This section will explain the main concepts involved in the construction of the hybrid model, highlighting elements present in each of its layers, techniques and training algorithms used in the model. It was initially proposed for pattern classification [

22], and in this paper, it will be applied in the identification of patients suitable for the treatment of cryotherapy and immunotherapy.

Figure 4 presents a pruning fuzzy neural network model where

A neurons indicate fuzzy neurons generated by the ANFIS (Adaptative Network based Fuzzy Inference Systems [

76]). method, which are linked to

Z neurons (unineuron) and which in turn are unified in a single neuron with a linear activation function. The first two layers of this model represent the fuzzy inference system. The third layer is the representation of a neural network of aggregation that has a single neuron.

#### 3.2. Second Layer

Fuzzy logical neurons form the second layer of the model. Type III neurons receive this assignment because they can construct diffusive if/then-like rules, extracting knowledge from the data. These logical neurons perform the aggregation of neurons of the first layer using fuzzy operators called t-norm and s-norm. These are operators of fuzzy sets so that representative values constructed for the neurons. A neuron that uses the t-norm is called an andneuron since the neuron that uses the s-norm as the operator is called the orneuron. Two neurons are efficient in solving problems, but there is a logical operator that allows using at one time a t-norm and at another time a s-norm. The operator is called uninorm [

77] and is the basis for the logical neuron used in this paper, proposed by [

78,

79] and use in [

80].

In this paper the uninorm is expressed as follows [

79]:

and

where

T is a t-norm,

S is an s-norm and

g is an identity element that varies between 0 and 1. In other words, uninorms can switch smoothly between an s-norm (if

g = 0) and a t-norm (if

g = 1). In this paper, reconsidered the

t-norm operator the product and as

s-norm operator the probabilistic sum.

The unineuron proposed in [

79] performs the following operations to compute its output:

each pair (${a}_{i}$, ${w}_{i}$) is transformed into a single value ${b}_{i}$ = **h**(${a}_{i}$, ${w}_{i}$)

calculate the unified aggregation of the transformed values **U** (${b}_{1},{b}_{2}\dots {b}_{n}$), where n is the number of inputs.

The function

p (relevancy transformation) is responsible for transforming the inputs and corresponding weights into individual transformed values. This function fulfills the requirement of monotonicity in value, which means if the input value increases, the transformed value must also increase. It also meets the requirement zero importance elements should have in effect and the normality of importance of one. Finally, function

p can bring consistency of effect of

${w}_{i}$. Formulation for the

p function can be described as [

79]:

using the weighted aggregation reported above the unineuron can be written as:

These neurons can create fuzzy rules that are the basis of the fuzzy inference system. They extract the knowledge from the database used in the test to compose a vital knowledge group for various problems. This rule-base can serve as business rules for building expert systems to assist practitioners in diverse areas, including clinicians who treat diseases and need information to define appropriate treatments for patients [

22].

#### 3.3. Third Layer

In the third layer of the model, a neural network is responsible for bringing the final responses to the model. A neural network of aggregation with a single neuron (Considered also a singleton). This neuron has an activation function capable of bringing answers according to the database involved in the training of the model. As it is a database for treatments of immunotherapy and cryotherapy, model will provide solutions on the suitability or not of the procedure according to the characteristics of the analyzed person [

22]. The equation responsible for the output of the model:

where

${z}_{0}$ = 1,

${v}_{0}$ is the bias,

${z}_{j}$ and

${v}_{j}$,

j = 1, …,

${l}_{p}$ are the output of each fuzzy neuron of the second layer and its corresponding weight, respectively. The activation function, in this case, is linear, which means the right weight of the second layer weights and the fuzzy inputs.

How fuzzy neural networks perform their training linked especially to techniques that update or generate fundamental parameters for the operation of the net. In the fuzzy neural network proposed by [

22], we use the concept of the extreme learning machine (ELM) proposed by [

21] and actively used in the literature to generate the weights analytically through the pseudo inverse concepts of Moore–Penrose [

81]. Weights and bias of the neurons of the first layer and the second layer generated randomly. Weights that connect the second layer to the neural network of aggregation created using the concepts of partial least squares. This approach is different from methods that use backpropagation [

82] to perform the performance of existing standards. Huang has proved that the generation of final network weights using the pseudo-inverse can generate a processing gain of the intelligent networks [

21]. A single step obtains the weights, and as it is not necessary to update the other parameters, intelligent model gains independence from the recurrent update of the network and generates answers with a high level of accuracy for the analyzed results. With the model defined in (

5), we can write

**y** as

**Z * v**, where

**v** is the vector of weights of the output layer,

**y** is the vector of outputs.

**Z** is determined to be [

21]:

The columns of the matrix

$\mathbf{Z}$, defined in (

6), correspond to the outputs of the hidden neurons of the single layer feed forward network (SLFN) with respect to the input

**a** $={[{a}_{1},{a}_{2},\dots ,{a}_{N}]}_{m\times N}^{T}$.

The ELM implements a random initialization of the weights of the hidden layer (based on a numerical range any),

${w}_{k}$. Then, weights of the output layer are obtained through the pseudo inverse according to the expression [

21]:

The approaches that use the ANFIS [

76] model to generate equally spaced membership functions have a relation of exponential neuron creation linked by the relationship between membership functions and problem dimensions. In fuzzy neural networks, same amount of neurons in the first layer is that of the second layer. To decrease this exponential relationship and facilitate pseudo-inverse calculations, in the models of [

5,

18,

20], resampling regularization techniques are used. Results obtained in [

20] were inferior to which originated the data used in the studies of cryotherapy and immunotherapy, but it should be noted that the results were obtained with a set of fuzzy rules. Work proposed in this paper differs from the work done in [

20] because of the way to select the most significant neurons, mainly because the technique dealt with in this paper does not need to choose the parameters to be used in the pruning of unnecessary information, is based entirely on the nature of the data. In the model of [

22], the concept of pruning is prominent of the F-scores proposed by [

83] initially to prune structures trained with ELM. The F-scores technique has two particularities where its numerator indicates the discrimination between the positive and negative sets, and the denominator is the sum of the deviation within each set of resources. A higher F-score indicates that the support has more discriminative power [

83]. The problem of choosing the most relevant hidden neurons is that a classic resource selection problem may occur. In this way, the F-scores metric is used to evaluate the discriminating power of the fuzzy neurons in the second layer about the classes of the patterns [

83]. All fuzzy rules with an F-score below a predefined eliminator limit are considered irrelevant to solve the problem. Model used defines this threshold from the training data, without requiring validation sets and computationally intensive cross-validation procedures as used in the procedures performed in the [

20] proposal. In one-step, pruning is complete, before adjusting the weights of the network output layer, giving high speed to the algorithm without loss of performance predictive.

The F-score is a simple metric, consisting of designing the input patterns for a high dimension space and then selecting the most relevant characteristics, those that contain maximum discriminatory information about the classes but useful in evaluating the discriminative power of the variables of the feature set [

84].

Given the

i-th feature vector (in the case of the FNN the

**z**-vector representing unineurons) with the number of positive instances

${n}_{+}$, negative instances

${n}_{+}$ and the number of all the instances

N, the F-score value of the

i-th feature is defined by [

83]:

where

${\overline{x}}_{i}^{(+)}$,

${\overline{x}}_{i}^{(-)}$ and

${\overline{x}}_{i}$ are the mean of the positive, negative and whole samples, respectively, and is the

k-th feature value in the

i-faith feature vector. Therefore all the neurons of the second layer

**z** have their calculated f-score and the new group of neurons is composed of all those that have the calculated value more significant than the average of the f-scores.

In this case, the

L neurons obtained from the first layer are evaluated for relevance, and the f-scores technique evaluates 50% of the most relevant neurons through the mean values of f-score, without using parameters in a single step [

83]. These pruned neurons are called in this paper by

${L}_{p}$.

The intelligent models act in the perception of the parameters involved in research through the collection of data from patients who are undergoing chemotherapy and immunotherapy treatment. In the study of [

3], Use of the database helped in the first evaluations on techniques capable of generating smart systems to support the definition of treatments to treat the curls properly. At first, existing models used in literature to aid in the prediction of treatment efficacy. Many models that are available on the Weka software [

85] employed to determine elements with high levels of accuracy to discover the patient profile. In this paper, we intend to compare the results of the pruned fuzzy neural network with other classifiers commonly used in the literature. Algorithm 1 presents the necessary steps for the execution of the classification of patterns on the treatments for warts. There are two parameters:

the number of membership functions, M

the type of fuzzy logic neuron, unineuron

**Algorithm 1:** Fuzzy neural network for detection of immunotherapy and cryotherapy treatments—fuzzy neural network (FNN) training. |

(1) Define the number os membership functions, M. (2) Calculate M neurons for each characteristic in the first layer using ANFIS. (3) Construct L fuzzy neurons with Gaussian membership functions constructed with center and $\sigma $ values derived from ANFIS. (4) Define the weights and bias of the fuzzy neurons randomly. (5) Construct L fuzzy logical neurons with random weights and bias on the second layer of the network by welding the L fuzzy neurons of the first layer. (6) Use f-scores to define the most significant neurons to the problem (${L}_{p}$). (7) For all K input do (7.1) Calculate the mapping ${z}_{k}\left({x}_{k}\right)$ using logical neurons (8) Estimate the weights of the output layer using Equation (7). (9) Calculate output **y** using Equation (5). |