Linear Regression from scratch

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These all questions implies the answer would be quantitative or continuous in nature. Linear regression is the fundamental topic for the predicting the continuous variable target variable on the basis of given explanatory or input variable.

In this article, I will cover topics related to linear regression. Here I am mentioning the topic of linear which will be cover in this article.

  1. Simple Linear Regression
  2. The fitness of Model or Ordinary least square
  3. Model evaluation
  4. Multiple linear regression
  5. Polynomial linear regression
Simple Linear Regression

Simple linear regression model the relationship between single features(explanatory variable) and a continuous response variable. Let suppose we have explanatory variable x and target variable y then the linear model with one explanatory variable defined as:

y=αx+β-------------------(1)

where β is known as the intercept of y, on x and α is slope or coefficient of the x.

To understand this let’s take an example of pizza price according to diameter is given below:

import matplotlib.pyplot as plt
x=[[5],[6],[7],[8],[9],[10],[13],[14],[15],[16]]
y=[[6],[8],[9],[10],[11],[12],[15],[16],[17],[18]]
plt.figure()
plt.xlabel("Diameter in inches")
plt.ylabel("price in dollar")
plt.title("pizza price against diameter")
plt.plot(x,y,'k.')
plt.show()

In this figure, we can see, there is the positive relationship between the diameter of pizza and it’s price. From the given above data, we can predict the price of the unseen diameter of pizza using linear regression.

from sklearn.linear_model import LinearRegression
lr=LinearRegression()
lr.fit(x,y)
lr.predict([[12],[15]])
array([[ 13.96431121],
       [ 17.07780157]])

Based on our simple linear regression model we got the 12-inch diameter pizza price is 13.964 and 15-inch diameter pizza is 17.0778 but in our training data the price of 15-inch pizza price is 17.00 this difference is an error and this error is called residual error or training error. The residual error is the error it is the difference between the observed price and price predicted.

Since linear regression model the relationship between the explanatory variable and target variable using linear surface called hyperplane.

The fitness of Model or Ordinary least square

We can find the best fitting line using minimize the sum of residuals that is model fit the value very close to observed value. This is the measure known as residual sum of squares.

The formula of residual sum of square is

Where y_i is observed value and y^_i is predicted value.

 

Now Recall the equation(1)  y=αx+β, we have to solve α and β so that minimize our residual sum of square.

For solving this equation we have to first eliminate the term α and the formula of α is Covariance(x,y)/var(x).

Covariance measure how much two variables changing together. It could be either positive or negative manner, if the value of both increases together called positive covariance or if one value increase and another decrease called negative covariance.

 

α=np.cov([5,6,7,8,9,10,13,14,15,16],[6,8,9,10,11,12,15,16,17,18]/np.var([5,6,7,8,9,10,13,14,15,16],ddof=1)
α=16.1555555556/15.5666666667
α=1.037830121342531

Now

β=y_bar - α.x_bar

Now solve the equation

β=12.2-1.037830121342531 x 10.3
β=1.510349750171931

 

For instances, if we want to predict the price of 12 inches pizza then the cost around will be $13.96 and our model also predicted same for 12 inches pizza. So we can say to us, Congratulations we have made it!

Model Evaluation

In the previous section, we have to discussed how to fit model for the regression model by estimating the model’s parameter from the training data. But it is crucial to test the model on data that hasn’t seen during training data. So we here an important question raises that how we know our model is a good representation of the real relationship or test datasets.

We can introduce MSE(Mean Squared Error) which is the average value of SSE cost function. It is very useful for comparing different regression models.

Another useful model evaluation techniques used called R² or r-squared.  R² measure how well our observed value predicted by the model. R² is also called coefficients of determination.  There are several methods for computing R² but in the case of linear regression, we preferred r squared is equal to Pearson product moment correlation coefficients or Pearson’s r.

Where, 
RSS= Residual sum of square
TSS= Total sum of square defined as,

R² explain how much of the data is explained by the model. R² is bounded between 0 to 1. If R² is 80% or 0.80 means 80% of variations residing in residuals means the fitted line is good. R² more than 60% assume good.

from sklearn.linear_model import LinearRegression
x=[[5],[6],[7],[8],[9],[10],[13],[14],[15],[16]]
y=[[6],[8],[9],[10],[11],[12],[15],[16],[17],[18]]
X_test = [[8], [9], [12], [15], [14]]
y_test = [[11], [8.5], [15], [18], [13]]
lr=LinearRegression()
lr.fit(x,y)
y_pred=lr.predict([[12],[15]])
print(lr.score(x,y))
print(y_pred)
print('R-squared: %.4f' % lr.score(X_test, y_test))
y predicted [[ 13.96431121]
             [ 17.07780157]]
R-squared: 0.6598

Multiple Linear Regression

R² on test dataset is 65 %, it is not too good so we have to improve our model. We can introduce a new feature by which price of pizza can be directly related to example the toppings of the pizza. The price of pizza can also depend upon toppings. So we added a new explanatory variable topping and now we can proceed multiple linear regression with two explanatory variables (1) dimension (2) toppings.

In general, multiple linear regression is simply a generalization process of single linear regression which uses more than one explanatory variables.

y=β+α1x1+α2x2+...+αnxn

Programming Implementation

from sklearn.linear_model import LinearRegression
X=[[5,1],[6,0],[7,1],[8,0],[9,0],[10,1],[13,1],[14,1],[15,0],[16,1]]
Y=[[6],[8],[9],[10],[11],[12],[15],[16],[17],[18]]
X_test = [[8,0], [9,1], [12,0], [15,1], [13,0]]
y_test = [[11], [8.5], [15], [18], [13]]
lr=LinearRegression()
lr.fit(X,Y)
print('R-squared: %.4f' % lr.score(X_test, y_test))
R-squared: 0.7629

Now we can see the advantages of generalization of linear model our r-squared increased. Multiple linear regression improved the performance of our model.

Polynomial Regression

In the previous sections, we discussed linear relationship-based model. A special type of multiple linear regression basically used for the nonlinearity assumptions by adding polynomial terms. It captured curvilinear relationship when adding polynomial terms.

y=β+α1x1+α2x²+...........+αnxª

Here a denotes the degree of the polynomial. Here I am going to define the quadratic form of regression with one explanatory variable.

import numpy as np
import matplotlib.pyplot as plt
from sklearn.linear_model import LinearRegression
from sklearn.preprocessing import PolynomialFeatures
X_train = [[6], [8], [9], [14], [18]]
y_train = [[7], [9], [13], [17.5], [18]]
X_test = [[6], [8], [11], [16]]
y_test = [[8], [12], [15], [18]]
regressor = LinearRegression()
regressor.fit(X_train, y_train)
xx = np.linspace(0, 26, 100)
yy = regressor.predict(xx.reshape(xx.shape[0], 1))
plt.plot(xx, yy)
quadratic_featurizer = PolynomialFeatures(degree=2)
X_train_quadratic = quadratic_featurizer.fit_transform(X_train)
X_test_quadratic = quadratic_featurizer.transform(X_test)
regressor_quadratic = LinearRegression()
regressor_quadratic.fit(X_train_quadratic, y_train)
xx_quadratic = quadratic_featurizer.transform(xx.reshape(xx.shape[0], 1))
plt.plot(xx, regressor_quadratic.predict(xx_quadratic), c='r',linestyle='--')
plt.title('Pizza price regressed on diameter')
plt.xlabel('Diameter in inches')
plt.ylabel('Price in dollars')
plt.axis([0, 25, 0, 25])
plt.grid(True)
plt.scatter(X_train, y_train)
plt.show()
print('Simple linear regression r-squared', regressor.score(X_test, y_test))
print('Quadratic regression r-squared', regressor_quadratic.score(X_test_quadratic, y_test))
Simple linear regression r-squared 0.84818661609
Quadratic regression r-squared 0.914832282464

Now we can clearly see that a polynomial degree of the model can give the better result that simple linear regression even on a single explanatory variable but sometimes if when we try to predict with the higher degree of polynomial it then fits training data exactly but doesn’t work with unseen data which causes overfitting occurred. Overfitting means that our model performs very well on training data but poor perform on testing data.

Last Note

This is all the basic you will need to get started with Linear regression this is the type of supervised learning and very useful for the prediction of the continuous target value. I recommend you to go through the whole process involved in linear regression and start your own project on linear regression from scratch.

There are also some additional topics come under while we learn linear regressions are Gradient Descent, Regularization, Bias-Variance tradeoff, Decision tree regression, Random forest regression and also different types of regression algorithms. I will cover all these topics in next articles.

If you liked this article be sure to like this article and you have any questions related to this answer I will do my best to answer.

 

About Mitra N Mishra 35 Articles
Mitra N Mishra is working as a full-stack data scientist.

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  109. I’m impressed, I have to admit. Seldom do I come across a blog that’s both equally educative and entertaining, and without a doubt, you’ve hit the nail on the head. The issue is something which too few men and women are speaking intelligently about. I am very happy that I stumbled across this in my hunt for something concerning this.

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  117. I’m impressed, I have to admit. Rarely do I encounter a blog that’s both educative and engaging, and let me tell you, you’ve hit the nail on the head. The issue is something which not enough folks are speaking intelligently about. I’m very happy that I came across this in my hunt for something regarding this.

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  131. An outstanding share! I’ve just forwarded this onto a co-worker who has been doing a little research on this. And he actually ordered me breakfast because I found it for him… lol. So let me reword this…. Thanks for the meal!! But yeah, thanks for spending some time to discuss this matter here on your web page.

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  136. A fascinating discussion is worth comment. I believe that you should write more on this subject, it might not be a taboo matter but usually folks don’t talk about these topics. To the next! Best wishes!!

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  143. An outstanding share! I have just forwarded this onto a coworker who had been doing a little research on this. And he in fact bought me breakfast simply because I found it for him… lol. So let me reword this…. Thank YOU for the meal!! But yeah, thanx for spending some time to discuss this matter here on your web site.

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  156. Howdy, I believe your web site might be having internet browser compatibility problems. When I take a look at your website in Safari, it looks fine but when opening in I.E., it’s got some overlapping issues. I merely wanted to give you a quick heads up! Other than that, excellent blog!

  157. An impressive share! I have just forwarded this onto a colleague who has been conducting a little homework on this. And he in fact bought me breakfast due to the fact that I discovered it for him… lol. So let me reword this…. Thank YOU for the meal!! But yeah, thanks for spending time to talk about this issue here on your site.

  158. Your style is so unique in comparison to other people I have read stuff from. Many thanks for posting when you’ve got the opportunity, Guess I’ll just book mark this blog.

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  162. Hi, I do believe this is a great blog. I stumbledupon it 😉 I’m going to come back once again since I bookmarked it. Money and freedom is the greatest way to change, may you be rich and continue to guide other people.

  163. Can I just say what a relief to discover an individual who genuinely knows what they’re discussing over the internet. You definitely realize how to bring an issue to light and make it important. More people should look at this and understand this side of your story. I was surprised you are not more popular given that you certainly possess the gift.

  164. I’m amazed, I have to admit. Seldom do I encounter a blog that’s both educative and amusing, and let me tell you, you’ve hit the nail on the head. The issue is something which not enough folks are speaking intelligently about. I am very happy I found this in my search for something regarding this.

  165. I have to thank you for the efforts you have put in penning this blog. I really hope to view the same high-grade blog posts from you in the future as well. In fact, your creative writing abilities has encouraged me to get my own blog now 😉

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  170. After I originally left a comment I appear to have clicked on the -Notify me when new comments are added- checkbox and now whenever a comment is added I recieve four emails with the same comment. There has to be a way you can remove me from that service? Cheers!

  171. Aw, this was a very nice post. Taking a few minutes and actual effort to create a really good article… but what can I say… I hesitate a lot and never manage to get nearly anything done.

  172. An interesting discussion is worth comment. I do think that you ought to write more about this subject matter, it might not be a taboo matter but generally people don’t speak about these subjects. To the next! Kind regards!!

  173. Hello there! I could have sworn I’ve visited this site before but after browsing through many of the articles I realized it’s new to me. Nonetheless, I’m certainly happy I stumbled upon it and I’ll be bookmarking it and checking back regularly!

  174. Having read this I thought it was really enlightening. I appreciate you taking the time and effort to put this information together. I once again find myself personally spending a lot of time both reading and posting comments. But so what, it was still worthwhile!

  175. May I simply say what a comfort to discover someone who really understands what they are discussing on the web. You certainly realize how to bring an issue to light and make it important. More people have to check this out and understand this side of the story. It’s surprising you’re not more popular given that you surely possess the gift.

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  177. You are so awesome! I do not suppose I’ve read through something like this before. So great to find somebody with a few original thoughts on this subject. Really.. many thanks for starting this up. This website is one thing that’s needed on the web, someone with some originality!

  178. Hi, I believe your website could be having internet browser compatibility problems. Whenever I look at your blog in Safari, it looks fine but when opening in IE, it’s got some overlapping issues. I just wanted to provide you with a quick heads up! Aside from that, fantastic blog!

  179. You’re so cool! I do not believe I’ve truly read through a single thing like this before. So nice to discover someone with some genuine thoughts on this subject matter. Really.. many thanks for starting this up. This site is something that is required on the web, someone with a little originality!

  180. I’m impressed, I must say. Rarely do I come across a blog that’s both educative and amusing, and without a doubt, you’ve hit the nail on the head. The issue is something not enough folks are speaking intelligently about. Now i’m very happy I found this during my hunt for something regarding this.

  181. I’m impressed, I have to admit. Seldom do I come across a blog that’s both equally educative and interesting, and without a doubt, you’ve hit the nail on the head. The issue is an issue that not enough people are speaking intelligently about. I am very happy I found this during my search for something concerning this.

  182. Next time I read a blog, I hope that it does not disappoint me as much as this one. After all, Yes, it was my choice to read, but I truly believed you would probably have something useful to talk about. All I hear is a bunch of complaining about something that you could fix if you weren’t too busy seeking attention.

  183. An outstanding share! I have just forwarded this onto a coworker who had been doing a little homework on this. And he in fact ordered me dinner due to the fact that I stumbled upon it for him… lol. So let me reword this…. Thanks for the meal!! But yeah, thanks for spending time to talk about this topic here on your internet site.

  184. I have to thank you for the efforts you’ve put in penning this website. I really hope to view the same high-grade content by you later on as well. In fact, your creative writing abilities has motivated me to get my very own site now 😉

  185. A motivating discussion is worth comment. I do think that you should publish more on this topic, it might not be a taboo matter but typically folks don’t speak about such subjects. To the next! Kind regards!!

  186. Greetings, There’s no doubt that your website may be having browser compatibility issues. When I look at your blog in Safari, it looks fine however when opening in I.E., it has some overlapping issues. I simply wanted to give you a quick heads up! Aside from that, wonderful blog!

  187. An outstanding share! I’ve just forwarded this onto a co-worker who was doing a little research on this. And he actually ordered me lunch because I discovered it for him… lol. So allow me to reword this…. Thank YOU for the meal!! But yeah, thanks for spending the time to talk about this matter here on your website.

  188. Hello! I could have sworn I’ve visited your blog before but after looking at a few of the posts I realized it’s new to me. Anyhow, I’m certainly pleased I discovered it and I’ll be bookmarking it and checking back often!

  189. I’d like to thank you for the efforts you’ve put in penning this website. I really hope to see the same high-grade content from you in the future as well. In fact, your creative writing abilities has motivated me to get my own, personal blog now 😉

  190. Can I simply just say what a relief to discover somebody that actually understands what they are talking about over the internet. You certainly realize how to bring an issue to light and make it important. A lot more people must check this out and understand this side of your story. It’s surprising you aren’t more popular because you definitely have the gift.

  191. The very next time I read a blog, I hope that it does not disappoint me just as much as this one. I mean, Yes, it was my choice to read through, nonetheless I really thought you’d have something useful to say. All I hear is a bunch of crying about something you could possibly fix if you weren’t too busy seeking attention.

  192. An impressive share! I’ve just forwarded this onto a friend who had been doing a little homework on this. And he actually ordered me breakfast simply because I found it for him… lol. So allow me to reword this…. Thanks for the meal!! But yeah, thanx for spending the time to discuss this subject here on your web page.

  193. Aw, this was an extremely good post. Finding the time and actual effort to make a top notch article… but what can I say… I procrastinate a lot and never seem to get nearly anything done.

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  195. I must thank you for the efforts you’ve put in penning this blog. I really hope to see the same high-grade blog posts from you in the future as well. In fact, your creative writing abilities has encouraged me to get my own, personal blog now 😉

  196. You’re so interesting! I do not believe I’ve read a single thing like that before. So wonderful to discover somebody with some original thoughts on this topic. Really.. thanks for starting this up. This web site is one thing that’s needed on the web, someone with a bit of originality!

  197. I’m amazed, I have to admit. Rarely do I encounter a blog that’s both equally educative and engaging, and let me tell you, you have hit the nail on the head. The problem is an issue that too few folks are speaking intelligently about. Now i’m very happy that I found this in my search for something concerning this.

  198. I’m amazed, I have to admit. Rarely do I encounter a blog that’s both educative and interesting, and without a doubt, you have hit the nail on the head. The issue is something which too few folks are speaking intelligently about. Now i’m very happy that I stumbled across this in my search for something concerning this.

  199. Hello there! I could have sworn I’ve visited this website before but after looking at many of the articles I realized it’s new to me. Anyways, I’m definitely pleased I stumbled upon it and I’ll be bookmarking it and checking back frequently!

  200. You are so interesting! I do not believe I have read through anything like this before. So nice to discover someone with original thoughts on this subject. Seriously.. thanks for starting this up. This website is something that is needed on the internet, someone with a bit of originality!

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  271. After I initially commented I seem to have clicked on the -Notify me when new comments are added- checkbox and from now on every time a comment is added I receive 4 emails with the same comment. Is there an easy method you are able to remove me from that service? Thanks!

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  274. Good post. I learn something new and challenging on websites I stumbleupon on a daily basis. It will always be helpful to read articles from other writers and use something from their websites.

  275. Aw, this was an incredibly nice post. Finding the time and actual effort to create a superb article… but what can I say… I put things off a lot and never manage to get anything done.

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  317. An outstanding share! I’ve just forwarded this onto a colleague who had been conducting a little research on this. And he actually ordered me breakfast due to the fact that I stumbled upon it for him… lol. So let me reword this…. Thank YOU for the meal!! But yeah, thanks for spending some time to talk about this issue here on your web site.

  318. Aw, this was a very nice post. Spending some time and actual effort to make a great article… but what can I say… I hesitate a lot and don’t manage to get nearly anything done.

  319. I must thank you for the efforts you’ve put in penning this blog. I am hoping to check out the same high-grade blog posts from you later on as well. In truth, your creative writing abilities has motivated me to get my very own site now 😉

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  334. When I originally left a comment I seem to have clicked the -Notify me when new comments are added- checkbox and from now on whenever a comment is added I receive 4 emails with the exact same comment. Perhaps there is an easy method you are able to remove me from that service? Thanks!

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  338. Hello! I could have sworn I’ve been to this website before but after looking at a few of the articles I realized it’s new to me. Anyways, I’m certainly pleased I stumbled upon it and I’ll be book-marking it and checking back frequently!

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