# Model Evaluation

August 2019

Thus far in our study of various machine learning models, we have often alluded to evaluating our models. Evaluation has been especially relevant when training and performing cross-validation.

Evaluation is also very important when we have successfully trained a model and are ready to say “Based on such-and-such measure, this is the quality of this model.”

Metrics in other situations usually refer to evaluating some measure of success that allows us to quantitatively compare one value against another.

For example when we are talking about stock analysis, we may refer to metrics such as the price-to-earnings ratio of a company’s stock, or we may refer to a company’s market capitalization.

What do such metrics look like for machine learning models? It turns out there are a few commonly-used evaluation metrics both for classification and regression. This lesson will introduce a few of these metrics and describe what they are meant to capture.

## Accuracy

is one of the most fundamental metrics used in classification. In its most basic terms, accuracy computes the ratio of the number of correct predictions to the total number of predictions:

$\textrm{Accuracy} = \frac{\# \textrm{Correct Predictions}}{\textrm{Total } \# \textrm{ Predictions}}$

For example, if we are doing a binary classification predicting the likelihood that a car is cheap or expensive, we may predict for a dataset of five cars (CHEAP, CHEAP, EXPENSIVE, CHEAP, EXPENSIVE).

If in reality the cars are (CHEAP, CHEAP, CHEAP, EXPENSIVE, EXPENSIVE), then our accuracy is $\frac{3}{5} = 60\%$. In this case our model is not doing a great job of solving the task.

Generally, accuracy is a good place to start with classification tasks. Sometimes, however, it is not a sufficient metric. When is that the case? To answer this question, we have to introduce a little more terminology.

In classification, a true positive is a positive label that our model predicts for a datapoint whose true label is also positive. For our running example, we can denote a CHEAP prediction as a positive one, so our model had 2 true positives.

A true negative is when our model accurately makes a negative prediction. In our running example, there are 2 EXPENSIVE cars of which our model labels 1 correctly, so the number of true negatives is 1.

A false positive is when our model predicts a positive label but the true label is negative. In the example, our model predicts CHEAP a single time when in fact the label was EXPENSIVE, so we have 1 false positive.

Similarly, a false negative is when our model predicts a negative label but the true label was positive. In our example, our model predicts EXPENSIVE once when in fact the label was CHEAP, so the number of false negatives is 1.

With these definitions in place, we can actually rewrite our definition of accuracy. Letting TP = True Positive, TN = True Negative, FN = False Negative, and FP = False Positive we have:

$\textrm{Accuracy} = \frac{TP + TN}{TP + TN + FP + FN}$

Given this definition, we see that our accuracy is $\frac{2 + 1}{2 + 1 + 1 + 1} = 60\%$ as we had before.

So why introduce all this terminology? All these new terms will help us to understand when accuracy is lacking as a metric.

Consider the example of car classification from several lessons ago, and imagine that we are classifying a dataset of 100 cars. Let’s say that our model has 2 true positives, 1 false positive, 7 false negatives, and 90 true negatives.

In this case, our model’s accuracy would be $\frac{2 + 90}{2 + 90 + 7 + 1} = 92\%$. That sounds great, right? Well, maybe at first.

But let’s think about the label distribution of our car dataset. It turns out we have 90 + 1 = 91 EXPENSIVE cars but only 2 + 7 = 9 CHEAP cars. Hence our model identified 90 of the 91 EXPENSIVE cars as EXPENSIVE, but only 2 of the 9 CHEAP cars as CHEAP.

This is clearly a problem, where our accuracy metric is giving us an incorrect signal about the quality of our model.

It turns out that when we have a large disparity in our label distribution (91 EXPENSIVE cars vs. 9 CHEAP cars), accuracy is not a great metric to use. In fact if our system had literally only predicted EXPENSIVE for every new car it received as an input, it would still have had a 91% accuracy.

But clearly not all cars are EXPENSIVE. It turns out we need a more fine-grained metric to deal with this issue.

## $F_1$ Score

The $F_1$ score does a better job of handling the label-disparity issue we encountered with accuracy. It does this by leveraging two measures: precision and recall. These two quantities are defined as follows:

$\textrm{Precision} = \frac{TP}{TP + FP}$ $\textrm{Recall} = \frac{TP}{TP + FN}$

For our 100-car dataset above, the precision of our model would be $\frac{2}{2 + 1} = 66.6\%$, while the recall would be $\frac{2}{2 + 7} = 22.2\%$. Ouch!

The $F_1$ score is then defined as:

$F_1 = 2\cdot \frac{\textrm{Precision} \cdot \textrm{Recall}}{\textrm{Precision} + \textrm{Recall}}$

Our current model would receive an $F_1$ score of $33.3\%$. All of a sudden our model seems MUCH worse. That’s a good thing because the model learned to completely disregard predicting one entire label in our dataset. This is really bad. Hence the $F_1$ score penalized it substantially.

In practice, when it comes to classification tasks, the $F_1$ score is more often used as a metric because it gives a more balanced measure of a model’s performance.

## Mean Absolute Error

Let’s shift gears a bit to discuss a commonly-used metric for regression tasks: mean absolute error. If on a 3 point dataset, we have a model outputting the values $Y_1$, $Y_2$, $Y_3$ and the true values are $G_1$, $G_2$, $G_3$, then the mean absolute error is defined as:

$\textrm{Mean Absolute Error} = \frac{\sum_{i=1}^3 |y_i - g_i|}{3}$

in other words, the average of the absolute errors. More concretely, if our model outputted $(0.1, -1.3, 0.8)$ and the true values are $(-0.4, -0.3, 1)$, then the mean absolute error would be:

$\textrm{MAE} = \frac{|0-1 - (-0.4)| + |-1.3 - (-0.3)| + |0.8 - 1|}{3}$ $\hspace{-1.15in}\approx 0.767$

While relatively straightforward, mean absolute error is a standard way of assessing the quality of regression models.

## Final Thoughts

The collection of metrics we discussed is only meant to provide a taste of the ways we can evaluate models. There are many, many more means of performing evaluation that are used by the scientific community.

While we introduced all these different metrics, we never discussed what a good score for a metric is. It turns out there is no one golden number for any metric. The score you should be aiming for is, of course, as close to perfect as possible. However, how reasonable perfection is depends mainly on your data, how complex it is, how learnable your problem is, etc.

Another important thing to remember is never to put all your eggs in one basket when evaluating a model and assume that excellent performance on a single metric definitely demonstrates the superiority of a model.

We’ll end this lesson with a relevant note of caution from economist Charles Goodhart: When a measure becomes a target, it ceases to be a good measure.

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