=Paper=
{{Paper
|id=Vol-2853/paper17
|storemode=property
|title=Method for Automatic Processing of Audit Content Based on Bidirectional Neural Network Mapping
|pdfUrl=https://ceur-ws.org/Vol-2853/paper17.pdf
|volume=Vol-2853
|authors=Tatiana Neskorodieva,Eugene Fedorov
|dblpUrl=https://dblp.org/rec/conf/intelitsis/NeskorodievaF21
}}
==Method for Automatic Processing of Audit Content Based on Bidirectional Neural Network Mapping==
https://ceur-ws.org/Vol-2853/paper17.pdf
Method for Automatic Processing of Audit Content Based on
Bidirectional Neural Network Mapping
Tatiana Neskorodievaa, Eugene Fedorova,b
a
Vasyl' Stus Donetsk National University, 600-richchia str., 21, Vinnytsia, 21021, Ukraine
b
Cherkasy State Technological University, Shevchenko blvd., 460, Cherkasy, 18006, Ukraine
Abstract
Currently, the analytical procedures used during the audit are based on data mining
techniques.
The object of the research is the process of the content auditing of the receipt of raw
materials for production and the manufactured products.
The aim of the work is to increase the effectiveness and efficiency of audit due to mapping
by full (bidirectional) counterpropagating neural network of content of the receipt of raw
materials for production and the manufactured products while automating procedures for
checking their compliance.
The vectors of feature for the objects of the sequences of the receipt of raw materials for
production and the manufactured products are generated, which are then used in the proposed
method. The created method, in contrast to the traditional one, provides for a batch mode,
which allows the method to increase the learning rate by an amount equal to the product of
the number of neurons in the hidden layer and the power of the training set, which is
critically important in the audit system for the implementation of multivariate intelligent
analysis, which involves enumerating various methods of forming subsets analysis.
The urgent task of increasing the audit efficiency was solved by automating the mapping of
audit indicators by full (bidirectional) counterpropagating neural network. A learning
algorithm based on ππ-means has been created, intended for implementation on a GPU using
CUDA technology, which increases the speed of identifying parameters of a neural network
model.
The neural network with the proposed training method based on the ππ-means rule can be used
to intellectualize the DSS audit. The prospects for further research are the application of the
proposed method by neural network mapping for a wide class of artificial intelligence tasks,
in particular, for creating a method for bidirectional mapping indicators of audit tasks.
Keywords 1
audit, mapping by neural network, full (bidirectional) counterpropagating neural network,
content of the receipt of raw materials for production and the manufactured products.
1. Introduction
In the process of development of international and national economies and industry of IT in
particular, it is possible to distinguish the following basic tendencies: realization of digital
transformations, forming of digital economy, globalization of socio-economic processes and of IT
accompanying them [1]. These processes result in the origin of global, multilevel hierarchical
structures of heterogeneous, multivariable, multifunction connections, interactions and cooperation of
managing subjects (objects of audit), the large volumes of information about them have been
accumulated in the informative systems of account, management and audit.
IntelITSISβ2021: 2nd International Workshop on Intelligent Information Technologies and Systems of Information Security, March 24β26,
2021, Khmelnytskyi, Ukraine
EMAIL: t.neskorodieva@donnu.edu.ua (T. Neskorodieva); fedorovee75@ukr.net (E. Fedorov)
ORCID: 0000-0003-2474-7697 (T. Neskorodieva); 0000-0003-3841-7373 (E. Fedorov)
Β© 2021 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
Consequently, nowadays the scientific and technical issue of the modern information technologies
in financial and economic sphere of Ukraine is forming of the methodology of planning and creation
of the decision support systems (DSS) at the audit of enterprises in the conditions of application of IT
and with the use of information technologies on the basis of the automated analysis of the large
volumes of data about financial and economic activity and states of enterprises with the multi-level
hierarchical structure of heterogeneous, multivariable, multifunction connections,
intercommunications and cooperation of objects of audit with the purpose of expansion of functional
possibilities, increase of efficiency and universality of IT-audit.
Currently, the analytical procedures used during the audit are based on data mining techniques [2-
4]. Automated DSS audit means the automatic forming of recommendable decisions, based on the
results of the automated analysis of data, that improves quality process of audit [5,6]. Unlike the
traditional approach, computer technologies of analysis of data in the system of audit accelerate and
promote the process accuracy of audit, that extremely critical in the conditions of plenty of associate
tasks on lower and middle levels, and also amounts of indexes and supervisions in every task [7,8].
When developing a decision-making system in audit based on data mining technologies, three
methods have been created: classifying variables, forming analysis sets, mapping analysis sets.
The peculiarity of the methodology for classifying indicators is that qualitatively different (by
semantic content) variables are classified: numerological, linguistic, quantitative, logical. The essence
of the second technique is determined by the qualitative meaning of the indicators. In accordance with
this, sets are formed with the corresponding semantic content: document numbers, the name of
indicators, quantitative estimates of the values of indicators, logical indicators.
The third technique is subordinated to the mappings of formed sets of the same type on each other
in order to determine equivalence in the following senses: numerological, linguistic, quantitative,
logical.
The most urgent problem is the mapping of quantitative indicators. The mapping of quantitative
indicators of the audit can be implemented through an ANN with associative memory. The main
ANNs with associative memory are presented in Table 1. The memory capacity was considered only
for ANNs with a binary or bipolar data type that perform reconstruction or classification. HAM stands
for hetero-associative memory, AAM stands for auto-associative memory.
As follows from Table 1, most neural networks have one or more disadvantages:
1. not used to reconstruct either the original or another sample;
2. do not work with real data.
3. do not have a high capacity of associative memory.
The aim of the work is to increase the efficiency of automatic data analysis in the audit DSS by
means of a bidirectional (forward and reverse) neural network mapping of sets of audit indicators in
order to identify systematic misstatements that lead to misstatement of reporting. In the audit system,
the topical task of the middle level is the automation of the analysis of the conformity of the content
of the supply of raw materials for production and the manufactured products (by the periods of
quantization of the verification period).
It is assumed that the audit indicators are noisy with Gaussian noise, which in turn simulates
random accounting errors (as opposed to systematic ones). It is also assumed that the residuals for the
quantization periods are distributed according to the normal law, the parameters of which can be
estimated from the accounting data. For the achievement of the aim it is necessary to solve the
following tasks:
β’ formalize the content of the audit process of the receipt of raw materials for production and
the manufactured products;
β’ choose a neural network model for mapping audit indicators (which are noisy with Gaussian
noise, which in turn simulates random accounting errors (as opposed to systematic ones, which
lead to distortion of reporting));
β’ choose a criterion for evaluating the effectiveness of a neural network model;
β’ propose a method for training a neural network model in batch mode;
β’ propose an algorithm for training a neural network model in batch mode for implementation
on a GPU;
β’ perform numerical studies.
Table 1
Basic ANNs with associative memory
Memory Memory Data
ANN capacity Purpose
type type
Forward-only HAM - Real Reconstruction of other
Counterpropagation Neural sample
Network [9, 10]
Full (bi-directional) AΠΠ, - Real Reconstruction of the
Counterpropagation Neural HΠΠ original or other sample
Network [11]
Sigmoid believe network [12, AΠΠ Medium Binary Reconstruction of the
13] original sample
Helmholtz machine [14] AΠΠ Medium Binary Reconstruction of the
original sample
Self-Organizing map [15, 16] AΠΠ - Real Clustering
Learning Vector Quantization AΠΠ - Real Clustering
NN [17]
Principal Component Analysis HAM - Real Dimension reduction
NN [18]
Independent Component HAM - Real Dimension reduction
Analysis NN [19, 20]
Cerebellar Model Articulation HAM - Real Coding
Controller [21]
Recurrent correlation AΠΠ High Bipolar Reconstruction of the
associative memory [22] original sample
Hopfield Neural Network [23] AΠΠ Low Bipolar Reconstruction of the
original sample
Gauss machine [24] AΠΠ Low Bipolar Reconstruction of the
original sample
Bidirectional associative AΠΠ, Low Bipolar Reconstruction,
memory [25] HAM classification
Brain State Model [26] AΠΠ - Real Clustering
Hamming neural network [27] AΠΠ High Bipolar Reconstruction of the
original sample
Boltzmann machine [28,29,30] AΠΠ, Medium Binary Reconstruction of the
HAM original or other sample
ART-2 [31] AΠΠ - Real Clustering
2. MATERIALS AND METHODS
2.1. Formalize the content of the audit process of the receipt of raw
materials for production and the manufactured products
Formalize the content of the supply of raw materials for production and the manufactured products
are formed on the basis of audit variables (Table 2). Elements of mapping sets β data of the receipt of
raw materials for production and the manufactured products by the periods of quantization of the
verification period. The vector of the receipt of raw materials features ππππ = (π₯π₯ππ1 , . . . , π₯π₯ππππ ) formed by
indicators of quantity of raw materials by type π’π’, π’π’ β ππ. The vector of manufactured products
features ππππ = (ππππ1 , . . . , ππππππ ) is formed by indicators of the quantity of manufactured products by type
ππ, ππ β ππ.
Table 2
Feature vectors for mapping raw material received - product produced
Input vector elements ππ Output vector elements D
designation sense designation sense
ππ type of operation (receipt ππ type of operation (production of
of raw materials for finished products (semi-finished
production) products))
π’π’ type of raw materials ππ type of product
ππ set of types of raw ππ set of types of product
materials
πππ’π’ quantity of raw materials ππππ quantity of product
(ππ)
π₯π₯π’π’ cost of raw material (ππ)
π₯π₯π’π’,ππ direct material costs for a product
type ππ of a raw material type π’π’
π₯π₯(ππ) total cost of raw material π₯π₯ππ
(ππ) direct material costs for a product
type ππ
To assess the dimension of the features vector, an analysis was made of the nomenclature of
purchases of raw materials (components) of large machine-building enterprises. So, based on this
analysis, we can conclude that the sections of the nomenclature are on average from 8 to 12, the
number of groups in each section is from 2 to 10.
We represent the implementation of the "generalized audit" in the form of a mapping (comparison)
of generalized quantitative features of the audited sets. The formation of generalized quantitative
features can be performed using ANN.
2.2. Choosing a neural network model for mapping audit sets
In the work, the Full (bidirectional) Counterpropagating Neural Network (FCPNN), which is a
non-recurrent static two-layer ANN, was chosen as a neural network. FCPNN output is linear.
FCPNN advantages:
1. unlike most ANNs are used to reconstruct another sample using auto-associative and hetero-
associative memory.
2. unlike bidirectional associative memory and the Boltzmann machine, it works with real data.
3. unlike bidirectional associative memory and the Boltzmann machine, it has less
computational complexity.
FCPNN model performing mapping of each input sample ππ = (π₯π₯1 , . . . , π₯π₯πππ₯π₯ ) to output sample ππ
(2) (2)
=(π€π€ππ β 1 , . . . , π€π€ππ β πππ¦π¦ ), is represented as
π₯π₯ (1)
ππ β = ππππππ ππππππ π§π§ππ , π§π§ππ = οΏ½βππ 2
ππ=1(π₯π₯ππ β π€π€ππππ ) , ππ β 1, ππ
(1) , (1)
ππ
(1)
where π€π€ππππ β connection weight from the ππ-th element of the input sample to the ππ-th neuron,
(2)
π€π€ππ β ππ β connection weight from the neuron-winner ππ β to ππ-th element of output sample,
(1)
ππ β the number of neurons in the hidden layer.
FCPNN model performing mapping of each output sample π
π
= (ππ1 , . . . , ππππππ ) to input sample
(2) (2) (2)
ππππ β = (π£π£ππ β 1 , . . . , π£π£ππβ πππ₯π₯ ), is represented as
ππ (1)
ππ β = ππππππ ππππππ π§π§ππ , π§π§ππ = οΏ½βππ 2
π π =1(πππ π β π£π£π π π π ) , ππ β 1, ππ
(1) ,
ππ
(1)
where π£π£π π π π β connection weight from the π π -th element of the input sample to the ππ-th neuron of
hidden layer,
(2)
π£π£ππ β ππ β connection weight from the neuron-winner ππ β of hidden layer to ππ-th element of output
sample,
ππ (1) β the number of neurons in the hidden layer.
2.3. Criterion choice for assessing the effectiveness of a neural network
model for mapping audit sets
In this work for training model FCPNN was chosen target function, that indicates selection of the
(1) (1) (2) (2)
vector of parameter values ππ = (π€π€11 , . . . , π€π€πππ₯π₯ ππ(1) , π€π€11 , . . . , π€π€ππ(1)πππ¦π¦ ), ππ =
(1) (1) (2) (2)
(π£π£11 , . . . , π£π£ππππ ππ(1) , π£π£11 , . . . , π£π£ππ(1)πππ₯π₯ ) which deliver the minimum mean square error (difference
between the model sample and the test sample)
1 1 (2) 2 1 (2) 2
πΉπΉ = οΏ½ππππππ βππππ=1 οΏ½ππππππβ β π
π
ππ οΏ½ + πππππ₯π₯ βππππ=1 οΏ½ππππππ β β ππππ οΏ½ οΏ½ β ππππππ (2)
2 ππ,ππ
(2)
where ππππππ β β ππ-th, output sample according to the model,
π
π
ππ β ππ-th test output sample,
(2)
ππππππ β β ππ-th, input sample according to the model,
ππππ β ππ-th test input sample,
ππ β training set power,
ππ ππ β is length of the sample π
π
,
ππ π₯π₯ β is length of the sample ππ.
2.4. Training method for neural network model in batch mode
The disadvantage of FCPNN is that it does not have a batch learning mode, which leads to
reducing of the learning speed. For FCPNN was used concurrent training (combination of training
with and without a teacher). This work proposes training FCPNN in batch mode.
First phase (training of the hidden layer) (steps 1-6).
(1)
The first phase allows you to calculate the weights of the hidden layer π€π€ππππ and consists of the
following blocks (Fig. 1).
1. Learning iteration number ππ = 0, initialization by uniform distribution on the interval (0,1) or
(1) (1)
[-0.5, 0.5] of weights π€π€ππππ (ππ), π£π£π π π π (ππ), ππ β 1, ππ π₯π₯ , ππ β 1, ππ (1), π π β 1, ππ ππ where ππ π₯π₯ β is length
of the sample π₯π₯ , ππ ππ β is length of the sample π
π
and ππ (1) β the number and the neurons in the
hidden layer.
π₯π₯ ππ
Training set is {(ππππ , π
π
ππ )|ππππ β π
π
ππ , π
π
ππ β π
π
ππ }, ππ β 1, ππ, where ππππ β ππ -th training input vector,
π
π
ππ β ππ -th training output vector, ππ β training set power.
Initial shortest distance π§π§Μ (0) =0.
2. Calculating the distance to all hidden neurons.
Distance π§π§ππππ from Β΅-th input sample to each i-th neuron and from each Β΅-th output sample to each
i- th neuron of the hidden basis is determined by the formula:
ππ π₯π₯ (1) ππ ππ (1)
π§π§ππππ = οΏ½βππ=1(π₯π₯ππππ β π€π€ππππ (ππ))2 + οΏ½βπ π =1(ππππππ β π£π£π π π π (ππ))2 , ππ β 1, ππ, ππ β 1, ππ (1) , (3)
(1)
where π€π€ππππ (ππ) β connection weight from k-th input sample to i-th neuron at time ππ,
(1)
π£π£π π π π (ππ) β pretrained connection weight from s-th element of output sample to i-th neuron of
hidden layer at time ππ.
1. Initializing the weights of the neurons of the hidden layer
2. Calculating the distance to all hidden neurons
3. Calculating the shortest distance and choosing the neuron
with the shortest distance
4. Setting the weights of the hidden layer neurons associated
with the neuron-winner and its neighbors
5. Calculating the average sum of the least distances
yes
6. z (n + 1) β z (n) > Ξ΅
no
7
Figure 1. The sequence of steps in training method of FCPNN in batch mode (the first phase)
3. Calculating the shortest distance and choosing the neuron with the shortest distance
Calculating the shortest distance
π§π§ππ = πππππππ§π§ππππ , ππ β 1, ππ, ππ β 1, ππ (1) (4)
ππ
and choosing the neuron-winner ππππβ , for which the distance π§π§ππππ is shortest
ππππβ = ππππππ ππππππ π§π§ππππ , ππ β 1, ππ, ππ β 1, ππ (1) . (5)
ππ
4. Setting the weights of the hidden layer neurons associated with the neuron-winner ππππβ and its
neighbors based on k-means rule
(1) βππ β
ππ=1 β(ππ,ππππ )π₯π₯ππππ
π€π€ππππ (ππ + 1) = βππ β , ππ β 1, ππ π₯π₯ , ππ β 1, ππ (1) , (6)
ππ=1 β(ππ,ππππ )
(1) βππ β
ππ=1 β(ππ,ππππ )ππππππ
π£π£π π π π (ππ + 1) = βππ β , π π β 1, ππ ππ , ππ β 1, ππ (1) , (6β)
ππ=1 β(ππ,ππππ )
where β(ππ, ππ β ) β rectangular topological neighborhood function,
1, ππ = ππ β
β(ππ, ππ β ) = οΏ½ .
0, ππ β ππ β
5. Calculating the average sum of the shortest distances
1
π§π§Μ (ππ + 1) = βππππ=1 π§π§ππ . (7)
ππ
6. Checking the termination condition
If |π§π§Μ (ππ + 1) β π§π§Μ (ππ)| β€ ππ, the finish, else ππ = ππ + 1, go to step 2.
Second phase (training the output layer) (steps 7-12). The second phase allows you to calculate the
(2) (2)
weights of the output layer π€π€ππππ and π£π£ππππ and consists of the following blocks (Fig. 2).
7. Learning iteration number ππ = 0, initialization by uniform distribution on the interval (0,1) or [-
(2) (2)
0.5, 0.5] of weights π€π€ππππ (ππ), π£π£ππππ (ππ), ππ β 1, ππ (1) , ππ β 1, ππ ππ , ππ β 1, ππ π₯π₯ , where ππ π₯π₯ β length of the
input sample ππ, ππ ππ β length of the output sample π
π
, ππ (1) β the number and the neurons in the hidden
layer.
π₯π₯ π¦π¦
Training set is {(ππππ , π
π
ππ )|ππππ β π
π
ππ , π
π
ππ β π
π
ππ }, ππ β 1, ππ, where ππππ β ππ-th training input vector, π
π
ππ
β ππ -th training output vector, ππ β training set power.
Initial shortest distance π§π§Μ (0) =0.
6
7. Initialization of the neuron weights of the output layer
8. Calculating the distance to all hidden neurons
9. Calculating the shortest distance and choosing the neuron
with the shortest distance
10. Calculating the distance to all output neurons
11. Setting the weights of the hidden layer neurons associated
with the neuron-winner and its neighbors
12. Calculating the average sum of the least distances z ( n + 1)
yes
13. z ( n + 1) β z ( n) > Ξ΅
no
14. Output the weights
Figure 2. Sequence of procedures for the FCPNN training method in batch mode (second phase)
8. Calculating the distance to all hidden neurons
Sum of distances π§π§ππππ from Β΅-th input sample in input layer from each i-th neuron of the hidden
layer and from each Β΅-th output sample in the input layer to each i-th neuron of the hidden layer is
determined by the formula:
π₯π₯ (1) ππ (1)
π§π§ππππ = οΏ½βππ 2 ππ
ππ=1(π₯π₯ππππ β π€π€ππππ ) + οΏ½βπ π =1(ππππππ β π£π£π π π π ) , ππ β 1, ππ, ππ β 1, ππ
2 (1) , (8)
(1)
where π€π€ππππ β pretrained connection weight from k-th element of input sample to i-th neuron at
time ππ,
(1)
π£π£π π π π β pretrained connection weight from s- th element of input sample to i- th neuron of hidden
layer at time ππ.
9. Calculating the shortest distance and choosing the neuron with the shortest distance.
Calculating the shortest distance
π§π§ππ = πππππππ§π§ππππ , ππ β 1, ππ, ππ β 1, ππ (1) (9)
ππ
and choosing the neuron-winner ππππβ , for which the distance π§π§ππππ is shortest.
ππππβ = ππππππ ππππππ π§π§ππππ , ππ β 1, ππ, ππ β 1, ππ (1) . (10)
ππ
10. Calculating the distance to all output neurons.
Distance π§π§ππ from the neuron-winner ππππβ to Β΅-th input and output sample in output layer is
determined by the formula:
ππ (2) π₯π₯ (2)
π§π§ππ = οΏ½βππ 2 ππ
ππ=1(ππππππ β π€π€ππ β ππ (ππ)) + οΏ½βππ=1(π₯π₯ππππ β π£π£ππ β ππ (ππ)) , ππ β 1, ππ,
2 (11)
(2)
where π€π€ππππβ ππ (ππ) β weight of connection from the winner neuron ππππβ of hidden layer to j-th element of
the input sample in output layer at time ππ,
11. Setting the weights of the output layer neurons associated with the neuron-winner ππππβ and its
neighbors based on k-means rule
(2) βππ β
ππ=1 β(ππ,ππππ )ππππππ
π€π€ππππ (ππ + 1) = βππ β , ππ β 1, ππ (1) , ππ β 1, ππ ππ , (12)
ππ=1 β(ππ,ππππ )
(2) βππ β
ππ=1 β(ππ,ππππ )π₯π₯ππππ
π£π£ππππ (ππ + 1) = βππ β , ππ β 1, ππ (1) , π π β 1, ππ π₯π₯ , (12β)
ππ=1 β(ππ,ππππ )
where β(ππ, ππ β ) β rectangular topological neighborhood function,
1, ππ = ππ β
β(ππ, ππ β ) = οΏ½ .
0, ππ β ππ β
12. Calculating the average sum of the shortest distances
1
π§π§Μ (ππ + 1) = βππππ=1 π§π§ππ . (13)
ππ
13. Checking the termination condition
If |π§π§Μ (ππ + 1) β π§π§Μ (ππ)| β€ ππ, the finish, else ππ = ππ + 1, go to step 8.
2.5. Algorithm for training neuron network model in batch mode for
implementation on GPU
For the proposed method of training FCPNN on audit data example, examines the algorithm for
implementation on a GPU with usage of CUDA parallel processing technology.
The first phase (training the hidden layer). The first phase based on formulas (1)-(7) is shown in
Fig. 3. This block diagram functions as follows.
Step 1 β Operator enters lengths ππ π₯π₯ of the sample ππ, the lengths ππ ππ of the sample π
π
, the
number and the neurons in the hidden layer ππ (1) , power of the training set ππ, training set
π₯π₯ ππ
{(ππππ , π
π
ππ )|ππππ β π
π
ππ , π
π
ππ β π
π
ππ }, ππ β 1, ππ.
Step 2 β Initialization by uniform distribution over the interval (0,1) or [-0.5, 0.5] of weights
(1) (1)
π€π€ππππ (ππ), π£π£π π π π (ππ), ππ β 1, ππ π₯π₯ , π π β 1, ππ ππ , ππ β 1, ππ (1) .
Step 3 β Calculation of distances to all hidden neurons of the ANN, using ππ β
ππ (1) threads on
GPU, which are grouped into P blocks. Each thread calculates the distance from Β΅-th input sample to
each i-th neuron π§π§ππππ .
Step 4 β Computation based on shortest distance reduction and determining the neurons with the
shortest distance using ππ β
ππ (1) threads on GPU, which are grouped into P blocks. The result of the
work of each block is a neuron-winner ππππβ with the smallest distance π§π§ππ .
Step 5 β Setting the weights of the output layer neurons associated with the neuron- winner ππππβ and
its neighbors based on reduction using ππ π₯π₯ β
ππ (1) β
ππ threads on GPU, which are grouped into ππ π₯π₯ β
(1)
ππ (1) blocks. The result of the work of each block is the weight π€π€ππππ (ππ + 1).
Step 6 β Setting the weights of the output layer neurons associated with the neuron- winner ππππβ and
its neighbors based on reduction using ππ ππ β
ππ (1) β
ππ threads on GPU, which are grouped into ππ ππ β
(1)
ππ (1) blocks. The result of the work of each block is the weight π£π£π π π π (ππ + 1).
Step 7 β Calculation based on reduction of the average sum of the shortest distances using ππ
threads on GPU, which are grouped into 1 block. The result of the block is the average sum of the
smallest distances π§π§Μ (ππ + 1).
Step 8 β If average sum of smallest distances of neighboring iterations are close, |π§π§Μ (ππ + 1) β
π§π§Μ (ππ)| β€ ππ, then finish, else β increasing number of iteration ππ = ππ + 1, go to step 3.
1
2
3
4
5
6
7
8
+ 9
Figure 3. Block diagram of the FCPNN learning algorithm in batch mode (first phase)
(1) (1)
Step 9 β Recording of weight π€π€ππππ (ππ + 1) and π£π£π π π π (ππ + 1) in the database.
Second phase (training of output layer). The second phase based on formulas (8)-(13) is showed in
Figure. 4. This flowchart operates as follows.
Step 1 β Operator enters lengths ππ π₯π₯ of the sample ππ, the lengths ππ ππ of the sample π
π
the number
and the neurons in the hidden layer ππ (1) , power of the training set ππ, training set
π₯π₯ ππ
οΏ½οΏ½ππππ , π
π
ππ οΏ½οΏ½ππππ β π
π
ππ , π
π
ππ β π
π
ππ οΏ½ , ππ β 1, ππ.
Step 2 β Initialization by uniform distribution over the interval (0,1) or [-0.5, 0.5] of weights
(2) (2)
π€π€ππππ (ππ), π£π£ππππ (ππ), ππ β 1, ππ (1) , ππ β 1, ππ ππ , ππ β 1, ππ π₯π₯ .
Step 3 β Calculation of distances to all hidden neurons of the ANN, using ππ β
ππ (1) threads on
GPU, which are grouped into P blocks. Each thread calculates the distance from Β΅-th input sample to
each ππ-th neuron π§π§ππππ .
Step 4 β Computation based on shortest distance reduction and determining the neurons with the
shortest distance using ππ β
ππ (1) threads on GPU, which are grouped into ππ blocks. The result of the
work of each block is a neuron-winner ππππβ with the smallest distance π§π§ππ .
Step 5 β Calculating distances from the neuron-winner ππππβ to Β΅-th output sample using ππ β
ππ (ππ)
threads on GPU, which are grouped into ππ blocks. Each thread calculates the distance from the
neuron-winner ππππβ to Β΅-th output sample π§π§ππ .
Step 6 β Setting the weights of the output layer neurons associated with the neuron-winner ππππβ and
its neighbors based on reduction using ππ (1) β
ππ ππ β
ππ threads on GPU, which are grouped into ππ (1) β
(2)
ππ ππ blocks. The result of the work of each block is the weight π€π€ππππ (ππ + 1).
Step 7 β Setting the weights of the output layer neurons associated with the neuron-winner ππππβ and
its neighbors based on reduction using ππ (1) β
ππ π₯π₯ β
ππ threads on GPU, which are grouped into ππ (1) β
(2)
ππ π₯π₯ block. The result of the block is the average sum of the smallest distances π£π£ππππ (ππ + 1).
Step 8 β Calculation based on reduction of the average sum of the shortest distances using ππ
threads on GPU, which are grouped into 1 block. The result of the block is the average sum of the
smallest distances π§π§Μ (ππ + 1).
Step 9 β If average sum of smallest distances of neighboring iterations are close, |π§π§Μ (ππ + 1) β
π§π§Μ (ππ)| β€ ππ, then finish, else β increasing number of iteration ππ = ππ + 1, go to step 3.
(2) (2)
Step 10 β Recording of weight π€π€ππππ (ππ + 1) and π£π£ππππ (ππ + 1), in the database.
1
2
3
4
5
6
7
8
9
+ 10
Figure 4. Block diagram of the FCPNN training algorithm in batch mode (second phase)
2.6. Numerical research
The results of the comparison of the proposed method using GPU and the traditional FCPNN
training method are presented in Table 3.
Table 3
Comparison of the computational complexity of the proposed and traditional training methods of
FCPNN
Method
Feature
proposed traditional
Computational
ππ(ππ1ππππππ + ππ2ππππππ ) ππ(ππππ(1) ππ1ππππππ + (ππππ(1) + ππ)ππ2ππππππ )
complexity
Evaluation of computational complexity of the proposed method using the GPU, and the
traditional method of teaching FOCPNN were based on the number of calculation distances,
computing of which is the most consuming part of method. Moreover, ππ1ππππππ β the maximum number
of iterations of the first training phase, ππ2ππππππ β the maximum number of iterations of the second
training phase, ππ (1) β the number of neurons in the hidden layer, ππ β the power of the training set.
2.7. Discussion
The traditional FCPNN learning method does not provide support for batch mode, which increases
computational complexity (Table 3). Proposed method eliminates this flaw and allows for
approximate increase of learning rate in ππππ (1) . By reducing the computational complexity, it is
possible to increase the accuracy of the method by decreasing the parameter Ξ΅ and increasing the
number of neurons in the hidden and output layers.
2.8. Conclusion
1. The urgent task of increasing the effectiveness of audit in the context of large volumes of
analyzed data and limited verification time was solved by automating the formation of generalized
features of audit sets and their mapping by means of a full bidirectional counterpropagating neural
network.
2. For increased learning rate of full bidirectional counterpropagating neural network, was
developed a method based on the ππ-means rule for training in batch mode. The proposed method
provides: approximately increase learning rate in ππππ (1), where ππ (1) is the number of neurons in the
hidden layer and ππ is the power of the learning set.
3. Created a learning algorithm based on ππ-means, intended for implementation on a GPU using
CUDA technology.
4. The proposed method of training based on the ππ-means rule can be used to intellectualize the
DSS audit.
Prospects for further research is the study of the proposed method for a wide class of artificial
intelligence tasks, as well as the creation of a method for bidirectional mapping audit features to solve
audit problems.
3. References
[1] The World Bank: World Development Report 2016: Digital Dividends, 2016. URL:
https://www.worldbank.org/en/publication/wdr2016.
[2] M. Schultz, and M. Tropmann-Frick. Autoencoder Neural Networks versus External Auditors:
Detecting Unusual Journal Entries in Financial Statement Audits. Proceedings of the 53rd
Hawaii International Conference on System Sciences (HICSS-2020), Maui, Hawaii, USA. 2021,
pp. 5421-5430. doi:10.24251/hicss.2020.666.
[3] J. Nonnenmacher, F. Kruse, G. Schumann, G. Marx. Using Autoencoders for Data-Driven
Analysis in Internal Auditing. In Proceedings of the 54th Hawaii International Conference on
System Sciences, Maui, Hawaii, USA, 2021, pp. 5748-5757. doi: 10.24251/hicss.2021.697.
[4] Y. Bodyanskiy, O. Boiko, Y. Zaychenko, G. Hamidov and A. Zelikman. The Hybrid GMDH-
Neo-fuzzy Neural Network in Forecasting Problems in Financial Sphere. Proceedings of 2nd
International Conference on System Analysis & Intelligent Computing (SAIC), Kyiv, Ukraine,
IEEE, 2020, pp. 1-6, doi: 10.1109/SAIC51296.2020.9239152.
[5] T. NeskorodΡeva, E. Fedorov, I. Izonin. Forecast Method for Audit Data Analysis by Modified
Liquid State Machine. Proceedings of the 1st International Workshop on Intelligent Information
Technologies & Systems of Information Security (IntelITSIS 2020), Khmelnytskyi, Ukraine, 10-
12 June, 2020: proceedings. β CEUR-WS vol. 2623, 2020, pp. 25-35.
[6] T. NeskorodΡeva, E. Fedorov. Method for Automatic Analysis of Compliance of Expenses Data
and the Enterprise Income by Neural Network Model of Forecast. Proceedings of the 2nd
International Workshop on Modern Machine Learning Technologies and Data Science
(MoMLeT&DS-2020), Lviv-Shatsk, Ukraine, 2-3 June, 2020: proceedings. β CEUR-WS,
Volume I: Main Conference. vol. 2631, 2020, pp. 145-158.
[7] A.V. Barmak, Y.V. Krak, E.A. Manziuk, V.S. Kasianiuk. Information technology separating
hyperplanes synthesis for linear classifiers. Journal of Automation and Information Sciences, vol.
51(5) (2019) 54-64. doi: 10.1615/JAutomatInfScien.v51.i5.50
[8] T.V. Prokopenko, O. Grigor. Development of the comprehensive method to manage risks in
projects related to information technologies. Eastern-European Journal of Enterprise
Technologies vol. 2, 2018, pp. 37-43. doi:10.15587/1729-4061.2018.128140.
[9] U.P. Singh, S. Jain, A. Tiwari, R.K. Singh. Gradient evolution-based counter propagation
network for approximation of noncanonical system. Soft Computing, 23 13, (2019) 4955β4967.
doi:10.1007/s00500-018-3160-7.
[10] A. Asokan, K. Gunavathi, R. Anitha. Classification of Melakartha ragas using neural networks.
Proceedings of the International Conference on Innovations in Information, Embedded and
Communication Systems (ICIIECS-2017), 17-18 March 2017, Coimbatore, India. IEEE. 2017,
pp.1-6. doi:10.1109/ICIIECS.2017.8276040.
[11] Sonika; A. Pratap; M. Chauhan; A. Dixit. New technique for detecting fraudulent transactions
using hybrid network consisting of full-counter propagation network and probabilistic network.
2016 International Conference on Computing, Communication and Automation (ICCCA), 29-30
April 2016, Greater Noida, India, IEEE. 2016, pp. 29-30. doi:10.1109/CCAA.2016.7813713.
[12] P.M. Baggenstoss. Applications of Projected Belief Networks (PBN). 27th European Signal
Processing Conference (EUSIPCO). 2-6 Sept. 2019, A Coruna, Spain, IEEE. 2019 pp. 1-5
doi:10.23919/EUSIPCO.2019.8902708.
[13] P. M. Baggenstoss, On the Duality Between Belief Networks and Feed-Forward Neural
Networks, in IEEE Transactions on Neural Networks and Learning Systems, vol. 30 (1). 2019,
pp. 190-200, doi: 10.1109/TNNLS.2018.2836662.
[14] P. Sountsov, P. Miller. Spiking neuron network Helmholtz machine. Frontiers in Computational
Neuroscience. Frontiers Media SA. vol. 9, 2015. doi:10.3389/fncom.2015.00046.
[15] T Kohonen Self-organizing and associative memory, 3rd edn. Springer, New York, (2012).
[16] T Kohonen. Essentials of the self-organizing map. Neural Networks Part of special issue:
Twenty-fifth Anniversay Commemorative Issue. vol. 37, 2013, 52-65.
doi:10.1016/j.neunet.2012.09.018.
[17] H. de Vries, R. Memisevic, A. Courville. Deep Learning Vector Quantization. ESANN 2016
proceedings, European Symposium on Artificial Neural Networks, Computational Intelligence
and Machine Learning. Bruges (Belgium), April 2016 (2016) 27-29. URL:
http://www.i6doc.com/en/
[18] J.-W. Cho, H.-M. Park, Independent vector analysis followed by hmm-based feature
enhancement for robust speech recognition, Sig. Process. 120 (2016) 200β208.
doi:10.1016/j.sigpro.2015.09.002.
[19] R. Chai, G.R. Naik, T.N. Nguyen, S.H. Ling, Y. Tran, A. Craig, H.T. Nguyen, Driver fatigue
classification with independent component by entropy rate bound minimization analysis in an
eeg-based system, IEEE J. Biomed. Health Inf. IEEE vol. 21 (3), 2017, pp. 715β724.
doi:10.1109/JBHI.2016.2532354.
[20] A. Tharwat, Principal component analysis-a tutorial, Int. J. Appl. Pattern Recognit. Inderscience
Publishers; vol. 3 (3), 2016, pp. 197β240. doi:10.1504/ijapr.2016.10000630.
[21] M. I. Achmad, H. Adinugroho and A. Susanto, "Cerebellar model articulation controller
(CMAC) for sequential images coding," 2014 The 1st International Conference on Information
Technology, Computer, and Electrical Engineering, Semarang, Indonesia, 2014, pp. 160-166,
doi: 10.1109/ICITACEE.2014.7065734.
[22] R.A. Lobo, M.E. Valle. Ensemble of Binary Classifiers Combined Using Recurrent Correlation
Associative Memories. In: Cerri R., Prati R.C. (eds) Intelligent Systems (BRACIS 2020). Lecture
Notes in Computer Science, vol 12320. Springer, Cham. doi:10.1007/978-3-030-61380-8_30.
[23] Kobayashi M. Quaternionic Hopfield neural networks with twin-multistate activation function.
Neurocomputing 267, 2017, pp. 304β310. doi:10.1016/j.neucom.2017.06.013.
[24] Du K.L., Swamy M.N.S. Neural Networks and Statistical Learning. Springer-Verlag London,
(2014). doi: 10.1007/978-1-4471-5571-3.
[25] E. Javidmanesh. Global Stability and Bifurcation in Delayed Bidirectional Associative Memory
Neural Networks With an Arbitrary Number of Neurons. Journal of Dynamic Systems,
Measurement, and Control. ASME International. 139(8) (2017). doi:10.1115/1.4036229.
[26] Y. Park. Optimal and robust design of brain-state-in-a-box neural associative memories. Neural
Networks. Elsevier BV. 23(2), 2010, pp. 210β8. doi: 10.1016/j.neunet.2009.10.008
[27] O.I. Khristodulo, A.A. Makhmutova, T.V. Sazonova Use algorithm Based at Hamming Neural
Network Method for Natural Objects Classification. Procedia Computer Science. Elsevier BV.
103, 2017, pp. 388β395. doi: 10.1016/j.procs.2017.01.126.
[28] A. Fischer, C. Igel. Training restricted Boltzmann machines: An introduction. Pattern
Recognition. Elsevier BV, 47(1) (2014) 25β39. doi:10.1016/j.patcog.2013.05.025.
[29] Q. Wang, X. Gao, K. Wan, F. Li, Z. Hu. A Novel Restricted Boltzmann Machine Training
Algorithm with Fast Gibbs Sampling Policy. Mathematical Problems in Engineering. Hindawi
Limited (2020) 1β19. doi:10.1155/2020/4206457T.
[30] A. Fischer, C. Igel. Training Restricted Boltzmann Machines: An Introduction Pattern
Recognition, 47 (2014) 25-39. doi.10.1016/j.patcog.2013.05.025.
[31] Barszcz, A. Bielecki, M. WΓ³jcik, M. Bielecka. ART-2 Artificial Neural Networks Applications
for Classification of Vibration Signals and Operational States of Wind Turbines for Intelligent
Monitoring. Advances in Condition Monitoring of Machinery in Non-Stationary Operations.
Springer Berlin Heidelberg. 2013, pp. 679β88. doi:10.1155/2020/4206457.