1FfUrl0

Chapter 7
Root mean squared error and prediction
The root mean squared error corresponds approximately to an
estimate of the standard deviation. Since most of the data is at the
most two standard deviations from the mean, we can double our
RMSE to obtain a rough confident interval. This is only completely
valid if the errors are normally distributed, but it is roughly correct
even if they are not.
Multidimensional regression
So far, we have only used a single variable for prediction, the number of
rooms per dwelling. We will now use all the data we have to fit a model using
multidimensional regression. We now try to predict a single output (the average
house price) based on multiple inputs.
The code looks very much like before:
x = boston.data
# we still add a bias term, but now we must use np.concatenate,
which
# concatenates two arrays/lists because we
# have several input variables in v
x = np.array([np.concatenate(v,[1]) for v in boston.data])
y = boston.target
s,total_error,_,_ = np.linalg.lstsq(x,y)
Now, the root mean squared error is only 4.7! This is better than what we had before,
which indicates that the extra variables did help. Unfortunately, we can no longer
easily display the results as we have a 14-dimensional regression.
Cross-validation for regression
If you remember when we first introduced classification, we stressed the importance
of cross-validation for checking the quality of our predictions. In regression, this is
not always done. In fact, we only discussed the training error model earlier. This is
a mistake if you want to confidently infer the generalization ability. Since ordinary
least squares is a very simple model, this is often not a very serious mistake (the
amount of overfitting is slight). However, we should still test this empirically, which
we will do now using scikit-learn. We will also use its linear regression classes as
they will be easier to replace for more advanced methods later in the chapter:
from sklearn.linear_model import LinearRegression
[ 151 ]