![]() ![]() I highly suggest that you read the article before continuing, as gradient descent, although a little complicated, is a very important part of polynomial regression.Īfter our regressor completes the gradient descent process, it will have reached optimal parameter values that best minimize the MSE cost function discussed in the previous section. Due to gradient descent’s complexity, I have written another article dedicated to explaining the math behind it. This is where gradient descent, another complex mathematical process, comes into play. Please make sure to check it out right away, as MSE is a large part of polynomial regression.īy using MSE, our model will be able to determine which parameter values create a better representation of the data than others.īut how do machine learning algorithms converge upon optimal parameter values in the first place? Gradient Descent This cost function is a little complex, so I wrote an article dedicated to explaining it. In order to do this, a polynomial regressor will implement what is called The Mean Squared Error (MSE) Cost Function, a mathematical formula that returns a numerical value representing the error of our model. In order to finalize a polynomial equation of the form discussed in the previous section, our model will need to be able to determine how well an equation represents the data given. Now that we know what our polynomial regression equation will look like, let’s discuss how our algorithm will create such an equation. If there is more than one independent variable, we will end up with a graph similar to the one below. This curve will be one that best represents the data being given. The parameter values ( b_0 - b_n) will be tuned by our polynomial regression algorithm such that we have a complete equation of a curve of best fit. I is the i’th product of a pair of features with a total degree less than or equal to dĭ+c_C_d is the number of unique pairs of features with a total degree less than or equal to dĪs we can see, the equation incorporates the polynomial transformation results that we discussed in the previous section. ![]() P is the product of a pair of features with a total degree less than or equal to d X_1 - x_c are the independent variables in the dataset Y represents the dependent variable (output value)ī_0 represents the y-intercept of the parabolic functionī_1 - b_dc - b_(d+c_C_d) represent parameter values that our model will tuneĭ represents the degree of the polynomial being tunedĬ represents the number of independent variables in the dataset before polynomial transformation Let’s talk about each variable in the equation: Let’s dive into it and break down each part of the equation. Unsurprisingly, the equation of a polynomial regression algorithm can be modeled by an (almost) regular polynomial equation. Much like the linear regression algorithms discussed in previous articles, a polynomial regressor tries to create an equation which it believes creates the best representation of the data given. Now that we’ve covered the basics of the polynomial transformation of datasets, let’s talk about the intuition behind the equation of polynomial regression. However, there’s a slight twist: not only will there be a column for each variable transformed to degree n, but there will be a column for the product of each unique pair of features that have a total degree less than or equal to n.įor example, if a dataset with two independent variables, x_1 and x_2, were to get transformed to the third degree, the combination x_1 ^ 2 * x_2 would be included since the total degree sum is equal to 3 on the other hand, the combination x_1 ^2 * x_2 ^2 would not be included since the total degree sum is 4, which is greater than 3. This process is repeated for each independent variable originally provided in the dataset. If we were to transform the dataset to degree 4, for example, we would have 3 new columns: x^2, x^3, and x^4. As we can see from the array above, new columns of x are created wherein the values of x are exponentiated to a certain power until the column of degree n is reached. ![]()
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