In this video, our final topic will be on prediction and decision making. How can we determine if our model is correct? The first thing you should do is make sure your model results make sense. You should always use visualization, numerical measures for evaluation and comparing between different models. Let's look at an example of prediction. If you recall, we train the model using the fit method. Now we want to find out what the price would be for a car that has a highway miles per gallon of 30. Plugging this value into the predict method gives us a resulting price of $13,771.30. This seems to make sense. For example; The value is nauts negative, extremely high or extremely low. We can look at the coefficients by examining the coef_attribute. If you recall the expression for the simple linear model that predicts price from highway miles per gallon. This value corresponds to the multiple of the highway miles per gallon feature, as such an increase of one unit in highway miles per gallon. The value of the car decreases approximately $821. This value also seems reasonable. Sometimes your model will produce values that don't make sense. For example, if we plot the model out for highway miles per gallon in the ranges of 0 to 100 we get negative values for the price. This could be because the values in that range are not realistic. The linear assumption is incorrect or we don't have data for cars in that range. In this case, it is unlikely that a car will have fuel mileage in that range. So our model seems valid. To generate a sequence of values in a specified range, import numpy, then use the numpy arrange function to generate the sequence. The sequence starts at one and increments by one till we reach 100. The first parameter is the starting point of the sequence. The second parameter is the endpoint plus one of the sequence. The final parameter is the step size between elements in the sequence. In this case, it's one. So we increment the sequence one step at a time. From one to two and so on. We can use the output to predict new values. The output is a numpy array, many of the values are negative. Using a regression plot to visualize your data is the first method you should try. See the labs for examples of how to plot polynomial regression. For this example, the effect of the independent variable is evident in this case. The data trends down as the dependent variable increases. The plot also shows some nonlinear behavior. Examining the residual plot, we see in this case the residuals have a curvature suggesting nonlinear behavior. A distribution plot is a good method for multiple linear regression. For example, we see the predictive values for prices in the range from 30,000 to 50,000 are inaccurate. This suggests a non-linear model may be more suitable or we need more data in this range. The mean square error is perhaps the most intuitive numerical measure for determining if a model is good or not. Let's see how different measures of mean square error impact the model. The figure shows an example of a mean square error of 3,495. This example has a mean square error of 3,652. The final plot has a mean square error of 12,870. As the square error increases, the targets get further from the predicted points. As we discussed, R-squared is another popular method to evaluate your model. In this plot, we see the target points in red and the predicted line in blue and R-squared of 0.9986. The model appears to be a good fit. This model has an R-squared of 0.9226. There still is a strong linear relationship in R- squared of 0.0806, the data is a lot more messy but the linear relation is evident. In R-squared 0.61, the linear function is harder to see but on closer inspection we see the data is increasing with the independent variable and acceptable value for R-squared depends on what field you're studying. Some authors suggest a value should be equal to or greater than 0.10. Comparing MLR and SLR is a lower MSE always implying a better fit? Not necessarily. MSE for an MLR model will be smaller than the MSE for an SLR model, since the errors of the data will decrease when more variables are included in the model. Polynomial regression will also have a smaller MSE than regular regression. A similar inverse relationship holds for R-squared. In the next section we'll look at better ways to evaluate the model.
