KR20120072291A - Method for imputating of missing values using incremental expectation maximization principal component analysis - Google Patents

Abstract

PURPOSE: A method for replacing missing values which uses an EM(Expectation Maximization) PCA is provided to maximize data accuracy and reduce computation complexity by measuring incremental EM PCA. CONSTITUTION: An input vector is received. If the input vector includes a value, the value is initialized as average of data matrix. A PCA(Principle Component Analysis) is performed about the data matrix. The data matrix is re-configured. A mixing value is replaced with the missing value of the reconfigured data matrix. An error rate about the data matrix is performed until error rate is less than a critical value.

Description

METHODS FOR IMPUTATING OF MISSING VALUES USING INCREMENTAL EXPECTATION MAXIMIZATION PRINCIPAL COMPONENT ANALYSIS}

The present invention relates to a method for replacing missing values using incremental (Expectation Maximization) EMP Principal Component Analysis (PCA). In order to solve the problem of inaccurate data when a large number of missing values are eliminated or measured or estimated using a simple method, the data accuracy is improved by finding and measuring missing values in real time data using a progressive EM PCA. A method of replacing missing values using incremental EM PCA to maximize.

Missing values refer to data elements that are not found in the data matrix, which can occur due to various reasons, for example technical problems or biological characteristics.

In general, EEG data is intermittently entered. Therefore, in an EEG data signal environment where noise, artifacts or other variations are present, such loss of EEG data occurs frequently. Thus, signal analysis in EEG can be adversely affected by missing data periods.

These missing values are mainly excluded from the analysis or replaced with measured values. If the ratio of missing values is above a certain level, there is a problem that a significant result error may occur due to a decrease in data reliability in data analysis.

The following two methods can be exemplified as a method of dealing with missing values that occur frequently in most physiological data.

First, a method of removing incomplete data as shown in FIG. Second, a method of measuring or estimating missing values from the remaining signal.

In FIG. 1, EEG data includes missing values. Thirty-one electrodes (columns), for about 130 time points, dark color indicates missing values that occur in writing. The dark cells and rows t in column n indicate missing values for a specific time (tth time point / row) and the number of specific electrodes (nth column).

The first method is clearly simple. However, if the cause of missing values is not arbitrary, removing missing values creates a biased model and causes loss of information.

The second method mainly uses simple replacement or multiple replacement. According to these replacement methods, the loss of information from the original data is relatively small compared to the case of eliminating incomplete data, and thus has been conventionally used and widely developed (Graham et al., 2007; Graham, 2009; Horton). and Lipsitz, 2001).

However, since the simple replacement method always uses the mean value of observations as the data to replace, there is a problem of underestimating the standard deviation. That is, the simple replacement method does not take into account the increase in uncertainty caused by missing values.

On the other hand, a multi-substitution method, for example, MLE (Maximum Likelihood Estimation), etc. have been proposed to solve the problem that occurs in the simple substitution method. The multiple imputation method considers multiple candidate values for missing value estimation.

In the multiple imputation method, missing values are replaced with candidate values by random replacement. Multiple imputation methods consist of performing statistical analysis of the complete data results and then integrating the measured coefficient values (Hardy and Robillard, 2005; Little and Rubin, 2002).

On the other hand, multivariate analysis techniques, such as principal component analysis (PCA), are applied to various analytical processes that exhibit nonlinear characteristics (Nuding and Zetzsche, 2007). This principal component analysis method finds and diagnoses abnormal values and solves the problem of missing values at the data collection stage (Xu et al., 2009; Gu et al., 2009).

For example, Adams has applied EM PCA using principal component analysis and EM algorithms to replace missing values in the manufacture of pharmaceuticals (Adams et al., 2002). In addition, Zhao applied robust EM PCA and modified EM PCA to the data acquired in the wastewater treatment process and compared the performance against missing values (Zhao et al., 2006).

Substitution methods as described above that replace missing values with appropriate values are known as efficient methods for controlling missing values and ensure significant performance. In particular, robust EM PCA is known to be the best of existing methods for measuring missing values.

However, these replacement methods require a lot of information and require a long time as execution time for the replacement. Moreover, in the case of correlation analysis, if the missing value is high, the data reliability is lowered and inevitably leads to inaccurate results. In addition, robust EM PCA can only guarantee performance when measuring missing values if it is not negative data.

Disclosure of Invention Problems to be Solved by the Invention The present invention has been made to overcome the above-mentioned problems. In the analysis of physiological multivariate data, the problem of inaccurate data is generated when a plurality of missing values are removed or measured or estimated using a simple method. will be.

Another problem to be solved by the present invention is to overcome the above problems, by maximizing data accuracy while reducing computational complexity by finding and measuring missing values in real time data using progressive EM PCA.

The present invention relates to a method for replacing missing values using a progressive EM PCA, comprising: (a) receiving an input vector x; (b) initializing the missing value with an average of the matrix of the data matrix X when the input input vector x includes the missing value; (c) performing PCA on the data matrix X; (d) reconstructing the data matrix X; And (e) replacing missing values in the data matrix X before reconstruction with missing values in the reconstructed data matrix X; It includes.

Preferably, after step (e), repeating steps (c) to (e) until (f) the error rate for the data matrix X in which missing values are replaced is less than or equal to a preset threshold. ; It characterized in that it further comprises.

Also preferably, after step (f), (g) calculating a hidden variable; (h) calculating the total hidden variable energy; And (i) calculating the total input data energy.

Also preferably, the steps (g) to (i) are each characterized by the following equation.

Where y is a hidden variable, T is a score vector, W _new is an updated weight matrix, x _replace is a determined missing value, E _hv is an existing or updated total hidden variable energy, and λ is an exponent Is an oblivion factor, E _x is the existing or updated total input data energy.)

Also preferably, if E _hv < fE _x (f is the number of principal components), after step (i), (j) adding a hidden variable.

And preferably, the exponential forgetting factor λ is selected from 0.96 to 0.98.

According to the present invention, in the analysis of physiological multivariate data, there is an effect of solving the problem that incorrect data is generated when a plurality of missing values are removed or measured or estimated using a simple method.

According to the present invention, the incremental EM PCA is used to find and measure missing values in real time data, thereby reducing calculation complexity and maximizing data accuracy.

1 is a view for explaining a conventional method of replacing missing values.
2 is a view for explaining the accuracy evaluation results of the method according to the present invention.
3 is a view for explaining an evaluation result regarding the detection of hidden parameters of the method according to the present invention.
4 is a view for explaining the reconstruction evaluation results of the method according to the present invention.
5 is a view for explaining a calculation complexity evaluation result of the method according to the present invention.

Prior to describing the specific details for carrying out the invention, the terminology or words used in this specification and claims may properly define the concept of a term in order for the inventor to best describe his or her invention. It should be interpreted as meaning and concept corresponding to the technical idea of the present invention on the basis of the principle.

In addition, when it is determined that the detailed description of the known function and its configuration related to the present invention may unnecessarily obscure the subject matter of the present invention, it should be noted that the detailed description is omitted.

In the present invention, a gradual EM PCA method as a method of replacing missing values, in order to solve the problem that incorrect data occurs when a plurality of missing values are removed or measured using a simple method in physiological multivariate data analysis such as EEG data. This is disclosed.

According to the progressive EM PCA completed by the present inventors, it is possible to recover missing values, find hidden variables, and predict trends, which can be inferred from other observations using the correlation between different results.

The complexity of the calculation is reduced by the use of interrelationships, and unnecessary data characteristics can be eliminated by the hidden variable discovery. Consequently, according to the present invention, the data accuracy is significantly improved as compared with the conventional missing value replacement method.

Hereinafter, the PCA will be described in detail before describing the specific contents for carrying out the present invention.

PCA is a technique for reducing the number of dimensions in data analysis. The main features can be summarized as dimension reduction, noise reduction, and orthogonal transformation.

Specifically, the PCA takes characteristic components with representative data between components and extracts unique components that imply data characteristics and reduces dimensions. That is, the PCA is for determining the axis of the principal component that fully represents the data, and the dimension is reduced by projecting the data matrix X of the axis of the principal component.

The axis of the principal component is computed using the following singular value decomposition (SVD) by the eigenvector, the eigenvalue, and the principal component.

Therefore, PCA can be expressed as the multiplication of two matrices as shown in Equation 1 below.

[Equation 1]

X = TV ^T

In Equation 1, X is a data matrix having characteristic values n and observation m, and if T is a score vector having an m × f matrix, V is a loading vector having an n × f matrix. f denotes the number of principal components, and in the case of time series data, m denotes a viewpoint.

However, if the data matrix X contains missing values, the traditional PCA lacks consistency.

Hereinafter, the progressive EM PCA according to the present invention will be described in detail.

EM PCA has been proposed to solve the problems of traditional PCA (AI-Deek et al., 2004; Roweis, 1998). The algorithm makes it possible to extract several extracted eigenvectors and eigenvalues from high-dimensional data.

EM PCA improves computational efficiency in terms of time and space, and does not require calculation of sample covariance of data, and naturally accepts missing values.

In the EM PCA, missing values are found by repeating the calculation by the two steps of [Equation 2] and [Equation 3]. Step E determines missing values based on the expected values of the data.

The initial value starts at the mean of the matrix. Step M maximizes the value obtained in step E and re-determines the missing value to the value expected by PCA by iteration of EM.

[Equation 2]

&Quot; (3) "

To replace missing values, the PCA first determines the value as the mean of the matrix and applies the completed data. The applied data is then reconstructed, and the reconstructed data is used to replace missing values until convergence.

Therefore, EM PCA can be summarized as follows.

First, the missing values are initialized with the average of the matrix of data matrix X.

Second, perform the PCA with a complete data set. The loading vector V and the score vector T are calculated iteratively by [Equation 2] and [Equation 3].

&Quot; (4) "

Y = V ^T X // Project X to the loading vector V _new

Third, reconstruct the data matrix X with a predefined number for the principal principal component.

[Equation 5]

// 재구성

// reconstruct

Fourth, replace the missing value present in the original matrix X with the reconstructed value for it. Observation value X remains unchanged.

&Quot; (6) "

// 결측값 대체

// replace missing values

[Equation 7]

// 에러율 평가

// error rate evaluation

Finally, repeat steps 2 through 4 until they converge.

The data matrix X containing missing values is replaced with the values expected from the EM PCA algorithm.

Next, the optimal value of the missing value is obtained during the iteration of the EM PCA.

Since EM PCA minimizes the sum of squares of the residuals as follows, the EM PCA parameter satisfies the minimum square criterion.

[Equation 8]

는 재구성된 i번째 데이터를 의미한다.
In Equation 8,

Denotes the reconstructed i-th data.

Hereinafter, the case of co-evolving EEG data for the progressive EM PCA for replacing missing values according to the present invention will be described in detail.

PCA is an analytical technique used to separate other components from multivariate dependent data, and is an efficient analytical technique that reduces the amount of information while representing the given data information.

를 k 차원(n〉k)의 데이터

로 변환하여 데이터 X의 최대 정보를 유지하며 각각 상관되어 있지는 않다. 간략하게, PCA는 변수들의 선형 조합을 통해 설명될 수 있으며 새로운 주성분들을 유도한다.PCA n-dimensional data

Is k-dimensional (n> k) data

The maximum information of the data X is maintained by converting to and not correlated with each other. Briefly, PCA can be described through a linear combination of variables and leads to new principal components.

In the present invention, the progressive EM PCA summarizes the data by the discovery of hidden variables that predict and replace missing values in the multivariate EEG time series data and characterize the data.

First, the EEG time series data is created by the data matrix of the variable n and the observation time t.

For example, data x is composed of n × t as shown in [Table 1].

Therefore, multivariate EEG time series data has a time dependency.

으로 n차원 특성 벡터를 표현한다. 이때 n은 데이터 측정을 위한 센서의 수이며 t는 시점이다.And including missing values

Represents an n-dimensional feature vector. Where n is the number of sensors for data measurement and t is the time point.

The general form of the principal component linear combination is expressed as Y = W ^T X. The characteristic vector of the reduced dimension is represented by Y = W _k ^T X. This represents the reduced size of the X matrix, where n> k.

The matrix with (k × t) feature vectors and k is the dimension of the feature vectors and consists of the W fundamental vectors (n × n).

The base vector gradually uses the standard base vector.

However, in EEG time series data, the standard base vector is inappropriate to represent the data.

A new fundamental vector is required to represent the characteristics by the time of measurement of the EEG time series data until the data is measured by time.

variable meaning

Y
y_t
d_i
m_k,t
e_i
P
λ
E_t,i n
k
W
w _i
X
x
x _t

x _replace
x _{t, miss}
x _mean

Y
y _t
d _i
m _{k, t}
e _i
P
λ
E _{t, i} Number of streams
Number of hidden variables
Weighting matrix
i relation weight vector
n-dimensional matrix X
Input vector
N stream input at time t

N stream input replaced by missing value at time t
N stream input containing missing values at time t
Average from x ₁ to x _{t -1}


reconstruction of x _i
Hidden Variable Matrix
vector of hidden variables at x _t
Energy measurement of the I hidden parameter
Energy accumulated by k at time t
Reconfiguration error
Potential variables
Exponential forgetting factor
Total energy of hidden variables up to time t

를 점차적으로 갱신하는 동안, 점진적 PCA는 새로운 벡터를 찾는다.The progressive PCA according to the present invention detects a new base value by a base vector w that is gradually updated while predicting the original high dimensional data to reflect the characteristics of the time series in existing principal component analysis. In other words, the base vector of the principal components to minimize the mean reconstruction error

While gradually updating, the progressive PCA finds a new vector.

으로 갱신한다.Where k is the number of principal components. Therefore, incremental PCA requires a base vector of _k and principal component of the values of w _{i, k} .

Update with

In the present invention, the number of hidden variables is k and the initial number of hidden variables is set to three.

Then, the vector x _t is converted into a new matrix 3 x t.

The value of the abbreviated feature vector y _i is the base vector represented by Equation 9

It is obtained from the multiplication of the original vector x _i by w _k and y _i .

[Equation 9]

The size of the components is gradually evaluated using the time using the base vector summarized in [Equation 9] to determine the number of principal components. Equation 10 is used for the size evaluation to determine the number of components.

&Quot; (10) "

Size thresholds range from a minimum of 95% to a maximum of 98% of standard values (Pan et al., 2004; Papadimitriou et al., 2005; Sun et al., 2005). In this specification, rather than energy, it is expressed as a magnitude threshold.

This energy range gives the upper and lower bounds of the energy and determines the components of the energy. In other words, the energy range determines the number of hidden variables.

If the energy of the concealed variable calculated in a particular timestamp is lower than the lower energy boundary, the concealed variables are increased or vice versa.

If we set the lower boundary energy lower, we will get fewer hidden variables. In that case, useful information may be lost from low energy concealment parameters.

Nevertheless, the first hidden variable is sufficient to summarize the entire EEG time series signal in the form of a distinct stream. The energy level recording of the concealment variable is necessary for sufficient results, and the energy range disclosed in the present invention is justified.

In Equation 11, the average reconstruction error ratio (ARE) is used to update the base vector.

&Quot; (11) "

은 k 차원 공간에서 투영 후 복원으로 표시되고 x_t의 복원은 [수학식 12]에 의한다.In Equation 11,

Is denoted as post-projection reconstruction in k-dimensional space and the reconstruction of x _t is given by Equation 12.

[Equation 12]

EEG EEG data includes missing values that occur in real time.

If a new sample x _{t + 1} contains missing values by a given time variable, the algorithm quickly adjusts the missing values and updates the base vector in a progressive EM PCA according to the present invention (see Table 2). Gradually change the principal component.

Progressive EM PCA, exponential forgetting factor, λ are used to apply to recent behavior in EEG data.

Common selections of exponential forgetting factors have values between 0.96 and 0.98 (Pan et al., 2004; Papadimitriou et al., 2005; Sun et al., 2005).

Compared to sliding windows, it does not require buffer space for the time series of EEG signals, thus reducing memory usage.

In the present invention, λ = 0.96, and this approach works gradually and is therefore particularly suitable for EEG time series data.

Progressive EM PCA to Replace Missing Values

[수학식 4] 내지 [수학식 7]에 의한 결측값 대체

// 은닉 변수 계산

// 전체 은닉 변수 에너지 계산

// 전체 입력 데이터 에너지 계산
만약 E_hv〈fE_x
k=k+1 // 은닉 변수 추가
만약 E_hv〉FE_x
k=k-1
끝
만약 입력 벡터 x가 결측값을 포함하지 않는다면,
[수학식 9] 내지 [수학식 12]를 사용한 I=1 내지 k까지의 가중치 벡터 갱신
끝
은닉 변수 k의 개수 조정
끝[Input value]
New input x
Existing Weight Matrix W
Traditional energy measurement by historical data d
Existing Full Concealed Variable Energy E _hv
Existing Full Input Data Energy E _x
Lower bound f of predefined energy
Upper boundary F of predefined energy
Exponential forget factor λ
[Output value]
Updated Weight Metrics W
Updated energy measurement d
Updated Total Hidden Variable Energy E _hv
Updated Total Input Data Energy E _x
Determined missing value x _replace
[algorithm]
If the input vector x contains missing values,
x _replace = x _{t, miss} = x _mean // Reset missing values by x _mean (keep x observations)
Error = MSE (x _t , x _replace ) // evaluate error rate
If missing values do not converge,

Replace missing values by Equations 4 to 7

// hidden variable calculation

// Calculate Total Hidden Variable Energy

// calculate total input data energy
If E _hv 〈fE _x
k = k + 1 // add a hidden variable
If E _hv 〉 FE _x
k = k-1
End
If the input vector x does not contain missing values,
Update weight vector from I = 1 to k using Equations 9 to 12
End
Adjust the number of hidden variables k
End

Summarizing the features of the progressive EM PCA described above, missing values are measured in EEG time series data including missing values. Patterns representing the characteristics of the data are derived by changes in time, and correlations between the data are found by finding hidden variables in the EEG time series data.

According to the present invention, it is possible to reduce the complexity of the calculation by reducing the memory usage, to derive the characteristics and to measure the missing value of the EEG time series data by the real-time process. In addition, the data is summarized by a lower dimension of the concealment function than the original input data.

Hereinafter, the experimental results for the progressive EM PCA according to the present invention will be described in detail.

First, the data sets used in the experiment will be described as follows.

As in Table 3, three sets of multivariate EEG time series data were used in the experiment.

The first set is EEG data for seizure epilepsy provided by Krug and is publicly available at http://www.epileptologie-bonn.de/cms/.

Epilepsy EEG data were measured using 100 channels for a time of 23.6 seconds for patients with epilepsy and were recorded by successive EEG multi-channels after visual confirmation of artificial eye movements or muscle movements.

Studies 1 and 2 were obtained from surface EEG records of five healthy volunteers who repeatedly closed their eyes. Data from both studies were measured between five patients in the interstitial zone (subject 4) and no seizure intervals in the hippocampal tissue of the opposite hemisphere of the brain (subject 3).

Study 5 included seizure behavior and was selected from all recorded positions representing seizure behavior. Studies 1 and 2 were recorded outside the skull but studies 3, 4 and 5 were recorded inside the skull.

Separate from the other recording electrodes, the recorded parameters were fixed. For this reason, different analysis results can be attributed to the dynamic properties of other brains. Therefore, the experiment uses three studies (1, 3 and 5), which are complete data sets.

Self Regulation EEG data set was provided by Klaus (http://www.bbci.de/competition/ii/). This experiment was conducted to determine the position of the computer screen (left, right, up, down).

If a target appears, the subject moves the cursor on the screen (from left to right, right to left) according to the idea of tracking the target, and the brain waves are recorded. Self-regulating EEG data includes EEG data collected using 64 channels from three subjects.

One subject was given ten periods per day, each of which was specified. Also, in this experiment, mu or beta rhythm amplitude was used to control the vertical cursor movement toward the vertical position of the target located in the right corner of the video screen.

Data was collected from 10 parts each 30 minutes. Each part consists of six runs, separated by one minute of rest, and each run consists of 32 individual tests. Each test starts from 1 second while the screen is black.

The target then appeared in one of four possible positions on the right edge.

A second later, one cursor appeared in the middle of the left edge of the screen and started moving from left to right at a constant speed.

The vertical position of the cursor was controlled by the subject's brain waves (update rate: 10 times per sec). The object's goal is to move the cursor to the correct target height. When the cursor reaches the right side, the screen becomes black.

This meant the end of the experiment, where 1 to 6 parts of each subject were used.

The final data set used in the experiment is the EEG data for the event-related potential (ERP).

Such data is provided at http://sccn.ucsd.edu/~arno/fam2data/publicly_available_EEG_data.html.

ERP EEG data was recorded using 31 channels. This data set was obtained from participants sitting 110 cm in a dimly lit room from a computer screen controlled by a computer.

These subjects performed two tasks: animal classification and cognition. Subjects had to press a button that was sensitive to touch. A small anchor point was drawn in the center of the black screen.

Thereafter, 8-bit color vertical photographs flashed for 20 ms using the programmed image plate. This short presentation time allowed subjects to avoid seeking eye movement responsibly. Participants responded according to the "go / no go" legend.

In animal classification, participants had to respond regardless of the time the animal appeared in the photograph. Cognitive work started at the learning stage.

The specific investigative image flashed 14 times for 20 ms, with two presentations mixed for 1000 ms after the fifth and tenth, and eyeball images were explored at 1000 ms stimulation intervals.

The two tasks consisted of a series of 100 images, with 50 target images containing 50 non-targets in animal classification. Fifty unique photographs were randomly mixed with fifty non-targets in cognitive work.

All EEG data is a complete data set with no missing values. Therefore, in order to evaluate the performance of the progressive EM PCA according to the present invention, missing values were generated at rates of 5%, 10% and 15%.

And, the accuracy evaluation results by the progressive EM PCA according to the present invention will be described.

Accuracy evaluation was performed by measuring the accuracy of the results according to the progressive EM PCA according to the present invention and the values measured by the original values. In the progressive EM PCA according to the present invention, a hidden variable automatically summarizes a large amount of data. Energy values from 0.95 to 0.98 were used in all data to extract the number.

For efficient accuracy assessment, the progressive EV PCA according to the present invention is compared with robust EMPCA (Zhao et al., 2006) and missing value SVD (MSVD, Troyanskaya et al., 2001), as shown in Table 4, respectively. The error rate of imputing missing values for was measured. Each data set contains 5%, 10% and 15% ratios of missing values generated repeatedly.

In Table 4, in the case of robust EMPCA and epilepsy EEG data MSVE, the mean of 22 and 80 concealment variables were extracted from a data set containing 5% missing rate, but the method according to the present invention successfully achieved 21 concealment parameters. Are summarized.

In epilepsy EEG data, the method according to the invention shows a lower missing value replacement error rate of 0.0016% and 0.0004% on average than robust EMPCA and MSVD.

The mean replacement error rate for MSVD was lower than for robust EMPCA.

Singular decomposition (SVD, Troyanskaya et al., 2001) is a tool that is capable of recovering missing values from a single result, limited to finding linear correlations through multiple results, and robust EMPCA is only available for non-negative data. Once again, this is an applicable tool.

In Table 4, for self-regulating EEG data, the application of the present invention results in a mean error rate of 0.00008% higher than the robust EM PCA.

Study # 1 is the first part of the first subject, Study # 2 is the first part of the second subject, and Study # 3 is the result of the third subject of the self-regulating EEG data. The robust EM PCA (Zhao et al., 2006) is known to be the best of the existing methods for measuring missing values for linear data.

In linear data, for example, robust EM PCA shows a relatively higher accuracy than the method according to the invention. Moreover, MSVDs exhibit lower error rates than robust EM PCAs.

Therefore, in linear data, MSVD shows high accuracy in missing value substitution. However, in nonlinear data, the method according to the invention as a whole shows a relatively lower error rate than the robust EM PCA.

In Fig. 2, data distributions (a) and (b) are interstitial EEG data and ERP EEG data, respectively. (a) and (b) show nonlinear distributions, and (c) is linear data of self-regulated EEG data. In (a), the 1-electrode has an amplitude of -200 to 200 μV and the 2-electrode has an amplitude of -300 to 200 μV. The 1- and 2-electrodes of (b) have values in the range of amplitudes -500 to 600 and -1000 to 3000 μV, respectively. (c) appears in the amplitude range of -20000 to 20000.

When applying the method according to the invention in ERP EEG data, the missing value replacement error rate shows an error rate of 0.00007% lower than the MSVD. Comparing robust EM PCA and MSVD, MSVD shows an average error rate of 0.00019% lower than robust EM PCA.

Hereinafter, the evaluation regarding the hidden variable discovery by the gradual EM PCA according to the present invention will be described in detail.

The discovery of hidden variables is important to summarize EEG time series data sets and to reduce data study time. For evaluating hidden variable findings, the method according to the invention for randomly generating data with missing values of 5%, 10% and 15% of each EEG data set to measure missing values and to find hidden variables And MSVD were applied respectively.

3 shows a comparison of the first variables found, where (a) represents epilepsy EEG data, (b) is the first hidden variable of self-regulated EEG data and (c) is the result of the first portion of ERP EEG data . Each data set contains a 5% missing rate.

In Fig. 3, the method according to the invention shows a relatively similar pattern for the hidden variables when applying the original complete data than when applying the MSVD. The top graph of FIG. 3 (a) is the first hidden variable found by applying the method according to the invention on a complete data set.

The middle part of Figure 3 (a) is the first hidden variable found after applying the method according to the invention. In this case, the pattern similar to the top of 3 (a) is shown. However, the pattern at the bottom of FIG. 3 (a) is different from the graph at the top.

The bottom graph is for MSVD. In this case, unlike the method according to the present invention, it is clear that an accurate value close to the original values cannot be expected.

3 (b) shows the results of the first portion of subject 1 in self-regulated EEG data. In the case of Figure 3 (b), it can be seen that when compared to the pattern of the first hidden variable for the method, PCA and MSVD according to the present invention, each method shows a similar pattern.

3 (c) is the first hidden variable of ERP EEG data. When comparing the pattern of hidden variables found by each of the three methods, it can be seen that the PCA and the method according to the invention are clearly separated into four different tasks, respectively.

However, in the case of MSVD, the task cannot be distinguished.

In contrast, the method according to the invention automatically finds hidden variables in real time. The method according to the invention is also applicable in various fields related to shape recognition for finding patterns and hidden variables.

Hereinafter, the reconstruction evaluation results for the progressive EM PCA according to the present invention will be described in detail.

Minimizing the reconstruction error can minimize the loss of information due to feature extraction or dimension reduction of the data, and thus can be widely applied in the field of computer vision related to image compression, image recovery and information recognition systems.

4 is a curve of reconstruction data and original data for three randomly selected sensors. The reconstruction data was reconstructed by the method according to the invention to include missing values of 5% rate.

4 (a) shows the sensor 30 and the sensor 90 in the epilepsy data recorded during each of the five types of tasks. Figure 4 (b) is a result of comparing the reconstruction data and the original data of the ERP EEG data. 4 (c) shows original and reconstructed data of three channels randomly selected from self-regulated EEG data, and shows original and reconstructed data of two and 35 channels of a first object.

In Fig. 4, bright dotted lines represent original data, and dark colored lines represent reconstruction data after applying the method according to the present invention. From Figure 4, it can be seen that the reconstruction data is similar to the graph of the original data.

Therefore, the method according to the present invention can minimize loss of information in data reduction of data including missing values, and thus can be applied to various fields for feature extraction, dimension reduction and data compression of data.

Hereinafter, the calculation complexity evaluation result for the progressive EM PCA according to the present invention will be described in detail.

이며, 이때 T는 각 시점이고 k는 은닉 변수들의 수이다. 그러나, MSVD의 복잡성은

이며, 이때 m은 표본의 수이고 n은 X의 차원이다.The complexity of the method according to the invention

Where T is each time point and k is the number of hidden variables. However, the complexity of MSVD

Where m is the number of samples and n is the dimension of X.

Therefore, the method according to the present invention reduces memory usage than robust EMPCA. 5 shows the run time of an algorithm at a ratio of various data sets and a series of lengths. In each run, 5% of data in each data set was treated as lacking. These data sets were used to compare the MSVD with the method according to the present invention.

Figure 5 (a) shows the study time of the interstitial EEG data, (b) shows the study time of the ERP EEG data, (c) shows the study time of the self-regulated EEG data, MSVD shows the highest run time . This is because when a new data t + 1 is added, all data from data 1 to data t must be used for the study.

However, the method according to the present invention uses the initial values and the weights of missing values at time t, so that even if t + 1 is added, much research time is not required. Therefore, the method according to the present invention can solve the limited memory problem to measure the progress time problem and missing data of real time data and can successfully summarize the data with the discovery of hidden variables.

As described above and described with reference to a preferred embodiment for illustrating the technical idea of the present invention, the present invention is not limited to the configuration and operation as shown and described as described above, it is a deviation from the scope of the technical idea It will be apparent to those skilled in the art that many changes and modifications can be made to the invention without departing from the scope of the invention. Accordingly, all such suitable changes, modifications, and equivalents should be considered to be within the scope of the present invention.

Claims (6)

In the method of replacing missing values using progressive EM (Principal Component Analysis),
(a) receiving an input vector x;
(b) initializing the missing value with an average of the matrix of the data matrix X when the input input vector x includes the missing value;
(c) performing PCA on the data matrix X;
(d) reconstructing the data matrix X; And
(e) replacing missing values in the data matrix X before reconstruction with missing values in the reconstructed data matrix X; Missing value replacement method using a gradual EM PCA comprising a.
The method of claim 1,
After step (e),
(f) repeating steps (c) to (e) until the error rate for the data matrix X with missing values is less than or equal to a preset threshold; Missing value replacement method using a gradual EM PCA, characterized in that it further comprises.
The method of claim 2,
After step (f),
(g) calculating a hidden variable;
(h) calculating the total hidden variable energy; And
(i) calculating the total input data energy; Missing value replacement method using a gradual EM PCA, characterized in that it further comprises.
The method of claim 3, wherein
Step (g) to step (i) is a method of replacing missing values using a gradual EM PCA, characterized in that each of the following equation.

Where y is a hidden variable, T is a score vector, W _new is an updated weight matrix, x _replace is a determined missing value, E _hv is an existing or updated total hidden variable energy, and λ is an exponent Is an oblivion factor, E _x is the existing or updated total input data energy.)
The method of claim 4, wherein
If E _hv < fE _x (f is the number of principal components), after step (i),
(j) adding a hidden variable; Missing value replacement method using a gradual EM PCA, characterized in that it further comprises.
The method of claim 4, wherein
The exponential forgetting factor [lambda] is selected from 0.96 to 0.98.

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