Method Detection Audit Data Anomalies on Basis Restricted Cauchy Machine

Abstract

The paper presents a method for the anomalies detection in waste-free production audit data based on the neural network model of Gauss-Bernoulli bidirectional restricted Cauchy machine (BRCM). The purpose of the work is to increase the efficiency of audit data analysis of wastefree production on the basis of the neural network model of anomalies detection without the use of the marked data that simplifies audit. To achieve this goal, the following tasks have been set and solved: offered model of generalized multiple transformations of audit data in the form of a two-layer neural network. Cauchy offered neural network model of Gauss-Bernoulli bidirectional restricted Cauchy machine possesses a heteroassociative memory; works real data; has no restrictions for memory capacity; provide high accuracy of anomalies detection; uses Cauchy's distribution that increases the speed of convergence of a method of parametrical identification. To increase the speed of Gauss-Bernoulli parametric identification of a bidirectional restricted Cauchy machine, a parametric identification method was developed to be implemented on a GPU using CUDA technology. The offered method allows increasing training speed by approximately proportional to the product of numbers of neurons in the hidden layer and power of a training set. The made experiments confirmed the operability of the developed software and allow to recommend it for use in practice in a subsystem of the automated analysis of DSS of audit for detection of anomalies.

Keywords

• 1 Audit
• mapping by neural network
• Gauss-Bernoulli bidirectional restricted Cauchy machine
• anomalies detection

1. Introduction

2. Problem Statement

Let for model of detection of anomalies the training set be set S  {(ximn , xomut ,dimn ,domut )} , m 1, M , where ximn is m - y raw materials vector, xomut is m - y vector of finished goods, dimn is m - y the expected reference vector of raw materials, domut is m - y the expected reference vector of finished goods.

Then a problem of increase in accuracy of detection of anomalies on Gauss-Bernoulli's model of the bidirectional limited machine of BRCM g(xin , xout , w) , where xin is raw materials vector, xout is the vector of finished goods, w is a vector of parameters, is represented as a stay problem for this model of 1 M such vector of parameters w* , which meets criterion F   (g(ximn , xomut , w* )  (dimn ,domut ))2  min . M m1

3. Literature Review

Currently, the analytical procedures used during the audit are based on data mining techniques [ 2, 3 ]. 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. 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 amounts of indexes and supervisions in every task.

The development of methods of estimation and prediction [ 4, 5 ], formation of generalized associative relationships [ 6 ] are described in the works of the authors of this article. The goals of creating these methods: reducing the computational complexity for simple tasks (a single mapping of elements or sub-elements of the audit subject area), automatic structural identification, increasing the accuracy for com-plex tasks (compositions of mappings of elements or sub-elements of the audit subject area) and the possibility of applying these methods for the generalized analysis of elements and sub-elements of the audit subject area (Table 1).

The choice of model in the audit DSS depends on: 1. Characteristics of the audit data type (time series data, spatial data as mappings). 2. Audit level (upper middle, lower). 3. Audit tasks (internal, external). 4. The type of analysis tasks (detection of anomalies, structural analysis, assessment of indicators). 5. The characteristics of the enterprise (large, medium, small) and the type of activity (industry) at the top level. 6. Characteristics of sets and subsets of operations at lower levels (numerological, quantitative, semantic, logical).

This choice is schematically formalized in the form of a binary decision tree for choosing a neural network data audit model (see Fig. 1).

The proposed logical-neural network method makes it possible to automate the process of data analysis in the audit DSS and optimize it depending on the characteristics of the audit process and the audit object. One of the main tasks of data analysis of the audit subject area is the identification of anomalies. Let's consider the existing types of anomalies and methods of their operation.

Types of anomalies [ 7–9 ]: ● Point (are provided by points in character space). ● Contextual (usually a point of a time series or the rarefied data which depends on the environment).

● Collective (the section of a time series or the rarefied data).

Methods of detection of anomalies [ 7–9 ]: 1. Approach on the basis of rules (logical approach):

I. methods on the basis of associative rules with classification and without classification (for example, the Apriori method); II. methods on the basis of a decision tree with classification (for example, a method of the isolated wood).

Payment - delivery of raw materials

Settlements with

suppliers-customer settlements Release of raw materials - posting of finished products (a composition of mappings between a set of input and output data)

Payment - delivery

of raw materials Settlements with suppliers settlements with customers (a composition of mappings between a set of input and output data)

Approach on the basis of ANN:

I. ANN without classification (for example, the one-class SVM (support vector machine), ANN associative memory (for example, the autoencoder, SOFM (selforganizing feature map), Hopfield neural network, Boltzmann machine), ANN of the forecast of a time series (for example, NARNN (non-linear autoregressive neural network), NARMANN (nonlinear autoregressivemoving average neural network), SRN (simple recurrent network), BRNN (bidirectional recurrent neural network), LSTM (long short-term memory), BiLSTM, GRU (gated recurrent unit), BiGRU));

II. ANN with classification (for example, MLP (multilayer perceptron), RBFNN (radial-basis function neural network)).

Approach on the basis of Bayes's networks with classification Approach on the basis of a clustering:

I. clustering on the basis of centroid (for example, a method of k-means) or distributions (for example, the EM (expectation–maximization) method); II. clustering on the basis of medoid (for example, the PAM (partitioning around medoids) methods, a subtractive clustering); III. density clustering (for example, DBSCAN (density-based spatial clustering of applications with noise) methods, OPTICS (ordering points to identify the clustering structure methods).

ANN

Mapp data

ANN with associative memory

no waste

CPNN

RCM do not correspond speed

CPNN

Shallow CPNN FOCPNN BCPNN

with waste, layers correspond to the production of semi-finished products

Deep SRN with

associative memory accuracy, layers do not correspond production RCM forward-only

Shallow

Gauss-Bernoulli FORCM bidirectional

Shallow Gauss-Bernoulli BRCM

Time series data

ANN for forecasting

прогноза layers correspond to the production of semi-finished products

Deep CPNN

FOCPNN forward-only Shallow FOCPNN bidirectional

Shallow BCPNN

Approach on the basis of the neighborhood (metric approach) (for example, methods of the k-nearest neighbors, LOF (local outlier factor)) Approaches on the basis of distributions: a. Parametrical approach on a basis:

I. Gaussian distributions (for example, MCD (minimum covariance determinant) method); II. mixtures of distributions (for example, HMM (hidden Markov models), GMM (Gaussian mixture models)). b. Nonparametric approach on a basis:

I. histograms;

II. functions of a kernel (for example, Parzen window method).

Approach on the basis of regression model (for example, the Box-Jenkins meth-od) 8. Approach on the basis of the spectral theory (matrix decomposition) (for example, PCA (principal component analysis) method) 9. Approach on the basis of information theory (entropy).

Now the most popular is approach of detection of anomalies based on neural networks.

Disadvantages of the one-class SVM is restriction for quantity of support vectors. A disadvantage of ANN of the forecast of a time series is that they require existence of a time series. A disadvantage of ANN with classification is the requirement to classify anomalies that is not always possible owing to labor input of obtaining the marked data on each type of anomalies. Therefore, in our work, we chose ANN with associative memory.

Traditional neural networks with an associative memory are: 1. Neural networks only with heteroassociative memory (for example, FOCPNN (forwardonly counter propagation neural network) [ 10 ], PCANN (principal com-ponent analysis neural network) [ 11 ], ICANN (independent component analysis neural network) [ 12 ], CMAC (cerebellar model articulation controller) [ 13 ]. 2. Neural networks only with an autoassociative memory (for example, the autoencoder [ 14 ], SBN (sigmoid belief network) [ 15 ], Helmholtz machine [ 16 ], SOFM [ 17 ], LVQNN (learning vector quantization neural network) [ 18 ], RCAM (recurrent correlation associative memory) [ 19 ], Hopfield neural network [ 20 ], Gauss machine [ 21 ], BSB (brain-state-in-box) [ 22 ], Hamming neural network [ 23 ], ART (Adaptive resonance theory) [ 24 ]. 3. ANN with a heteroassociative and autoassociative memory (for example, BCPNN (bidirectional counter propagation neural network) [ 25 ], BAM (bidirectional associative memory) [ 26 ], Boltzmann machine [ 27 ]).

The majority of neural networks with an associative memory possess some or more shortcomings: 1. Do not possess at the same time autoassociative and heteroassociative memory, 2. Do not work with material data. 3. Have no high capacity of an associative memory. 4. Have no high accuracy.

5. Have high computing complexity.

In this regard, creation of a neural network which will allow to eliminate the specified defects is relevant.

The purpose of work is increase in efficiency of data analysis of audit of waste-free production on the basis of neural network model of detection of anomalies with-out use of the marked data that simplifies audit [ 28–30 ].

For achievement of the goal, it is necessary to solve the following problems: ● Offer neural network model of detection of anomalies. ● Select criterion for evaluation of efficiency of neural network model of detection of anomalies. ● Offer a method of parametrical identification of neural network model of detection of anomalies.

● Execute numerical researches. 4. Block Diagram of Neural Network Model of Detection of Anomalies

In this paper, the structure of the data transformation model is determined based on the production structure. It is assumed that the transformation of raw materials into finished products in one step without waste without intermediate products. Each type of raw material is used in the production of one or more types of finished products. The production structure for each planning period (month, quarter, year) is deter-mined on the basis of long-term contracts and short-term (in particular urgent) orders. The production plan is decomposed into quantization periods of the planning period, taking into account the production capacity for different types of products.

In this case, the transformation of these raw materials into finished products for the planning period can be represented in the form of a two-layer neural network. The number of neurons in the input layer is equal to the number of raw materials used in production. The number of neurons in the output layer is equal to the number of types of finished products. The input values are the amount of raw materials by type, the output of the network is the finished product values for the planning period or the quantization period.

When taking into account the supply of raw materials and the release of finished products for the quantization period, the amount of recorded released raw materials and released finished products is made up of the actual values of indicators and the difference in residuals at the beginning of the quantization period. These balances are not calculated at the end (beginning) of each quantization period and not reflected in accounting systems, balances are determined only at the beginning and end of the planning period).

The detecting anomalies problem during the release (write-off) of raw materials for the production of finished products for the periods of quantization of the verification period (in particular, the periods of quantization can be chosen equal to the production cycle - shift, day, several days) is formulated as follows. Determine the quantization periods for which the structure of consumption of raw materials is significant (the level of materiality is set by the decision maker) is higher or lower than the average values for the period according to the data of the release of raw materials into pro-duction and the posting of products.

This data transformation model is used to create an anomaly detection model that can be represented as a two-layer neural network. The number of neurons in the visible layer is equal to the sum of the number of raw materials used in production and the number of types of finished products. Thus, the number of raw materials and finished products is supplied to the visible layer by type for the planning period or the quantization period of the verification period. To train the neural network, the "correct" data are used (the formation of which has been verified). Data that are subject to verification are used as control data.

The block diagram of model of Gauss-Bernoulli BRCM (bidirectional restricted Cauchy machine) [ 6 ] which is recurrent ANN and contains one visible layer and one hidden layer (Fig. 2). vector of raw materials vector of finished goods … … … … … … visible neurons hidden neurons where  j is mathematical expectation, which characterizes the average value of indicators of supplied raw materials or capitalized products,  j is a mean square deviation (if the training a vector are normalized and centered, then  j =1), which characterizes the variance of the difference in residuals at the beginning of the quantization period, N(0,1) is the function returning standard normally distributed random number.

Transition probability j -th a stochastic neuron in a state  is defined in a form ●

1  1    j   .

Pj   j 2 exp  2   j   The stochastic hidden neurons which state is described on the basis of Bernoulli's distribution in a form with probability Pj , with probability 1  Pj .

Transition probability j -th a stochastic neuron in a state 1 is defined in a form

1 1 Pj   arctan  E j  ,

2  where E j is increment of energy of ANN at state change j -th a stochastic neuron with 0 on 1. Gauss-Bernoulli's BRCM advantages: 1. Unlike the majority ANN possesses at the same time autoassociative and heteroassociative memory. 2. Unlike a bidirectional associative memory and Boltzmann machine works real data. 3. Unlike a bidirectional associative memory and Boltzmann machine has no restrictions for memory capacity. 4. Unlike a bidirectional associative memory provide big accuracy.

5. Unlike Boltzmann machine has smaller computing complexity.

5. Neural Network Model of Detection of Anomalies

Positive phase (Step 1–3).

1. Initialization of a state of the visible neurons corresponding to raw materials x1in  xin 2. Initialization of a state of the visible neurons corresponding to finished goods x1out  xout 3. Calculation of a state of the hidden neurons  j 1, N h  .

Pj  12  1 arctan  bhj  Nii1n wiijnh x1iiiinn  Nio1ut wiojuth x1ioiouutt  .

1, x1hj   0,

Pj  U (0,1),

Pj  U (0,1). where U (0,1) is the function returning uniform distributed random number in the range [ 0,1 ] . Negative phase (Steps 4 and 5).

4. Calculation of a state of the visible neurons corresponding to raw materials  j 1, N in  where bhj is bias for j-th of a neuron of the hidden layer, bijn is bias for j-th of a neuron of the visible layer corresponding to raw materials, bojut is bias for j-th of a neuron of the visible layer corresponding to finished goods, winh is connection weight from the neuron i-th in a visible layer corresponding to ij raw materials to j-th to a neuron of the hidden layer, wouth is connection weight from the neuron i-th ij in a visible layer corresponding to finished goods to j-th to a neuron of the hidden layer, N h is number of neurons in the hidden layer, N in is the number of the neurons in a visible layer corresponding to raw materials, N out is the number of the neurons in a visible layer corresponding to finished goods.

N h  ijn  bijn  ijn  winh x1ih ,

ij i1 x2ijn   ijn  ijn N (0,1) .

N h  ojut  bojut  ojut  wiojuth x1ih ,

i1 x2ojut   ojut  ojut N (0,1) , 5. Calculation of a state of the visible neurons corresponding to finished goods  j 1, N out 

In work for training of the BRCM model the function of the purpose which means the choice of such values of a vector of parameters is selected w  (w1in1h ,..., wNininhNh , w1o1uth ,..., wNouotutNh h ) , which deliver a minimum of a root mean square error (the differences of a sample on model and a test sample)

F  M ( N in1 N out ) mM1 x2imn  dimn 2  x2omut  domut 2   mwin , where x2imn is m -th an evaluation vector of raw materials on model, dimn is m -th raw materials vector, x2omut is m -th an evaluation vector of finished goods on model, dout is m -th vector of finished goods. m

Method of Parametrical Identification

Detection of Anomalies on the basis

Contrastive Divergence) of of

Neural Network Model of

Algorithm CD-1 (One-Step

The method of parametrical identification of neural network model of detection of anomalies on the basis of algorithm CD-1 consists of the following blocks (Fig. 3).

1. Initialization 2. Initialization of the state of visible neurons corresponding to raw materials (positive phase) 3. Initialization of the state of visible neurons corresponding to the finished product (positive phase) 4. Calculation of the state of hidden neurons (positive phase) 5. Calculation of the state of visible neurons corresponding to to raw materials (negative phase) 6. Calculation of the state of visible neurons corresponding to the finished product (negative phase) 7. Calculation of the state of hidden neurons (negative phase) 8. Setting synaptic weights based on the stochastic rule 9. Continue?

No

Yes and weights winh (n) , i 1, N in , j 1, N h , wouth (n) , i 1, N out , j 1, N h , winh (n)  0 , ij ij ii wouth (n)  0 , winh (n)  winh (n) , wouth (n)  wouth (n) .

ii ij ji ij ji

The training set is set {(ximn , xomut ) | ximn (0,1)Nin , xomut (0,1)Nout } , m 1, M , where ximn is mth raw materials vector, xout – m -th vector of finished goods, M – power of a training m set, vector of mean square deviations for a raw materials vector σin  ( in ,..., Ninin ) ; vector j of mean square deviations for a vector of finished goods σout  ( out ,..., Nouotut ) . j Positive phase (Step 2–4) 2. Initialization of a state of the visible neurons corresponding to raw materials x1imn  ximn , 3. Initialization of a state of the visible neurons corresponding to finished goods x1omut  xout m , 4. Calculation of a state of the hidden neurons  j 1, N h 

Pmj 

 1

 2  1  in

i  Nin Nout arctan  bhj (n)   wiijnh (n) x1imni   wiojuth (n) x1oomuuitt  , m 1, M , i1 i1 i

 1, x1hmj   0, Pmj  U (0,1)

Pmj  U (0,1) . m 1, M Negative phase (Steps 5–7) 5. Calculation of a state of the visible neurons corresponding to raw materials  j 1, N in   minj  bijn (n)  ijn  wiijnh (n)x1hmi , m 1, M , x2imnj   in  ijn N (0,1) , m 1, M .

mj  out  bout (n)  out  wout h (n)x1hmi , m 1, M , mj j j ij x2imnj   in  ijn N (0,1) , m 1, M

mj 6. Calculation of a state of the visible neurons corresponding to finished goods  j 1, N out  7. Calculation of a state of the hidden neurons  j 1, N h 

Pmj 

 1

 2  1  in

i  Nin Nout arctan  bhj (n)   wiijnh (n) x2imni   wiojuth (n) x2oomuutit  , m 1, M , i1 i1 i

 Nh i1 N h i1 x1omuit 8. Setup of bias and synoptic weights on the basis of the stochastic rule x2hmj   0, 1,

P  U (0,1) mj P  U (0,1) mj

M x1imni biin (n  1)  biin (n)   1   M m1  in 2  i

M biout (n  1)  biout (n)   1     M m1  out 2  i

, m 1, M .   M m1  out 2 

i  x2out  mi  , i 1, N in ,

, i 1, N out ,  1 M bih (n  1)  bih (n)    x1hmi   M m1 1 M

 x2mhi  , i 1, N h ,

M m1  ij  M1 mM1 x1imni iixn1hmj , i 1, N in , j 1, N h , ij  M1 mM1 x2imni iixn2hmj , i 1, N in , j 1, N h , winh (n 1)  wiijnh (n)  (ij  ij ) , i 1, N in , j 1, N h , ij ij  M1 mM1 x1omuitioxut1mhj , i 1, N out , j 1, N h , ij  1 M x2omuit x2hmj , i 1, N out , j 1, N h ,

M m1  iout wouth (n 1)  wiojuth (n)  (ij  ij ) , i 1, N out , j 1, N h

ij 9. Check of termination condition If 1 M Nin | x1imni  x2imni |  Nout | x1omuit  x2omuit |   then n  n  1 , go to 2.

M (N in  N out ) m1 i1 i1 

8. Experiments and Results

The offered method was investigated on indicators of delivery and payment of stocks of manufacturing enterprise with a two-year depth of sample with daily time intervals.

Results of comparison of the offered neural network model (BRCM) with neural network model are presented by a bidirectional counter propagation neural network (BCPNN) on the basis of criterion of a root mean square error in Table 2.

According to Table 2, use of BRCM reduces a root mean square error and by that increases the accuracy of detection of anomalies

Results of comparison of the offered method of parametrical identification with use and without use of GPU and technology of parallel processing information of CUDA are provided in Table 3.

According to tab. 2, use of GPU reduces computing complexity approximately in 2N hM  log2 (N hM ) time and by that increases the speed of parametrical identification. 9. Conclusions 1. The relevant problem of increase in efficiency of detection of anomalies in data of audit of wastefree production was solved by means of neural network model of Gauss-Bernoulli bidirectional restricted Cauchy machine. 2. The proposed neural network model of Gauss-Bernoulli bidirectional restricted Cauchy machine possesses at the same time autoassociative and heteroassociative memory; real data; has no restrictions for memory capacity; provide high accuracy of anomalies detection; uses Cauchy's distribution that increases the speed of convergence of a method of parametrical identification. 3. For increase speed of parametrical identification of Gauss-Bernoulli bidirectional restricted Cauchy machine, the method of parametrical identification intended for implementation on GPU by means of CUDA technology was developed. The offered method allows to increase training speed approximately in 2N hM  log2 (N hM ) time, where N h is number of neurons in the hidden layer, M is power of a training set. 4. The made experiments confirmed operability of the developed software and allow to recommend it for use in practice in a subsystem of the automated analysis of DSS of audit for anomalies detection.

Prospects of further researches are in checking the offered methods on broader set of test databases. 10.

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