Skip to main content

Polynomial regression and model comparison

Welcome back, in my previous post I described how we can perform linear regression using normal equation in Dynamo and I left with a question "what if the input and output are not linearly dependent?” Let’s say we have a hypothesis that the housing price doesn’t depend on floor area and number of rooms linearly but it has a following relationship as y = a_{0} * x_{0} + a_{1} * x_{1} + a_{2} * x_{2} + a_{3} * x_{1} * x_{2} + a_{4} * x_{1}^2 + a_{5} * x_{2}^2 \: where x_{0} = 1, \: x_{1} is floor area and x_{2} is number of rooms. Then we can introduce few new parameters x_{3} = x_{1} * x_{2}, \: x_{4} = x_{1}^2, \: x_{5} = x_{2}^2 and then perform the linear regression to find the coefficient matrix.

The Dynamo graph to setup the feature vector looks as follows. Once the feature vector is setup rest all is same as previous linear regression example.

Note that the price prediction for a given floor area and rooms we need to again construct the same feature vector. The price prediction based on the complete dataset and this new hypothesis for a 3 room house of area 1650 sq. ft. is 307,176 whereas the price predicted based on previous linear regression model is 293,081. Now the sellers can argue that the polynomial model is good because it is fetching them the higher price and similarly buyers can argue that linear regression model is better because it gives a lower price for his desired house. To identify the best model we need to build a test set and use that test set to figure out which model gives less error. We can use 80% of the dataset to evaluate the regression model and rest 20% we can use to test both the models. While splitting the dataset into a training set and test set we must shuffle the data before the split just to avoid any bias present in the dataset.

The Dynamo graph to split the dataset into training and testing is shown below. We shuffle the data row-wise and then extract sub matrix based on the percentage of the dataset to be used for training.

Now we can solve the linear regression model using Matrix.SolveLinearEquation node, which uses the same technique to evaluate the regression model as discussed in the previous blog post on Linear Regression. Once we have computed the two models (linear and polynomial) we can now run this model with the test set to get the error value. The total error for a model is the sum of the absolute error for each of the predictions as compared to the actual price data. The comparison of the two errors shows that the linear regression model performs better on the test set hence we should choose linear regression as compared to the polynomial regression hypothesis we discussed here. The error computation graph in Dynamo is shown below.

The final graph to run both the models and error comparison is shown below and it is shared at Github.

Conclusion: In this article, I have demonstrated how we can model polynomial equation based hypothesis and solve for the prediction model using linear regression and compare the two hypothesis based on the error value on a test set which is different than the training set. As always, please do share your feedback and suggestions on this article as well as the DynamoAI package on this page or on Github.


Popular posts from this blog

Linear Regression in Dynamo

Today we are collecting more data than ever but making sense out of the data is very challenging. Machine learning helps us analyze the data and build an analytical model to help us with future prediction. To create a machine learning prediction model we usually develop a hypothesis based on our observation of the collected data and fine tune the model to reduce the cost function by training the model. One of the very simple hypothesis we can develop by assuming a linear relationship between input and output parameters.

Suppose we have data on housing price and we assume that housing prices are linearly related to the floor area of the house then we can use linear regression to predict the price of a house with specific floor area. One of the most important steps towards building the hypothesis is being able to visualize the data and understand the trend. So let first draw a scatter plot of our data as shown in the figure below. I am using Grapher package to draw the scatter plot.

Gradient Descent in Dynamo

Hello and Welcome back. So far we have been able to develop a simple regression model and solved it using linear algebra. In this article, I am going to explore another alternative approach called Gradient Descent. It’s an iterative method to optimize a cost function by finding a local/global minimum by taking steps proportional to the gradient of the cost function. To apply the Gradient Descent algorithm we need to first define our cost function for our machine learning problem. We have a set of features for which the actual output is given we want to develop a prediction model to predict the output for the given set of features. We can represent our feature vectors for the complete training set as matrix X, the corresponding output values as vector y, and the predicted values as vector \hat{y}. So our prediction model is \hat{y} = X*a and the error vector for the training set in the model is err = \hat{y} - y. We would like to choose a cost function such that the overall error of t…