Business Intelligence

The larger is the second one so our prediction is that vector: the churner.

The values are not complementary because we are maximizing the numerator forgetting P(x)

since it does not depend on y. Moreover, the notation is not saying equal but proportional.

We have a lot of vectors so everyone of them has a very small probability of x. The confidence

is expressed as the ratio between the value maximising the expression and the sum of the two

(~ 3): the confidence level is approximately 2.5/3=5/6 and 1/6 is the probability of error.

Bayesian network

Last remark on this method: is it possible to make a less strong assumption about

independence? Some partial dependence? Yes: which allow to define

Bayesian network,

dependency relationships between (small) subset of attributes.

The idea is a partial dependence: reticular hierarchical links are in fact introduced though

which it is possible to assign selected stochastic dependencies.

The two core elements are:

Acyclic oriented graph

- in which the nodes correspond to predictive variables and the

arc indicates relationships of stochastic dependence.

Conditional probability table

- that combine effects of target variables: the idea is making

small limited combinations of variables.

The main application is in health care: people think they are good in knowing the relationship

between data (e.g. relationship with smoke or family history and cancer). The consequence of

partial relationship: a small subset of variables there is a dependence but in a large group of

variables is not possible; we can combine groups of two or three variables together (joined

conditional distribution is possible for small subset of data).

LOGISTIC REGRESSION

It is another technique which coverts binary classification problems into linear regression ones.

Regression is pretty much like classification but for numerical target variables. Logistic

regression is a way to adapt regression to classification, it is based on parametric assumptions

(all the others were not).

The idea is to calculate posterior probability, but differently from Bayesian method, it is

parametrical (we make an assumption on the shape of probability).

Sigmoid function:

logistic function,

A also called

This kind of expression is a sigmoid (or logistic function), for

example to describe technology development: when we have

early technology the efficiency increases slowly, then the

improvement is bigger because of the technology, finally the

technology becomes mature and further improvement become

very hard. The behaviour is S shaped. 47

The logistic regression postulates that the posterior probability of the response variable

logistic function:

follows a

Given this assumption, it is possible to calculate the odds ratios:

Instead of explaining a binary variable, we explain da logarithm which in continuous variable

so we moved from categorical variable (1,0: binary) to a numerical quantity which is the ratio

above. At this point, we have a standard linear regression problem.

Logistic regression is a good technique with a very small number of variables. When the number

of variables increases it becomes unstable and it is difficult to obtain a solution.

Example: explosion of space shuttle.

Basing on past data was observed that when the temperature was lower, there was a failure in

the chains blocking the gasoline which came out before having consumed it: the risk of burning

is higher.

From the data: with lower temperature, Y was more likely. But there was pressure to start the

mission and NASA wanted more financial support. They decided to start the mission and there

was an explosion which led to a trial. One of the element considered was the regression model:

the single explanatory variable is the temperature. The odds ratios have been calculated and

the parameters have been esteemed.

The temperature at the time of explosion was very low: what was the probability of having a

failure? Very close to 100%. With such a high probability of the failure, the damage happened

at the very beginning and the gasoline started to come out from the very beginning.

In this case we have one single variable, so the model is applicable and can lead to successful

results. 48

NEURAL NETWORKS

It is a technique introduced very early (1956) which has been used for predictive purposes.

Basically, a neural network is an oriented graph in which nodes represent neurons and are

connected by arc, which represent synapses. Every arc has associated a weight and each node

an activation function.

Through the perceptron, represented in the figure below, we can simulate the electric

behaviour of a single neuron simulating the level of energy and the output; so we are far from

the real human brain. The analogy is based on the fact that there are n values entering

input layer. weights

(independent variables: input node for each variable): Then there are

vectors provided by coefficient associated to each input connection (linear combination even if

weighted

our neurons do not perform linear combinations of the inputs): then we calculate

sum. distortion activation function

We have a constant