# Example

Christoph Molnar

In this example, we use the linear regression model to predict the number of rented bikes on a particular day, given weather and calendar information. For the interpretation, we examine the estimated regression weights. The features consist of numerical and categorical features. For each feature, the table shows the estimated weight, the standard error of the estimate (SE), and the absolute value of the t-statistic (|t|).

Weight | SE | |t| | |
---|---|---|---|

(Intercept) | 2399.4 | 238.3 | 10.1 |

seasonSPRING | 899.3 | 122.3 | 7.4 |

seasonSUMMER | 138.2 | 161.7 | 0.9 |

seasonFALL | 425.6 | 110.8 | 3.8 |

holidayHOLIDAY | -686.1 | 203.3 | 3.4 |

workingdayWORKING DAY | 124.9 | 73.3 | 1.7 |

weathersitMISTY | -379.4 | 87.6 | 4.3 |

weathersitRAIN/SNOW/STORM | -1901.5 | 223.6 | 8.5 |

temp | 110.7 | 7.0 | 15.7 |

hum | -17.4 | 3.2 | 5.5 |

windspeed | -42.5 | 6.9 | 6.2 |

days_since_2011 | 4.9 | 0.2 | 28.5 |

Interpretation of a numerical feature (temperature): An increase of the temperature by 1 degree Celsius increases the predicted number of bicycles by 110.7, when all other features remain fixed.

Interpretation of a categorical feature (“weathersit”): The estimated number of bicycles is -1901.5 lower when it is raining, snowing or stormy, compared to good weather – again assuming that all other features do not change. When the weather is misty, the predicted number of bicycles is -379.4 lower compared to good weather, given all other features remain the same.

All the interpretations always come with the footnote that “all other features remain the same”. This is because of the nature of linear regression models. The predicted target is a linear combination of the weighted features. The estimated linear equation is a hyperplane in the feature/target space (a simple line in the case of a single feature). The weights specify the slope (gradient) of the hyperplane in each direction. The good side is that the additivity isolates the interpretation of an individual feature effect from all other features. That is possible because all the feature effects (= weight times feature value) in the equation are combined with a plus. On the bad side of things, the interpretation ignores the joint distribution of the features. Increasing one feature, but not changing another, can lead to unrealistic or at least unlikely data points. For example increasing the number of rooms might be unrealistic without also increasing the size of a house.