Please find instructions for this order in the uploaded file, answer all questions carefully. Give...

Question

Please find instructions for this order in the uploaded file, answer all questions carefully. Give clear explanations on short-answer questions and submit all in a word document. For coding parts, submit your codes in ipynb files. pdf files uploaded are for your reference.

For the following assignments, please provide as much evidence of the results as possible, including the code, screenshots (only plots – not text or code) and documentation. Submit code in one ipynb file, for explanations please use word for documentation. 1.a. Choose one of the cleaned datasets at https://www.kaggle.com/annavictoria/ml-friendly-public-datasets. Split it into training and test data. Write from scratch and apply any ML algorithm that you learned to this dataset. You can use Python to implement it. For the implementation, you may use any classes, modules, and functions in Python libraries such as NumPy to do various math / linear algebra operations, but not use the ML classes or functions directly. Apply another algorithm that you learned to the same dataset. For this one, you are free to implement it from scratch or use the ML class and functions directly from the ML packages. Which one of the algorithm fares better? Use as many evaluation metrics as possible to discuss the performance of the algorithms. Write down your comments in your script. 1.b. Derive an equation for accuracy in terms of Specificity and Sensitivity. The equation can include metrics such as number of True Positives or number of False Positives, etc. in addition to accuracy, Specificity and Sensitivity. Give an interpretation of the equation. 2.(a) Refer to online tutorials on regularization such as https://medium.com/coinmonks/regularization-of-linear-models-with-sklearn-f88633a93a2 and https://towardsdatascience.com/ridge-and-lasso-regression-a-complete-guide-with-python-scikit-learn-e20e34bcbf0b Apply the techniques from the above tutorial to the student dataset at https://archive.ics.uci.edu/ml/datasets/student+performance Does regularization help improve the accuracy of predicting the final Math grade of the students? 2.(b) For regularization, we added the regularizer to the loss function. Does it make sense to multiply or subtract the term, instead? Explain. 3.(a) Manually generate the decision tree (as much as possible) for the following subset from a large dataset using the ID3 algorithm. Show the information gain computation at each stage. 3.(b) Then generate the decision tree programmatically using Python. Submit code and the decision tree so generated. Dataset for "Play Tennis" Decision Tree that you will generate 4/18/2020 Regularization of Linear Models with SKLearn - Coinmonks - Medium https://medium.com/coinmonks/regularization-of-linear-models-with-sklearn-f88633a93a2 1/5 Regularization of Linear Models with SKLearn Robert Thas John Follow Jul 11, 2018 · 3 min read An over�t model Linear models are usually a good starting point for training a model. However, a lot of datasets do not exhibit linear relationships between the independent and the dependent variables. As a result, it is frequently necessary to create a polynomial model. However, these models are usually prone to overfitting. One method of reducing overfitting in polynomial models is through the use of regularization. Let’s start by building a baseline model to determine the required improvement. We will make use of the popular Boston Housing dataset which is available on Kaggle here. Let’s import the necessary libraries and load up our training dataset. 1 2 3 #imports import numpy as np import pandas as pd https://medium.com/@robertjohn_15390?source=post_page-----f88633a93a2---------------------- https://medium.com/@robertjohn_15390?source=post_page-----f88633a93a2---------------------- https://medium.com/coinmonks/regularization-of-linear-models-with-sklearn-f88633a93a2?source=post_page-----f88633a93a2---------------------- https://www.kaggle.com/c/boston-housing 4/18/2020 Regularization of Linear Models with SKLearn - Coinmonks - Medium https://medium.com/coinmonks/regularization-of-linear-models-with-sklearn-f88633a93a2 2/5 Let’s split our data into a training set and a validation set. We will hold out 30% of the data for validation. We will use a random state to make our experiment reproducible. view raw 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 regularization_imports.py hosted with ❤ by GitHub import math from sklearn.model_selection import train_test_split from sklearn.linear_model import LinearRegression from sklearn.linear_model import Ridge from sklearn.linear_model import Lasso from sklearn.linear_model import ElasticNet from sklearn.metrics import mean_squared_error from sklearn.preprocessing import PolynomialFeatures from sklearn.pipeline import Pipeline from sklearn.preprocessing import StandardScaler import matplotlib.pyplot as plt import seaborn as sns sns.set() %matplotlib inline #import training dataset train_df = pd.read_csv('train.csv', index_col='ID') #see the columns in our data train_df.info() # take a look at the head of the dataset train_df.head() view rawtrain_test_split_boston.py hosted with ❤ by GitHub 1 2 3 4 5 #create our X and y X = train_df.drop('medv', axis=1) y = train_df['medv'] X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=42, test_size=0.3) https://gist.github.com/securetorobert/99ef19b2566805cc28ee187d86ccad73/raw/5574b69eadbe9c3173d1c56313f5f1327c7d29e7/regularization_imports.py https://gist.github.com/securetorobert/99ef19b2566805cc28ee187d86ccad73#file-regularization_imports-py https://github.com/ https://gist.github.com/securetorobert/fecf3de2aa804a1ad4a2310629537c0e/raw/5314ce11edd6e05a4a94d852de8bdf09befc20a2/train_test_split_boston.py https://gist.github.com/securetorobert/fecf3de2aa804a1ad4a2310629537c0e#file-train_test_split_boston-py https://github.com/ 4/18/2020 Regularization of Linear Models with SKLearn - Coinmonks - Medium https://medium.com/coinmonks/regularization-of-linear-models-with-sklearn-f88633a93a2 3/5 Let’s establish a baseline by training a linear regression model. The model above should give us a training accuracy and a test accuracy of about 72%. We should also get an RMSE of about 4.587. The next models we train should outperform this model with higher accuracy scores and a lower RMSE. We need to engineer new features. Specifically, we need to create polynomial features by taking our individual features and raising them to a chosen power. Thankfully, scikit- learn has an implementation for this and we don’t need to do it manually. Something else we would like to do is standardize our data. This scales our data down to a range between 0 and 1. This serves the purpose of letting us work with reasonable numbers when we raise to a power. Finally, because we need to carry out the same operations on our training, validation, and test sets, we will introduce a pipeline. This will let us pipe our process so the same steps get carried out repeatedly. To summarize, we will scale our data, then create polynomial features, and then train a linear regression model. After running our code, we will get a training accuracy of about 94.75%, and a test accuracy of 46.76%. This is a sign of overfitting. It’s normally not a desirable feature, but that is exactly what we were hoping for. view raw 1 2 3 4 5 6 7 8 9 10 11 linear_regression_boston.py hosted with ❤ by GitHub lr_model = LinearRegression() lr_model.fit(X_train, y_train) print('Training score: {}'.format(lr_model.score(X_train, y_train))) print('Test score: {}'.format(lr_model.score(X_test, y_test))) y_pred = lr_model.predict(X_test) mse = mean_squared_error(y_test, y_pred) rmse = math.sqrt(mse) print('RMSE: {}'.format(rmse)) https://gist.github.com/securetorobert/d8fa6ff57ccafad6236f1a8b5459a254/raw/20ad6b303ed513e7e5074b3d4b123c39136ec904/linear_regression_boston.py https://gist.github.com/securetorobert/d8fa6ff57ccafad6236f1a8b5459a254#file-linear_regression_boston-py https://github.com/ 4/18/2020 Regularization of Linear Models with SKLearn - Coinmonks - Medium https://medium.com/coinmonks/regularization-of-linear-models-with-sklearn-f88633a93a2 4/5 We will now apply regularization to our new data. l2 Regularization or Ridge Regression To understand Ridge Regression, we need to remind ourselves of what happens during gradient descent, when our model coefficients are trained. During training, our initial weights are updated according to a gradient update rule using a learning rate and a gradient. Ridge regression adds a penalty to the update, and as a result shrinks the size of our weights. This is implemented in scikit-learn as a class called Ridge. We will create a new pipeline, this time using Ridge. We will specify our regularization strength by passing in a parameter, alpha. This can be really small, like 0.1, or as large as you would want it to be. The larger the value of alpha, the less variance your model will exhibit. By executing the code, we should have a training accuracy of about 91.8%, and a test accuracy of about 82.87%. That is an improvement on our baseline linear regression model. Let’s try something else. l1 Regularization or Lasso Regression By creating a polynomial model, we created additional features. The question we need to ask ourselves is which of our features are relevant to our model, and which are not. l1 regularization tries to answer this question by driving the values of certain coefficients down to 0. This eliminates the least important features in our model. We will create a pipeline similar to the one above, but using Lasso. You can play around with the value of alpha, which can range from 0.1 to 1. The code above should give us a training accuracy of 84.8%, and a test accuracy of 83%. This is an even better model than the one we trained earlier. At this point, you can evaluate your model by finding the RMSE. Don’t forget to read the documentation for everything we used. 4/18/2020 Regularization of Linear Models with SKLearn - Coinmonks - Medium https://medium.com/coinmonks/regularization-of-linear-models-with-sklearn-f88633a93a2 5/5 I hope you found this tutorial useful. Until next time. Machine Learning Regularization Ridge Regression Lasso Regression Sklearn About Help Legal https://medium.com/tag/machine-learning https://medium.com/tag/regularization https://medium.com/tag/ridge-regression https://medium.com/tag/lasso-regression https://medium.com/tag/sklearn https://medium.com/?source=post_page-----f88633a93a2---------------------- https://medium.com/about?autoplay=1&source=post_page-----f88633a93a2---------------------- https://help.medium.com/?source=post_page-----f88633a93a2---------------------- https://medium.com/policy/9db0094a1e0f?source=post_page-----f88633a93a2---------------------- 4/18/2020 Ridge and Lasso Regression: L1 and L2 Regularization https://towardsdatascience.com/ridge-and-lasso-regression-a-complete-guide-with-python-scikit-learn-e20e34bcbf0b 1/11 Ridge and Lasso Regression: L1 and L2 Regularization Complete Guide Using Scikit-Learn Saptashwa Bhattacharyya Follow Sep 26, 2018 · 8 min read Moving on from a very important unsupervised learning technique that I have discussed last week, today we will dig deep in to supervised learning through linear regression, specifically two special linear regression model — Lasso and Ridge regression. As I’m using the term linear, first let’s clarify that linear models are one of the simplest way to predict output using a linear function of input features. Linear model with n features for output prediction In the equation (1.1) above, we have shown the linear model based on the n number of features. Considering only a single feature as you probably already have understood that w[0] will be slope and b will represent intercept. Linear regression looks for optimizing w and b such that it minimizes the cost function. The cost function can be written as Cost function for simple linear model https://towardsdatascience.com/@saptashwa?source=post_page-----e20e34bcbf0b---------------------- https://towardsdatascience.com/@saptashwa?source=post_page-----e20e34bcbf0b---------------------- https://medium.com/m/signin?operation=register&redirect=https%3A%2F%2Ftowardsdatascience.com%2Fridge-and-lasso-regression-a-complete-guide-with-python-scikit-learn-e20e34bcbf0b&source=-9a3c3c477239-------------------------follow_byline- https://towardsdatascience.com/ridge-and-lasso-regression-a-complete-guide-with-python-scikit-learn-e20e34bcbf0b?source=post_page-----e20e34bcbf0b---------------------- https://towardsdatascience.com/dive-into-pca-principal-component-analysis-with-python-43ded13ead21 4/18/2020 Ridge and Lasso Regression: L1 and L2 Regularization https://towardsdatascience.com/ridge-and-lasso-regression-a-complete-guide-with-python-scikit-learn-e20e34bcbf0b 2/11 In the equation above I have assumed the data-set has M instances and p features. Once we use linear regression on a data-set divided in to training and test set, calculating the scores on training and test set can give us a rough idea about whether the model is suffering from over-fitting or under-fitting. The chosen linear model can be just right also, if you’re lucky enough! If we have very few features on a data-set and the score is poor for both training and test set then it’s a problem of under-fitting. On the other hand if we have large number of features and test score is relatively poor than the training score then it’s the problem of over-generalization or over-fitting. Ridge and Lasso regression are some of the simple techniques to reduce model complexity and prevent over-fitting which may result from simple linear regression. . . . Ridge Regression : In ridge regression, the cost function is altered by adding a penalty equivalent to square of the magnitude of the coefficients. Cost function for ridge regression This is equivalent to saying minimizing the cost function in equation 1.2 under the condition as below Supplement 1: Constrain on Ridge regression coe�cients So ridge regression puts constraint on the coefficients (w).

instructions-sngpfppp.docx regularization-of-linear-models-with-sklearn-coinmonks-medium-w0yzbx02.pdf ridge-and-lasso-regression-l1-and-l2-regularization-thhfhc2l.pdf student-hvf22lgn.pdf student-e2cg5oex.zip

Anandkumar · Accepted Answer

95363/1/1.a.docx
import pandas as pd
from sklearn.model_selection import train_test_split
from sklearn.tree import DecisionTreeClassifier 
from sklearn import metrics
from sklearn.neighbors import KNeighborsClassifier
data = pd.read_csv(r'C:\Users\ANAND\Desktop\Greynodes\95363\iris.csv')
train, test = train_test_split(data, test_size = 0.4, stratify = data['Species'], random_state = 42)
X_train = train[['SepalLengthCm','SepalWidthCm','PetalLengthCm','PetalWidthCm']]
y_train = train.Species
X_test = test[['SepalLengthCm','SepalWidthCm','PetalLengthCm','PetalWidthCm']]
y_test = test.Species
mod_dt = DecisionTreeClassifier(max_depth = 3, random_state = 1)
mod_dt.fit(X_train,y_train)
prediction=mod_dt.predict(X_test)
print('The accuracy of the Decision Tree is',"{:.3f}".format(metrics.accuracy_score(prediction,y_test)))
mod_knn = KNeighborsClassifier(n_neighbors=10)
mod_knn.fit(X_train,y_train)
prediction=mod_knn.predict(X_test)
print('The accuracy of the K neigbour Tree is',"{:.3f}".format(metrics.accuracy_score(prediction,y_test)))
# Comments:  We can notice Descision tree performs better than Knearest algorithm. 
Output:
95363/1/1.b.docx
import pandas as pd
from sklearn.model_selection import train_test_split
from sklearn.tree import DecisionTreeClassifier 
from sklearn import metrics
data = pd.read_csv(r'C:\Users\ANAND\Desktop\Greynodes\95363\iris.csv')
train , test = train_test_split(data,train_size=0.7,  test_size = 0.3, stratify = data['Species'], random_state = 100)
X_train = train[['SepalLengthCm','SepalWidthCm','PetalLengthCm','PetalWidthCm']]
y_train = train.Species
X_test = test[['SepalLengthCm','SepalWidthCm','PetalLengthCm','PetalWidthCm']]
y_test = test.Species
mod_dt = DecisionTreeClassifier(max_depth = 3, random_state = 1)
mod_dt.fit(X_train,y_train)
prediction=mod_dt.predict(X_train)
confusion = metrics.confusion_matrix(y_train,prediction)
print("-------------- Confusion Matrix printed ----------------")
print(confusion)
print("")
print("-------------- Accuracy score check ----------------")
print(metrics.accuracy_score(y_train,prediction))
# So, there are two false negatives in ‘Iris-versicolor’ no other false negatives from the given algorithm. 
Output:
95363/1/Iris.csv
Id,SepalLengthCm,SepalWidthCm,PetalLengthCm,PetalWidthCm,Species
1,5.1,3.5,1.4,0.2,Iris-setosa
2,4.9,3.0,1.4,0.2,Iris-setosa
3,4.7,3.2,1.3,0.2,Iris-setosa
4,4.6,3.1,1.5,0.2,Iris-setosa
5,5.0,3.6,1.4,0.2,Iris-setosa
6,5.4,3.9,1.7,0.4,Iris-setosa
7,4.6,3.4,1.4,0.3,Iris-setosa
8,5.0,3.4,1.5,0.2,Iris-setosa
9,4.4,2.9,1.4,0.2,Iris-setosa
10,4.9,3.1,1.5,0.1,Iris-setosa
11,5.4,3.7,1.5,0.2,Iris-setosa
12,4.8,3.4,1.6,0.2,Iris-setosa
13,4.8,3.0,1.4,0.1,Iris-setosa
14,4.3,3.0,1.1,0.1,Iris-setosa
15,5.8,4.0,1.2,0.2,Iris-setosa
16,5.7,4.4,1.5,0.4,Iris-setosa
17,5.4,3.9,1.3,0.4,Iris-setosa
18,5.1,3.5,1.4,0.3,Iris-setosa
19,5.7,3.8,1.7,0.3,Iris-setosa
20,5.1,3.8,1.5,0.3,Iris-setosa
21,5.4,3.4,1.7,0.2,Iris-setosa
22,5.1,3.7,1.5,0.4,Iris-setosa
23,4.6,3.6,1.0,0.2,Iris-setosa
24,5.1,3.3,1.7,0.5,Iris-setosa
25,4.8,3.4,1.9,0.2,Iris-setosa
26,5.0,3.0,1.6,0.2,Iris-setosa
27,5.0,3.4,1.6,0.4,Iris-setosa
28,5.2,3.5,1.5,0.2,Iris-setosa
29,5.2,3.4,1.4,0.2,Iris-setosa
30,4.7,3.2,1.6,0.2,Iris-setosa
31,4.8,3.1,1.6,0.2,Iris-setosa
32,5.4,3.4,1.5,0.4,Iris-setosa
33,5.2,4.1,1.5,0.1,Iris-setosa
34,5.5,4.2,1.4,0.2,Iris-setosa
35,4.9,3.1,1.5,0.1,Iris-setosa
36,5.0,3.2,1.2,0.2,Iris-setosa
37,5.5,3.5,1.3,0.2,Iris-setosa
38,4.9,3.1,1.5,0.1,Iris-setosa
39,4.4,3.0,1.3,0.2,Iris-setosa
40,5.1,3.4,1.5,0.2,Iris-setosa
41,5.0,3.5,1.3,0.3,Iris-setosa
42,4.5,2.3,1.3,0.3,Iris-setosa
43,4.4,3.2,1.3,0.2,Iris-setosa
44,5.0,3.5,1.6,0.6,Iris-setosa
45,5.1,3.8,1.9,0.4,Iris-setosa
46,4.8,3.0,1.4,0.3,Iris-setosa
47,5.1,3.8,1.6,0.2,Iris-setosa
48,4.6,3.2,1.4,0.2,Iris-setosa
49,5.3,3.7,1.5,0.2,Iris-setosa
50,5.0,3.3,1.4,0.2,Iris-setosa
51,7.0,3.2,4.7,1.4,Iris-versicolor
52,6.4,3.2,4.5,1.5,Iris-versicolor
53,6.9,3.1,4.9,1.5,Iris-versicolor
54,5.5,2.3,4.0,1.3,Iris-versicolor
55,6.5,2.8,4.6,1.5,Iris-versicolor
56,5.7,2.8,4.5,1.3,Iris-versicolor
57,6.3,3.3,4.7,1.6,Iris-versicolor
58,4.9,2.4,3.3,1.0,Iris-versicolor
59,6.6,2.9,4.6,1.3,Iris-versicolor
60,5.2,2.7,3.9,1.4,Iris-versicolor
61,5.0,2.0,3.5,1.0,Iris-versicolor
62,5.9,3.0,4.2,1.5,Iris-versicolor
63,6.0,2.2,4.0,1.0,Iris-versicolor
64,6.1,2.9,4.7,1.4,Iris-versicolor
65,5.6,2.9,3.6,1.3,Iris-versicolor
66,6.7,3.1,4.4,1.4,Iris-versicolor
67,5.6,3.0,4.5,1.5,Iris-versicolor
68,5.8,2.7,4.1,1.0,Iris-versicolor
69,6.2,2.2,4.5,1.5,Iris-versicolor
70,5.6,2.5,3.9,1.1,Iris-versicolor
71,5.9,3.2,4.8,1.8,Iris-versicolor
72,6.1,2.8,4.0,1.3,Iris-versicolor
73,6.3,2.5,4.9,1.5,Iris-versicolor
74,6.1,2.8,4.7,1.2,Iris-versicolor
75,6.4,2.9,4.3,1.3,Iris-versicolor
76,6.6,3.0,4.4,1.4,Iris-versicolor
77,6.8,2.8,4.8,1.4,Iris-versicolor
78,6.7,3.0,5.0,1.7,Iris-versicolor
79,6.0,2.9,4.5,1.5,Iris-versicolor
80,5.7,2.6,3.5,1.0,Iris-versicolor
81,5.5,2.4,3.8,1.1,Iris-versicolor
82,5.5,2.4,3.7,1.0,Iris-versicolor
83,5.8,2.7,3.9,1.2,Iris-versicolor
84,6.0,2.7,5.1,1.6,Iris-versicolor
85,5.4,3.0,4.5,1.5,Iris-versicolor
86,6.0,3.4,4.5,1.6,Iris-versicolor
87,6.7,3.1,4.7,1.5,Iris-versicolor
88,6.3,2.3,4.4,1.3,Iris-versicolor
89,5.6,3.0,4.1,1.3,Iris-versicolor
90,5.5,2.5,4.0,1.3,Iris-versicolor
91,5.5,2.6,4.4,1.2,Iris-versicolor
92,6.1,3.0,4.6,1.4,Iris-versicolor
93,5.8,2.6,4.0,1.2,Iris-versicolor
94,5.0,2.3,3.3,1.0,Iris-versicolor
95,5.6,2.7,4.2,1.3,Iris-versicolor
96,5.7,3.0,4.2,1.2,Iris-versicolor
97,5.7,2.9,4.2,1.3,Iris-versicolor
98,6.2,2.9,4.3,1.3,Iris-versicolor
99,5.1,2.5,3.0,1.1,Iris-versicolor
100,5.7,2.8,4.1,1.3,Iris-versicolor
101,6.3,3.3,6.0,2.5,Iris-virginica
102,5.8,2.7,5.1,1.9,Iris-virginica
103,7.1,3.0,5.9,2.1,Iris-virginica
104,6.3,2.9,5.6,1.8,Iris-virginica
105,6.5,3.0,5.8,2.2,Iris-virginica
106,7.6,3.0,6.6,2.1,Iris-virginica
107,4.9,2.5,4.5,1.7,Iris-virginica
108,7.3,2.9,6.3,1.8,Iris-virginica
109,6.7,2.5,5.8,1.8,Iris-virginica
110,7.2,3.6,6.1,2.5,Iris-virginica
111,6.5,3.2,5.1,2.0,Iris-virginica
112,6.4,2.7,5.3,1.9,Iris-virginica
113,6.8,3.0,5.5,2.1,Iris-virginica
114,5.7,2.5,5.0,2.0,Iris-virginica
115,5.8,2.8,5.1,2.4,Iris-virginica
116,6.4,3.2,5.3,2.3,Iris-virginica
117,6.5,3.0,5.5,1.8,Iris-virginica
118,7.7,3.8,6.7,2.2,Iris-virginica
119,7.7,2.6,6.9,2.3,Iris-virginica
120,6.0,2.2,5.0,1.5,Iris-virginica
121,6.9,3.2,5.7,2.3,Iris-virginica
122,5.6,2.8,4.9,2.0,Iris-virginica
123,7.7,2.8,6.7,2.0,Iris-virginica
124,6.3,2.7,4.9,1.8,Iris-virginica
125,6.7,3.3,5.7,2.1,Iris-virginica
126,7.2,3.2,6.0,1.8,Iris-virginica
127,6.2,2.8,4.8,1.8,Iris-virginica
128,6.1,3.0,4.9,1.8,Iris-virginica
129,6.4,2.8,5.6,2.1,Iris-virginica
130,7.2,3.0,5.8,1.6,Iris-virginica
131,7.4,2.8,6.1,1.9,Iris-virginica
132,7.9,3.8,6.4,2.0,Iris-virginica
133,6.4,2.8,5.6,2.2,Iris-virginica
134,6.3,2.8,5.1,1.5,Iris-virginica
135,6.1,2.6,5.6,1.4,Iris-virginica
136,7.7,3.0,6.1,2.3,Iris-virginica
137,6.3,3.4,5.6,2.4,Iris-virginica
138,6.4,3.1,5.5,1.8,Iris-virginica
139,6.0,3.0,4.8,1.8,Iris-virginica
140,6.9,3.1,5.4,2.1,Iris-virginica
141,6.7,3.1,5.6,2.4,Iris-virginica
142,6.9,3.1,5.1,2.3,Iris-virginica
143,5.8,2.7,5.1,1.9,Iris-virginica
144,6.8,3.2,5.9,2.3,Iris-virginica
145,6.7,3.3,5.7,2.5,Iris-virginica
146,6.7,3.0,5.2,2.3,Iris-virginica
147,6.3,2.5,5.0,1.9,Iris-virginica
148,6.5,3.0,5.2,2.0,Iris-virginica
149,6.2,3.4,5.4,2.3,Iris-virginica
150,5.9,3.0,5.1,1.8,Iris-virginica
95363/2/2.a.docx
import pandas as pd
from sklearn.model_selection import train_test_split
from sklearn.linear_model import LinearRegression
from sklearn.preprocessing import StandardScaler
from sklearn.preprocessing import PolynomialFeatures
from sklearn.pipeline import Pipeline
df = pd.read_csv(r'C:\Users\ANAND\Desktop\Greynodes\95363\2\student-mat.csv')
X = df[['G1','G2']]
Y = df[['G3']]
X_train, X_test, y_train, y_test = train_test_split(X, Y, test_size = 1/3, random_state = 0)
regressor = LinearRegression()
regressor.fit(X_train, y_train)
y_pred = regressor.predict(X_test)
print("--------- Score Before Regularisation -------------------")
print('Training score: {}'.format(regressor.score(X_train, y_train)))
print('Test score: {}'.format(regressor.score(X_test, y_test)))
print("")
print("---------- Score After Regularisation -------------------")
steps = [
    ('scalar', StandardScaler()),
    ('poly', PolynomialFeatures(degree=4)),
    ('model', LinearRegression())
]
pipeline = Pipeline(steps)
pipeline.fit(X_train, y_train)
print('Training score: {}'.format(pipeline.score(X_train, y_train)))
print('Test score: {}'.format(pipeline.score(X_test, y_test)))
Output:
Note: We notice a slight increase in the scores when regularization is applied on the original data. Though this denotes that there is a slight overfitting on the original data by means of normal fitting process. Some degree of regularisation may help to adjust and ensure the fit is more precise. But we need to be careful that larges degrees of regularisation will have a impact on the data score and prediction outcome itself.
95363/2/2.b.docx
For regularization, we added the regularized to the loss function. Does it make sense to multiply or subtract the term, instead? Explain.
Some among the reason why we cannot follow subtraction or multiplication are listed below,
1) If the object which is unregularized is convex then adding a regularized item will yet maintain it as convex. Whereas in the case of multiplying or subtracting will impact its convexity.
2) The problem with minmin and multiple is that if the error is 0, then the regularized penalty will end up being 0. This allows the model to overfit.
3) It may lead producing more errors and may lead deviation from the expectation of  θ=0.
95363/2/student/student.txt
# Attributes for both student-mat.csv (Math course) and student-por.csv (Portuguese language course) datasets:
1 school - student's school (binary: "GP" - Gabriel Pereira or "MS" - Mousinho da Silveira)
2 sex - student's sex (binary: "F" - female or "M" - male)
3 age - student's age (numeric: from 15 to 22)
4 address - student's home address type (binary: "U" - urban or "R" - rural)
5 famsize - family size (binary: "LE3" - less or equal to 3 or "GT3" - greater than 3)
6 Pstatus - parent's cohabitation status (binary: "T" - living together or "A" - apart)
7 Medu - mother's education (numeric: 0 - none,  1 - primary education (4th grade), 2 – 5th to 9th grade, 3 – secondary education or 4 – higher education)
8 Fedu - father's education (numeric: 0 - none,  1 - primary education (4th grade), 2 – 5th to 9th grade, 3 – secondary education or 4 – higher education)
9 Mjob - mother's job (nominal: "teacher", "health" care related, civil "services" (e.g. administrative or police), "at_home" or "other")
10 Fjob - father's job (nominal: "teacher", "health" care related, civil "services" (e.g. administrative or police), "at_home" or "other")
11 reason - reason to choose this school (nominal: close to "home", school "reputation", "course" preference or "other")
12 guardian - student's guardian (nominal: "mother", "father" or "other")
13 traveltime - home to school travel time (numeric: 1 - 1 hour)
14 studytime - weekly study time (numeric: 1 - 10 hours)
15 failures - number of past class failures (numeric: n if 11 hour)
14 studytime - weekly study time (numeric: 1 - 10 hours)
15 failures - number of past class failures (numeric: n if 1 more generalization, coefficients are 
barely
# restricted and in this case linear and ridge regression resembles
rr.fit(X_train, y_train)
rr100 = Ridge(alpha=100) #  comparison with alpha value
rr100.fit(X_train, y_train)
4/18/2020 Ridge and Lasso Regression: L1 and L2 Regularization
https://towardsdatascience.com/ridge-and-lasso-regression-a-complete-guide-with-python-scikit-learn-e20e34bcbf0b 4/11
train_score=lr.score(X_train, y_train)
test_score=lr.score(X_test, y_test)
Ridge_train_score = rr.score(X_train,y_train)
Ridge_test_score = rr.score(X_test, y_test)
Ridge_train_score100 = rr100.score(X_train,y_train)
Ridge_test_score100 = rr100.score(X_test, y_test)
print "linear regression train score:", train_score
print "linear regression test score:", test_score
print "ridge regression train score low alpha:", Ridge_train_score
print "ridge regression test score low alpha:", Ridge_test_score
print "ridge regression train score high alpha:", 
Ridge_train_score100
print "ridge regression test score high alpha:", Ridge_test_score100
plt.plot(rr.coef_,alpha=0.7,linestyle='none',marker='*',markersize=5
,color='red',label=r'Ridge; $\alpha = 0.01$',zorder=7) # zorder for 
ordering the markers
plt.plot(rr100.coef_,alpha=0.5,linestyle='none',marker='d',markersiz
e=6,color='blue',label=r'Ridge; $\alpha = 100$') # alpha here is for 
transparency
plt.plot(lr.coef_,alpha=0.4,linestyle='none',marker='o',markersize=7
,color='green',label='Linear Regression')
plt.xlabel('Coefficient Index',fontsize=16)
plt.ylabel('Coefficient Magnitude',fontsize=16)
plt.legend(fontsize=13,loc=4)
plt.show()
4/18/2020 Ridge and Lasso Regression: L1 and L2 Regularization
https://towardsdatascience.com/ridge-and-lasso-regression-a-complete-guide-with-python-scikit-learn-e20e34bcbf0b 5/11
Figure 1: Ridge regression for di�erent values of alpha is plotted to show linear regression as limiting case of
ridge regression.
Let’s understand the figure above. In X axis we plot the coefficient index and, for Boston
data there are 13 features (for Python 0th index refers to 1st feature). For low value of α
(0.01), when the coefficients are less restricted, the magnitudes of the coefficients are
almost same as of linear regression. For higher value of α (100), we see that for
coefficient indices 3,4,5 the magnitudes are considerably less compared to linear
regression case. This is an example of shrinking coefficient magnitude using Ridge
regression.
. . .
Lasso Regression : The cost function for Lasso (least absolute shrinkage and selection
operator) regression can be written as
Cost function for Lasso regression
Supplement 2: Lasso regression coe�cients; subject to similar constrain as Ridge, shown before.
4/18/2020 Ridge and Lasso Regression: L1 and L2 Regularization
https://towardsdatascience.com/ridge-and-lasso-regression-a-complete-guide-with-python-scikit-learn-e20e34bcbf0b 6/11
Just like Ridge regression cost function, for lambda =0, the equation above reduces to
equation 1.2. The only difference is instead of taking the square of the coefficients,
magnitudes are taken into account. This type of regularization (L1) can lead to zero
coefficients i.e. some of the features are completely neglected for the evaluation of
output. So Lasso regression not only helps in reducing over-fitting but it can help us
in feature selection. Just like Ridge regression the regularization parameter (lambda)
can be controlled and we will see the effect below using cancer data set in sklearn .
Reason I am using cancer data instead of Boston house data, that I have used before, is,
cancer data-set have 30 features compared to only 13 features of Boston house data. So
feature selection using Lasso regression can be depicted well by changing the
regularization parameter.
Figure 2: Lasso regression and feature selection dependence on the regularization parameter value.
The code I used to make these plots is as below
4/18/2020 Ridge and Lasso Regression: L1 and L2 Regularization
https://towardsdatascience.com/ridge-and-lasso-regression-a-complete-guide-with-python-scikit-learn-e20e34bcbf0b 7/11
import math 
import matplotlib.pyplot as plt 
import pandas as pd
import numpy as np
# difference of lasso and ridge regression is that some of the 
coefficients can be zero i.e. some of the features are 
# completely neglected
from sklearn.linear_model import Lasso
from sklearn.linear_model import LinearRegression
from sklearn.datasets import load_breast_cancer
from sklearn.cross_validation import train_test_split
cancer = load_breast_cancer()
#print cancer.keys()
cancer_df = pd.DataFrame(cancer.data, columns=cancer.feature_names)
#print cancer_df.head(3)
X = cancer.data
Y = cancer.target
X_train,X_test,y_train,y_test=train_test_split(X,Y, test_size=0.3, 
random_state=31)
lasso = Lasso()
lasso.fit(X_train,y_train)
train_score=lasso.score(X_train,y_train)
test_score=lasso.score(X_test,y_test)
coeff_used = np.sum(lasso.coef_!=0)
print "training score:", train_score 
print "test score: ", test_score
print "number of features used: ", coeff_used
lasso001 = Lasso(alpha=0.01, max_iter=10e5)
lasso001.fit(X_train,y_train)
train_score001=lasso001.score(X_train,y_train)
test_score001=lasso001.score(X_test,y_test)
coeff_used001 = np.sum(lasso001.coef_!=0)
print "training score for alpha=0.01:", train_score001 
print "test score for alpha =0.01: ", test_score001
print "number of features used: for alpha =0.01:", coeff_used001
lasso00001 = Lasso(alpha=0.0001, max_iter=10e5)
lasso00001.fit(X_train,y_train)
4/18/2020 Ridge and Lasso Regression: L1 and L2 Regularization
https://towardsdatascience.com/ridge-and-lasso-regression-a-complete-guide-with-python-scikit-learn-e20e34bcbf0b 8/11
train_score00001=lasso00001.score(X_train,y_train)
test_score00001=lasso00001.score(X_test,y_test)
coeff_used00001 = np.sum(lasso00001.coef_!=0)
print "training score for alpha=0.0001:", train_score00001 
print "test score for alpha =0.0001: ", test_score00001
print "number of features used: for alpha =0.0001:", coeff_used00001
lr = LinearRegression()
lr.fit(X_train,y_train)
lr_train_score=lr.score(X_train,y_train)
lr_test_score=lr.score(X_test,y_test)
print "LR training score:", lr_train_score 
print "LR test score: ", lr_test_score
plt.subplot(1,2,1)
plt.plot(lasso.coef_,alpha=0.7,linestyle='none',marker='*',markersiz
e=5,color='red',label=r'Lasso; $\alpha = 1$',zorder=7) # alpha here 
is for transparency
plt.plot(lasso001.coef_,alpha=0.5,linestyle='none',marker='d',marker
size=6,color='blue',label=r'Lasso; $\alpha = 0.01$') # alpha here is 
for transparency
plt.xlabel('Coefficient Index',fontsize=16)
plt.ylabel('Coefficient Magnitude',fontsize=16)
plt.legend(fontsize=13,loc=4)
plt.subplot(1,2,2)
plt.plot(lasso.coef_,alpha=0.7,linestyle='none',marker='*',markersiz
e=5,color='red',label=r'Lasso; $\alpha = 1$',zorder=7) # alpha here 
is for transparency
plt.plot(lasso001.coef_,alpha=0.5,linestyle='none',marker='d',marker
size=6,color='blue',label=r'Lasso; $\alpha = 0.01$') # alpha here is 
for transparency
plt.plot(lasso00001.coef_,alpha=0.8,linestyle='none',marker='v',mark
ersize=6,color='black',label=r'Lasso; $\alpha = 0.00001$') # alpha 
here is for transparency
plt.plot(lr.coef_,alpha=0.7,linestyle='none',marker='o',markersize=5
,color='green',label='Linear Regression',zorder=2)
plt.xlabel('Coefficient Index',fontsize=16)
plt.ylabel('Coefficient Magnitude',fontsize=16)
plt.legend(fontsize=13,loc=4)
plt.tight_layout()
plt.show()
#output 
4/18/2020 Ridge and Lasso Regression: L1 and L2 Regularization
https://towardsdatascience.com/ridge-and-lasso-regression-a-complete-guide-with-python-scikit-learn-e20e34bcbf0b 9/11
training score: 0.5600974529893081
test score:  0.5832244618818156
number of features used:  4
training score for alpha=0.01: 0.7037865778498829
test score for alpha =0.01:  0.664183157772623
number of features used: for alpha =0.01: 10
training score for alpha=0.0001: 0.7754092006936697
test score for alpha =0.0001:  0.7318608210757904
number of features used: for alpha =0.0001: 22
LR training score: 0.7842206194055068
LR test score:  0.7329325010888681
Let’s understand the plot and the code in a short summary.
The default value of regularization parameter in Lasso regression (given by α) is 1.
With this, out of 30 features in cancer data-set, only 4 features are used (non zero
value of the coefficient).
Both training and test score (with only 4 features) are low; conclude that the model
is under-fitting the cancer data-set.
Reduce this under-fitting by reducing alpha and increasing number of iterations.
Now α = 0.01, non-zero features =10, training and test score increases.
Comparison of coefficient magnitude for two different values of alpha are shown in
the left panel of figure 2. For alpha =1, we can see most of the coefficients are zero
or nearly zero, which is not the case for alpha=0.01.
Further reduce α =0.0001, non-zero features = 22. Training and test scores are
similar to basic linear regression case.
In the right panel of figure, for α = 0.0001, coefficients for Lasso regression and
linear regression show close resemblance.
. . .
How Lasso Regularization Leads to Feature Selection?
4/18/2020 Ridge and Lasso Regression: L1 and L2 Regularization
https://towardsdatascience.com/ridge-and-lasso-regression-a-complete-guide-with-python-scikit-learn-e20e34bcbf0b 10/11
So far we have gone through the basics of Ridge and Lasso regression and seen some
examples to understand the applications. Now, I will try to explain why the Lasso
regression can result in feature selection and Ridge regression only reduces the
coefficients close to zero, but not zero. An illustrative figure below will help us to
understand better, where we will assume a hypothetical data-set with only two features.
Using the constrain for the coefficients of Ridge and Lasso regression (as shown above in
the supplements 1 and 2), we can plot the figure below
Figure 3: Why LASSO can reduce dimension of feature space? Example on 2D feature space. Modi�ed from
the plot used in ‘The Elements of Statistical Learning’ by me.
For a two dimensional feature space, the constraint regions (see supplement 1 and 2) are
plotted for Lasso and Ridge regression with cyan and green colours. The elliptical
contours are the cost function of linear regression (eq. 1.2). Now if we have relaxed
conditions on the coefficients, then the constrained regions can get bigger and
eventually they will hit the centre of the ellipse. This is the case when Ridge and Lasso
4/18/2020 Ridge and Lasso Regression: L1 and L2 Regularization
https://towardsdatascience.com/ridge-and-lasso-regression-a-complete-guide-with-python-scikit-learn-e20e34bcbf0b 11/11
regression resembles linear regression results. Otherwise, both methods determine
coefficients by finding the first point where the elliptical contours hit the region of
constraints. The diamond (Lasso) has corners on the axes, unlike the disk, and
whenever the elliptical region hits such point, one of the features completely
vanishes! For higher dimensional feature space there can be many solutions on the axis
with Lasso regression and thus we get only the important features selected.
Finally to end this meditation, let’s summarize what we have learnt so far
1. Cost function of Ridge and Lasso regression and importance of regularization term.
2. Went through some examples using simple data-sets to understand Linear regression
as a limiting case for both Lasso and Ridge regression.
3. Understood why Lasso regression can lead to feature selection whereas Ridge can
only shrink coefficients close to zero.
For further reading I suggest “The element of statistical learning”; J. Friedman et.al.,
Springer, pages- 79-91, 2008.
Hope you have enjoyed the post and stay happy ! Cheers !
P.S: Please see the comment made by Akanksha Rawat for a critical view on
standardizing the variables before applying Ridge regression algorithm.
Machine Learning Python Scikit Learn Data Science Linear Regression
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95363/Requirement/student-e2cg5oex.zip
student-mat.csv
school;sex;age;address;famsize;Pstatus;Medu;Fedu;Mjob;Fjob;reason;guardian;traveltime;studytime;failures;schoolsup;famsup;paid;activities;nursery;higher;internet;romantic;famrel;freetime;goout;Dalc;Walc;health;absences;G1;G2;G3
"GP";"F";18;"U";"GT3";"A";4;4;"at_home";"teacher";"course";"mother";2;2;0;"yes";"no";"no";"no";"yes";"yes";"no";"no";4;3;4;1;1;3;6;"5";"6";6
"GP";"F";17;"U";"GT3";"T";1;1;"at_home";"other";"course";"father";1;2;0;"no";"yes";"no";"no";"no";"yes";"yes";"no";5;3;3;1;1;3;4;"5";"5";6
"GP";"F";15;"U";"LE3";"T";1;1;"at_home";"other";"other";"mother";1;2;3;"yes";"no";"yes";"no";"yes";"yes";"yes";"no";4;3;2;2;3;3;10;"7";"8";10
"GP";"F";15;"U";"GT3";"T";4;2;"health";"services";"home";"mother";1;3;0;"no";"yes";"yes";"yes";"yes";"yes";"yes";"yes";3;2;2;1;1;5;2;"15";"14";15
"GP";"F";16;"U";"GT3";"T";3;3;"other";"other";"home";"father";1;2;0;"no";"yes";"yes";"no";"yes";"yes";"no";"no";4;3;2;1;2;5;4;"6";"10";10
"GP";"M";16;"U";"LE3";"T";4;3;"services";"other";"reputation";"mother";1;2;0;"no";"yes";"yes";"yes";"yes";"yes";"yes";"no";5;4;2;1;2;5;10;"15";"15";15
"GP";"M";16;"U";"LE3";"T";2;2;"other";"other";"home";"mother";1;2;0;"no";"no";"no";"no";"yes";"yes";"yes";"no";4;4;4;1;1;3;0;"12";"12";11
"GP";"F";17;"U";"GT3";"A";4;4;"other";"teacher";"home";"mother";2;2;0;"yes";"yes";"no";"no";"yes";"yes";"no";"no";4;1;4;1;1;1;6;"6";"5";6
"GP";"M";15;"U";"LE3";"A";3;2;"services";"other";"home";"mother";1;2;0;"no";"yes";"yes";"no";"yes";"yes";"yes";"no";4;2;2;1;1;1;0;"16";"18";19
"GP";"M";15;"U";"GT3";"T";3;4;"other";"other";"home";"mother";1;2;0;"no";"yes";"yes";"yes";"yes";"yes";"yes";"no";5;5;1;1;1;5;0;"14";"15";15
"GP";"F";15;"U";"GT3";"T";4;4;"teacher";"health";"reputation";"mother";1;2;0;"no";"yes";"yes";"no";"yes";"yes";"yes";"no";3;3;3;1;2;2;0;"10";"8";9
"GP";"F";15;"U";"GT3";"T";2;1;"services";"other";"reputation";"father";3;3;0;"no";"yes";"no";"yes";"yes";"yes";"yes";"no";5;2;2;1;1;4;4;"10";"12";12
"GP";"M";15;"U";"LE3";"T";4;4;"health";"services";"course";"father";1;1;0;"no";"yes";"yes";"yes";"yes";"yes";"yes";"no";4;3;3;1;3;5;2;"14";"14";14
"GP";"M";15;"U";"GT3";"T";4;3;"teacher";"other";"course";"mother";2;2;0;"no";"yes";"yes";"no";"yes";"yes";"yes";"no";5;4;3;1;2;3;2;"10";"10";11
"GP";"M";15;"U";"GT3";"A";2;2;"other";"other";"home";"other";1;3;0;"no";"yes";"no";"no";"yes";"yes";"yes";"yes";4;5;2;1;1;3;0;"14";"16";16
"GP";"F";16;"U";"GT3";"T";4;4;"health";"other";"home";"mother";1;1;0;"no";"yes";"no";"no";"yes";"yes";"yes";"no";4;4;4;1;2;2;4;"14";"14";14
"GP";"F";16;"U";"GT3";"T";4;4;"services";"services";"reputation";"mother";1;3;0;"no";"yes";"yes";"yes";"yes";"yes";"yes";"no";3;2;3;1;2;2;6;"13";"14";14
"GP";"F";16;"U";"GT3";"T";3;3;"other";"other";"reputation";"mother";3;2;0;"yes";"yes";"no";"yes";"yes";"yes";"no";"no";5;3;2;1;1;4;4;"8";"10";10
"GP";"M";17;"U";"GT3";"T";3;2;"services";"services";"course";"mother";1;1;3;"no";"yes";"no";"yes";"yes";"yes";"yes";"no";5;5;5;2;4;5;16;"6";"5";5
"GP";"M";16;"U";"LE3";"T";4;3;"health";"other";"home";"father";1;1;0;"no";"no";"yes";"yes";"yes";"yes";"yes";"no";3;1;3;1;3;5;4;"8";"10";10
"GP";"M";15;"U";"GT3";"T";4;3;"teacher";"other";"reputation";"mother";1;2;0;"no";"no";"no";"no";"yes";"yes";"yes";"no";4;4;1;1;1;1;0;"13";"14";15
"GP";"M";15;"U";"GT3";"T";4;4;"health";"health";"other";"father";1;1;0;"no";"yes";"yes";"no";"yes";"yes";"yes";"no";5;4;2;1;1;5;0;"12";"15";15
"GP";"M";16;"U";"LE3";"T";4;2;"teacher";"other";"course";"mother";1;2;0;"no";"no";"no";"yes";"yes";"yes";"yes";"no";4;5;1;1;3;5;2;"15";"15";16
"GP";"M";16;"U";"LE3";"T";2;2;"other";"other";"reputation";"mother";2;2;0;"no";"yes";"no";"yes";"yes";"yes";"yes";"no";5;4;4;2;4;5;0;"13";"13";12
"GP";"F";15;"R";"GT3";"T";2;4;"services";"health";"course";"mother";1;3;0;"yes";"yes";"yes";"yes";"yes";"yes";"yes";"no";4;3;2;1;1;5;2;"10";"9";8
"GP";"F";16;"U";"GT3";"T";2;2;"services";"services";"home";"mother";1;1;2;"no";"yes";"yes";"no";"no";"yes";"yes";"no";1;2;2;1;3;5;14;"6";"9";8
"GP";"M";15;"U";"GT3";"T";2;2;"other";"other";"home";"mother";1;1;0;"no";"yes";"yes";"no";"yes";"yes";"yes";"no";4;2;2;1;2;5;2;"12";"12";11
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