We introduce a smoothed variant of the Smoothly Clipped Absolute Deviation (SCAD) thresholding rule for wavelet denoising by replacing its piecewise linear transition with a raised cosine. The resulting shrinkage function is odd, continuous on R, and continuously differentiable away from the main threshold, yet retains the hallmark SCAD properties of sparsity for small coefficients and near unbiasedness for large ones. This smoothness places the rule within the continuous thresholding class for which Stein’s unbiased risk estimate (SURE) is valid. As a result, unbiased risk computation, stable data-driven threshold selection, and the asymptotic theory of Kudryavtsev and Shestakov apply. A corresponding nonconvex prior is obtained whose posterior mode coincides with the estimator, yielding a transparent Bayesian interpretation. We give an explicit SURE risk expression, discuss the oracle scale of the optimal threshold, and describe both global and level-dependent adaptive versions. The smooth SCAD rule therefore offers a tractable refinement of SCAD, combining low bias, exact sparsity, and analytical convenience in a single wavelet shrinkage procedure.

Breast histology images carry diagnostic information in color, texture, orientation, and tissue architecture across a range of scales. In H E microscopy this information is inherently chromatic and is not fully recovered when the red, green, and blue (RGB) channels are reduced to grayscale or transformed as independent scalar images. We propose an interpretable quaternion nondecimated wavelet framework for breast histology classification. Each RGB image is encoded as a pure quaternion field, and a quaternion nondecimated wavelet transform in two dimensions (QNDWT2D) produces multiscale, directional, color-coupled coefficient fields on the original image grid, keeping color as a single vector quantity rather than three separate channels. From these coefficients we build interpretable feature families summarizing stain balance, wavelet energy, amplitude heterogeneity, quaternion phase concentration, color-axis geometry, directional anisotropy, orientation entropy, and scale-dependent energy decay, each tied to a histopathological property such as nuclear density or glandular organization. We evaluate the descriptors on the BreAst Cancer Histology (BACH) challenge, a balanced four-class set of normal, benign, in situ, and invasive tissue, using a radial-kernel support vector machine (SVM) with repeated nested cross-validation. The descriptors yield balanced recognition across classes, with errors concentrated among adjacent categories while normal and invasive are rarely reversed. Permutation importance shows that directional, phase-concentration, anisotropy, scale, and amplitude-variability groups all contribute, indicating that the classifier draws on genuine quaternion and multiscale geometry rather than global color alone. The framework uses no pretrained networks, learned filters, or external databases, offering a reproducible, interpretable baseline for computational pathology.

Contrast-enhanced spectral mammography (CESM) acquires two images of each breast, a low-energy image and a recombined contrast image, but two questions central to building a classifier on them remain unsettled: whether the two image types carry comparable malignancy signal, and how a patient's several images should be combined into a single decision. Both are hard to answer reliably, because most published CESM classifiers split cross-validation folds at the image level, letting images of the same patient fall in both training and test sets and inflating reported performance. We pair a masked complex non-decimated wavelet feature bank with an elastic-net logistic classifier, evaluated under repeated patient-grouped nested cross-validation with patient-cluster bootstrap inference on the CDD-CESM dataset (1,880 images, 308 patients); under this leakage-free evaluation the inflation from testing on previously seen patients is negligible. On normal-versus-malignant detection, the two acquisitions are statistically indistinguishable in patient-level AUC under the proposed evaluation framework. Under single-image fusion the contrast image reaches a patient-level AUC of 0.874 (95

We introduce a smooth variant of the SCAD thresholding rule for wavelet denoising by replacing its piecewise linear transition with a raised cosine. The resulting shrinkage function is odd, continuous on R, and continuously differentiable away from the main threshold, yet retains the hallmark SCAD properties of sparsity for small coefficients and near unbiasedness for large ones. This smoothness places the rule within the continuous thresholding class for which Stein's unbiased risk estimate is valid. As a result, unbiased risk computation, stable data-driven threshold selection, and the asymptotic theory of Kudryavtsev and Shestakov apply. A corresponding nonconvex prior is obtained whose posterior mode coincides with the estimator, yielding a transparent Bayesian interpretation. We give an explicit SURE risk expression, discuss the oracle scale of the optimal threshold, and describe both global and level-dependent adaptive versions. The smooth SCAD rule therefore offers a tractable refinement of SCAD, combining low bias, exact sparsity, and analytical convenience in a single wavelet shrinkage procedure.

Incomplete reporting of diagnostic accuracy data remains a persistent problem in medical research. In many studies, only part of the 2x2 diagnostic table is reported, leaving denominators for diseased and non-diseased groups unknown and preventing direct calculation of sensitivity, specificity, predictive values, and related operating characteristics. To address this limitation, we develop hierarchical Bayesian models for reconstructing incomplete 2x2 diagnostic tables from such partial information. Two motivating scenarios are considered: one in which only a single test-outcome row is observed, and another in which true positives, false positives, and the total sample size are reported but the remaining cells are missing. The proposed models are illustrated on a benchmark breast MRI study with complete counts, treated as partially observed in order to assess reconstruction performance under controlled missingness. The framework yields posterior inference for the missing cell counts and associated diagnostic measures, together with uncertainty quantification in weakly identified settings.

Data encountered in practice are frequently contaminated by additive noise, and wavelet shrinkage remains a fundamental tool for recovering underlying signals in nonparametric estimation. Classical procedures such as hard and soft thresholding decide whether to retain a wavelet coefficient almost entirely from its magnitude. Although effective in many settings, these rules can be too rigid for coefficients whose magnitudes fall in an intermediate region where the distinction between signal and noise is uncertain. We propose MLShrink, a two-threshold wavelet denoising procedure that combines wavelet shrinkage with machine learning. Coefficients below a lower threshold are discarded, coefficients above an upper threshold are retained, and coefficients in the intermediate band are classified using local wavelet-domain features. In this way, MLShrink preserves the simplicity of classical thresholding away from the decision boundary while allowing data-adaptive decisions for ambiguous coefficients. The paper also develops a theoretical framework tailored to this architecture. We show that MLShrink is a nonexpansive support-selection rule, derive an oracle-based risk decomposition showing that excess denoising risk is determined by classification errors on the undecided band, and establish an oracle-consistency result under suitable assumptions on classifier performance. Simulation experiments on standard benchmark signals indicate that MLShrink is competitive with several established wavelet shrinkage methods and is especially effective for signals with irregular, edge-rich, or non-smooth structure. These findings suggest that learned decisions on the intermediate threshold band provide a useful and interpretable connection between classical wavelet denoising and modern statistical learning.

ABSTRACT The Hurst exponent () plays a key role in understanding long‐range dependence and self‐similarity in time series data. Wavelet‐based methods have gained popularity for estimating because they efficiently capture patterns across multiple scales. This review explores how these methods have developed from their theoretical roots to real‐world applications in fields like biology, engineering, and telecommunications. The review aims to highlight key techniques, compare their strengths and limitations, and point out challenges that still need to be addressed, such as handling noise and non‐stationary data. The field is well‐established, but there is growing interest in combining traditional wavelet‐based models with modern machine learning to push the boundaries even further. This review offers a clear starting point and roadmap for future exploration. We critically evaluate methodological robustness, computational scalability, and practical adoption, identifying key challenges that define the current frontier of self‐similarity estimation.

Orthogonal wavelet transforms are a cornerstone of modern signal and image denoising because they combine multiscale representation, energy preservation, and perfect reconstruction. In this paper, we show that these advantages can be retained and substantially enhanced by moving beyond classical single-basis wavelet filterbanks to a broader class of composite wavelet-like matrices. By combining orthogonal wavelet matrices through products, Kronecker products, and block-diagonal constructions, we obtain new unitary transforms that generally fall outside the strict wavelet filterbank class, yet remain fully invertible and numerically stable. The central finding is that such composite transforms induce stronger concentration of signal energy into fewer coefficients than conventional wavelets. This increased sparsity, quantified using Lorenz curve diagnostics, directly translates into improved denoising under identical thresholding rules. Extensive simulations on Donoho-Johnstone benchmark signals, complex-valued unitary examples, and adaptive block constructions demonstrate consistent reductions in mean-squared error relative to single-basis transforms. Applications to atmospheric turbulence measurements and image denoising of the Barbara benchmark further confirm that composite transforms better preserve salient structures while suppressing noise.

Kolmogorov complexity of a finite binary word reflects both algorithmic structure and the empirical distribution of symbols appearing in the word. Words with symbol frequencies far from one half belong to smaller combinatorial classes and therefore appear less complex under the standard definition. In this paper, an entropy-normalized complexity measure is introduced that divides the Kolmogorov complexity of a word by the empirical entropy of its observed distribution of zeros and ones. This adjustment isolates intrinsic descriptive complexity from the purely combinatorial effect of symbol imbalance. For Martin–Löf random sequences under constructive exchangeable measures, the adjusted complexity grows linearly and converges to one. A pathological construction shows that regularity of the underlying measure is essential. The proposed framework connects Kolmogorov complexity, empirical entropy, and randomness in a natural manner and suggests applications in randomness testing and in the analysis of structured binary data.

This paper develops a unified framework for quantum wavelet shrinkage, extending classical denoising ideas into the quantum domain. Shrinkage is interpreted as a completely positive trace-preserving process, so attenuation of coefficients is carried out through controlled decoherence rather than nonlinear thresholding. Phase damping and ancilla-driven constructions realize this behavior coherently and show that statistical adaptivity and quantum unitarity can be combined within a single circuit model. The same physical mechanisms that reduce quantum coherence, such as dephasing and amplitude damping, are repurposed as programmable resources for noise suppression. Practical demonstrations implemented with Qiskit illustrate how circuits and channels emulate coefficientwise attenuation, and all examples are provided as Jupyter notebooks in the companion GitHub repository. Encoding schemes for amplitude, phase, and hybrid representations are examined in relation to transform coherence and measurement feasibility, and realizations suited to current noisy intermediate-scale quantum devices are discussed. The work provides a conceptual and experimental link between wavelet-based statistical inference and quantum information processing, and shows how engineered decoherence can act as an operational surrogate for classical shrinkage.

The nondecimated or translation-invariant wavelet transform (NDWT) is a central tool in classical multiscale signal analysis, valued for its stability, redundancy, and shift invariance. This paper develops two complementary quantum formulations of the NDWT that embed these classical properties coherently into quantum computation. The first formulation is based on the epsilon-decimated interpretation of the NDWT and realizes all circularly shifted wavelet transforms simultaneously by promoting the shift index to a quantum register and applying controlled circular shifts followed by a wavelet analysis unitary. The resulting construction yields an explicit, fully unitary quantum representation of redundant wavelet coefficients and supports coherent postprocessing, including quantum shrinkage via ancilla-driven completely positive trace preserving maps. The second formulation is based on the Hadamard test and uses diagonal phase operators to probe scale-shift wavelet structure through interference, providing direct access to shift-invariant energy scalograms and multiscale spectra without explicit coefficient reconstruction. Together, these two approaches demonstrate that redundancy and translation invariance can be exploited rather than avoided in the quantum setting. Applications to denoising, feature extraction, and spectral scaling illustrate how quantum NDWTs provide a flexible and physically meaningful foundation for multiscale quantum signal processing.

Stochastic resonance (SR), a phenomenon originally introduced in climate modeling, enhances signal detection by leveraging optimal noise levels within non-linear systems. Traditional SR techniques, mainly based on single-threshold detectors, are limited to signals whose behavior does not depend on time. Often large amounts of noise are needed to detect weak signals, which can distort complex signal characteristics. To address these limitations, this study explores multi-threshold systems and the application of SR in multiscale applications using wavelet transforms. In the multiscale domain signals can be analyzed at different levels of resolution to better understand the underlying dynamics. We propose a double-threshold detection system that integrates two single-threshold detectors to enhance weak signal detection. We evaluate it both in the original data domain and in the multiscale domain using simulated and real-world signals and compare its performance with existing methods. Experimental results demonstrate that, in the original data domain, the proposed double-threshold detector significantly improves weak signal detection compared to conventional single-threshold approaches. Its performance is further improved in the frequency domain, requiring lower noise levels while outperforming existing detection systems. This study advances SR-based detection methodologies by introducing a robust approach to weak signal identification, with potential applications in various disciplines.

Detecting early-stage ovarian cancer accurately and efficiently is crucial for timely treatment. Various methods for early diagnosis have been explored, including a focus on features derived from protein mass spectra, but these tend to overlook the complex interplay across protein expression levels. We propose an innovative method to automate the search for diagnostic features in these spectra by analyzing their inherent scaling characteristics. We compare two techniques for estimating the self-similarity in a signal using the scaling behavior of its wavelet packet decomposition. The methods are applied to the mass spectra using a rolling window approach, yielding a collection of self-similarity indexes that capture protein interactions, potentially indicative of ovarian cancer. Then, the most discriminatory scaling descriptors from this collection are selected for use in classification algorithms. To assess their effectiveness for early diagnosis of ovarian cancer, the techniques are applied to two datasets from the American National Cancer Institute. Comparative evaluation against an existing wavelet-based method shows that one wavelet packet-based technique led to improved diagnostic performance for one of the analyzed datasets (95.67% vs. 96.78% test accuracy, respectively). This highlights the potential of wavelet packet-based methods to capture novel diagnostic information related to ovarian cancer. This innovative approach offers promise for better early detection and improved patient outcomes in ovarian cancer.

The wavelet spectra is a common starting point for estimating the Hurst exponent of a self-similar signal using wavelet-based techniques. The decay of the log_2 average energy of the detail wavelet coefficients as a function of the level of signal decomposition can be used to construct estimators for this parameter. In this paper, we expand on previous work which introduced the “dual" wavelet spectra, where decomposition levels are instead treated as a function of energy values, and propose a relationship between its slope and the Hurst exponent by inverting the standard wavelet spectra, thereby creating a new estimator. The effectiveness of this estimator and its sensitivity to several settings are demonstrated through a simulation study. Finally, we show how the technique performs as a feature extraction method by applying it to the task of detecting the presence of breast cancer in mammogram images. Dual spectra wavelet features had a statistically significant effect on the log-odds of Cancer.

SummaryEstimating the parameter when is known or simultaneous estimation of and of the binomial distribution based on independent observations has been considered by many authors over the last several decades. A range of estimators have been proposed, and questions regarding asymptotic and small sample properties received adequate treatment. In this paper, we provide an extensive review and a comprehensive performance comparison of the estimators from the literature. We propose a conceptually simple estimator of that uses the marginal likelihood when is integrated out by simultaneous optimisation w.r.t. and the hyperparameters. We compare the proposed estimator with various existing estimators and find its performance competitive and, in some scenarios, superior.

A heart murmur is an atypical sound produced by blood flow through the heart. It can indicate a serious heart condition, so detecting heart murmurs is critical for identifying and managing cardiovascular diseases. However, current methods for identifying murmurous heart sounds do not fully utilize the valuable insights that can be gained by exploring different properties of heart sound signals. To address this issue, this study proposes a new discriminatory set of multiscale features based on the scaling and complexity properties of heart sounds, as characterized in the wavelet domain. Scaling properties are characterized by examining fractal behaviors, while complexity is explored by calculating wavelet entropy. We evaluated the diagnostic performance of these proposed features for detecting murmurs using a set of classifiers. When applied to a publicly available heart sound dataset, our proposed wavelet-based multiscale features achieved 76.61% accuracy using support vector machine classifier, demonstrating competitive performance with existing deep learning methods while requiring significantly fewer features. This suggests that scaling nature and complexity properties in heart sounds could be potential biomarkers for improving the accuracy of murmur detection.

Many natural processes are characterized by complex patterns of self-similarity, where repetitive structures occur across different resolutions. The Hurst exponent is a key parameter used to quantify this self-similarity. While wavelet-based techniques are effective in estimating the Hurst exponent, their performance can be compromised by noise, outliers, and modeling assumptions. This study makes a dual contribution by introducing a novel method for estimating the Hurst exponent under standard modeling assumptions and applying this method to a significant study on gait data. The novel method leverages wavelet transforms (WT) to refine the traditional assessment of self-similarity, which typically depends on the regular decay of signal energies at various resolutions. Our method integrates the standard fractional Brownian motion (fBm) model with exact probability distributions of wavelet coefficients, combining estimates of the Hurst exponent from pairs of wavelet decomposition levels into a single estimate, named ALPHEE, that offers a more precise measure of self-similarity. The study investigates the use of self-similarity features in machine learning algorithms for identifying elderly adults who have had unintentional falls. By analyzing linear acceleration (LA) and angular velocity (AV) in 147 subjects (79 fallers, 68 non-fallers), the study finds higher regularity in LA and AV for fallers. The performance of classification models is compared with and without self-similarity features, suggesting these features enhance the detection of fallers versus non-fallers. The results show that integrating self-similarity features significantly improves performance, with the proposed method achieving 89.65% accuracy, compared to 82.75% using the standard method. This improvement surpasses existing studies based on the same dataset, suggesting that the proposed method more accurately captures self-similar properties, leading to better performance in gait data analysis.

Wavelet Transforms are a widely used technique for decomposing a signal into coefficient vectors that correspond to distinct frequency/scale bands while retaining time localization. This property enables an adaptive analysis of signals at different scales, capturing both temporal and spectral patterns. By examining how correlations between two signals vary across these scales, we obtain a more nuanced understanding of their relationship than what is possible from a single global correlation measure. In this work, we expand on the theory of wavelet-based correlations already used in the literature and elaborate on wavelet correlograms, partial wavelet correlations, and additive wavelet correlations using the Pearson and Kendall definitions. We use both Orthogonal and Non-decimated discrete Wavelet Transforms, and assess the robustness of these correlations under different wavelet bases. Simulation studies are conducted to illustrate these methods, and we conclude with applications to real-world datasets.

A number of approaches have dealt with statistical assessment of self-similarity, and many of those are based on multiscale concepts. Most rely on certain distributional assumptions which are usually violated by real data traces, often characterized by large temporal or spatial mean level shifts, missing values or extreme observations. A novel, robust approach based on Theil-type weighted regression is proposed for estimating self-similarity in two-dimensional data (images). The method is compared to two traditional estimation techniques that use wavelet decompositions; ordinary least squares (OLS) and Abry-Veitch bias correcting estimator (AV). As an application, the suitability of the self-similarity estimate resulting from the the robust approach is illustrated as a predictive feature in the classification of digitized mammogram images as cancerous or non-cancerous. The diagnostic employed here is based on the properties of image backgrounds, which is typically an unused modality in breast cancer screening. Classification results show nearly 68% accuracy, varying slightly with the choice of wavelet basis, and the range of multiresolution levels used.

The objective of this research focuses on the development of a statistical methodology able to answer the question of whether variation in the intake of sulfur amino acids (SAA) affects the metabolic process. Traditional approaches, which evaluate specific biomarkers after a series of preprocessing procedures, have been criticized as not being fully informative, as well as inappropriate for translation of methodology. Rather than focusing on particular biomarkers, our proposed methodology involves the multifractal analysis that measures the inhomogeneity of regularity of the proton nuclear magnetic resonance (1H-NMR) spectrum by wavelet-based multifractal spectrum. With two different statistical models (Model-I and Model-II), three different geometric features of the multifractal spectrum of each 1H-NMR spectrum (spectral mode, left slope, and broadness) are employed to evaluate the effect of SAA and discriminate 1H-NMR spectra associated with different treatments. The investigated effects of SAA include group effect (high and low doses of SAA), depletion/repletion effect, and time over data effect. The 1H-NMR spectra analysis outcomes show that group effect is significant for both models. The hourly variation in time and depletion/repletion effects does not show noticeable differences for the three features in Model-I. However, these two effects are significant for the spectral mode feature in Model-II. The 1H-NMR spectra of the SAA low groups exhibit highly regular patterns with more variability than that of the SAA high groups for both models. Moreover, the discriminatory analysis conducted using the support vector machine and the principal components analysis shows that the 1H-NMR spectra of SAA high and low groups can be easily discriminatory for both models, while the spectra of depletion and repletion within these groups are discriminatory for Model-I and Model-II. Therefore, the study outcomes imply that the amount of SAA is important and that SAA intake affects mostly the hourly variation of the metabolic process and the difference between depletion and repletion each day. In conclusion, the proposed multifractal analysis of 1H-NMR spectra provides a novel tool to investigate metabolic processes.