1 Introduction
Factor models are important tools for reducing the dimension of observed data and extracting the relevant information. They are used for modeling a large number of variables through a small number of unobserved variables to be estimated in many applications. With the emergence of big data in many fields, especially the increasing data dimensionality, extensive studies on the estimation of highdimensional factor models have been conducted.
Bai and Ng bai2002determining proposes using information criteria for estimating the number of factors, which is developed under the framework of high data dimensions (), seriously different from the previous methods lewbel1991rank ; connor1993test ; cragg1997inferring ; forni1998let developed under the assumption that the data dimension is fixed or small. A critical assumption made in the work is the factors’ cumulative effect on grows proportionally to . Stock and Watson stock2002forecasting suggests using principal components for estimating factors in highdimensional datasets. Kapetanios kapetanios2004new ; kapetanios2010testing first proposes exploiting a structure of residual terms in the approximate factor models. Based on Kapetanios’s work, Onatski onatski2010determining relaxes the restrictions on the covariance structure of the residual terms and develops a new consistent estimator for estimating the number of factors. Harding harding2013estimating
imposes restrictions on the spatialtemporal correlation patterns of the residual terms, and proposes an estimation method for the number of factors by relating the moments of the empirical spectral density (ESD) of covariance matrices of observed data to the parameters regarding the spatialtemporal correlations. Yeo and Papanicolaou
yeo2016random presents a new approach to estimate the number of factors by connecting the factor model estimation problem to the limiting spectral density (LSD) of covariance matrices of the residuals, in which two strict assumptions are made: one is the spatial correlation of the real residuals can be completely eliminated by removing the estimated number of factors; the other is the residuals follow an AR(1) process.Based on Yeo and Papanicolaou’s work, in this paper, we relax the restrictions for the structure of the residuals and propose approaching the LSD of covariance matrices of the residuals through a multiplicative covariance structure model with an controllable parameter. This allows the proposed approach being more flexible and practical in analyzing the realworld data with complex correlation structure residuals. Take the power flow data for example, the classical physical model in matrix form is as follows,
(1)  
where denote the variations of regarding variables and is the inverse of the Jacobian matrix. is the observed data (e.g., voltage amplitude and phase angle), are considered as the signals (e.g., active and reactive power), and represents small random fluctuations or measuring errors. Since a lot of measurement noise is contained in the residual term and the spatialtemporal correlations among its entries are complex, it is unreasonable to model the residuals from power data based on crude assumptions and simplifications as in Yeo and Papanicolaou’s work.
Inspired by the idea of decomposing the observed data into systemic components (factors) and idiosyncratic components (residuals), we consider an approximate factor model for variables and observations as follows,
(2) 
where is an observed data matrix, is an ( is the number of factors) factor loading matrix, is an matrix of factors, and is an residual matrix.
One simple way to estimate is using the principal components and assuming as pure noise. However, our approach mainly focuses on and we estimate the number of factors and the ESD of covariance matrix of simultaneously. The main advantages of the proposed approach can be summarized as follows:
(i) It relaxes restrictions on the structure of the residuals . pure noise or just temporalcorrelation assumption for the residuals is crude and unreasonable in practice. Instead of modeling with strict structure item, the proposed approach prefers approaching the ESD of covariance matrix of through a multiplicative covariance structure model with an controllable parameter, which makes the approach more flexible and practical.
(ii) The proposed approach uses free probability techniques in RMT to derive the LSD of the built multiplicative covariance model, which greatly simplifies the calculation process and ensures the efficiency of the approach.
(iii) It relates the estimation of the number of factors to the ESD of covariance matrix of , which allows controlling both the number of factors and the spectral shape of the residuals.
(iv) The theoretical studies on the synthetic data generated from Monte Carlo experiments show the proposed approach is robust to noise and sensitive to the weak factors, and the built multiplicative covariance structure can fit the ESD of covariance matrices of the autocross(weak)correlation structure residuals better than the AR(1) model in Yeo and Papanicolaou’s approach.
(v) By using the power data generated from IEEE 118bus test system, the estimators in the proposed approach are proved to be sensitive in indicating the number and scale of anomaly events occurred in the power system.
(vi) With the realworld online monitoring data from a power grid, the estimators in the proposed approach are found to be successful in indicating the system states.
The rest of this paper is organized as follows. In Section 2, we apply the MarchenkoPastur law for the residuals from both synthetic data and realworld power data. In Section 3, we present our approach for the estimation of highdimensional factor models. In Section 4, by using the synthetic data generated from Monte Carlo experiment, we evaluate the performance of our approach and compare it with that developed by Yeo and Papanicolaou in terms of detecting weak factors and convergence rate. Section 5 shows the applications of our approach to power data analysis. In Section 6, conclusions are presented.
2 Motivation Example
MarchenkoPastur law (MP law): Let be an random matrix, whose entries are independent identically distributed (i.i.d.) variables with the mean
and the variance
. The corresponding covariance matrix is defined as . As but , according to the MP law marvcenko1967distribution , the ESD ofconverges to the limit with probability density function (PDF)
(3) 
where , .
In this section, we first apply the MP law for the residuals from the synthetic data generated by the following model,
(4) 
where , , and are independent. The true number of factors is set to be 4. As is shown in Fig. 1, with the factors removed continuously, the ESD of covariance matrices of the residuals converges to the MP law.
In contrast, we apply the MP law for the residuals from the realworld online monitoring data in a power grid. Let matrix be the sampling data with , and is the residual matrix obtained by subtracting principal components from . We convert into the standard form through
(5) 
where , , and . As is shown in Fig. 2, no matter how many factors are removed, the ESD of covariance matrices of the residuals from the realworld data does not fit to the MP law. Therefore, it is necessary to build a new model to fit the ESD from real residuals in estimating factor models.
3 Methodology
In this section, we propose our approach for the estimation of highdimensional factor models. In Section 3.1, we provide preliminaries that will be used in the proposed approach. In Section 3.2, we introduce a new factor model estimation approach, which connects the estimation of the number of factors to the ESD of covariance matrices of the residuals. Considering a lot of measurement noise is contained in the residuals and the complex correlation structure of the residuals from power data, an approaching way is proposed for deriving the LSD of covariance matrices of the residuals. Specific steps of the proposed approach are given in Section 3.3, during which free probability techniques are used for deriving the spectral distribution of the built multiplicative covariance structure model.
3.1 Preliminaries
Definition 1
For a random matrix , the empirical spectral density of is defined as,
(6) 
where for
denote the eigenvalues of
, and is the Dirac delta function centered at .Definition 2
The limiting spectral density of is defined as the limit of (6) as .
Definition 3
The Stieltjes Transform (Green’s Function) of is defined as,
(7) 
and can be reconstructed through
(8) 
Definition 4
The th moment of is defined as,
(9) 
Definition 5
The moment generating function as a power series at zero is defined as,
(10) 
and its relation to the Green’s function is
(11) 
Definition 6
Let () be a unital algebra with a unital linear functional. Suppose are unital subalgebras, then are freely independent (or just free) mingo2017free with respect to if whenever for and such that

for

with for

Definition 7
Given the functional inverse of the moment generating function , the Stransform speicher1994multiplicative ; voiculescu1992free is defined as,
(12) 
Theorem 1
Let and are two freely invariant random matrices, the Stransform of the product is simply the product of their Stransforms
(13) 
3.2 Factor model estimation
The proposed estimation approach aims to match the LSD calculated from the modeled multiplicative covariance matrix to the ESD of covariance matrix of the real residuals that are obtained by subtracting principal components. By minimizing the distance between the two spectrums, the estimators are obtained.
The first step is to obtain the ESD of covariance matrix of the real residuals. For highdimensional data, the principal components are able to approximately mimic all true factors stock2002forecasting . Here, we use principal components to represent factors and the real residuals are obtained by subtracting factors from the observed data, which is defined as
(14) 
where is the number of factors, is an
matrix which is given as eigenvectors corresponding to the
largest eigenvalues of , and is an matrix which is estimated by . The covariance matrix of the real residuals can be calculated as,(15) 
where the subscript indicates it is constructed from the real residuals. Thus we can obtain the ESD of , which is denoted as .
The next step is to model the covariance matrix of the real residuals. Here, we factorize into crosscovariances and autocovariances, namely,
(16) 
the coefficients and are respectively collected into an crosscovariance matrix and a autocovariance matrix , both are symmetric and positivedefinite. The crosscovariance matrix is a way to model the weak spatial (cross) correlation of the residuals, because the main spatial correlations can be effectively eliminated by removing factors (principal components). The autocovariance matrix is used to model the temporal (auto) correlation of the residuals. In order to obtain the LSD of , one simple way is to consider
as an identity matrix
and model as the covariance AR(1) matrix based on the crude assumptions that the spatial correlations of the residuals can be completely removed from factors and the residuals follow an AR(1) process. However, for the power data, a lot of measurement noise (which is usually considered to be random) is contained in the residuals and the spatialtemporal correlations of the residuals are uncertain. Here, instead of modeling and directly, we prefer approaching the LSD of through a multiplicative covariance structure with an controllable parameter ,namely,(17) 
where the subscript denotes it is constructed from the modeled multiplicative covariance matrix, , is an random Gaussian matrix, and which ensures the spectral distribution of converges to a nonrandom limit as . The LSD of
can be derived by using free probability theory (FRT) in Section
3.3, which is denoted as .The last step is to search for the optimal parameter set by minimizing the distance between and , which is denoted as,
(18) 
where is a spectral distance measure. In yeo2016random
, several distance metrics are tested and JensenShannon divergence is proved to be the most sensitive to the presence of spikes (i.e., the deviating eigenvalues in the spectrum) as well as correctly reflecting the distribution of the bulk (i.e., the grouped eigenvalues in the spectrum). Here, we choose JensenShannon divergence as the spectral distance measure, which is a symmetrized version of KullbackLeibler divergence and defined as,
(19) 
where . It can be seen that becomes smaller as approaches , and vice versa. Therefore, we can match to by minimizing , through which the optimal parameter set is obtained.
3.3 free probability theory for the calculation of
As is discussed in Section 3.2, is easily obtained by removing principal components from the real data, but the implementation of calculating from the Stieltjes transform for the multiplicative covariance structure is difficult. Here, free probability theory is used to derive the LSD of . The prescription is shown as follows:

Obtain the LSDs of , denoted as . Consider the case that involved in (17) are zeromean with variance and , we can obtain by using the MP law, namely,
(20) where , , and .

Calculate the Stieltjes transform for according to (7), denoted as .

From , deduce the corresponding moment generating function according to (11).

From , deduce the corresponding Stransform according to (12).

Since and are two freely invariant random matrices, according to Theorem 1, the Stransform for is calculated as,
(21) 
Obtain the LSD from through (8).
In order to approximate as much as possible, we allow an controllable parameter in the built multiplicative covariance model: the radio rate regarding . Fig. 3 illustrates the spectrum distribution of with different . For small , the spectral density resembles the MP law. As increases, the shape of the spectrum becomes ‘thinner’ and more heavily tailed, which resembles the inverse process of continuously removing factors from the realworld online monitoring data in Section 2. By controlling and simultaneously, our approach is more flexible and accurate in estimating highdimensional factor models.
Combining Section 3.2, the proposed factor model estimation approach is summarized as in Algorithm 1.
Algorithm 1: Procedure of factor model estimation 

The observed data matrix . 
The estimated number of factors , and the ratio rate . 

4 Theoretical Studies
In this section, we first evaluate the performance of the proposed approach by using the synthetic data generated from Monte Carlo experiment, in which different correlation structures are set for the synthetical residuals. Then we compare the performance of our approach with that proposed by Yeo and Papanicolaou in terms of detecting weak factors and convergence rate.
4.1 Data Generation
The synthetic data is generated from the model used in Yeo and Papanicolaou’s work yeo2016random . This model is also used in many other literatures, like Bai and Ng bai2002determining , Onatski onatski2010determining , and Ahn and Horenstein ahn2013eigenvalue , etc. The model is written as,
(23) 
where
(24) 
and
(25) 
with . The explanations for this model are as follows:

, which makes the residual level controlled only by .

, where represents the signaltonoise radio and it is defined as .

controls the degree of autocorrelations in the residuals.

controls the magnitudes of crosscorrelations in the residuals.

controls the affecting ranges of the crosscorrelations in the residuals. Considering the local crosscorrelations can be broader with the increase of data dimensions, is usually set to be proportional to .
Combining the characteristics of the data from power system, our simulation experiments have several perspectives. Firstly, since the signaltonoise rate for power data is usually at a low level, is set to be small values in the experiments. Next, considering the main crosscorrelations in the residuals can be eliminated by removing factors, is set to be much smaller than , and the effects of different combinations of them are tested. Lastly, different sample sizes are set to test the performance of the proposed approach and is set to be . Parameter configurations in the Monte Carlo experiment are shown in Table 1
Sample sizes  {50,100,200,300,500}  

Number of factors  {2,3,4}  
1/SNR  {1/10000,1/1000,1/100,1/10,1}  
Correlations in residuals  {(0,0,0),(0.5,0,0),(0,0.05,/10),(0.5,0.05,/10)} 
4.2 Performance of Our Approach
The performance of our approach is tested by using the generated data in Section 4.1. There are four different residual correlation structures, i.e., no correlation (), autocorrelationonly (), cross(weak)correlationonly (), autocross(weak)correlation (). The true number of factors is set to be . Average values of the estimated and over simulations are shown in Table 2.
It can be observed that the average estimator is almost equal to the true number of factors for a broad range of and for the cases . For the case , the number of estimated factors is about , because several weak factors caused by the weak crosscorrelation of the residuals are presented. It indicates our approach has powerful ability to identify weak factors. It can also be observed that the estimators become more accurate with the increase of the sample size. Meanwhile, varied correlation structures of the residuals are tested in the experiments and the corresponding examples of the fitting results of our approach for the synthetical residuals are shown in Fig. 4. controls the autocorrelation magnitude for the residuals and measures the crosscorrelation within the range of in the residuals. As seen from column in Table 2, It can be concluded that the estimator is affected both by the auto and crosscorrelations of the residuals, while the estimator is affected by the crosscorrelation of the residuals.
3.000  0.5851  3.000  0.7405  10.010  0.6395  2.948  0.7564  
3.000  0.5910  2.998  0.7435  10.000  0.6366  3.000  0.7534  
3.000  0.6019  3.010  0.7366  10.045  0.6494  3.061  0.7682  
3.006  0.5930  3.007  0.7415  10.047  0.6831  2.924  0.7484  
3.011  0.5999  3.033  0.7435  10.045  0.6702  3.199  0.7257  
3.000  0.5772  3.099  0.7524  10.030  0.6399  3.017  0.7445  
3.000  0.5801  3.031  0.7524  10.005  0.6380  3.274  0.7484  
3.000  0.5811  2.900  0.7583  10.031  0.6330  3.101  0.7544  
3.000  0.5801  3.000  0.7564  10.010  0.6399  3.382  0.7494  
3.002  0.5891  3.045  0.7425  10.023  0.6380  3.300  0.7405  
3.000  0.6366  3.000  0.7187  10.003  0.6633  3.000  0.7405  
3.000  0.6247  2.998  0.7088  10.000  0.6534  2.996  0.7474  
3.000  0.6286  3.002  0.7316  10.000  0.6435  3.132  0.7465  
3.000  0.6207  2.999  0.7227  10.003  0.6593  2.946  0.7395  
3.000  0.6336  3.000  0.7118  10.005  0.6583  3.161  0.7286  
3.000  0.5841  3.000  0.7653  10.000  0.6310  3.000  0.7702  
3.000  0.5712  3.000  0.7613  10.005  0.6310  3.099  0.7663  
3.000  0.5782  2.998  0.7603  10.000  0.6390  3.099  0.7732  
3.000  0.5792  3.010  0.7712  10.000  0.6320  3.000  0.7603  
3.000  0.5722  3.004  0.7672  10.001  0.6300  3.099  0.7752 
4.3 Comparison with other approaches
In Yeo and Papanicolaou’s work yeo2016random , the estimators from their approach are compared with the BIC3 estimator of Bai and Ng bai2002determining , the ED estimator of Onatski onatski2010determining , and the ER estimator of Ahn and Horenstein ahn2013eigenvalue
in detail. It shows Yeo and Papanicolaou’s approach converges the fastest when the noise level is high and has more powerful ability to identify weak factors than other methods. In this section, we mainly compare the performance of our free probability (FP) approach with that of Yeo and Papanicolaou’s free random variable (FRV) method.
Fig. 5 shows the JensenShannon (JS) divergences of and regarding the sample size and the signaltonoise radio , calculated through FRV and FP approach respectively. In the simulations, the true number of factors was set to be , and . Combining the characteristics of the real residuals from power data, autocross(weak)correlation structure is set for the synthetical residuals, i.e., , and . As is seen from the figure, the optimal JS divergences calculated though FP approach are smaller than those from FRV, which indicates that our built multiplicative covariance model can fit the residuals better than that based on FRV. What’s more, our estimation approach has a faster convergence rate than FRV, especially for the small sample size. When the sample size is large, both FRV and FP approaches converge very well, regardless of the noise levels.
5 Empirical Studies
In this section, we illustrate the proposed approach by using the realworld online monitoring data collected from a power grid and the power flow data generated from IEEE 118bus test system. We first check how well our built model can fit the residuals from the real data. Then, implications of and are explored by using the power flow data, in which we track the evolutions of and by moving a window on the data at continuous sampling times.
5.1 Fit of our model to real data
The realworld online monitoring data are threephase voltages collected from monitoring devices installed on the low voltage side of distribution transformers within one feeder. The data is sampled every minutes and the sampling time is from 2017/3/1 00:00:00 to 2017/3/31 23:45:00. Thus, a data set is formulated. Instead of taking the entire matrix for analysis, we move a window on the data set at continuous sampling times. Fig. 6 shows several sample fitting results of our built multiplicative covariance model to the real residuals. It can be observed that our built multiplicative covariance model can fit the residuals well, while the MP law does not. What’s more, it is noted that the estimated and are different for the data sampled at different sampling moments, which explains why the estimators in our approach can be used to indicate the system states.
5.2 Implication of
The power flow data generated from IEEE 118bus test system zimmerman2011matpower is used to explore the implication of . The IEEE 118bus test system represents a portion of the U.S. Midwest Electric Power System, and it is edited into IEEE Common Data Format and PECO PSAP Format by Richard Christie from the University of Washington richard1993 . In the early 2000’s, researchers from the Illinois Institute Technology (IIT) work with the system and add some line characteristics IIT pena2018extended . The oneline diagram of the IEEE 118bus test system is shown in Fig. 7. It consists of buses, branches, load sides and generators with a total installed capacity of 7220MW.
In the data generation process, a sudden change of the active load at one bus is considered as an anomaly event and a little white and autoregressive (AR(1)) noise is introduced to represent random fluctuations and measuring errors. The anomaly events can cause the variation of the data’s crosscorrelations. From Section 4.2, we know that is mainly affected by the crosscorrelation of the data. Here, in order to explore the relations between the number of anomaly events and , different number of anomaly events are set, which is shown in Table 3. The generated data contains voltage measurement variables with sampling times, which is shown in Fig. 8. Thus, a data set is formulated. In the experiment, we move a window at continuous sampling times on the data set, which enables us to track the temporal evolutions of .
Bus  Sampling Time  Active Load(MW) 

20  
30  
60  
Others  Unchanged 

is the signaltonoise ratio, which is set to be .

represents random white gaussian noise.

represents the autoregressive noise, and the correlation coefficient is set to be .
The timeseries of generated with continuously moving windows is shown in Fig. 9. The relations between the number of anomaly events and the parameter are stated as follows:
I. From to , the estimated remains almost constant at . The fitting result of our built model to the residuals during this period of time (such as ) is shown in Fig. 10(a). In the experiment, no strong factors are observed during this period of time. The most likely explanation is that the proposed approach is sensitive to the weak factors caused by random fluctuations or measuring errors and can identify them effectively.
II. From to , two strong factors are observed in the experiment and the average estimated is between and , during which one anomaly event is contained in the moving window. The fitting result of our built model to the residuals during this period of time (such as ) is shown in Fig. 10(b). From to , three strong factors are observed and the average number of estimated factors is between and , during which two anomaly events are contained in the moving window. The fitting result of our built model to the residuals during this period of time (such as ) is shown in Fig. 10(c). From , four strong factors are observed and the average estimated is about , during which three anomaly events are contained in the moving window. The fitting result of our built model to the residuals during this period of time (such as ) is shown in Fig. 10(d). It can be concluded that is driven by the number of anomaly events.
III. From to , the value of decreases by every other sampling times, because the width of the moving window is and the number of anomaly events contained in the moving window decreases by every sampling times. It validates the conclusion that is driven by the number of anomaly events.
IV. From , no strong factors are observed and remains nearly , which validates that the proposed approach is sensitive to the weak factors caused by random fluctuations or measuring errors.
5.3 Implication of
From Section 4.2, we know that is affected both by the cross and autocorrelation of the data in our approach. The number of anomaly events can cause the variation of the data’s crosscorrelations. In this section, we first explore how the number of anomaly events affects by using the generated data in Fig. 8. In the experiment, a window is moved on the data set at continuous sampling times and the generated curve is shown in Fig. 11(a). The relations between the number of anomaly events and are stated as follows:
I. From to , no anomaly events occur and remains almost constant.
II. From to , increases by every other sampling times for the number of anomaly events contained in the moving window increases by every sampling times. From to , decreases by every other sampling times for the number of anomaly events contained in the moving window decreases by every sampling times. It shows is positively affected by the number of anomaly events contained in the moving window, because the crosscorrelations of the residuals varies with the number of anomaly events. It validates our assumption that the crosscorrelation of the residuals can not be completely eliminated by removing factors, i.e., weak crosscorrelation structure assumption for the residuals.
III. From , no anomaly events are contained in the moving window and returns to a constant and remains afterwards.
Meanwhile, the scale of anomaly events can affect the variation of the data’s autocorrelations. Here, we explore how the scale of anomaly events affects . Assumed events with different scales are set for bus , which is shown in Table 4. The generated data contains voltage measurements with sampling times. A window is moved on the data set at continuous sampling times and the generated curve is shown in Fig. 11(b). The relations between the scale of anomaly events and are stated as follows:
Bus  Sampling Time  Active Power(MW) 
20  
Others  Unchanged 

The parameters are set the same as in Table 3.
I. From to , the estimated remains almost constant, which indicates no anomaly events occur and the system operates in normal state.
II. From to , the curves are almost inverted Ushaped, because anomaly events in Table 4 are set and the delay lags of anomaly events to are equal to the moving window’s width. It is noted that the estimated corresponding to the anomaly event of the active power (AP) from to has the largest value and that of the AP from to has the smallest value, which indicates increases with the scale of anomaly events. Because the scale of anomaly events is positively related to the autocorrelation of the residuals from the power data.
III. From , the estimated returns to constant and remains afterwards, which indicates the system has returned to normal state.
6 Conclusions
The spectrum from realworld power data is complex and cannot be trivially dissected by the MP law. In this paper, we propose a new approach to estimate factor models by connecting the estimation of the number of factors to fitting the ESD of covariance matrices of the residuals. Considering a lot of measurement noise is contained in the power data and the uncertain correlation structure of the residuals from the power data, our approach prefers approaching the ESD of covariance matrices of the residuals by using a multiplicative covariance structure model, which avoids making crude assumptions or simplifications on the complex correlation structure of the data. The free probability techniques in random matrix theory ensure the efficiency of the proposed approach.
Theoretical studies show that the proposed approach is robust to noise and has powerful ability to identify weak factors. The built multiplicative covariance structure model can fit the ESD of covariance matrices of the residuals better and has a faster convergence rate compared with that developed by Yeo and Papanicolaou. Empirical studies show that the estimators in our approach effectively characterize the number and scale of anomaly events in a power system, which can be used to indicate the system states.
Appendix A
Let be an random matrix, whose entries are independent identically distributed (i.i.d) variables with the mean and the variance . The covariance matrix of is calculated as,
(26) 
As but , according to the MP law, the spectral density of is obtained as
(27) 
where , , and .
References
References
 (1) J. Bai, S. Ng, Determining the number of factors in approximate factor models, Econometrica 70 (1) (2002) 191–221.
 (2) A. Lewbel, The rank of demand systems: theory and nonparametric estimation, Econometrica: Journal of the Econometric Society (1991) 711–730.
 (3) G. Connor, R. A. Korajczyk, A test for the number of factors in an approximate factor model, the Journal of Finance 48 (4) (1993) 1263–1291.
 (4) J. G. Cragg, S. G. Donald, Inferring the rank of a matrix, Journal of econometrics 76 (12) (1997) 223–250.
 (5) M. Forni, L. Reichlin, Let’s get real: a factor analytical approach to disaggregated business cycle dynamics, The Review of Economic Studies 65 (3) (1998) 453–473.
 (6) J. H. Stock, M. W. Watson, Forecasting using principal components from a large number of predictors, Journal of the American statistical association 97 (460) (2002) 1167–1179.
 (7) G. Kapetanios, A new method for determining the number of factors in factor models with large datasets, Tech. rep., Working Paper, Department of Economics, Queen Mary, University of London (2004).
 (8) G. Kapetanios, A testing procedure for determining the number of factors in approximate factor models with large datasets, Journal of Business & Economic Statistics 28 (3) (2010) 397–409.
 (9) A. Onatski, Determining the number of factors from empirical distribution of eigenvalues, The Review of Economics and Statistics 92 (4) (2010) 1004–1016.
 (10) M. Harding, Estimating the number of factors in large dimensional factor models, Journal of Econometrics.
 (11) J. Yeo, G. Papanicolaou, Random matrix approach to estimation of highdimensional factor models, arXiv preprint arXiv:1611.05571.
 (12) V. A. Marčenko, L. A. Pastur, Distribution of eigenvalues for some sets of random matrices, Mathematics of the USSRSbornik 1 (4) (1967) 457.
 (13) J. A. Mingo, R. Speicher, Free probability and random matrices, Vol. 4, Springer, 2017.
 (14) R. Speicher, Multiplicative functions on the lattice of noncrossing partitions and free convolution, Mathematische Annalen 298 (1) (1994) 611–628.
 (15) D. V. Voiculescu, K. J. Dykema, A. Nica, Free random variables, no. 1, American Mathematical Soc., 1992.
 (16) S. C. Ahn, A. R. Horenstein, Eigenvalue ratio test for the number of factors, Econometrica 81 (3) (2013) 1203–1227.
 (17) R. D. Zimmerman, C. E. MurilloSánchez, R. J. Thomas, et al., Matpower: Steadystate operations, planning, and analysis tools for power systems research and education, IEEE Transactions on power systems 26 (1) (2011) 12–19.
 (18) R. Christie, Power systems test case archive, Aug. 1993, [Accessed Feb. 4, 2015]. [online]. Available:http://www.ee.washington.edu/research/pstca/pf118/pg_tca118bus.htm.
 (19) IIT, Index of data illinois institute of technology, Illinois Inst. Technol.,Chicago, IL, USA,[online]. Available:http://motor.ece.iit.edu/data/.
 (20) I. Pena, C. B. MartinezAnido, B.M. Hodge, An extended ieee 118bus test system with high renewable penetration, IEEE Transactions on Power Systems 33 (1) (2018) 281–289.
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