Report Wine Quality White

0.46866667• P(y=medium-high) =0.18666667

Conditional probabilities:

In this case the predictive variables are numerical, so the conditional probabilities can be calculated assuming that the probabilities follow a certain probability distribution. In our case it is assumed that it is Gaussian. Therefore the values reported by the classification correspond to the μ (mean) and the σ (variance) of the Gaussian which represents the distribution of each predictive variable with respect to the target.

In this case:

1. 0.1999496free.solfur.dioxide.scaledY
   • [,1] [,2]
     • low 0.08115404 0.07915199
     • medium-low 0.11983218 0.06302392
     • medium 0.11612254 0.05455907
     • medium-high 0.11191202 0.04639108
2. Applying the Bayes theorem:
   • |P(x y)P(y)|P(y x) = where:
   • P(x)P(y)
     • • a priori probability 82(xj − μ )jhn 1 − 22(σ )∏ || | P(x y = v ) = eP(x y) = P(x y) jhj hj
     • • 2π σj=1 jh
3. Thanks to the confusion matrices it is possible to define some performance indices such as accuracy and precision. The reported values were obtained by iterating the classification process 50 times in order to obtain average performance indices that are as close as possible to the real ones.
   • In this case:
     • • 46.62% the model A average accuracy is
     • • the precision is:
       • - 23.3% for "low" quality wines
       • - 51.34% for "medium-low" quality wines
       • - 53.9% for "medium" quality wines
       • - 37.85% for "medium-high" quality wines
4. From the data shown, it can be seen that the model is not very

accurate in estimating the target.

Analyzing the values of the conditional probabilities obtained from the classification, we can see how the mean and variance values of each variable are similar for adjacent quality levels. As an example, Figure 8 shows the Gaussian distributions of alcohol corresponding to the different quality levels. Looking at the figure in fact is possible to see how the blue and red curves for example are very similar.

Therefore, to increase the accuracy and performance of the process, Model B is introduced in which a discretization of the "quality" target is applied.

Figure 8

94.2.2 Model B

In this second model, the classification process is preceded by a discretization of the target variable "quality". For the process, a discretization by "size" was chosen. The selected classes are:

• "Low" class: includes observations rated with a score of 4 and 5 by the testers
• "Medium" class: includes the observations rated

with a score of 6 and 7 by the testers

Analyzing the distribution of the new dataset (

figure9

and

figure10

) and comparing with the previous distribution in

Figure6

is possible to see that the solution obtained can be reasonably used since it does not present a massive loss of information.

Figure9

Figure10

As already reported in METHOD A, the objective of the classification method is to predict the target given a new observation (P (y | x)).

Also in this case the classification phase follows a phase of subdivision into training set and test set by sampling as in the previous case (hold out method): 3000 observations in the training set and 1698 in the set set.

At the end of the process the confusion matrix shown in

Table 3 is obtained. The numerical values are shown as an example and refer to a single random iteration.

	Actual
	Low	Medium
Predicted	Low	346	248
	Medium	242	862

By applying the classification method, the a priori probabilities P(y) and the

conditional probabilities P (x | y) are also obtained. In particular:

A priori probabilities:

• P (y = low) = 0.34410
• P (y = medium) = 0.656

Conditional probabilities:

As in the previous case (model A), the predictive variables are numerical, so the conditional μ probabilities can be calculated assuming that the probabilities follow a Gaussian. The mean and σ standard deviation values follow:

fixed.acidity.scaled	pH.scaled	Y
low	0.3025073	0.08249329
medium	0.2902742	0.07908142

volatile.acidity.scaled	sulphates.scaled	Y
low	0.