**2. Related work**

PCA was first introduced into mechanics by [22], as an analogue of the axis theorem. It was later named "PCA" by [23]. The range of applications in finance and economics is extensive. Take as an example [24], who used PCA to document three factor structures. Stock and Watson [25] used PCA to monitor economic development and activity, as well as the inflation index. Egloff et al. [26] used PCA as a way to analyze the dimensions of inconsistent dynamics. Volatility is a statistical measure that can be used to determine these inconsistencies using a two-factor volatility model. This includes long-term and short-term fluctuations in the volatility structure. Baker and Wurgler [27] used PCA to measure investors sentiment, i.e., their positive or negative view. This was done according to the principle of the number of sentiment proxies before Baker, [28] created the policy uncertainty index. This index represents potential risks in the near future.

The most important item in the construction of PCA is the estimation of the eigenvalues of the covariance matrix sample. Anderson and Weeks [29] and Anderson [30] showed that sample eigenvalues were consistent when dealing with asymptomatic sentiment proxy results. Waternaux [31] proved that similar results are obtained with simple eigenvalues as long as there is a fourth moment in the data. In addition to the discussions in the [32] book, [33] was able to establish the asymptotic distribution of eigenvectors using generalized assumptions.

However, this PCA approach to eigenvalues has some downsides. The first problem is certainly dimensionality, which can be noticed when the cross sectional dimension grows simultaneously with the sample in the same period. Then inconsistencies occur. Another problem arises from linear data types that do not include nonlinear patterns. A third problem [34] arises from the dependence of the asymptotic theory on fixed assumptions for the analysis. For these reasons, we have a problem when we use PCA for reimbursement data. Most of the time, we need years of data to make an assumption, which in turn leads to other problems, such as permanence and consistency of non-fixed parameters. This type of data has backlogs and volatility times often vary.

These problems stimulate the improvement in this field and motivate the development of tools for PCA methods. The approach to the problem, where the number of occurances grows in fixed time periods, touches all the listed downsides. Theoretically, it is known that as the frequency of the sample increases, the estimated variance and covariance increase. This is true until the microstructure of the market begins to take effect. Incidentally, this is not a serious problem if we choose a sampling frequency of minutes, which we use as opposed to the below one second time interval most often used for liquid stocks. A high frequency asymptotic analysis with the crosssectional dimension is expected as the time interval increases sharply. This high frequency asymptotic framework allows us to perform non-parametric analysis as well as independent, non-static and analysis without underlying parameters as is the case with low frequency processes.

Asymptotic theory is very common in many contexts. Jacod et al. [13] and Jacod and Podolskij [35] also dealt with one problem that we deal with in this paper, where the cross sectional dimensions are invariant and the process is continuous. Mykland and Zhang [36] designed an alternative theory to the one put forward by [37], that discuss inference for volatility function dependence. It is based on the aggregation of local estimates and uses a finite number of blocks. Saha et al. [38] considered the expected values of the integrated covariance matrix under conditions where there is an error measure and the matrix is large containing high frequency data. Tao et al. [39] addressed work on the convergence rate. Jacod and Rosenbaum [40] analyzed estimators, composed of aggregating functions of estimates. They did so using integrated quarticity estimation. Heinrich and Podolskij [41] discussed empirical covariate matrices of Brownian integrals. Here is discussed the measurement of the leverage effect and its evaluation by the integrated correlation method [42].

PCA analysis can be used in analysis of financial data for different purposes. For example [43] used it to identify the type of impact on grouped impact factors, such as assessing the quality of accounting information and facilitating the process of financial analysis conducted by different users. On the other hand, [44] used PCA to assess the impact of the evolution of Finnish standards on IFRS (International Financial Reporting Standards). Finally [45] used PCA analysis to determine the macroeconomic impact on the profitability of Romanian listed companies, using data from 1997 to 2007, and identified following indicators: liquidity, solvency, and firm's dimension.

When it comes to the use of PCA analysis in financial statements analysis, four papers that focus on Romanian listed companies will be reviewed first. All papers emphasize the importance of using PCA analysis in the analysis of key financial ratios. In the first paper author [46] analyzed the data of 16 initial variables which he grouped into 3 new variables (general efficiency indicator, indicator in correlation with historical debts of companies and development indicator (given long-term debt and deferred income). Those three variables where able to explain 96.72% of initial variability. In the second paper, [47] analyzed data for 2010 including initially seven indicators of standard financial analysis and they reduced them to only two (which explain 94% of initial variability). In third paper, [48] used data from the stock exchange in the period 2006–2011 to identify the main components of financial statements which explain 79.08% of initial variability. The same group of indicators has been used by [43] on research sample that consisted of 111 companies from Madrid stock exchange and 32 companies from Eurostoxx50 for reporting periods 2005–2007. Research results showed that those six indicators explained 87% of total variance, with the first two indicators at app 44% of total variance.
