6.1 What Is Machine Learning?

Supervised Learning

Supervised learning is analogous to a student learning from an instructor. At first, the student may answer many questions incorrectly, but the instructor is there to correct the student each time. When the student gives (mostly) correct responses, there is evidence that the student has sufficiently learned the material and would be able to answer similar questions correctly out in the real world where it really matters.

The goal of supervised learning is to produce a model that gives a mapping from the set of inputs or features to a set of output values or labels (see Figure 6.2). The model is function $y=f\left(X\right)$, where $X$ is the input array and $y$ is the output value or label. The way that it does this is through training and testing. When training, the algorithm creates a correspondence between input features and the output value or label, often going through many iterations of trial and error until a desired level of accuracy is achieved. The model is then tested on additional data, and if the accuracy remains high enough, then the model may be deployed for use in real-world applications. Otherwise, the model may need to be adjusted substantially or abandoned altogether.

Figure 6.2 The Supervised Learning Cycle. The supervised learning cycle consists of gathering data, training on some part of the data, testing on another part of the data, and deployment to solve problems about new data.

Examples of supervised learning models include linear regression (discussed in Correlation and Linear Regression Analysis), logistic regression, naïve Bayes classification, decision trees, random forests, and many kinds of neural networks. Linear regression may not seem like a machine learning algorithm because there are no correction steps. A formula simply produces the line of best fit. However, the regression formula itself represents an optimization that reduces the error in the values predicted by the regression line versus the actual data. If the linear regression model is not accurate enough, additional data could be collected to improve the accuracy of the model, and once the regression line is found, it may then be used to predict values that were not in the initial dataset. Linear regression has been developed in previous chapters, and you will see logistic regression, Bayes methods, decision trees, and random forests later in this chapter.

Unsupervised Learning

Unsupervised learning is like building a puzzle without having the benefit of a picture to compare with. It is not impossible to build a puzzle without the picture, but it may be more difficult and time-consuming.

The main advantage of unsupervised learning is that the training data does not need to be labeled. All the data is considered input, and the output would typically be information about the inherent structure of the data. Common tasks are to determine clusters, patterns, and shapes within the dataset. For example, if the current locations in latitude and longitude of every person in the world were somehow known and collected into a dataset, then an algorithm may discover locations of high densities of people—hence learn of the existence of cities!

Unsupervised learning algorithms include k-means clustering, DBScan, and some kinds of neural networks. More generally, most clustering algorithms and dimension-reduction methods such as principal component analysis (PCA) and topological data analysis (TDA) fall into the category of unsupervised learning. You will encounter the k-means and DBScan algorithms in this chapter. The more advanced topics of PCA and TDA are outside the scope of this text.

Variations and Hybrid Models

Although we will only cover supervised and unsupervised models, it is important to note that there are some additional models of machine learning.

When some of the data in a dataset has labels and some does not, a semi-supervised learning algorithm may be appropriate. The labeled data can be run through a supervised learning algorithm to generate a predictive model that can then be used to label the unlabeled data, called pseudo-data. From this point, the dataset can be regarded as completely labeled data, which can then be analyzed using supervised learning.

Reinforcement learning introduces rewards for accuracy while also penalizing incorrectness. This model most closely follows the analogy of a student-teacher relationship. Many algorithms in artificial intelligence use reinforcement learning. While reinforcement generally improves accuracy, there can be issues in locking the model into a rigid state. For example, have you ever noticed that when you search online for a type of product, you begin to see advertisements for those kinds of products everywhere—even after you’ve purchased it? It’s as if the algorithm has decided that the only product you ever wish to purchase is what you have just searched for!

Model Building

Suppose we are given a dataset with inputs or features $X$ and labels or values $y$. Our goal is to produce a function that maps $y=f\left(X\right)$. In practice, we do not expect that our model can capture the function $f$ with 100% accuracy. Instead, we should find a model $\hat{f}$ such that predicted values $\hat{y}=\hat{f}\left(X\right)$ are close enough to $y$. In most cases, the metric we use to determine the fitness of our model $\hat{f}$ is some measure of error between $\hat{y}$ and $y$ (e.g., see the calculation of residuals of a linear regression in Correlation and Linear Regression Analysis). The way that error is measured is specific to the type of model that we are building and the nature of the outputs. When the outputs are labels, we may use a count or percentage of misclassifications. When the outputs are numerical, we may find the total distances (absolute, squared, etc.) between predicted and actual output values.

First, the data are separated into a training set and testing set. Usually about 60–80% of the data is used for training with the remaining amount used for testing. While training, parameters may be adjusted to reduce error. The result of the training phase will be the model $\hat{f}$. Then, the model is tested on the testing set and error computed between predicted values and known outputs. If the model meets some predefined threshold of accuracy (low enough error), then the model can be used with some confidence to make predictions about new data.

Measures of Accuracy

As mentioned previously, it is important to measure how good our machine learning models are. The choice of a specific measure of accuracy depends on the nature of the problem. Here are some common accuracy measures for different machine learning tasks:

1. Classification
   • Accuracy is the ratio of correct predictions to the total number of predictions.
   • Precision $\left(p\right)$ is the ratio of true positive predictions $\left(TP\right)$ to the total number of positive (true positive plus false positive) predictions $(TP+FP)$. This is useful when the response variable is true/false and we want to minimize false positives. $p=\frac{TP}{TP+FP}$
   • Recall $\left(r\right)$ is the ratio of true positive predictions $\left(TP\right)$ to the total number of actual positives (true positive plus false negative $(TP+FN)$ predictions. This is useful when the response variable is true/false and we want to minimize false negatives. $r=\frac{TP}{TP+FN}$
   • F1 Score is a combination of precision $\left(p\right)$ and recall $\left(r\right)$. $F1=\frac{2\left(p\right)\left(r\right)}{p+r}=\frac{2TP}{2TP+FP+FN}$
2. Regression (comparison of the predicted values, ${\hat{y}}_i$, and the actual values, $y_i$)
   • Mean absolute error (MAE): $\frac{1}{n}\sum _{i=1}^n|y_i-{\hat{y}}_i|$
   • Mean absolute percentage error (MAPE): $\frac{1}{n}\sum _{i=1}^n|\frac{y_i-{\hat{y}}_i}{y_i}|$
   • Mean squared error (MSE): $\frac{1}{n}\sum _{i=1}^n(y_i-{\hat{y}}_i{)}^2$
   • Root mean squared error (RMSE): $\sqrt{\frac{1}{n}\sum _{i=1}^n(y_i-{\hat{y}}_i{)}^2}$
   • R-squared is the proportion of the variance in the response variable that is predictable from the input variable(s).

Note that MAE and RMSE, also discussed in Forecast Evaluation Methods, have the same units as the data itself. MSE and RMSE can be very sensitive to outliers because the differences are squared. MAPE (also introduced in Forecast Evaluation Methods) is often represented in percent form, which you can easily find by multiplying the raw MAPE score by 100%. The R-squared measure, which is discussed in Inferential Statistics and Regression Analysis in the context of linear regression, can be computed by statistical software packages, though we will not make significant use of it in this chapter. In all cases, the smaller the value of the error term, the more accurate the modeling would be.

Due to the tedious calculations involved in these measures, statistical software packages are typically used to compute them in practice. But it is important for every data scientist to understand how the formulas work so that results can be evaluated and interpreted correctly. Next, we will work out a small example by hand so you can have experience with the formulas.

Overfitting

It might at first seem desirable to produce a model that makes no mistakes on the training data; however, such “perfect” models are usually terrible at making predictions about data not in the training set. It is best to explain this idea by example.

Suppose we want to create a model for the relationship between x and y based on the given dataset,

How would we go about creating the model? We might first split S into a training set, $S_{\text{train}}=\{(2,10),(4,9),(6,6),(8,7)\}$, and testing set, $S_{\text{test}}=\left\{\right(10,4),(12,5\left)\right\}$. It can be shown that the cubic equation, $\hat{y}=\frac{1}{8}{x}^3-\frac{7}{4}{x}^2+\frac{13}{2}x+3$, fits the training data perfectly. However, there is no reason to expect that the relationship between x and y is truly cubic. Indeed, the predictions of the cubic model on $S_{\text{test}}$ are terrible! See Table 6.2.

The linear model (linear regression) that represents a best fit for the first four data points is $\hat{y}=-\mathrm{0.\; 6}x+11$. Figure 6.3 shows the data points along with the two models.

Figure 6.3 Data Points along a Cubic Model and a Linear Model. The four data points in $S_{\text{train}}$ can be fitted onto a cubic curve (0% error), but it does poorly on additional data from the testing data, $S_{\text{test}}$. The cubic curve overfits the training data. The regression line does not fit the training data exactly; however, it does a better job predicting additional data.

To get a good sense of how much better the linear model is compared to the cubic model in this example, we will compute the MAPE and RMSE for each model, as shown in Table 6.3.

• MAPE for cubic model: $\frac{1}{6}(0/10+0/9+0/6+0/7+14/4+40/5)=1.917$, or $191.7\%$
• RMSE for cubic model: $\sqrt{\frac{1}{6}(0^2+0^2+0^2+0^2+{14}^2+{40}^2)}=17.3$
• MAPE for linear model: $\frac{1}{6}(0.2/10+0.4/9+1.4/6+0.8/7+1/4+1.2/5)=0.15$, or $15\%$
• RMSE for linear model: $\sqrt{\frac{1}{6}({0.2}^2+{0.4}^2+{1.4}^2+{0.8}^2+1^2+{1.2}^2)}=0.93$

The MAPE and RMSE values for the linear model are much lower than their respective values for the cubic model even though the latter predicts four values with 100% accuracy.

Underfitting

A model that is underfit may perform well on some input data and poorly on other data. This is because an underfit model fails to detect some important feature(s) of the dataset.

Example 6.3

Problem

A student recorded their time spent on each homework set in their statistics course. Each HW set was roughly similar in length, but the student noticed that they spent less and less time completing them as the semester went along. Find the linear regression model for the data in Table 6.4 and discuss its effectiveness in making further predictions.

HW Set	1	2	3	4	5	6
Time (hr)	9.8	5.3	3.4	2.6	1.9	1.7

Table 6.4 Homework Set Completion Times

Solution

There is a definitive downward trend in the data. A simple linear regression model for this data is:

$$\hat{y}=-1.47143x+9.26667$$

However, the line of best fit does not seem to fit the data all that well. There is no way that a linear function can bend the way the data points seem to do in this example (see Figure 6.4).

Figure 6.4 Linear Model That Best Fits the Data in Example 6.3. This graph contains a scatterplot of the six data points together with the linear model that best fits the data. This is a clear case of underfitting.

Another issue with using the linear model for the data in Table 6.4 is that eventually the values of $\hat{y}$ become negative! According to this model, it will take about $-1$ hours to complete the 7th homework set, which is absurd. Perhaps a decreasing exponential model, shown in Figure 6.5, would work better here. Proper model selection is the most effective way of dealing with underfitting.

Figure 6.5 Exponential Model That Best Fits the Data in Example 6.3. This graph contains a scatterplot of the six data points together with the exponential model that best fits the data. Under this model, the prediction of $\hat{y}=1$ when $x=7$ seems much more reasonable.