Mieko Tanaka-Yamawaki

Cross Correlation of Intra-Day Stock Prices in Comparison to Random Matrix Theory

We propose and apply a new algorithm of principal component analysis which is suitable for a large sized, highly random time series data, such as a set of stock prices in a stock market. This algorithm utilizes the fact that the major part of the time series is random, and compare the eigenvalue spectrum of cross correlation matrix of a large set of random time series, to the spectrum derived by the random matrix theory (RMT) at the limit of large dimension (the number of independent time series) and long enough length of time series. We test this algorithm on the real tick data of American stocks at different years between 1994 and 2002 and show that the extracted principal components indeed reflects the change of leading stock sectors during this period.

Testing Randomness by Means of RMT Formula

We propose a new method of testing randomness by applying the method of RMT-PCA, which was originally used for extracting principal components from a massive price data in the stock market. The method utilizes RMT formula derived in the limit of infinite dimension and infinite length of data strings, and can be applied to test the randomness of very long, highly random data strings. Although level of accuracy is not high in a rigorous sense, it is expected to be a convenient tool to test the randomness of the real-world numerical data. In this paper we will show the result of applying this method (RMT-test) on two machine-generated random numbers (LCG, MT), as well as artificially distorted random numbers, and examine its effectiveness.

Testing Randomness by Means of Random Matrix Theory

Random matrix theory (RMT) derives, at the limit of both the dimension N and the length of sequences L going to infinity, that the eigenvalue distribution of the cross correlation matrix with high random nature can be expressed by one function of Q = L/N .U sing this fact, we propose a new method of testing randomness of a given sequence. Namely, a sequence passes the test if the eigenvalue distribution of the cross correlation matrix made of the pieces of a given sequence matches the corresponding theoretical curve derived by RMT, and fails otherwise. The comparison is quantified by employing the moments of the eigenvalue distribution to its theoretical counterparts. We have tested its performance on five kinds of test data including the Linear Congruential Generator (LCG), the Mersenne Twister (MT), and three physical random number generators, and confirmed that all the five pass the test. However, the method can distinguish the difference of randomness of the derivatives of random sequences, and the initial part of LCG, which are distinctly less random than the original sequences.

Moment Approach for Quantitative Evaluation of Randomness Based on RMT Formula

We develop in this article a quantitative formulation of the randomness-test based on the random matrix theory (RMT-test), in order to compare a subtle difference of randomness between given random sequences. Namely, we compare the moments of the actual eigenvalue distribution to the corresponding theoretical expression that we derive from the formula theoretically derived by the random matrix theory. We employ the moment analysis in order to compare the eigen-value distribution of the cross correlation matrix between pairs of sequences. Using this method, we compare the randomness of five kinds of random data generated by two pseudo-random generators (LCG and MT) and three physical generators. Although the randomness of the individual sequence can be quantified in a precise manner using this method, we found that the measured values of randomness fluctuate significantly. Taking the average over 100 independent samples each, we conclude that the randomness of the random data generated by the five generators are indistinguishable by the proposed method, while the same method can detect the randomness of the derivatives of the sequences, or the initial part of LCG, which are distinctly lower.

Extracting principal components from pseudo-random data by using random matrix theory

We develop a methodology to grasp temporal trend in a stock market that changes year to year, or sometimes within a year depending on numerous factors. For this purpose, we employ a new algorithm to extract significant principal components in a large dimensional space of stock time series. The key point of this method lies in the randomness and complexity of the stock time series. Here we extract significant principal components by picking a few distinctly large eigenvalues of cross correlation matrix of stock pairs in comparison to the known spectrum of corresponding random matrix derived in the random matrix theory (RMT). The criterion to separate signal from noise is the maximum value of the theoretical spectrum of We test the method using 1 hour data extracted from NYSE-TAQ database of tickwise stock prices, as well as daily close price and show that the result correctly reflect the actual trend of the market.

Predicting the Security Levels of Stock Investment by Using the RMT-test

Abstract The authors propose to use the degree of randomness of high frequency price time series for the purpose of measuring the security levels of stock investments. The RMT-test is employed as a tool to measure the randomness. The data to be analyzed are the tick-wise price time series of selected stocks in the Tokyo Stock Exchange Market for three years from 2007 to 2009. The result shows that the stock of the highest randomness is a stable stock that belongs to the sector of electric/gas power supply, which turns out to be more profitable than the Nikkei Average Price throughout the following year. This indicates that the suitable stocks to invest under a bear market have higher randomness that belongs to the category of ‘defensive’ stocks, according to the new classification method introduced by Tanaka-Yamawaki, et. al., while the suitable stocks to invest under a bull market have lower randomness that belong to the category of ‘outer demand’ and ‘market sensitive’ stocks in the same classification method.

Short-Term Price Prediction and the Selection of Indicators

Although the prediction of the future price is known to be hard due to the strong randomness inherent in the price fluctuation, intra-day price movements are expected to be predicted by reading out the patterns observed in tick-wise price motions. Our first task on this line of thought is to identify the set of effective variables suitable for studying the problem. We have first constructed a price prediction generator that computes the best prediction by reading the data tick by tick. We report in this article the effect of the adaptive choice of the best combination of technical indicators out of ten popular indicators, and also the result of using a set of novel dimensionless dynamical indicators constructed from the local values of derivatives and the second derivatives of the price times series. We have obtained a good performance of nearly 70 percent of correctly predicted direction of motion at 10 ticks ahead of the prediction time by means of adaptive choice of the technical indicators, and even better performance in the second attempt of using the two dimensionless dynamical indicators.

Effective indices to characterize short sequences of human random generations

Brain impediments such as dementia are a serious problem today. It would be very useful if software for private diagnosis were available. In this paper, we show the effectiveness of the human random generation test (HRG) for such software, and propose a set of four indices to be used for classifying the HRG data. Human-generated random numbers have strong characteristics compared to computer-generated random numbers, and these are known to be correlated to the individual characters of the subjects. However, analysis using the correlation dimension or HMM requires a long data sequence, and thus is not suitable for diagnoses.We therefore focus on short sequences of HRG and search for effective indices to detect signs of brain disability hidden in the HRG data. We studied data from subjects of different age groups, and successfully differentiated the data from the different groups.

Trend-Extraction of Stock Prices in the American Market by Means of RMT-PCA

We apply the RMT-PCA, recently developed PCA in order to grasp temporal trends in a stock market, on the daily-close stock prices of American Stocks in NYSE for 16 years from 1994 to 2009 and show the effectiveness and consistency of this method by analyzing the whole data at once, as well as analyzing the cut data in various partitions, such as two files of 8 year length, four files of 4 year length, and eight files of 2year length. The result shows a good agreement to the actual historical trends of the markets. We also discuss on the internal consistency among the results of different time intervals.

Effect of Reputation on the Formation of Cooperative Network of Prisoners

We consider in this paper the effect of the player’s reputation implemented in a multi-agent model of iterated prisoner’s dilemma to develop cooperative networks. Our model assumes two separate strategies per agent to apply upon a cooperative partner and a defective partner. Starting from a randomly selected pair of strategies, (SD , SC) where SD and SC being the strategy on the defective partner and the cooperative partner, the agent autonomously learns a set of better strategies by imitating the better performing agent. Reputation is defined to be the rate of cooperative choice that the agent has chosen in the course of iteration. Each agent is given a fixed criterion on the minimum reputation to require upon the partner, so that this agent refuses to play the game with agents of bad reputation. We show by simulations that the model successfully develop cooperative networks of players by means of selecting the reputable partners as well as updating the strategies.