Hey everyone, welcome to my first blog post! Now that we understand what the parameter \(m\) is responsible for, let's take a look at the \(y\)-intercept \(b\) and set it to \(1\): The steepness of the line is the same as the previous line since we haven't modified \(m\). No one likes that. If you heard someone trying to "fit a line through the data" that person most likely worked with a Linear Regression model. Nodding along we confirm that we'll dive deeper into this topic and hang up the telephone in sheer excitement! Given that we're dealing with 2 dimensions (the number of claims and the issued payments) one of the potential diagrams we can create is a so called scatter plot which uses (Cartesian) coordinates to display the values of a given data set. Let's answer all those questions by implementing Linear and Multiple Regression from scratch! Welcome to one more tutorial! Let's translate this idea into Math. We could for example go through each individual \((x, y)\) pair in our data set and subtract its \(y\) value from the \(y\) value our line "predicts" for the corresponding \(x\). Today I will focus only on multiple regression and will show you how to calculate the intercept and as many slope coefficients as you need with some linear algebra. If you take a moment to think about what your model should do automatically for the user, you’ll probably end up with the list of two things (or more): In case you don’t do so, your model will fail. Let's translate the slope-intercept form into a function we call predict (we'll use this function for our predictions later on): Let's put the theory into practice and try to guesstimate a line which best describes our data. Let's put all the pieces together and implement the Gradient Descent algorithm to find the best fitting line: Running this algorithm results in a best estimate for the \(m\) and \(b\) values. Note: Throughout this post we'll be using the "Auto Insurance in Sweden" data set which was compiled by the "Swedish Committee on Analysis of Risk Premium in Motor Insurance". Let’s say you want to make a prediction for the first row of X: Everything works. At the same time it predicts large negative numbers near the end of the \(x\)-axis although those values should be positive. It’s not hard, but upon completion, you’ll be more confident in why everything works. Learn how to implement your own spam filter with the help of Bayes Theorem. The following code captures what we've just described: Repeating this process multiple times should help us find the \(m\) and \(b\) values for our line for which any given prediction \(y\) calculated by that line results in the smallest error possible. Multiplying the vector by \(-1\) will let it point into the opposite direction, the direction of greatest decrease (remember that we want to find a local minimum). There are two main types of Linear Regression models: 1. Let's call our co-worker and share the good news. For the simple linear regression this was fairly easy as we were essentially just drawing the line of best fit on a scatter chart. It's ok if you just skim through this section to get a high-level overview. Here's the linear equation we've used so far: Having multiple \(x\) values means that we'll also have multiple \(m\) values (one for each \(x\)). The first coefficient represents the intercept or the bias term, and all the others will need to be multiplied with the respective value of X. In my last post I demonstrated how to obtain linear regression parameter estimates in R using only matrices and linear algebra. Taking those observations into account we guess the following description for our line: Not too bad for our first guess! Summing up these differences results in a number we can use to compare different lines against each other. 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