Time Series Of Eigenvalues Research Articles

Spectral analysis of time-dependent market-adjusted return correlation matrix

We present an adjusted method for calculating the eigenvalues of a time-dependent return correlation matrix in a moving window. First, we compare the normalized maximum eigenvalue time series of the market-adjusted return correlation matrix to that of the logarithmic return correlation matrix on an 18-year dataset of 310 S&P 500-listed stocks for small and large window or memory sizes. We observe that the resulting new eigenvalue time series is more stationary than the time series obtained without the adjustment. Second, we perform this analysis while sweeping the window size τ∈{5,…,100}∪{500} in order to examine the dependence on the choice of window size. This approach demonstrates the multi-modality of the eigenvalue distributions. We find that the three dimensional distribution of the eigenvalue time series for our market-adjusted return is significantly more stationary than that produced by classic method. Finally, our model offers an approximate polarization domain characterized by a smooth L-shaped strip. The polarization with large amplitude is revealed, while there is persistence in agreement of individual stock returns’ movement with market with small amplitude most of the time.

Cross-Correlation Dynamics in Financial Time Series

The dynamics of the equal-time cross-correlation matrix of multivariate financial time series is explored by examination of the eigenvalue spectrum over sliding time windows. Empirical results for the S&P 500 and the Dow Jones Euro Stoxx 50 indices reveal that the dynamics of the small eigenvalues of the cross-correlation matrix, over these time windows, oppose those of the largest eigenvalue. This behaviour is shown to be independent of the size of the time window and the number of stocks examined. A basic one-factor model is then proposed, which captures the main dynamical features of the eigenvalue spectrum of the empirical data. Through the addition of perturbations to the one-factor model, (leading to a ‘market plus sectors’ model), additional sectoral features are added, resulting in an Inverse Participation Ratio comparable to that found for empirical data. By partitioning the eigenvalue time series, we then show that negative index returns, (drawdowns), are associated with periods where the largest eigenvalue is greatest, while positive index returns, (drawups), are associated with periods where the largest eigenvalue is smallest. The study of correlation dynamics provides some insight on the collective behaviour of traders with varying strategies.

Spectral Analysis of Time-Dependent Market-Adjusted Return Correlation Matrix

We present an adjusted method for calculating the eigenvalues of a time-dependent return correlation matrix that produces a more stationary distribution of eigenvalues. First, we compare the normalized maximum eigenvalue time series of the market-adjusted return correlation matrix to that of logarithmic return correlation matrix on an 18-year dataset of 310 S&P 500-listed stocks for two (small and large) window or memory sizes. We observe that the resulting new eigenvalue time series is more stationary than time series obtained through the use of existing method for each memory. Later, we perform this analysis while sweeping the window size τ e {5, ..., 100} in order to examine the dependence on the choice of window size. We find that the three dimensional distribution of the eigenvalue time series for our market-adjusted return is significantly more stationary than that produced by classic method. Moreover, our model offers an approximate polarization domain of smooth L-shaped strip. The polarization with large amplitude is revealed, while there is persistence in agreement with small amplitude most of the time.

MULTISCALED CROSS-CORRELATION DYNAMICS IN FINANCIAL TIME-SERIES

The cross-correlation matrix between equities comprises multiple interactions between traders with varying strategies and time horizons. In this paper, we use the Maximum Overlap Discrete Wavelet Transform to calculate correlation matrices over different time–scales and then explore the eigenvalue spectrum over sliding time-windows. The dynamics of the eigenvalue spectrum at different times and scales provides insight into the interactions between the numerous constituents involved. Eigenvalue dynamics are examined for both medium, and high-frequency equity returns, with the associated correlation structure shown to be dependent on both time and scale. Additionally, the Epps effect is established using this multivariate method and analyzed at longer scales than previously studied. A partition of the eigenvalue time-series demonstrates, at very short scales, the emergence of negative returns when the largest eigenvalue is greatest. Finally, a portfolio optimization shows the importance of time–scale information in the context of risk management.

Cross-correlation dynamics in financial time series

The dynamics of the equal-time cross-correlation matrix of multivariate financial time series is explored by examination of the eigenvalue spectrum over sliding time windows. Empirical results for the S&P 500 and the Dow Jones Euro Stoxx 50 indices reveal that the dynamics of the small eigenvalues of the cross-correlation matrix, over these time windows, oppose those of the largest eigenvalue. This behaviour is shown to be independent of the size of the time window and the number of stocks examined. A basic one-factor model is then proposed, which captures the main dynamical features of the eigenvalue spectrum of the empirical data. Through the addition of perturbations to the one-factor model, (leading to a ‘market plus sectors’ model), additional sectoral features are added, resulting in an Inverse Participation Ratio comparable to that found for empirical data. By partitioning the eigenvalue time series, we then show that negative index returns, ( drawdowns), are associated with periods where the largest eigenvalue is greatest, while positive index returns, ( drawups), are associated with periods where the largest eigenvalue is smallest. The study of correlation dynamics provides some insight on the collective behaviour of traders with varying strategies.

Multivariate Analysis of Multichannel Surface Myoelectric Signals to Determine Muscular Fatigue

Multivariate analysis would be effective in finding the functional change from several variables obtained from multivariate biological signals. We applied this idea to the discrimination of sustained fatiguing contraction from negative ramp contraction. The time-series of eigenvalues were obtained from multidimensional biological variables by the Karhunen-Loève expansion. The results showed that, the first and second eigenvalues came close to each other during fatiguing contraction, whereas only the first eigenvalue was dominant during negative ramp contraction. Moreover, the factor loadings showed considerable difference between fatiguing contraction and negative ramp contraction. As a result, the muscular-fatigue-related functional change could be represented clearly by the time-series of proportions and factor loadings.

Multivariate analysis at switching points between different types of muscle contractions

The biomedical signal often is affected by the anatomical structure, the method of measurement and the analytical approach which makes it difficult to determine the feature of the change only by observing the signal. Then it is important to analyze the biological activity comprehensively from the biomedical signals measured through multiple channels or from the multiple parameters estimated from a single biomedical signal. From the viewpoint of the analysis of a long-term biomedical signal, one of the most important indices is the switching point of biological activity. In the analysis of the economical indices, the prediction of earthquakes, and the weather forecasts, it is usual in practice to estimate the switching point for the trend from the time-series data. The goal of this study is to estimate the switching point of the behavior of the biological system from the multiple time-series. The statistical properties of the switching point are investigated as the preliminary study. Four evaluation parameters are estimated from the surface electromyogram of the tibialis anterior muscle and the eigenvalue time-series is determined from the multiple evaluation parameter time-series by Karhunen-Loéve (KL) expansion. The results showed that two eigenvalues λ1, and λ2 approached each other near the switching point between the different muscle contractions and only λ1 exhibited a large value at other phases. The distribution of the principal components also showed a change near the switching point. The forementioned tendency was observed in approximately 55 percent of five trials for 10 subjects.