## 1.2 What is *Predictive Modeling*?

Predictive modeling is the process of developing a mathematical tool or model that generates an accurate prediction about a random quantity of interest.

In predictive modeling we are interested in predicting a random variable, typically denoted by \(Y,\) from a set of related variables \(X_1,\ldots,X_p.\) The focus is on learning what is the probabilistic model that relates \(Y\) with \(X_1,\ldots,X_p,\) and use that acquired knowledge for predicting \(Y\) given an observation of \(X_1,\ldots,X_p.\) Some concrete examples of this are:

- Predicting the wine quality (\(Y\)) from a set of environmental variables (\(X_1,\ldots,X_p\)).
- Predicting the number of sales (\(Y\)) from a set of marketing investments (\(X_1,\ldots,X_p\)).
- Modeling the average house value in a given suburb (\(Y\)) from a set of community-related features (\(X_1,\ldots,X_p\)).
- Predicting the probability of failure (\(Y\)) of a rocket launcher from the ambient temperature (\(X_1\)).
- Predicting students academic performance (\(Y\)) according to education resources and learning methodologies (\(X_1,\ldots,X_p\)).
- Predicting the fuel consumption of a car (\(Y\)) from a set of driving variables (\(X_1,\ldots,X_p\)).

The process of predictive modeling can be statistically abstracted in the following way. We believe that \(Y\) and \(X_1,\ldots,X_p\) are related by a *regression model* of the form

\[\begin{align} Y=m(X_1,\ldots,X_p)+\varepsilon,\tag{1.1} \end{align}\]

where \(m\) is the *regression function* and \(\varepsilon\) is a random error with zero mean that accounts for the uncertainty of knowing \(Y\) if \(X_1,\ldots,X_p\) are known. The function \(m:\mathbb{R}^p\rightarrow\mathbb{R}\) is unknown in practice and its estimation is the objective of predictive modeling: **\(m\) encodes the relation**^{2} **between \(Y\) and \(X_1,\ldots,X_p\)**. In other words, \(m\) captures the *trend* of the relation between \(Y\) and \(X_1,\ldots,X_p,\) and \(\varepsilon\) represents the *stochasticity* of that relation. Knowing \(m\) allows to predict \(Y.\) This course is devoted to *statistical models*^{3} that allow us to come up with an estimate of \(m,\) denoted by \(\hat m,\) that can be used to predict \(Y.\)

Let’s see a concrete example of this with an artificial dataset. Suppose \(Y\) represents average fuel consumption (l/100km) of a car and \(X\) is the average speed (km/h). It is well-known from physics that the energy and speed have a quadratic relationship, and therefore we may assume that \(Y\) and \(X\) are *truly* quadratically-related for the sake of exposition:

\[\begin{align*} Y=a+bX+cX^2+\varepsilon. \end{align*}\]

Then \(m:\mathbb{R}\rightarrow\mathbb{R}\) (\(p=1\)) with \(m(x)=a+bx+cx^2.\) Suppose the following data consists of measurements from a given car model, measured in different drivers and conditions (we do not have data for accounting for all those effects, which go to the \(\varepsilon\) term^{4}):

```
x <- c(64, 20, 14, 64, 44, 39, 25, 53, 48, 9, 100, 112, 78, 105, 116, 94, 71,
71, 101, 109)
y <- c(4, 6, 6.4, 4.1, 4.9, 4.4, 6.6, 4.4, 3.8, 7, 7.4, 8.4, 5.2, 7.6, 9.8,
6.4, 5.1, 4.8, 8.2, 8.7)
plot(x, y, xlab = "Speed", ylab = "Fuel consumption")
```

From this data, we can estimate \(m\) by means of a polynomial model^{5}:

```
# Estimates for a, b, and c
lm(y ~ x + I(x^2))
##
## Call:
## lm(formula = y ~ x + I(x^2))
##
## Coefficients:
## (Intercept) x I(x^2)
## 8.512421 -0.153291 0.001408
```

Then the estimate of \(m\) is \(\hat m(x)=\hat{a}+\hat{b} x+\hat{c} x^2=8.512-0.153x+0.001x^2\) and its fit to the data is pretty good. As a consequence, we can use this precise mathematical function to predict the \(Y\) from a particular observation \(X.\) For example, the estimated fuel consumption at speed 90 km/h is `8.512421 - 0.153291 * 90 + 0.001408 * 90^2 = 6.1210`

.

```
plot(x, y, xlab = "Speed", ylab = "Fuel consumption")
curve(8.512421 - 0.153291 * x + 0.001408 * x^2, add = TRUE, col = 2)
```

There are a number of generic issues and decisions to take when building and estimating regression models that are worth to highlight:

The

**prediction accuracy versus interpretability**trade-off.*Prediction accuracy*is key in any predictive model: of course, the better the model is able to predict \(Y,\) the more useful it will be. However, some models achieve this predictive accuracy in exchange of a clear*interpretability*of the model (the so-called*black boxes*). Interpretability is key in order to gain insights on the prediction process, to know exactly which variables are most influential in \(Y,\) to be able to interpret the parameters of the model, and to translate the prediction process to non-experts. In essence, interpretability allows to**explain precisely**how and why the model behaves when predicting \(Y\) from \(X_1,\ldots,X_p.\) Most of models seen in**these notes clearly favor interpretability**^{6}and hence they may make a sacrifice in terms of their prediction accuracy when compared with more convoluted models.**Model correctness versus model usefulness**. Correctness and usefulness are two different concepts in modeling. The first refers to the model being*statistically*correct, this is, it translates to stating that the**assumptions**on which the model relating \(Y\) with \(X_1,\ldots,X_p\) is built are satisfied. The second refers to the model being**useful for explaining or predicting**\(Y\) from \(X_1,\ldots,X_p.\) Both concepts are certainly related (if the model is correct/useful, then*likely*it is useful/correct) but neither is implied by the other. For example, a regression model might be correct but useless if the variance of \(\varepsilon\) is large (too much noise). And yet if the model is not completely correct, it may give useful insights and predictions, but inference may be completely spurious^{7}.**Flexibility versus simplicity**.*The best model*is the one which is very simple (low number of parameters), highly interpretable, and delivers great predictions. This is often unachievable in practice. What can be achieved is*a good model*: the one that balances the simplicity with the prediction accuracy, which is often increased the more flexible the model is. However,**flexibility comes at a price**: more flexible (hence more complex) models use more parameters that need to be estimated from a finite amount of information – the sample. This is problematic, as overly flexible models are more dependent on the sample, up to the point in which they end up not estimating the true relation between \(Y\) and \(X_1,\ldots,X_p,\) \(m,\) but merely*interpolating*the observed data. This well-known phenomenon is called**overfitting**and it can be avoided by splitting the dataset in two datasets^{8}: the*training*dataset, used for estimating the model; and the*testing*dataset, used for evaluating the fitted model predictive performance. On the other hand, excessive simplicity (**underfitting**) is also problematic, since the true relation between \(Y\) and \(X_1,\ldots,X_p\) may be overly simplified. Therefore, a trade-off in the degree of flexibility has to be attained for having a good model. This is often referred to as the**bias-variance trade-off**(low flexibility increases the bias of the fitted model, high flexibility increases the variance). An illustration of this transversal problem in predictive modeling is given in Figure 1.3.

The relation is encoded in

*average*by means of the*conditional expectation*.↩︎That can be regarded as “structures for \(m\)”.↩︎

This is an alternative useful view of \(\varepsilon\): the aggregation of the effects what we cannot account for predicting \(Y.\)↩︎

Note we use the information that \(m\)

*has to be*of a particular form (in this case quadratic) which is an unrealistic situation for other data applications.↩︎Not only that but they are

*neatly*interpretable.↩︎Particularly, it usually happens that the inference based on erroneous assumptions

*underestimates*variability, as the assumptions tend to ensure that the information of the sample is maximal for estimating the model at hand. Thus, inference based on erroneous assumptions results in a false sense of confidence: a larger error is made in reality than the one the model theory states.↩︎Three datasets if we are fitting

*hyperparameters*or*tuning parameters*in our model: the*training*dataset for estimating the model parameters; the*validation*dataset for estimating the hyper-parameters; and the*testing*dataset for evaluating the final performance of the fitted model.↩︎