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logistic and linear regression.txt
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logistic and linear regression.txt
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>> Linear Regression
from sklearn.linear_model import LogisticRegression
import sklearn.dataset
reg=LinearRegression()
reg.fit(X_rooms,y)
prediction_space=np.linspace(min(X_rooms),max(X_rooms)).reshape(-1,1)
plt.scatter(X_rooms,y)
#X_train, X_test,y_train,y_test=train_test_split(X_rooms,y,test_size=0.3,random_state=42)
plt.ylabel('house price in $1000s')
plt.xlabel('number of rooms')
plt.plot(prediction_space, reg.predict(prediction_space),color='black', linewidth=3)
plt.show()
#regression.score(X_test,y_test)
>>> Logistic Regression
from sklearn.linear_model import LogisticRegression
import sklearn.datasets
import matplotlib.pyplot as plt
import numpy as np
wine=sklearn.datasets.load_wine()
reg=LogisticRegression()
print(wine.feature_names)
X=wine.data
y=wine.target
X_train, X_test,y_train,y_test=train_test_split(X,y,test_size=0.3,random_state=42)
reg.fit(X_train,y_train)
print(reg.score(X,y))
row1=np.array(X_train[:1])
print (reg.predict_proba(alcohol))
#prediction_space=np.linspace(min(alcohol),max(alcohol)).reshape(-1,1)
>>Linear SVC
svm=LinearSVC()
svm.fit(X_train,y_train)
print(svm.score(X_train,y_train))
>>non linear SVM
svm=SVC()
svm.fit(X_train,y_train)
print(svm.score(X_train,y_train))
# Instantiate logistic regression and train
lr = LogisticRegression()
lr.fit(X, y)
# Predict sentiment for a glowing review
review1 = "LOVED IT! This movie was amazing. Top 10 this year."
review1_features = get_features(review1)
print(review1_features)
print("Review:", review1)
print("Probability of positive review:", lr.predict_proba(review1_features)[0,1])
# Predict sentiment for a poor review
review2 = "Total junk! I'll never watch a film by that director again, no matter how good the reviews."
review2_features = get_features(review2)
print(review2_features)
print("Review:", review2)
print("Probability of positive review:", lr.predict_proba(review2_features)[0,1])
>>classification
1. when the y values are categories
2. decision boundaries are surfaces separating different predicted classes
3. linear classifier learns linear decisions boundaries
4. linear separable if it can be perfectly explained by a linear classifier
5. multi class classification has three or more categories
>>plotting the linear boundaries
from sklearn.linear_model import LogisticRegression
from sklearn.svm import SVC, LinearSVC
from sklearn.neighbors import KNeighborsClassifier
# Define the classifiers
classifiers = [LogisticRegression(), LinearSVC(),
SVC(), KNeighborsClassifier()]
# Fit the classifiers
for c in classifiers:
c.fit(X, y)
# Plot the classifiers
plot_4_classifiers(X, y, classifiers)
plt.show()
>>Logistic regression: Positive or negative to determine what side of the boundary your on
lr.coef_
lr.intercept_
raw model output (negative or positive) predicting either 0 or 1
changing the intercept change the y intercept of the boundary
changing the coefficients change the slope of the boundary
>>change the y intercept and the slope of the boundary
# Set the coefficients
model.coef_ = np.array([[-1,1]])
model.intercept_ = np.array([-3])
# Plot the data and decision boundary
plot_classifier(X,y,model)
# Print the number of errors
num_err = np.sum(y != model.predict(X))
print("Number of errors:", num_err)
>>Loss functions
LinearRegression used a minimize loss function
loss function tells us how poorly the model is doing at predicting on the training data
fit is running code to minimize the loss
A natural loss for classification problems it the number of errors made
0-1 loss 0 for a correct prediction and 1 for and incorrect prediction
from scipy.optimize import minimize
minimize(np.square,0).x
what coefficients make my squared error as small as possible
# The squared error, summed over training examples
def my_loss(w):
s = 0
for i in range(y.size):
# Get the true and predicted target values for example 'i'
y_i_true = y[i]
y_i_pred = w@X[i]
s = s + (y_i_true - y_i_pred)**2
return s
# Returns the w that makes my_loss(w) smallest
w_fit = minimize(my_loss, X[0]).x
print(w_fit)
# Compare with scikit-learn's LinearRegression coefficients
lr = LinearRegression(fit_intercept=False).fit(X,y)
print(lr.coef_)
>>Loss is the difference between the true and the predicted
1. sin products a raw model in either positive or negative halves
a. raw model -1 (incorrect) predict +1 (correct)
logistic loss diagram
1. loss decreases as the corrections are predicted correctly
hinge loss in svm
# Mathematical functions for logistic and hinge losses
def log_loss(raw_model_output):
return np.log(1+np.exp(-raw_model_output))
def hinge_loss(raw_model_output):
return np.maximum(0,1-raw_model_output)
# Create a grid of values and plot
grid = np.linspace(-2,2,1000)
plt.plot(grid, log_loss(grid), label='logistic')
plt.plot(grid, hinge_loss(grid), label='hinge')
plt.legend()
plt.show()
>>log loss
# The logistic loss, summed over training examples
def my_loss(w):
s = 0
for i in range(y.size):
raw_model_output = w@X[i]
s = s + log_loss(raw_model_output * y[i])
return s
# Returns the w that makes my_loss(w) smallest
w_fit = minimize(my_loss, X[0]).x
print(w_fit)
# Compare with scikit-learn's LogisticRegression
lr = LogisticRegression(fit_intercept=False, C=1000000).fit(X,y)
print(lr.coef_)
>>regularization combats overfitting by making the coeffients smaller
coefficient value by coefficient index
small c means more regularization
large c means less regularization
lr_weak_reg=LogisticRegression(C=100)
lr_strong_reg=LogisticRegression(C=0.01)
lr_weak_reg.fit(X_train,y_train)
lr_strong_reg.fit(X_train,y_train)
lr_weak_reg.score(X_train,y_train)
lr_strong_reg.score(X_train,y_train)
regularization penalizes large values of the coefficients
regularized loss= original loss + large coeffient penalty
* more regularization equals lower training accuracy
regularizing causes you to fit less reducing overfitting of the data.
lasso = linear regression with L1 regularization
ridge= linear regression with L2 regularization
L1 helps with feature selection
>>
lr_L1 = LogisticRegression(penalty='l1')
lr_l2 =LogisticRegression()
lr_L1.fit(X_train,y_train)
lr_L2.fit(X_train,y_train)
plt.plot(lr_L1.coef_.flatten())
plt.plot(lr_L2.coef_.flatten())
>>
# Train and validaton errors initialized as empty list
train_errs = list()
valid_errs = list()
# Loop over values of C_value
for C_value in [0.001, 0.01, 0.1, 1, 10, 100, 1000]:
# Create LogisticRegression object and fit
lr = LogisticRegression(C=C_value)
lr.fit(X_train, y_train)
# Evaluate error rates and append to lists
train_errs.append( 1.0 - lr.score(X_train, y_train) )
valid_errs.append( 1.0 - lr.score(X_valid, y_valid) )
# Plot results
plt.semilogx(C_values, train_errs, C_values, valid_errs)
plt.legend(("train", "validation"))
plt.show()
>>Lasso
# Specify L1 regularization
lr = LogisticRegression(penalty='l1')
# Instantiate the GridSearchCV object and run the search
searcher = GridSearchCV(lr, {'C':[0.001, 0.01, 0.1, 1, 10]})
searcher.fit(X_train, y_train)
# Report the best parameters
print("Best CV params", searcher.best_params_)
# Find the number of nonzero coefficients (selected features)
best_lr = searcher.best_estimator_
coefs = best_lr.coef_
print("Total number of features:", coefs.size)
print("Number of selected features:", np.count_nonzero(coefs))
>>coefficents
# Get the indices of the sorted cofficients
inds_ascending = np.argsort(lr.coef_.flatten())
inds_descending = inds_ascending[::-1]
# Print the most positive words
print("Most positive words: ", end="")
for i in range(5):
print(vocab[inds_descending], end=", ")
print("\n")
# Print most negative words
print("Most negative words: ", end="")
for i in range(5):
print(vocab[inds_ascending], end=", ")
print("\n")
>>Logistic regression as a probablity
1. The decision boundary is 0.5 red is >= 0.5 and blue is <0.5
when regularization is turned on C=1 then the model coefficients are closer to .5
old
model coeffients=[[1.55,1.57]] and model intercept[-0.64]
new
model coeffients=[[0.45,0.64]] and model intercept[-0.26]
two features equates to two coefficients
the magnitude give us our confidence level
the coefficients give the slope of the line
The sigmoid function squashes the raw model output to be between 0 and 1 and .5 we are at the boundary
>>
# Set the regularization strength
model = LogisticRegression(C=1)
# Fit and plot
model.fit(X,y)
plot_classifier(X,y,model,proba=True)
# Predict probabilities on training points
prob = model.predict_proba(X)
print("Maximum predicted probability", np.max(prob))
>>
# Set the regularization strength
model = LogisticRegression(C=0.1)
# Fit and plot
model.fit(X,y)
plot_classifier(X,y,model,proba=True)
# Predict probabilities on training points
prob = model.predict_proba(X)
print("Maximum predicted probability", np.max(prob))
>>
lr = LogisticRegression()
lr.fit(X,y)
# Get predicted probabilities
proba = lr.predict_proba(X)
# Sort the example indices by their maximum probability
proba_inds = np.argsort(np.max(proba,axis=1))
# Show the most confident (least ambiguous) digit
show_digit(proba_inds[-1], lr)
# Show the least confident (most ambiguous) digit
show_digit(proba_inds[0], lr)
>>multi class classifiers
lr0.fit(X,y==0)
lr1.fit(X,y==1)
lr2.fit(X,y==2)
lr0.decision_function(X)[0]
lr1.decision_function(X)[0]
lr2.decision_function(X)[0]
or
lr.fit(X,y)
lr.predict(X)[0]
one vs rest
Multinomial or softmax
>>one-vs-rest
lr_ovr=LogisticRegression()
lr_ovr.fit(X,y)
lr_ovr.coef_.shape
>>
lr_mn=LogisticRegression(
multi_class="multinomial",
solver="lbfgs")
lr_mn.fit(X,y)
lrmmn.coef_shape
>>
# Fit one-vs-rest logistic regression classifier
lr_ovr = LogisticRegression()
lr_ovr.fit(X_train, y_train)
print("OVR training accuracy:", lr_ovr.score(X_train, y_train))
print("OVR test accuracy :", lr_ovr.score(X_test, y_test))
# Fit softmax classifier
lr_mn = LogisticRegression(multi_class="multinomial", solver="lbfgs")
lr_mn.fit(X_train, y_train)
print("Softmax training accuracy:", lr_mn.score(X_train, y_train))
print("Softmax test accuracy :", lr_mn.score(X_test, y_test))
>>
# Print training accuracies
print("Softmax training accuracy:", lr_mn.score(X_train, y_train))
print("One-vs-rest training accuracy:", lr_ovr.score(X_train, y_train))
# Create the binary classifier (class 1 vs. rest)
lr_class_1 = LogisticRegression(C=100)
lr_class_1.fit(X_train, y_train==1)
# Plot the binary classifier (class 1 vs. rest)
plot_classifier(X_train, y_train==1, lr_class_1)
>> Non-linear classification
# We'll use SVC instead of LinearSVC from now on
from sklearn.svm import SVC
# Create/plot the binary classifier (class 1 vs. rest)
svm_class_1 = SVC()
svm_class_1.fit(X_train, y_train==1)
plot_classifier(X_train, y_train==1, svm_class_1)