Skip to main content

Linear Regression, Maths behind it and Best Reason for using least squares

 Linear Regression

Linear Regression is quite a way that will attempt to fit a line through observed variables (x,y). If we look into the graphs it will try to find the line which is more touching to the point (x^i,y^i) for all I in the dataset, y as a linear function of x and it will seem like

Image for post

As it will try to prophesy a line most possible it will do by the least-squares method which is going to decrease the squares of the distance from the line for all the points.

Catholic rule of Least Squares

As the method will try to find the line Y = MX+ b and for solving it by math just find the distance from the line by putting the point in the equation and all the distances and just minimize it by taking derivative and assigning it to the zero. The general equation looks like

Image for post

and to view it on the graph it will look like

Image for post

Why we use least squares

One purpose is that the equations required in solving for the best-fit line are easy to solve. The only satisfying reason for using it will be proved by the gauss theorem which states

Suppose y=f(x). We measure values x^i,y^i and compute errors y^i-f(x^i). If these errors are independent and normally distributed, then consider, for any possible linear function f the probability P^f​ of getting the measurements (x^i​,y^i​) if y=f(x) were the correct model. The least-squares line is the line for which P^f​ is maximized.

This is the only best reason why we used it over the other methods.

Future Readings

  • Björck, Å. (1996). Numerical Methods for Least Squares Problems. SIAM.
  • Kariya, T.; Kurata, H. (2004). Generalized Least Squares. Hoboken

I hope this helps you in understanding you, linear regression easy way.

Labels:

Comments

Post a Comment

Popular posts from this blog

Random Forest and how it works

  Random Forest Random Forest is a Machine Learning Algorithm based on Decision Trees. Random forest works on the ensemble method which is very common these days. The ensemble method means that to make a decision collectively based on the decision trees. Actually, we make a prediction, not simply based on One Decision Tree, but by an unanimous Prediction, made by ‘ K’  Decision Trees. Why should we use There are four reasons why should we us e  the random forest algorithm. The one is that it can be used for both  classification and regression  businesses. Overfitting is one critical problem that may make the results worse, but for the Random Forest algorithm, if there are enough trees in the forest, the classifier  won’t overfit  the model. The third reason is the classifier of Random Forest can handle  missing values , and the last advantage is that the Random Forest classifier can be modeled for  categorical values. How does the Random...
1 comment
Read more

DBSCAN Clustering Algorithm-with maths

  DBSCAN is a short-form of   D ensity- B ased   S patial   C lustering of   A pplications with   N oise. It is an unsupervised algorithm that will take the set of points and make them into some sets which have the same properties. It is based on the density-based clustering and it will mark the outliers also which do not lie in any of the cluster or set. There are some terms that we need to know before we proceed further for algorithm: Density Reachability A point “p” is said to be   density reachable from a point “q” if point “p” is within ε distance from point “q” and “q” has a sufficient number of points in its neighbors which are within distance ε. Density Connectivity A point “p” and “q” are said to be density connected if there exists a point “r” which has a sufficient number of points in its neighbors and both the points “p” and “q” is within the ε distance. This is a chaining process. So, if “q” is neighbor of “r”, “r” is neighbor of “s”, “s” ...
2 comments
Read more

Neural Network theory and implementation for Regression

Introduction and background In this article, we are going to build the regression model from neural networks for predicting the price of a house based on the features. Here is the implementation and the theory behind it. The neural network is basically if you see is derived from the logistic regression, as we know that in the logistic regression: Formulae for Logistic Regression:  y = ax+b so for every node in each layer, we will apply it and after this output is from the activation function which will have the input from logistic regression and the output is output from the activation function. So now  w e will implement the neural  network  with 5 hidden layers. Implementation 1. Import the libraries which we will going to use 2. Import the dataset and check the types of the columns 3. Now build your training and test set from the dataset. 4. Now we have our data we will now make the model and I will describe to you how it will predict the price. Here we are making...
Post a Comment
Read more