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In the paper we advocate the use of wavelet analysis in classifying assets and sectors as defensive (less dependent on the business cycle and the state of the market) or cyclical (highly influenced by the general level of economic activity or the variability of broad market indices). We demonstrate that such tools of bivariate wavelet analysis like wavelet gain, wavelet coherency and wavelet-based phase-locking value together with risk profiles defined via univariate wavelet spectra serve better the purpose of sector classification than the traditional approach based on beta coefficients, correlations and variance comparison. Not only are the wavelet quantities better suited to examine (often time-varying) dependences in the frequency bands of interest, i.e., frequencies associated with business cycles, but they also correct the information provided by the coefficients computed according to the classical approach for possible phase shifts and, in the case of the phase-locking value, also changing amplitudes. The suggested wavelet approach is illustrated with an analysis of the EURO STOXX Total Market sector and supersector indices.

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In this study, the relation between the cyclical behaviors of stock market indices of industry, service, finance and technology sectors at Istanbul Stock Exchange and gross domestic product of Turkey between the 1998 January and 2011 September, is analyzed. The results suggest that stock exchange indices move in the same direction with economic activity and stock market leads the economy by about one quarter. However, when the sectoral differences are considered, movements in technology sector index are transmitted to economy in two months whereas it is three months for the industrial and service sector. The slowest sector is the financial sector for which pass-through speed is four months. (C) 2012 Published by Elsevier Ltd. Selection and/or peer review under responsibility of Prof. Dr. Huseyin Arasli

We investigate the use of Hilbert wavelet pairs (HWPs) in the non-decimated discrete wavelet transform for the time-varying spectral analysis of multivariate time series. HWPs consist of two high-pass and two low-pass compactly supported filters, such that one high-pass filter is the Hilbert transform (approximately) of the other. Thus, common quantities in the spectral analysis of time series (e.g., power spectrum, coherence, phase) may be estimated in both time and frequency. Compact support of the wavelet filters ensures that the frequency axis will be partitioned dyadically as with the usual discrete wavelet transform. The proposed methodology is used to analyze a bivariate time series of zonal (u) and meridional (v) winds over Truk Island.

Wavelets have proved particularly effective for extracting discriminative features in ECG signal classification. In this paper, we show that wavelet performances in terms of classification accuracy can be pushed further by customizing them for the considered classification task. A novel approach for generating the wavelet that best represents the ECG beats in terms of discrimination capability is proposed. It makes use of the polyphase representation of the wavelet filter bank and formulates the design problem within a particle swarm optimization (PSO) framework. Experimental results conducted on the benchmark MIT/BIH arrhythmia database with the state-of-the-art support vector machine (SVM) classifier confirm the superiority in terms of classification accuracy and stability of the proposed method over standard wavelets (i.e., Daubechies and Symlet wavelets).

We characterize empirically the financial cycle using two approaches: analysis of turning points and frequency-based filters. We identify the financial cycle with the medium-term component in the joint fluctuations of credit and property prices; equity prices do not fit this picture well. We show that financial cycle peaks are very closely associated with financial crises and that the length and amplitude of the financial cycle have increased markedly since the mid-1980s. We argue that this reflects, in particular, financial liberalization and changes in monetary policy frameworks. So defined, the financial cycle is much longer than the traditional business cycle. Business cycle recessions are much deeper when they coincide with the contraction phase of the financial cycle. We also draw attention to the "unfinished recession" phenomenon: policy responses that fail to take into account the length of the financial cycle may help contain recessions in the short run but at the expense of larger recessions down the road.

This paper takes a fresh look at the nature of financial and real business cycles in OECD countries using annual data series and shorter quarterly and monthly economic indicators. It first analyses the main characteristics of the cycle, including the length, amplitude, asymmetry and changes of these parameters during expansions and contractions. It then studies the degree of economic and financial cycle synchronization between OECD countries but also of economic and financial variables within a given country, and gauges the extent to which cycle synchronization changed over time. Finally, the paper provides some new evidence on the drivers of the great moderation and analyses the banking sector’s pro-cyclicality by using aggregate and bank-level data. The main findings show that the amplitude of the real business cycle was becoming smaller during the great moderation, but asset price cycles were becoming more volatile. In part this was linked to developments in the banking sector which tended to accentuate pro-cyclical behavior.

This paper introduces a new method, single-trial phase locking statistics (S-PLS) to estimate phase locking in single trials of brain signals between two electrodes. The possibility of studying single trials removes an important limitation in the study of long-range synchrony in brain signals. S-PLS is closely related to our previous method, phase locking statistics (PLS) that estimates phase locking over a set of trials. The S-PLS method is described in detail and applied to human surface recordings during the task of face-recognition. We compare these results with those provided by PLS and show that they are qualitatively very similar, although S-PLS provides better discrimination of synchronic episodes.

This paper investigates the hedging effectiveness of a dynamic moving window
OLS hedging model, formed using wavelet decomposed time-series. The wavelet
transform is applied to calculate the appropriate dynamic minimum-variance
hedge ratio for various hedging horizons for a number of assets. The
effectiveness of the dynamic multiscale hedging strategy is then tested, both
in- and out-of-sample, using standard variance reduction and expanded to
include a downside risk metric, the time horizon dependent Value-at-Risk.
Measured using variance reduction, the effectiveness converges to one at longer
scales, while a measure of VaR reduction indicates a portion of residual risk
remains at all scales. Analysis of the hedge portfolio distributions indicate
that this unhedged tail risk is related to excess portfolio kurtosis found at
all scales.

The wide acceptance of Hedge Funds by Institutional Investors and Pension Funds has led to an explosive growth in assets under management. These investors are drawn to Hedge Funds due to the seemingly low correlation with traditional investments and the attractive returns.
The correlations and market risk (the Beta in the Capital Asset Pricing Model) of Hedge Funds are generally calculated using monthly returns data, which may produce misleading results as Hedge Funds often hold illiquid exchange-traded securities or difficult to price over-the-
counter securities. In this paper, the Maximum Overlap Discrete Wavelet Transform (MODWT) is applied to measure the scaling properties of Hedge Fund correlation and market risk with respect to the S&P 500. It is found that the level of correlation and market risk varies greatly
according to the strategy studied and the time scale examined. Finally, the effects of scaling properties on the risk profile of a portfolio made up of Hedge Funds is studied using correlation matrices calculated over different time horizons.

In this article we test the random walk hypothesis for weekly stock market returns by comparing variance estimators derived
from data sampled at different frequencies. The random walk model is strongly rejected for the entire sample period (1962–1985)
and for all subperiods for a variety of aggregate returns indexes and size-sorted portfolios. Although the rejections are
due largely to the behavior of small stocks, they cannot be attributed completely to the effects of infrequent trading or
time-varying volatilities. Moreover, the rejection of the random walk for weekly returns does not support a mean-reverting
model of asset prices.

This paper considers the design of pairs of wavelet bases where the wavelets form a Hilbert transform pair. The derivation is based on the limit functions defined by the infinite product formula. It is found that the scaling filters should be offset from one another by a half sample. This gives an alternative derivation and explanation for the result by Kingsbury (1999), that the dual-tree DWT is (nearly) shift-invariant when the scaling filters satisfy the same offset.

Assessing Nonstationary Time Series Using Wavelets by Brandon J Whitcher Chairperson of Supervisory Committee: Professor Peter Guttorp & Professor Donald B. Percival Statistics & Applied Physics Laboratory The discrete wavelet transform has be used extensively in the field of Statistics, mostly in the area of "denoising signals" or nonparametric regression. This thesis provides a new application for the discrete wavelet transform, assessing nonstationary events in time series -- especially long memory processes. Long memory processes are those which exhibit substantial correlations between events separated by a long period of time. Departures from stationarity in these heavily autocorrelated time series, such as an abrupt change in the variance at an unknown location or "bursts" of increased variability, can be detected and accurately located using discrete wavelet transforms -- both orthogonal and overcomplete. A cumulative sum of squares method, utilizing a Kolomogorov--Smirnov-type ...

In this article, we formulate a time-scale decomposition of an international version of the CAPM that accounts for both market and exchange rate risk. In addition, we derive an analytical formula for time-scale value at risk (VaR) and time-scale marginal VaR of a portfolio. We apply our methodology to stock indices of seven emerging economies belonging to Latin America and Asia, for the sample period 1990-2004. Our main conclusions are the following. First, the estimation results hinge upon the choice of the world market portfolio. In particular, the stock markets of the sampled countries appear to be more integrated with other emerging countries than with developed ones. Second, value at risk depends on the investor's time horizon. In the short run, potential losses are greater than in the long run. Third, additional exposure to some specific stock indices will increase value at risk to a greater extent, depending on the investment horizon. Our results go in line with recent research in asset pricing that stresses the importance of heterogeneous investors.

This chapter discusses the problem of selecting optimal security portfolios by risk-averse investors who have the alternative of investing in risk-free securities with a positive return or borrowing at the same rate of interest and who can sell short if they wish. It presents alternative and more transparent proofs under these more general market conditions for Tobin's important separation theorem that “ … the proportionate composition of the non-cash assets is independent of their aggregate share of the investment balance … and for risk avertere in purely competitive markets when utility functions are quadratic or rates of return are multivariate normal. The chapter focuses on the set of risk assets held in risk averters' portfolios. It discusses various significant equilibrium properties within the risk asset portfolio. The chapter considers a few implications of the results for the normative aspects of the capital budgeting decisions of a company whose stock is traded in the market. It explores the complications introduced by institutional limits on amounts that either individuals or corporations may borrow at given rates, by rising costs of borrowed funds, and certain other real world complications.

1. Introduction to wavelets 2. Review of Fourier theory and filters 3. Orthonormal transforms of time series 4. The discrete wavelet transform 5. The maximal overlap discrete wavelet transform 6. The discrete wavelet packet transform 7. Random variables and stochastic processes 8. The wavelet variance 9. Analysis and synthesis of long memory processes 10. Wavelet-based signal estimation 11. Wavelet analysis of finite energy signals Appendix. Answers to embedded exercises References Author index Subject index.

Wavelet methods have recently undergone a rapid period of development with important implications for a number of disciplines including statistics. This book has three main objectives: (i) providing an introduction to wavelets and their uses in statistics; (ii) acting as a quick and broad reference to many developments in the area; (iii) interspersing R code that enables the reader to learn the methods, to carry out their own analyses, and further develop their own ideas. The book code is designed to work with the freeware R package WaveThresh4, but the book can be read independently of R. The book introduces the wavelet transform by starting with the simple Haar wavelet transform, and then builds to consider more general wavelets, complex-valued wavelets, non-decimated transforms, multidimensional wavelets, multiple wavelets, wavelet packets, boundary handling, and initialization. Later chapters consider a variety of wavelet-based nonparametric regression methods for different noise models and designs including density estimation, hazard rate estimation, and inverse problems; the use of wavelets for stationary and non-stationary time series analysis; and how wavelets might be used for variance estimation and intensity estimation for non-Gaussian sequences. The book is aimed both at Masters/Ph.D. students in a numerate discipline (such as statistics, mathematics, economics, engineering, computer science, and physics) and postdoctoral researchers/users interested in statistical wavelet methods.

This paper introduces a new family of portmanteau tests for serial correlation. Using the wavelet transform, we decompose the variance of the underlying process into the variance of its low frequency and of its high frequency components and we design a variance ratio test of no serial correlation in the presence of dependence. Such decomposition can be carried out iteratively, each wavelet filter leading to a rich family of tests whose joint limiting null distribution is a multivariate normal. We illustrate the size and power properties of the proposed tests through Monte Carlo simulations.

A body of work using the continuous wavelet transform has been growing. We provide a self-contained summary on its most relevant theoretical results, describe how such transforms can be implemented in practice, and generalize the concept of simple coherency to partial wavelet coherency and multiple wavelet coherency, moving beyond bivariate analysis. We also describe a family of wavelets, which emerges as an alternative to the popular Morlet wavelet, the generalized Morse wavelets. A user-friendly toolbox, with examples, is attached to this paper.

The size effect is sensitive to the length of the return interval used in estimating betas. Beta changes with the return interval because an asset's covariance with the market and the market's variance do not change proportionately as the return interval is changed. We document beta sensitivity to the return interval. Evidence from cross-sectional regressions of returns on monthly and annual betas is inconsistent with beta changes stemming only from the higher standard errors of the longer-interval betas. We provide evidence that the size effect becomes statistically insignificant when risk is measured by betas estimated using annual returns.

In this article, wavelet tools and economic sentiment indicators are used to study the similarity and synchronization of economic cycles in the eurozone. The time‐varying and frequency‐varying patterns of business cycles synchronization are assessed and the impact of the creation of the European monetary union (EMU) in 1999 is tested. Among several results, it is found that: the EMU is associated with a significant increase in the similarity and synchronization of the economic sentiment in the eurozone; and the hard‐peg of its currency to the euro led to a comparable effect on Denmark's economic sentiment after 1999, different from what happened in the United Kingdom.

Many physical processes are an amalgam of components operating on different scales, and scientific questions about observed data are often inherently linked to understanding the behavior at different scales. We explore time-scale properties of time series through the variance at different scales derived using wavelet methods. The great advantage of wavelet methods over ad hoc modifications of existing techniques is that wavelets provide exact scale-based decomposition results. We consider processes that are stationary, nonstationary but with stationary dth order differences, and nonstationary but with local stationarity. We study an estimator of the wavelet variance based on the maximal-overlap (undecimated) discrete wavelet transform. The asymptotic distribution of this wavelet variance estimator is derived for a wide class of stochastic processes, not necessarily Gaussian or linear. The variance of this distribution is estimated using spectral methods. Simulations confirm the theoretical results. The utility of the methodology is demonstrated on two scientifically important series, the surface albedo of pack ice (a strongly non-Gaussian series) and ocean shear data (a nonstationary series).

A great many people provided comments on early versions of this paper which led to major improvements in the exposition. In addition to the referees, who were most helpful, the author wishes to express his appreciation to Dr. Harry Markowitz of the RAND Corporation, Professor Jack Hirshleifer of the University of California at Los Angeles, and to Professors Yoram Barzel, George Brabb, Bruce Johnson, Walter Oi and R. Haney Scott of the University of Washington.

We analyze mutual fund industry selectivity—the performance of a fund’s industry allocation relative to the market. We find
that industry selection accounts for a full third of fund performance based on two-digit standard industrial classification
(SIC) codes, with the remaining attributable to the performance of individual stocks relative to their own industries. More
importantly, we find that industry-selection skill drives persistence in relative performance. Unlike stock-selection ability,
industry selectivity is not eroded by increasing fund assets. Our results suggest that accounting for a manager’s ability
to pick outperforming industries provides information beyond standard performance measures that can enhance a fund investor’s
future performance. (JEL G11, G14, G23)

This study presents a model to select the optimal hedge ratios of a portfolio composed of an arbitrary number of commodities. In particular, returns dependency and heterogeneous investment horizons are accounted for by copulas and wavelets, respectively. A portfolio of London Metal Exchange metals is analyzed for the period July 1993–December 2005, and it is concluded that neglecting cross correlations leads to biased estimates of the optimal hedge ratios and the degree of hedge effectiveness. Furthermore, when compared with a multivariate-GARCH specification, our methodology yields higher hedge effectiveness for the raw returns and their short-term components. © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 28:182–207, 2008

Consider the situation when we have training data containing many time series having known group membership and testing data with unknown group membership. The goals are to find timescale features (using training data) that can best separate the groups, and to use these highly discriminant features to classify test data. We propose a method for classification using a bias-corrected nondecimated wavelet transform. Wavelets are ideal for identifying highly discriminant local time and scale features. The observed signals will be treated as realizations of locally stationary wavelet processes, under which we define and rigorously estimate the evolutionary wavelet spectrum (timescale decomposition of variance). The evolutionary wavelet spectrum, which contains the second-moment information on the signals, is used as the classification signature. For each test time series, we compute the empirical wavelet spectrum and its divergence from the wavelet spectrum of each group. The test time series is then assigned to the group to which it is the least dissimilar. Under the locally stationary wavelet framework, we rigorously demonstrate that the classification procedure is consistent (i.e., misclassification probability goes to zero at the rate that is inversely proportional to divergence between the evolutionary wavelet spectra). The method is illustrated using,seismic signals (earthquake vs. explosion events) and is demonstrated to work very well in simulation studies.

Stock indices related to specific economic sectors play a major role in portfolio diversification. Notwithstanding its importance,
the traditional sector classification shows several flaws and it may not be able to properly discriminate the risk-return
profile of financial assets. We propose a latent class approach in order to correctly classify the stock companies into homogenous
groups under risk-return profile and to obtain sector indices which are consistent with the standard portfolio theory. Our
results allow to introduce a methodological dimension in the stock’s classification and to improve the reliability of sector
portfolio diversification.

Mallat's book is the undisputed reference in this field - it is the only one that covers the essential material in such breadth and depth. - Laurent Demanet, Stanford University The new edition of this classic book gives all the major concepts, techniques and applications of sparse representation, reflecting the key role the subject plays in today's signal processing. The book clearly presents the standard representations with Fourier, wavelet and time-frequency transforms, and the construction of orthogonal bases with fast algorithms. The central concept of sparsity is explained and applied to signal compression, noise reduction, and inverse problems, while coverage is given to sparse representations in redundant dictionaries, super-resolution and compressive sensing applications. Features: * Balances presentation of the mathematics with applications to signal processing * Algorithms and numerical examples are implemented in WaveLab, a MATLAB toolbox * Companion website for instructors and selected solutions and code available for students New in this edition * Sparse signal representations in dictionaries * Compressive sensing, super-resolution and source separation * Geometric image processing with curvelets and bandlets * Wavelets for computer graphics with lifting on surfaces * Time-frequency audio processing and denoising * Image compression with JPEG-2000 * New and updated exercises A Wavelet Tour of Signal Processing: The Sparse Way, third edition, is an invaluable resource for researchers and R&D engineers wishing to apply the theory in fields such as image processing, video processing and compression, bio-sensing, medical imaging, machine vision and communications engineering. Stephane Mallat is Professor in Applied Mathematics at École Polytechnique, Paris, France. From 1986 to 1996 he was a Professor at the Courant Institute of Mathematical Sciences at New York University, and between 2001 and 2007, he co-founded and became CEO of an image processing semiconductor company. Includes all the latest developments since the book was published in 1999, including its application to JPEG 2000 and MPEG-4 Algorithms and numerical examples are implemented in Wavelab, a MATLAB toolbox Balances presentation of the mathematics with applications to signal processing.

This paper examines the relationship between the Australian stock and futures markets over various time horizons. In contrast to methods employed in previous studies, wavelet analysis allows us to decompose data into various time scales. Using this technique and the Hurst exponent, we find that the Australian stock and futures markets are antipersistent. The wavelet correlation between the two markets varies over investment horizons, but remains very high. Furthermore, the magnitude of the correlation increases as the time scale increases, indicating that the stock market and the futures market of the All Ordinaries Index are found to be not fundamentally different. The hedge ratio increases as the wavelet time scale increases. In addition, the effectiveness of hedging strategies initially increases with the hedging horizon.

This essay analyzes the long-term lessons of the recent upturn and downturn in the telecommunications industry. It concludes that volatility and cyclicality will be an inherent part of the telecom sector in the future. To deal with such instabilities, companies and investors seek consolidation and cooperation. Government, too, is likely to stress stability more than before. Hence, an oligopoly is likely to emerge as the equilibrium market structure, and with it some regulation.

This paper describes a form of discrete wavelet transform, which generates complex coefficients by using a dual tree of wavelet filters to obtain their real and imaginary parts. This introduces limited redundancy (2m:1 for m-dimensional signals) and allows the transform to provide approximate shift invariance and directionally selective filters (properties lacking in the traditional wavelet transform) while preserving the usual properties of perfect reconstruction and computational efficiency with good well-balanced frequency responses. Here we analyze why the new transform can be designed to be shift invariant and describe how to estimate the accuracy of this approximation and design suitable filters to achieve this. We discuss two different variants of the new transform, based on odd/even and quarter-sample shift (Q-shift) filters, respectively. We then describe briefly how the dual tree may be extended for images and other multi-dimensional signals, and finally summarize a range of applications of the transform that take advantage of its unique properties.

The assessment of the comovement among international stock markets is of key interest, for example, for the international portfolio diversification literature. In this paper, we re-examine such comovement by resorting to a novel approach, wavelet analysis. Wavelet analysis allows one to measure the comovement in the time-frequency space. In this way, one can characterize how international stock returns relate in the time and frequency domains simultaneously, which allows one to provide a richer analysis of the comovement. We focus on Germany, Japan, UK and US and the analysis is done at both the aggregate and sectoral levels.

SUMMARY The wavelet variance decomposes the variance of a time series into components associated with different scales. We consider
two estimators of the wavelet variance: the first based upon the discrete wavelet transform, and the second, called the maximal-overlap
estimator, based upon a filtering interpretation of wavelets. We determine the large sample distribution for both estimators
and show that the maximal-overlap estimator is more efficient for a class of processes of interest in the physical sciences.
We discuss methods for determining an approximate confidence interval for the wavelet variance. We demonstrate through Monte
Carlo experiments that the large sample distribution for the maximal-overlap estimator is a reasonable approximation even
for the moderate sample size of 128 observations. We apply our proposed methodology to a series of observations related to
vertical shear in the ocean.

The US economy is arguably following an unsustainable trajectory. The main indicators of this are a large current account deficit, a large federal budget deficit and trend-wise increasing costs of Social Security and Medicare. In this chapter, we will discuss these observations and to what extent the financial and economic crisis may have changed the outlook. Before this, we need to define what we mean by sustainability. An often used definition of sustainability is that the inter-temporal budget restriction is satisfied.

This paper shows that increases in the minimum wage rate can have ambiguous effects on the working hours and welfare of employed workers in competitive labor markets. The reason is that employers may not comply with the minimum wage legislation and instead pay a lower subminimum wage rate. If workers are risk neutral, we prove that working hours and welfare are invariant to the minimum wage rate. If workers are risk averse and imprudent (which is the empirically likely case), then working hours decrease with the minimum wage rate, while their welfare may increase.

There is now considerable evidence suggesting that estimated betas of unconditional capital asset pricing models (CAPMs) exhibit statistically significant time variation. Therefore, many have advocated the use of conditional CAPMs. If we succeed in capturing the dynamics of beta risk, we are sure to outperform constant beta models. However, if the beta risk is inherently misspecified, there is a real possibility that we commit serious pricing errors, potentially larger than with a constant traditional beta model. In this paper we show that this is indeed the case, namely that pricing errors with constant traditional beta models are smaller than with conditional CAPMs. Copyright The American Finance Association 1998.

Most empirical studies of the static capital asset pricing model (CAPM) assume that betas remain constant over time and that the return on the value-weighted portfolio of all stocks is a proxy for the return on aggregate wealth. The general consensus is that the static CAPM is unable to explain satisfactorily the cross-section of average returns on stocks. The authors assume that the CAPM holds in a conditional sense, i.e., betas and the market risk premium vary over time. They include the return on human capital when measuring the return on aggregate wealth. The authors' specification performs well in explaining the cross-section of average returns. Copyright 1996 by American Finance Association.

Several authors have demonstrated that significant improvements
can be obtained in wavelet-based signal processing by utilizing a pair
of wavelet transforms where the wavelets form a Hilbert transform pair.
This paper describes design procedures, based on spectral factorization,
for the design of pairs of dyadic wavelet bases where the two wavelets
form an approximate Hilbert transform pair. Both orthogonal and
biorthogonal FIR solutions are presented, as well as IIR solutions. In
each case, the solution depends on an allpass filter having a flat delay
response. The design procedure allows for an arbitrary number of
vanishing wavelet moments to be specified. A Matlab program for the
procedure is given, and examples are also given to illustrate the
results

A new implementation of the Discrete Wavelet Transform is presented, suitable for a range of signal and image processing applications. It employs a dual tree of wavelet filters to obtain the real and imaginary parts of complex wavelet coefficients. This introduces limited redundancy (4:1 for 2-dimensional signals) and allows the transform to provide approximate shift invariance and directionally selective filters (properties lacking in the traditional wavelet transform) while preserving the usual properties of perfect reconstruction and computational efficiency. An application to texture synthesis is presented. 1. INTRODUCTION Although the Discrete Wavelet Transform (DWT) in its maximally decimated form (Mallat's dyadic filter tree [1]) has established an impressive reputation as a tool for image compression, its use for other signal analysis and reconstruction tasks has been hampered by two main disadvantages: ffl Lack of shift invariance, which means that small shifts in the input...

this paper we study the scale analysis of bivariate time series through use of the wavelet cross-covariance. The estimation of this quantity is carried out using the maximaloverlap discrete wavelet transform (MODWT). The MODWT (Greenhall, 1991; Percival and Guttorp, 1994) is also called the undecimated or translation invariant or shift invariant discrete wavelet transform (Shensa, 1992; Nason and Silverman, 1995; Coifman and Donoho, 1995). The MODWT carries out the same filtering steps as the ordinary discrete wavelet transform, but does not subsample. We need immediately to see what is meant by a scale analysis. Let

- M Vermorken
- A Szafarz
- H Pirotte

Vermorken M., Szafarz A., Pirotte H. (2010), Sector Classification Through Non-Gaussian
Similarity, Applied Financial Economics, 20, 861–878.

Defensive Stock Price Betas Rise, Economic and Management Perspective, Wells Capital Management

- J W Paulsen

Paulsen J. W. (2013), Defensive Stock Price Betas Rise, Economic and Management
Perspective, Wells Capital Management, January 23, 2013.

- R F Gençay
- F Selçuk
- B Whitcher

Gençay R. F., Selçuk F., Whitcher B. (2005), Multiscale Systematic Risk, Journal of
International Money and Finance, 24, 55–70.

- G H Moore

Moore G. H. (1983), Business Cycles, Inflation, and Forecasting, National Bureau of
Economic Research Studies in Business Cycles no. 24, Ballinger, Cambridge, 2 nd edition.

Comparison of Hilbert Transform and Wavelet Methods for the Analysis of Neuronal Synchrony

J. (2001), Comparison of Hilbert Transform and Wavelet Methods for the Analysis of
Neuronal Synchrony, Journal of Neuroscience Methods, 111, 83–98.

The Dual-Tree Complex Wavelet Transform. A Coherent Framework for Multiscale Signal and Image Processing

- I W Selesnick
- R G Baraniuk
- N G Kingsbury

Selesnick I. W., Baraniuk R. G., Kingsbury N. G. (2005), The Dual-Tree Complex Wavelet
Transform. A Coherent Framework for Multiscale Signal and Image Processing, IEEE
Signal Processing Magazine, 22, 123–149.