            wt_b=w
            break

norms = sorted([n for n in length_Wvector_list])
wt_b=norms[0][1]
b=norms[0][2]

# Predict using the optimized values of w and b    
y_predict=[]
for i in range (X.shape[0]):
    y_hat=np.dot(wt_b,X[i])+b
    y_predict.append(y_hat)        

print('Root Mean Squared Error:', np.sqrt(metrics.mean_squared_error(y, y_predict)))
print('Coefficient of determination (R2):', metrics.r2_score(y, y_predict))    

# plot 
fig,ax = plt.subplots(1, 1,figsize=(8, 5.2))
ax.scatter(y, y_predict, cmap='K', edgecolor='b',linewidth='0.5',alpha=1, label='testing points',marker='o', s=12)
ax.set_xlabel('Observed Area(km $^{2}$)',fontsize=14)
ax.set_ylabel('Simulated Area(km $^{2}$)',fontsize=14)

# find support vectors
positive_instances=[]
negative_instances=[]

for i in range(X.shape[0]):
    y_pre=(np.dot(wt_b,X[i]))+b
    if  y[i]-y_pre<=epsilon:
        positive_instances.append([y[i]-y_pre,[X[i],y[i]]])
    elif y[i]-y_pre>=-epsilon:
        negative_instances.append([y[i]-y_pre,[X[i],y[i]]]) 

len(positive_instances)+len(negative_instances)

sort_positive=sorted([n for n in positive_instances])
sort_negative=sorted([n for n in negative_instances])

positive_support_vector=sort_positive[0][1]
negative_support_vector=sort_negative[-1][1]

model_support_vectors=np.stack((positive_support_vector,negative_support_vector),axis=-1)

# visualize the data-set
colors = {1:'r',-1:'b'}
fig = plt.figure()
ax = fig.add_subplot(1,1,1)

plt.scatter(X,y,marker='o',c=y)

# plot support vectors
ax.scatter(model_support_vectors[0, :],model_support_vectors[1, :],s=200, linewidth=1,facecolors='none', edgecolors='b')

# hyperplane = x.w+b
# 0 = x.w+b
# psv = epsilon
# nsv = -epsilon
# dec = 0

def hyperplane_value(x,w,b,e):
    return (np.dot(w,x)+b+e)

datarange = (min_feature_value*1.,max_feature_value*1.)
hyp_x_min = datarange[0]
hyp_x_max = datarange[1]

# (w.x+b) = epsilon
# positive support vector hyperplane
psv1 = hyperplane_value(hyp_x_min, wt_b, b, epsilon)
psv2 = hyperplane_value(hyp_x_max, wt_b, b, epsilon)
ax.plot([hyp_x_min,hyp_x_max],[psv1,psv2], 'k')

# (w.x+b) = -epsilon
# negative support vector hyperplane
nsv1 = hyperplane_value(hyp_x_min, wt_b, b, -epsilon)
nsv2 = hyperplane_value(hyp_x_max, wt_b, b, -epsilon)
ax.plot([hyp_x_min,hyp_x_max],[nsv1,nsv2], 'k')

# (w.x+b) = 0
# positive support vector hyperplane
db1 = hyperplane_value(hyp_x_min, wt_b, b, 0)
db2 = hyperplane_value(hyp_x_max, wt_b, b, 0)
ax.plot([hyp_x_min,hyp_x_max],[db1,db2], 'y--')

#plt.axis([-5,10,-12,-1])
plt.show()
Mar 23 in Machine Learning by Dev
• 6,000 points
28 views

1 answer to this question.

0 votes

The problem was solved after I improved the program. The decision borders and support vectors are drawn correctly, as you can see

Hyperplane, support vectors, and decision boundaries

The code is shared below:

import pandas as pd
import numpy as np
from pandas import DataFrame
from sklearn import metrics

Data = pd.read_csv("Data.txt",delimiter="\t")

X=Data['waterlevel(x)'].values
y=Data['Area(y)'].values

# Plot the Data
import matplotlib.pyplot as plt
fig,ax = plt.subplots(1, 1,constrained_layout=True,figsize=(8, 4))
ax.plot(X, y,'k.')
ax.set_title('Urmia lake Area versus Level')
ax.set_xlabel('Water level (M)',fontsize=15)
ax.set_ylabel('Area (km^2)',fontsize=15)
#plt.axis([0, 25, 0, 25])
plt.grid(True)
plt.show()

# find max and min values of predictor variables (here X) to use it to specify initial values of w and b 
max_feature_value=np.amax(X)
min_feature_value=np.amin(X)
w_optimum = max_feature_value*0.5

w = [w_optimum for i in range(1)]   # w shoulb be a vector with dimension of the independent features (here:1)
wt_b=w
b_sum=0
for i in range(X.shape[0]):
    b_sum+=y[i]-np.dot(wt_b,X[i])

b_ini=b_sum/len(X)
b_step_size_lower = 0.9
b_step_size_upper = 0.1
b_multiple = 500   # step size for b
b_range = np.arange((b_ini*b_step_size_lower), -b_ini*b_step_size_upper, b_multiple)
print(len(b_range))

# Estimate w and b using stochastic gradient descent and trial and error
l_rate=0.1
n_epoch = 250
epsilon=500 # acceptable error
length_Wvector_list=[]
for i in range (len(b_range)):
    print(i)
    optimized = False
    while not optimized:
        correctly_regressed = True
        for j in range(X.shape[0]):
            # every data point should be satisfies the constraint  yi-(np.dot(w_t,xi)+b) <=epsilon or yi-(np.dot(w_t,xi)+b)>=-epsilon 
            if (y[j]-(np.dot(wt_b,X[j])+b_range[i]) > epsilon) or (y[j]-(np.dot(wt_b,X[j])+b_range[i]) < -epsilon)==True:
                correctly_regressed = False
                wt_b = np.asarray(wt_b) - l_rate      

        if correctly_regressed==True:
            length_Wvector_list.append([wt_b[0],wt_b,b_range[i]]) #store w, b for minimum magnitude , magnitude or length of a vector w_t is called the norm
            optimized = True
        if wt_b[0] < 0:
            optimized = True           

    wt_b_temp=wt_b
    wt_b=w

norms = sorted([n for n in length_Wvector_list])
wt_b=norms[0][1]
b=norms[0][2]

# Predict using the optimized values of w and b    
y_predict=[]
for i in range (X.shape[0]):
    y_hat=np.dot(wt_b,X[i])+b
    y_predict.append(y_hat)        

print('Root Mean Squared Error:', np.sqrt(metrics.mean_squared_error(y, y_predict)))
print('Coefficient of determination (R2):', metrics.r2_score(y, y_predict))    

# plot 
fig,ax = plt.subplots(1, 1,figsize=(8, 5.2))
ax.scatter(y, y_predict, cmap='K', edgecolor='b',linewidth='0.5',alpha=1, label='testing points',marker='o', s=12)
ax.set_xlabel('Observed Area(km $^{2}$)',fontsize=14)
ax.set_ylabel('Simulated Area(km $^{2}$)',fontsize=14)
ax.set_xlim([min(y)-100, max(y)+100])
ax.set_ylim([min(y)-100, max(y)+100])

# find support vectors
positive_instances=[]
negative_instances=[]

for i in range(X.shape[0]):
    y_pre=(np.dot(wt_b,X[i]))+b
    if  ((y[i]-y_pre>0) and (y[i]-y_pre<=epsilon))==True:
        positive_instances.append([y[i]-y_pre,[X[i],y[i]]])
    elif ((y[i]-y_pre<0) and (y[i]-y_pre>=-epsilon))==True:
        negative_instances.append([y[i]-y_pre,[X[i],y[i]]]) 

len(positive_instances)+len(negative_instances)

sort_positive=sorted([n for n in positive_instances])
sort_negative=sorted([n for n in negative_instances])

positive_support_vector=sort_positive[-1][1]
negative_support_vector=sort_negative[0][1]
model_support_vectors=np.stack((positive_support_vector,negative_support_vector),axis=-1)

# visualize the data-set
colors = {1:'r',-1:'b'}
fig = plt.figure()
ax = fig.add_subplot(1,1,1)
plt.scatter(X,y,marker='o',c=y)

# plot support vectors
ax.scatter(model_support_vectors[0, :],model_support_vectors[1, :],s=200, linewidth=1,facecolors='none', edgecolors='b')

# hyperplane = x.w+b
# 0 = x.w+b
# psv = epsilon
# nsv = -epsilon
# dec = 0

def hyperplane_value(x,w,b,e):
    return (np.dot(w,x)+b+e)

datarange = (min_feature_value*1.,max_feature_value*1.)
hyp_x_min = datarange[0]
hyp_x_max = datarange[1]

# (w.x+b) = epsilon
# positive support vector hyperplane
psv1 = hyperplane_value(hyp_x_min, wt_b, b, epsilon)
psv2 = hyperplane_value(hyp_x_max, wt_b, b, epsilon)
ax.plot([hyp_x_min,hyp_x_max],[psv1,psv2], 'k')

# (w.x+b) = -epsilon
# negative support vector hyperplane
nsv1 = hyperplane_value(hyp_x_min, wt_b, b, -epsilon)
nsv2 = hyperplane_value(hyp_x_max, wt_b, b, -epsilon)
ax.plot([hyp_x_min,hyp_x_max],[nsv1,nsv2], 'k')

# (w.x+b) = 0
# positive support vector hyperplane
db1 = hyperplane_value(hyp_x_min, wt_b, b, 0)
db2 = hyperplane_value(hyp_x_max, wt_b, b, 0)
ax.plot([hyp_x_min,hyp_x_max],[db1,db2], 'y--')

#plt.axis([-5,10,-12,-1])
plt.show()
answered Mar 25 by Nandini
• 5,480 points

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