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Also, you will need to append one more column at the end of it, which is the vector with all of its elements raised to the m th power. If you want to have this reversed, you'd need to call fliplr on that output matrix. A small note is that vander in MATLAB is returned in reverse order. X itself is a very popular matrix, which is known as the Vandermonde matrix and MATLAB has a command called vander to help you compute that matrix. All you have to do is solve for x, which is x = A^ is the pseudo-inverse. With regards to above, we can decompose the problem by solving a linear system: Ax = b. Specifically, let's re-arrange the two equations above so that it's in matrix form: This will jointly minimize the cost function which finds the best line of fit for our data points.ĭoing some re-arranging, we can isolate m and b on one side of the equations and the rest on the other sides:Īs you can see, we can formulate this into a 2 x 2 system of equations to solve for m and b. Now, we need to simultaneously solve for m and b with the above two equations. Knowing that is simply n, we can simplify the above to: We can again drop the factor of 2 and distribute the -1 throughout the expression: We can drop the factor 2 from the derivative as the other side of the equation is equal to 0, and we can also do some distribution of terms by multiplying the -x_i term throughout: The intuition behind this is that we are simultaneously finding m and b such that the cost function is jointly minimized by these two parameters. The minimum can be determined by finding the derivative with respect to each parameter, and setting these equal to 0. This function is convex, so there is an optimal minimum that we can determine. M and b are our slope and intercept for this best fit line, while x and y are a vector of x and y co-ordinates that form our data set. In other words, we wish to minimize the cost function F(m,b,x,y): , ) (that is, we have n data points), we want to minimize the sum of squared residuals between this line and the data points. If you recall from linear regression theory, we wish to find the best slope m and intercept b such that for a set of points (,. This will require some basic Calculus as well as some linear algebra for solving a 2 x 2 system of equations. You also want to do this from first principles. Judging from the link you provided, and my understanding of your problem, you want to calculate the line of best fit for a set of data points.