Abstract In this article, a comprehensive analysis of the approach known as Split-Step Wavelet Parabolic Equation (SSW-PE) in modeling radio wave propagation is presented. The SSW-PE introduces innovations, such as the application of narrow-angle and wideangle approaches, referred to as NAPE and WAPE, respectively. Furthermore, the SSW-PE demonstrates the incorporation of refractivity variations, modifications in terrain modeling for better representation, and considerations of surface boundary conditions. In addition to its innovative aspects, this study aims to provide a complete guide for effectively replicating the algorithm, thereby promoting the advancement of propagation studies using wavelets. The effectiveness and applicability of this approach are validated through comparative studies with well-established solutions, including the Discrete Mixed Fourier Transform (DMFT) version of the Split-Step Parabolic Equation (SSPE) method. Comparisons with measurements from real propagation cases are also conducted. Statistical analysis confirms the innovative potential of the SSW-PE algorithm, which also offers computational efficiency for rapid and consistent simulations. Thus, this article contributes to a comprehensive and innovative analysis, providing tangible resources for the research community interested in expanding this methodology. SplitStep Split Step SSWPE SSW PE (SSW-PE presented innovations narrowangle narrow angle approaches WAPE respectively Furthermore variations representation conditions aspects algorithm wavelets wellestablished well established solutions DMFT (DMFT SSPE (SSPE method conducted simulations Thus methodology

ABSTRACT An analytical version of the discrete ordinates method (ADO) is used to establish a concise and accurate solution for the Reverse Temperature Gradient problem in rarefied gas binary mixtures. The kinetic equations used are based in the McCormack kinetic model. It presents the complete development of an analytical solution in the spatial variable, in addition to numerical results for the critical values of latent heat for inversion from the direction of the temperature gradient and the heat flux of each gas in the mixtures Neon-Argon and Helium-Xenon. The algorithm is considered simple to elaborate and the code developed (in FORTRAN) requires around the one second to execute in a computer with Intel Core i3 - 2.30 GHz processor. ADO (ADO model variable NeonArgon Neon Argon HeliumXenon. HeliumXenon Helium Xenon. Xenon Helium-Xenon FORTRAN i 230 2 30 2.3 processor 23 3 2.

RESUMO Uma versão analítica do método de ordenadas discretas (ADO) é usada para determinar uma solução precisa e consistente para o problema do Reverso de Temperatura em misturas binárias de gases rarefeitos. As equações cinéticas usadas são baseadas no modelo cinético McCormack. Apresenta-se o desenvolvimento completo de uma solução analítica na variável espacial, além de resultados numéricos para os valores críticos do calor latente para a inversão de sentido do gradiente de temperatura e do fluxo de calor de cada gás nas misturas Neônio-Argônio e Hélio-Xenônio. O algoritmo é considerado simples de ser elaborado e o código desenvolvido (em FORTRAN) requer em torno de um segundo para ser executado em um computador com processador Intel Core i3 - 2.30 GHz. ADO (ADO rarefeitos McCormack Apresentase Apresenta se espacial NeônioArgônio Neônio Argônio HélioXenônio. HélioXenônio Hélio Xenônio. Xenônio Hélio-Xenônio FORTRAN i 230 2 30 2.3 GHz 23 3 2.

ABSTRACT In the present work, a calculation methodology is proposed to determine the filling and / or emptying times of the tanks in the pumping source systems. It focuses on solving a discrete calculation model that takes into account the main hydraulic variables that characterize three schemes or cases of design and operation in these systems. Two discrete approach calculation models are presented as a function of height and time, which are implemented in the MATLAB version R2013b numerical computing system for the comparison of different variants of the design schemes described. Both analytical procedures are used to solve a practical example where the three cases of operation are illustrated for different variants and calculation alternatives.

RESUMEN En el presente trabajo se propone una metodología de cálculo para determinar los tiempos de llenado y/o vaciado de los depósitos en los sistemas fuente por bombeo. La misma, se centra en la resolución de un modelo de cálculo discreto que tiene en cuenta las principales variables hidráulicas que caracterizan tres esquemas o casos de diseño y operación en dichos sistemas. Se presentan dos modelos de cálculo de enfoque discreto en función de la altura y el tiempo, los cuales se implementan en el sistema de cómputo numérico matlab versión r2013b para la comparación de diferentes variantes de los esquemas de diseño descritos. Ambos procedimientos analíticos se emplean en la resolución de un ejemplo práctico donde se ilustran los tres casos de operación para distintas variantes y alternativas de cálculo.

Abstract: In a mixed-integer nonlinear programming problem, integer restrictions divide the feasible region into discontinuous feasible parts with different sizes. Meta-heuristic optimization algorithms quickly lose diversity in such scenarios and get trapped in local optima. In this work, we propose an Estimation of Distribution Algorithm (EDA) with two modifications from its previous version ( E D A m v). The first modification consists in establishing the exploration and exploitation components for the histogram of discrete variables, aimed at improving the performance of the algorithm during the evolution. The second modification is a repulsion operator to overcome the population stagnation in discontinuous parts, so as continuing the search for possible solutions in other regions. From a comparative study on 16 test problems, the individual contribution of each modification was verified. According to statistical test results, the new proposal shows a significantly better performance than the other competitors tested.

Abstract This study evaluated the association between sense of coherence (SoC) and oral health-related quality of life (OHRQoL) among conscripts of the Brazilian Army, in two cities of southern Brazil. A cross-sectional study included all 18-19-year-old adolescents who joined the Brazilian Army as draftees for mandatory military service in the cities of Itaqui, RS, and Santiago, RS (n = 505). Data collection was conducted from 2019 to 2021, and included the application of questionnaires and a clinical oral examination to record gingivitis, malocclusion, and dental caries. OHRQoL was collected through the Brazilian short version of the Oral Health Impact Profile (OHIP), composed of 14 questions. The adolescents’ SoC was assessed using the validated Brazilian version of the SOC-13 scale. The primary outcome of this study was OHRQoL, modeled as a discrete variable (OHIP-14 scores). The main predictor variable was SoC, categorized as low, moderate, or high. The association between predictor variables and OHRQoL was assessed by Poisson regression models using a hierarchical approach. Unadjusted and adjusted rate ratios (RR), and 95% confidence intervals (CI) were estimated. All the analyses were performed using STATA software version 14.2. Adolescents with a moderate and high SoC had 27% (RR = 0.73, 95%CI = 0.64–0.84) and 51% (RR = 0.49, 95%CI = 0.41–0.58) lower mean OHIP-14 scores, respectively, than those with a low SoC score, after the inclusion of behavioral and clinical variables. This study showed a significant association between SoC and OHRQoL among 18–19-year-old southern Brazilian adolescents. Strengthening the SoC as a psychosocial resource may improve the well-being and OHRQoL of adolescents. (SoC healthrelated health related (OHRQoL Brazil crosssectional cross sectional 1819yearold yearold 18 19 year old Itaqui Santiago n 505. 505 . 505) 201 2021 gingivitis malocclusion caries OHIP, OHIP , (OHIP) 1 questions SOC13 SOC 13 SOC-1 scale OHIP14 (OHIP-1 scores. scores scores) approach RR, RR (RR) 95 CI (CI estimated 142 2 14.2 27 073 0 73 0.73 95CI 0.64–0.84 064084 64 84 51 049 49 0.49 0.41–0.58 041058 41 58 OHIP-1 respectively score 18–19yearold 18–19 wellbeing well being 50 20 202 (OHIP SOC1 SOC- OHIP1 (OHIP- 9 14. 07 7 0.7 0.64–0.8 06408 6 8 5 04 4 0.4 0.41–0.5 04105 OHIP- 19yearold 1819 18–1 0. 0.64–0. 0640 0.41–0. 0410 181 18– 0.64–0 064 0.41–0 041 0.64– 06 0.41– 0.64 0.41 0.6

Abstract: The Associated linear equations (ALEs) are parametric models of the Volterra operators. With them, a Volterra inverse is constructed to be used as open loop control of continuous nonlinear systems. However, most of the actual control systems are of discrete nature, this work introduces the novel discrete version of the ALEs. This discrete version is a series of ARX models of the Volterra operators for both the direct and the inverse series. These discrete models of the ALEs are used for an equalization strategy to create an open loop control on a reported simulated Duffing Oscillator and a physical Duffing system constructed by analog circuits.

ABSTRACT The use of the Multicriteria Decision Support Hybrid Algorithm for Decision Making Processes with Discrete Alternatives, acronym THOR, requires from the decision maker, during the judgment insertion stage, a significant amount of information that needs to be valued, which may cause the decision maker great cognitive fatigue. Therefore, this article aims to reformulate the THOR algorithm, in the judgment insertion stage, based on the inclusion of a fuzzy measurement scale, allowing the decision maker to express only a single value judgment. This reformulation follows the three steps of a fuzzy system: fuzzification, fuzzy inference and defuzzification. In addition, a comparative analysis is performed between the THOR algorithm and its new version based on the construction of the fuzzy scale. It should be noted that its reformulation does not compromise the methodological efficiency of the THOR algorithm, it only reduces the complexity of decision making.

Abstract The study of teachers' emotions has been gaining prominence, and emotions are currently recognized as an essential dimension affecting teachers’ work. Despite having gained relevance in the educational area, there is a need for valid instruments that measure teachers’ discrete emotions. This study aimed to translate the Teacher Emotion Questionnaire (TEQ) to the Portuguese language and to adapt it to the Brazilian culture. Fifty-one participants were involved: four translators, seven specialists, 40 teachers, and also the researchers and the main author of the original TEQ. The current methodological study followed six stages: (1) translations; (2) synthesis of the translated versions; (3) committee of experts; (4) evaluation by the target audience; (5) reverse translation; (6) pilot study. The translation and synthesis steps were carried out properly, and some changes were made after the assessment by the experts and target audience. The pilot study showed that the adapted instrument is easy to understand and fill in. Finally, after some adjustments, the cultural adaptation was approved by the main author of the original TEQ. This study was the first step to provide a reliable and valid instrument on the emotions of teachers. The next step will be the exploration of the psychometric properties of TEQ - Brazilian version.

Resumen El estudio de las emociones de los profesores ha ganado destaque, y actualmente es reconocido como una dimensión esencial del trabajo docente. Sin embargo, a pesar de haber ganado relevancia en el área educativa, es visible que hay una laguna de instrumentos cuantitativos válidos para medir las emociones discretas del profesor. Este estudio tuvo como objetivo traducir al portugués y adaptar el Teacher Emotion Questionnaire (TEQ) para la cultura brasileña. Participaron 51 individuos, de ellos cuatro traductores, siete especialistas, 40 profesores, y también las investigadoras y autora principal del TEQ original. El presente estudio metodológico de adaptación cultural contó con las etapas a continuación: (1) traducciones; (2) síntesis de las versiones traducidas; (3) comité de expertos; (4) evaluación por el público objetivo; (5) traducción inversa; (6) estudio piloto. Las etapas de traducción y síntesis fueron realizadas adecuadamente, introduciendo algunas modificaciones después del análisis de los especialistas y del público objetivo. El estudio piloto mostró que el instrumento adaptado es de fácil comprensión y llenado. Finalmente, la adaptación cultural fue aprobada por la autora principal del TEQ original, después de algunos ajustes. El presente estudio dio el primer paso para colocar a disposición un instrumento confiable y válido para analizar las emociones de los profesores. El próximo paso será la exploración de las propiedades psicométricas del TEQ - versión brasileña.

Resumo O estudo das emoções dos professores tem vindo a ganhar destaque, sendo atualmente reconhecido como uma dimensão essencial do trabalho docente. Contudo, apesar de ter vindo a ganhar relevância na área educacional, verifica-se uma lacuna de instrumentos quantitativos válidos para mensurar as emoções discretas do professor. Este estudo teve como objetivo traduzir para português e adaptar o Teacher Emotion Questionnaire (TEQ) para a cultura brasileira. Foram envolvidos, 51 participantes, dos quais quatro tradutores, sete especialistas, 40 professores, e ainda as pesquisadoras e autora principal do TEQ original. O presente estudo metodológico de adaptação cultural contou com as seguintes etapas: (1) traduções; (2) síntese das versões traduzidas; (3) comitê de especialistas; (4) avaliação pelo público-alvo; (5) tradução reversa; (6) estudo-piloto. As etapas tradução e síntese foram realizadas adequadamente, tendo sido introduzidas algumas alterações após a apreciação pelos especialistas e público-alvo. O estudo piloto mostrou que o instrumento adaptado é de fácil compreensão e preenchimento. Por fim, após alguns ajustes, a adaptação cultural foi aprovada pela autora principal do TEQ original. O presente estudo realizou o primeiro passo para disponibilizar um instrumento confiável e válido sobre as emoções dos professores. O próximo passo será a exploração das propriedades psicométricas do TEQ - Versão brasileira.

RESUMO Os fenômenos que envolvem a dinâmica de gases rarefeitos para fluxo de gases em micro-canais têm sido fonte de estudo em várias pesquisas. A Equação de Boltzmann descreve tais fenômenos sendo que a partir da sua simplificação surge a teoria cinética para a dinâmica dos gases rarefeitos. Os modelos cinéticos apresentam resultados sólidos e é abordado, neste estudo, em uma versão analítica através do método de ordenadas discretas. Dessa maneira, exploramos um modelo mais próximo da realidade, comparando os valores numéricos gerados por quatro modelos cinéticos derivados de maneira unificada da Equação Linearizada de Boltzmann. Em termos de viscosidade e condutividade térmica os modelos são confrontados baseados na estatística não paramétrica para problemas da dinâmica de gases rarefeitos resultantes do movimento de transferência de massa. Além disso, utiliza-se a interação gás-superfície definida pelo núcleo de Cercignani-Lampis ao longo de um canal para placas heterogêneas.

ABSTRACT The phenomena that involve the rarefied gas dynamics for gas flow in micro-channels have been a source of study in several studies. The Boltzmann Equation describes such phenomena, and from its simplification comes the kinetic theory for the dynamics of rarefied gases. The kinetic models present reliable results and are analyzed in this study under an analytical version using the discrete ordinate method. Therefore, we explore a model closer to reality, comparing the numerical values generated by four kinetic models derived in a unified way from Boltzmann’s Linearized Equation. In terms of viscosity and thermal conductivity, the models are compared based on non-parametric statistics for problems with rarefied gas dynamics resulting from the mass transfer movement. Also, the gas-surface interaction defined by the Cercignani-Lampis nucleus along a channel for heterogeneous plates is used.

The Discrete Pulse Transform (DPT) makes use of LULU smoothing to decompose a signal into block pulses. The most recent and effective implementation of the DPT is an algorithm called the Roadmaker's Pavage, which uses a graph-based algorithm that produces a hierarchical tree of pulses as its final output, shown to have important applications in artificial intelligence and pattern recognition. Even though the Roadmaker's Pavage is an efficient implementation, the theoretical structure of the DPT results in a slow, deterministic algorithm. This paper examines the use of the spectral domain of graphs and designing graph filter banks to downsample the algorithm. We investigate the extent to which this speeds up the algorithm and allows parallel processing. Converting graph signals to the spectral domain can also be a costly overhead, so methods of estimation for filter banks are examined, as well as the design of a good filter bank that may be reused without needing recalculation. The sampled version requires different hyperparameters in order to reconstruct the same textures of the image as the original algorithm, selected previously either through trial and error (subjective) or grid search (costly) which prevented studying the results on many images effectively. Here an objective and efficient way of deriving similar results between the original Roadmaker's Pavage and our proposed Filtered Roadmaker's Pavage is provided. The method makes use of the Ht-index which separates the distribution of information in the graph at scale intervals by recursively calculating averages on decreasing subsections of the scale data stored. This has enabled empirical research using benchmark datasets providing improved results. The results of these empirical tests showed that using the Filtered Roadmaker's Pavage algorithm consistently runs faster, using less computational resources, while having a positive SSIM (structural similarity) with low variance. This provides an informative and faster approximation to the nonlinear DPT, a property which is not standardly achieveable.CATEGORIES: • Mathematics of Computing ~ Probability and Statistics • Probability and Statistics ~ Probabilistics Algorithms

Abstract: This paper presents an alternative for noise reduction in electrical signals produced by partial discharges, PDs. The method is based on the local polynomial approximation (LPA) and its combination with the intersection of confidence intervals (ICI) algorithm. The method differs from other techniques by providing the best version of a signal from a mathematical and statistical analysis that does not depend on the calculation of its representation in the time-frequency plane. Likewise, an evaluation of the proposed method is done from experimentally obtained PD signals. Finally, the results are compared with those obtained by different filters (Fourier, FIR and discrete wavelets transform-DWT). This comparison shows that, although the methods based on the DWT provide good results, the LPA-ICI method provides signals with high signal-to-noise ratio (>70%), a high correlation (>0.899) and a low amplitude distortion (<3.5%).

Resumen: Este artículo presenta una alternativa para la reducción de ruido en señales eléctricas producidas por descargas parciales (DPs). El método se basa en la aproximación local polinomial (LPA) y su combinación con el algoritmo de intersección de intervalos de confianza (ICI), y se diferencia de otras técnicas en que proporciona la mejor versión de una señal a partir de un análisis matemático y estadístico que no depende del cálculo de su representación en el plano tiempo-frecuencia. Asimismo, se realiza una evaluación del método propuesto a partir de señales de DPs obtenidas experimentalmente. Finalmente, los resultados alcanzados son comparados con los obtenidos por diversos filtros (Fourier, FIR y transformación de ondas discreta-DWT). Esta comparación muestra que, aunque los métodos basados en la DWT entregan buenos resultados, el método LPA-ICI provee señales con alta relación señal-a-ruido (>70%), una alta correlación (>0.899) y una baja distorsión de amplitud (<3.5%).

Resumen En esta nota anunciamos los resultados de nuestra investigación sobre las propiedades L p de operadores pseudodiferenciales multilineales periódicos y/o discretos. Primero, revisaremos el análisis multilineal de tales operadores mostrando versiones análogas de los teoremas clásicos disponibles en el análisis multilineal euclidiano (debidos a Coifman y Meyer, Tomita, Miyachi, Fujita, Grafakos, Tao, etc.), pero, en el contexto de operadores periódicos y/o discretos. Se caracterizará la s-nuclearidad, 0 < s ≤ 1, para operadores multilineales pseudodiferenciales periódicos y/o discretos. Para cumplir este objetivo se clasificarán aquellos operadores lineales s-nucleares, 0 < s ≤ 1, multilineales con núcleo, sobre espacios de Lebesgue arbitrarios definidos en espacios de medida σ-finitos. Finalmente, como aplicación de los resultados presentados se obtiene la versión periódica de la desigualdad de Kato-Ponce, y se examina la s-nuclearidad de potenciales de Bessel lineales y multilineales, como también la s-nuclearidad de operadores integrales de Fourier periódicos admitiendo símbolos con tipos adecuados de singularidad.

Abstract In this note we announce our investigation on the L p properties for periodic and discrete multilinear pseudo-differential operators. First, we review the periodic analysis of multilinear pseudo-differential operators by showing classical multilinear Fourier multipliers theorems (proved by Coifman and Meyer, Tomita, Miyachi, Fujita, Grafakos, Tao, etc.) in the context of periodic and discrete multilinear pseudo-differential operators. For this, we use the periodic analysis of pseudo-differential operators developed by Ruzhansky and Turunen. The s-nuclearity, 0 < s ≤ 1, for the discrete and periodic multilinear pseudo-differential operators will be investigated. To do so, we classify those s-nuclear, 0 < s ≤ 1, multilinear integral operators on arbitrary Lebesgue spaces defined on σ-finite measures spaces. Finally, we present some applications of our analysis to deduce the periodic Kato-Ponce inequality and to examine the s-nuclearity of multilinear Bessel potentials as well as the s-nuclearity of periodic Fourier integral operators admitting suitable types of singularities. MSC2010: 58J40, 47B10, 47G30, 35S30.

Abstract A new discrete distribution is introduced. The distribution involves the negative binomial and size biased negative binomial distributions as sub-models among others and it is a weighted version of the two parameter discrete Lindley distribution. The distribution has various interesting properties, such as bathtub shape hazard function along with increasing/decreasing hazard rate, positive skewness, symmetric behavior, and over- and under-dispersion. Moreover, it is self decomposable and infinitely divisible, which makes the proposed distribution well suited for count data modeling. Other properties are investigated, including probability generating function, ordinary moments, factorial moments, negative moments and characterization. Estimation of the model parameters is investigated by the methods of moments and maximum likelihood, and a performance of the estimators is assessed by a simulation study. The credibility of the proposed distribution over the negative binomial, Poisson and generalized Poisson distributions is discussed based on some test statistics and four real data sets.

RESUMO Objetivo Disfunção cognitiva é comum em pacientes com esclerose múltipla. Por isto, a Brief Repeatable Battery of Neuropsychological Tests (BRB–N) foi desenvolvida para avaliar as funções cognitivas mais frequentemente alteradas na doença. Entretanto, estão faltando dados normativos desta bateria no Brasil. Assim, nosso objetivo foi fornecer valores normativos contínuos e discretos da BRB–N para a população brasileira. Métodos Foram recrutados 285 indivíduos sadios da comunidade em 10 centros do Brasil e aplicada a versão A em 237 e a versão B em 48 sujeitos. Normas contínuas foram calculadas com análise de regressão múltipla. Resultados Escores brutos médios e 5°percentil para cada subteste são fornecidos, estratificados por idade e nível educacional. Os escores brutos dos sujeitos sadios foram convertidos em escores de escalas e postos em regressão quanto a idade, sexo e educação, fornecendo equações que podem ser usadas para calcular escores previsíveis. Conclusão Nossos dados normativos permitem um uso mais amplo da BRB–N na prática clínica e na pesquisa, fornecendo normas para dados discretos e contínuos. Normas para dados discretos deveriam ser usadas com cuidado e escores demograficamente ajustados são geralmente preferidos quando interpretando dados neuropsicológicos.

ABSTRACT Objective Cognitive dysfunction is common in multiple sclerosis. The Brief Repeatable Battery of Neuropsychological Tests (BRB–N) was developed to assess cognitive functions most–frequently impaired in multiple sclerosis. However, normative values are lacking in Brazil. Therefore, we aimed to provide continuous and discrete normative values for the BRB–N in a Brazilian population sample. Methods We recruited 285 healthy individuals from the community at 10 Brazilian sites and applied the BRB–N version A in 237 participants and version B in 48 participants. Continuous norms were calculated with multiple–regression analysis. Results Mean raw scores and the 5th percentile for each neuropsychological measure are provided, stratified by age and educational level. Healthy participants' raw scores were converted to scaled scores, which were regressed on age, sex and education, yielding equations that can be used to calculate predicted scores. Conclusion Our normative data allow a more widespread use of the BRB–N in clinical practice and research.

RESUMO Devido sua habilidade de localização tempo-frequência, a Transformada wavelet tem sido aplicada em várias áreas de pesquisa envolvendo análise e processamento de dados, frequentemente substituindo a convencional Transformada de Fourier. A TransformadaWavelet Discreta tem um grande potencial de aplicação¸ destacando-se como uma importante ferramenta na compressão de sinal, processamento de imagem e sinal, suavização e filtragem de ruídos em dados. Ela também apresenta vantagens sobre a versão contínua por causa de sua fácil implementação, bom desempenho computacional e reconstrução perfeita do sinal após inversão. No entanto, a decimação requerida no cálculo da Transformada Wavelet Discreta a torna variante à translação e não apropriada para algumas aplicaões, tais como análise de sinais ou séries temporais. Por outro lado, a Transformada Wavelet Discreta Não Decimada é invariante à translação, porque elimina o processo de decimação, e consequentemente, é mais apropriada para identificar comportamentos estacionários e não estacionários presentes no sinal. No entanto, a Transformada Wavelet Não Decimada tem sido pouco usada na literatura. Esse artigo pretende mostrar as vantagens do uso na Transformada Wavelet Não Decimada na análise de sinais. Os principais aspectos teóricos e práticos da análise multiescala de séries temporais a partir das wavelets não decimadas, em termos de sua formulação usando o mesmo algoritmo piramidal da Transformada Wavelet Decimada, são apresentados. Por fim, aplicações com séries temporais simuladas e reais comparam o desempenho das transformadas wavelet decimada e não decimada, demonstrando a superioridade da wavelet não decimada, principalmente devido à análise invariante a translação, detecção de padrões e uma reconstrução mais perfeita do sinal.

ABSTRACT Due to the ability of time-frequency location, the wavelet transform has been applied in several areas of research involving signal analysis and processing, often replacing the conventional Fourier transform. The discrete wavelet transform has great application potential, being an important tool in signal compression, signal and image processing, smoothing and de-noising data. It also presents advantages over the continuous version because of its easy implementation, good computational performance and perfect reconstruction of the signal upon inversion. Nevertheless, the downsampling required in the computation of the discrete wavelet transform makes it shift variant and not appropriated to some applications, such as for signals or time series analysis. On the other hand, the Non-Decimated Discrete Wavelet Transform is shift-invariant because it eliminates the downsampling and, consequently, is more appropriate for identifying both stationary and non-stationary behaviors in signals. However, the non-decimated wavelet transform has been underused in the literature. This paper intends to show the advantages of using the non-decimated wavelet transform in signal analysis. The main theoretical and practical aspects of the multi-scale analysis of time series from non-decimated wavelets in terms of its formulation using the same pyramidal algorithm of the decimated wavelet transform was presented. Finally, applications with a simulated and real time series compare the performance of the decimated and non-decimated wavelet transform, demonstrating the superiority of non-decimated one, mainly due to the shift-invariant analysis, patterns detection and more perfect reconstruction of a signal.