Fractal geometry of financial time series

Abstract – A simple quantitative measure of the self-similarity in time-series in general and in the stock market in particular is the scaling behavior of the absolute size of the jumps across lags of size k. A stronger form of self-similarity entails not only that this mean absolute value, but also the full distributions of lag-k jumps have a scaling behavior characterized by the above Hurst exponent. In 1963 Benoit Mandelbrot showed that cotton prices have such a strong form of (distributional) self-similarity, and for the first time introduced Lévy’s stable random variables in the modeling of price records. This paper discusses the analysis of the self-similarity of high-frequency DEM-USD exchange rate records and the 30 main German stock price records. Distributional selfsimilarity is found in both cases and some of its consequences are discussed.

Self-similarity of high-frequency USD-DEM exchange rates

High frequency DEM-USD exchange rate data (resolution > 2 seconds) are analyzed for their scaling behavior as a function of the time lag. Motivated by the finding that the distribution of 1-quote returns is rather insensitive to the physical time duration between successive quotes, lags are measured in units of quotes. The mean absolute returns over lags of different sizes, shows three different regimes. The smallest time scales show no scaling, followed by two scaling regimes characterized by Hurst exponents H = 0.45 and H = 0.56, with a crossover occuring at lags of # 500 quotes. The up-down correlation coefficient, defined here, shows strong anticorrelations on scales smaller than 500. The lack of convergence to a large deviation rate function, convex tails in the logarithm of the probability distributions, strong up-down correlations and H < 0.5, show that the dynamics on small scales is more complicated than random walk models with i.i.d. increments. Nevertheless, for both scaling re..

A Local Multiscale Characterization of Edges applying the Wavelet Transform

The multiscale detection and characterization of edges in images according to their strength, their scale and their Holder exponent, is presented. The method is based on both older ideas involving scale-space filtering in the field of human vision, and on more recent developments in the crossroads of wavelet theory, singular functions and the analysis of fractal time series. Edges and their properties on all scales of observation are looked for, at the same time avoiding a proliferation of redundant information and computation. This is achieved by only looking a modulus maxima lines in the wavelet transform. In the end three descriptive parameters are left for each pixel in the image at which a modulus maxima line starts. These three edge-parameters make it possible to select edges according to the largest distance at which they are still visible, their strength, and their sharpness. In generic cases, this makes it possible to, e.g., separate edges due to noise from real ones. The nume..

Distribution of Local-connected Fractal Dimension and the degree of Liver fattiness from Ultrasound

Introduction Central to fractal geometry[3] is the concept of self-similarity. Fractal and multifractal[1] analysis provide tools for the quantitative analysis and description of self-similarity in natural and mathematical sets and distributions. The arsenal of tools for fractal analysis which is growing both in diversity and sophisication provides fresh new ways to analyse geometries, even those which are non-generically fractal. This contribution to Fractals in Medicine discusses an application of a hybrid of fractal and multifractal analysis to a problem in medical ultrasound image processing. Our method is a continuation of work by Richard Voss[7] which he applied to the X-ray detection of malignant breast tumors and the classification of chineese landscape paintings. Ultrasound imaging of the liver is a standard part of a checkup by the specialist in internal medicine. Besides being a harmless tool for diagnosing several life threathening diseases ultrasou

Effect of soft-copy display supported by CAD on mammography screening performance.

Contains fulltext : 50237.pdf (publisher's version ) (Closed access)Diagnostic performance and reading speed for conventional mammography film reading is compared to reading digitized mammograms on a dedicated workstation. A series of mammograms judged negative at screening and corresponding priors were collected. Half were diagnosed as cancer at the next screening, or earlier for interval cancers. The others were normal. Original films were read by fifteen experienced screening radiologists. The readers annotated potential abnormalities and estimated their likelihood of malignancy. More than 1 year later, five radiologists reread a subset of 271 cases (88 cancer cases having visible signs in retrospect and 183 normals) on a mammography workstation after film digitization. Markers from a computer-aided detection (CAD) system for microcalcifications were available to the readers. Performance was evaluated by comparison of A(z)-scores based on ROC and multiple-Reader multiple-case (MRMC) analysis, and localized receiver operating characteristic (LROC) analysis for the 271 cases. Reading speed was also determined. No significant difference in diagnostic performance was observed between conventional and soft-copy reading. Average A(z)-scores were 0.83 and 0.84 respectively. Soft-copy reading was only slightly slower than conventional reading. Using a mammography workstation including CAD for detection of microcalcifications, soft-copy reading is possible without loss of quality or efficiency