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See detailA multifractal formalism for non-concave and non-increasing spectra: the leaders profile method
Esser, Céline ULg; Kleyntssens, Thomas ULg; Nicolay, Samuel ULg

in Applied & Computational Harmonic Analysis (in press)

We present an implementation of a multifractal formalism based on the types of histogram of wavelet leaders. This method yields non-concave spectra and is not limited to their increasing part. We show ... [more ▼]

We present an implementation of a multifractal formalism based on the types of histogram of wavelet leaders. This method yields non-concave spectra and is not limited to their increasing part. We show both from the theoretical and from the applied points of view that this approach is more e cient than the wavelet-based multifractal formalisms previously introduced. [less ▲]

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See detailDoes Belgian Holocene speleothem record solar forcing and cold events?
Allan, Mohammed ULg; Deliège, Adrien ULg; Nicolay, Samuel ULg et al

in Climate of the Past Discussions (2017)

We present a decadal-centennial scale Holocene climate record based on trace elements contents from a 65 cm stalagmite (Père Noël) from Belgian Père Noël cave. Père Noël (PN) stalagmite covers the last 12 ... [more ▼]

We present a decadal-centennial scale Holocene climate record based on trace elements contents from a 65 cm stalagmite (Père Noël) from Belgian Père Noël cave. Père Noël (PN) stalagmite covers the last 12.7 ka according to U/Th dating. High spatial resolution measurements of trace elements (Sr, Ba, Mg and Al) were done by Laser-Ablation Inductively Coupled Plasma Mass Spectrometry (LA-ICP-MS). Trace elements profiles were interpreted as environmental and climate changes in the Han-sur-Lesse region. Power spectrum estimators and continuous wavelet transform were applied on trace elements time series to detect any statistically significant periodicities in the PN stalagmite. Spectral analyses reveal decadal to millennial periodicities (i.e., 68–75, 133–136, 198–209, 291–358, 404–602, 912–1029 and 2365–2670 yr) in the speleothem record. Results were compared to reconstructed sunspot number data to determine whether solar signal is presents in PN speleothem. The occurrence of significant solar periodicities (i.e., cycles of Gleissberg, de Vries, unnamed 500 years, Eddy and Hallstat) supports for an impact of solar forcing on PN speleothem trace elements contents. Moreover, several intervals of significant rapid climate change were detected during the Holocene at 10.3, 9.3–9.5, around 8.2, 6.4–6.2, 4.7–4.5, and around 2.7 ka BP. Those intervals are similar to the cold events evidenced in different natural paleoclimate archivers, suggesting common climate forcing mechanisms related to changes in solar irradiance. [less ▲]

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See detailDoes Belgian Holocene speleothem record solar forcing and cold events?
Allan, Mohammed ULg; Deliège, Adrien ULg; Nicolay, Samuel ULg et al

in Climate of the Past Discussions (2017)

We present a decadal-centennial scale Holocene climate record based on trace elements contents from a 65 cm stalagmite (Père Noël) from Belgian Père Noël cave. Père Noël (PN) stalagmite covers the last 12 ... [more ▼]

We present a decadal-centennial scale Holocene climate record based on trace elements contents from a 65 cm stalagmite (Père Noël) from Belgian Père Noël cave. Père Noël (PN) stalagmite covers the last 12.7 ka according to U/Th dating. High spatial resolution measurements of trace elements (Sr, Ba, Mg and Al) were done by Laser-Ablation Inductively Coupled Plasma Mass Spectrometry (LA-ICP-MS). Trace elements profiles were interpreted as environmental and climate changes in the Han-sur-Lesse region. Power spectrum estimators and continuous wavelet transform were applied on trace elements time series to detect any statistically significant periodicities in the PN stalagmite. Spectral analyses reveal decadal to millennial periodicities (i.e., 68–75, 133–136, 198–209, 291–358, 404–602, 912–1029 and 2365–2670 yr) in the speleothem record. Results were compared to reconstructed sunspot number data to determine whether solar signal is presents in PN speleothem. The occurrence of significant solar periodicities (i.e., cycles of Gleissberg, de Vries, unnamed 500 years, Eddy and Hallstat) supports for an impact of solar forcing on PN speleothem trace elements contents. Moreover, several intervals of significant rapid climate change were detected during the Holocene at 10.3, 9.3–9.5, around 8.2, 6.4–6.2, 4.7–4.5, and around 2.7 ka BP. Those intervals are similar to the cold events evidenced in different natural paleoclimate archivers, suggesting common climate forcing mechanisms related to changes in solar irradiance. [less ▲]

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See detailDoes Belgian Holocene speleothem record solar forcing and cold events?
Allan, Mohammed ULg; Deliège, Adrien ULg; Nicolay, Samuel ULg et al

in Climate of the Past Discussions (2017)

We present a decadal-centennial scale Holocene climate record based on trace elements contents from a 65 cm stalagmite (Père Noël) from Belgian Père Noël cave. Père Noël (PN) stalagmite covers the last 12 ... [more ▼]

We present a decadal-centennial scale Holocene climate record based on trace elements contents from a 65 cm stalagmite (Père Noël) from Belgian Père Noël cave. Père Noël (PN) stalagmite covers the last 12.7 ka according to U/Th dating. High spatial resolution measurements of trace elements (Sr, Ba, Mg and Al) were done by Laser-Ablation Inductively Coupled Plasma Mass Spectrometry (LA-ICP-MS). Trace elements profiles were interpreted as environmental and climate changes in the Han-sur-Lesse region. Power spectrum estimators and continuous wavelet transform were applied on trace elements time series to detect any statistically significant periodicities in the PN stalagmite. Spectral analyses reveal decadal to millennial periodicities (i.e., 68–75, 133–136, 198–209, 291–358, 404–602, 912–1029 and 2365–2670 yr) in the speleothem record. Results were compared to reconstructed sunspot number data to determine whether solar signal is presents in PN speleothem. The occurrence of significant solar periodicities (i.e., cycles of Gleissberg, de Vries, unnamed 500 years, Eddy and Hallstat) supports for an impact of solar forcing on PN speleothem trace elements contents. Moreover, several intervals of significant rapid climate change were detected during the Holocene at 10.3, 9.3–9.5, around 8.2, 6.4–6.2, 4.7–4.5, and around 2.7 ka BP. Those intervals are similar to the cold events evidenced in different natural paleoclimate archivers, suggesting common climate forcing mechanisms related to changes in solar irradiance. [less ▲]

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See detailDoes Belgian Holocene speleothem record solar forcing and cold events?
Allan, Mohammed ULg; Deliège, Adrien ULg; Nicolay, Samuel ULg et al

in Climate of the Past Discussions (2017)

We present a decadal-centennial scale Holocene climate record based on trace elements contents from a 65 cm stalagmite (Père Noël) from Belgian Père Noël cave. Père Noël (PN) stalagmite covers the last 12 ... [more ▼]

We present a decadal-centennial scale Holocene climate record based on trace elements contents from a 65 cm stalagmite (Père Noël) from Belgian Père Noël cave. Père Noël (PN) stalagmite covers the last 12.7 ka according to U/Th dating. High spatial resolution measurements of trace elements (Sr, Ba, Mg and Al) were done by Laser-Ablation Inductively Coupled Plasma Mass Spectrometry (LA-ICP-MS). Trace elements profiles were interpreted as environmental and climate changes in the Han-sur-Lesse region. Power spectrum estimators and continuous wavelet transform were applied on trace elements time series to detect any statistically significant periodicities in the PN stalagmite. Spectral analyses reveal decadal to millennial periodicities (i.e., 68–75, 133–136, 198–209, 291–358, 404–602, 912–1029 and 2365–2670 yr) in the speleothem record. Results were compared to reconstructed sunspot number data to determine whether solar signal is presents in PN speleothem. The occurrence of significant solar periodicities (i.e., cycles of Gleissberg, de Vries, unnamed 500 years, Eddy and Hallstat) supports for an impact of solar forcing on PN speleothem trace elements contents. Moreover, several intervals of significant rapid climate change were detected during the Holocene at 10.3, 9.3–9.5, around 8.2, 6.4–6.2, 4.7–4.5, and around 2.7 ka BP. Those intervals are similar to the cold events evidenced in different natural paleoclimate archivers, suggesting common climate forcing mechanisms related to changes in solar irradiance. [less ▲]

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See detailBelgian speleothem records Holocene cold events?
Allan, Mohammed ULg; Deliège, Adrien ULg; Nicolay, Samuel ULg et al

Poster (2017, May 12)

Speleothem is now regarded as valuable archive of climatic conditions on the continents, offering a number of advantages relative to other continental climate proxy recorders such as lake sediments and ... [more ▼]

Speleothem is now regarded as valuable archive of climatic conditions on the continents, offering a number of advantages relative to other continental climate proxy recorders such as lake sediments and peat cores. High spatial resolution measurements of Mg, Al, Sr, Ba were realized by using laser-ablation inductively coupled plasma mass spectrometry in the Belgian Pere Noel cave. A stalagmite from the Pere Noel (PN) cave representing 12000 years dated by U/Th method. Trace element variations in speleothem are a reflection of hydrochemical conditions. These changes were interpreted as indications of changes in climate in the Han-sur-Lesse region. The similar patterns found in δ 18O, δ 13C and chemical composition along the Pere Noel stalagmite suggests that trace elements in speleothems have the potential to provide high resolution insights into palaeoclimatic variability during the Holocene. A deeper analysis reveals several periods of significant rapid climate change during the Holocene (at 10.7-9.2 ka, 8.2-7.9 ka, 7.2-6.2 ka, 4.8-4.5 ka, and 3-2.4 ka BP), which are similar to the cold events detected from different natural paleoclimate archivers. A comparison between the geochemical analysis of Père Noël speleothem and solar activity (sunspot number) reveals a significant correlation. Spectral analysis methods reveal common solar periodicities (Gleissberg cycle, de Vries cycle, unnamed 500 year, Eddy cycles, and Hallstatt cycle). The geochemical analyses have the potential to prove that PN speleothem is sensitive to changes in solar activity on centennial and millennial timescales during the Holocene. [less ▲]

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See detailAnalysis and indications on long-term forecasting of the Oceanic Niño Index with wavelet-induced components
Deliège, Adrien ULg; Nicolay, Samuel ULg

in Pure and Applied Geophysics (2017), 174(4), 1815-1826

The present paper provides an analysis and a long-term forecasting scheme of the Oceanic Niño Index (ONI) using the continuous wavelet transform. First, it appears that oscillatory components with main ... [more ▼]

The present paper provides an analysis and a long-term forecasting scheme of the Oceanic Niño Index (ONI) using the continuous wavelet transform. First, it appears that oscillatory components with main periods of about 17, 31, 43, 61 and 140 months govern most of the variability of the signal, which is consistent with previous works. Then, this information enables us to derive a simple algorithm to model and forecast ONI. The model is based on the observation that the modes extracted from the signal are generally phased with positive or negative anomalies of ONI (El Niño and La Niña events). Such a feature is exploited to generate locally stationary curves that mimic this behavior and which can be easily extrapolated to form a basic forecast. The wavelet transform is then used again to smooth out the process and finalize the predictions. The skills of the technique described in this paper are assessed through retroactive forecasts of past El Niño and La Niña events and via classic indicators computed as functions of the lead time. The main asset of the proposed model resides in its long-lead prediction skills. Consequently, this approach should prove helpful as a complement to other models for estimating the long-term trends of ONI. [less ▲]

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See detailA Weak Local Irregularity Property in Sν Spaces
Clausel, Marianne; Nicolay, Samuel ULg

in Mediterranean Journal of Mathematics (2017), 14(3), 102

It has been shown that, from the prevalence point of view, elements of the Sν spaces are almost surely multifractal, while the Hölder exponent at almost every point is almost surely equal to the maximum ... [more ▼]

It has been shown that, from the prevalence point of view, elements of the Sν spaces are almost surely multifractal, while the Hölder exponent at almost every point is almost surely equal to the maximum Hölder exponent. We show here that typical elements of Sν are very irregular by proving that they almost surely satisfy a weak irregularity property: there exists a local irregularity exponent which is constant for almost every element of Sν and equal to the lowest Hölder exponent. [less ▲]

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See detailMars Topography Investigated Through the Wavelet Leaders Method: a Multidimensional Study of its Fractal Structure
Deliège, Adrien ULg; Kleyntssens, Thomas ULg; Nicolay, Samuel ULg

in Planetary and Space Science (2017), 136C

This work examines the scaling properties of Mars topography through a wavelet-based formalism. We conduct exhaustive one-dimensional (both longitudinal and latitudinal) and two-dimensional studies based ... [more ▼]

This work examines the scaling properties of Mars topography through a wavelet-based formalism. We conduct exhaustive one-dimensional (both longitudinal and latitudinal) and two-dimensional studies based on Mars Orbiter Laser Altimeter (MOLA) data using the multifractal formalism called Wavelet Leaders Method (WLM). This approach shows that a scale break occurs at approximately 15 km, giving two scaling regimes in both 1D and 2D cases. At small scales, these topographic profiles mostly display a monofractal behavior while a switch to multifractality is observed in several areas at larger scales. The scaling exponents extracted from this framework tend to be greater at small scales. In the 1D context, these observations are in agreement with previous works and thus suggest that the WLM is well-suited for examining scaling properties of topographic fields. Moreover, the 2D analysis is the first such complete study to our knowledge. It gives both a local and global insight on the scaling regimes of the surface of Mars and allows to exhibit the link between the scaling exponents and several famous features of the Martian topography. These results may be used as a solid basis for further investigations of the scaling laws of the Red planet and show that the WLM could be used to perform systematic analyses of the surface roughness of other celestial bodies. [less ▲]

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See detailRaffinement de l'estimation asymptotique de l'exposant de Hölder
Kleyntssens, Thomas ULg; Nicolay, Samuel ULg

Conference (2016, September 19)

Dans cet exposé, nous proposons une nouvelle approche basée sur les ondelettes (via les espaces Snu) permettant d'étudier plus finement la régularité Höldérienne d'une fonction

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See detailAbout the multifractal nature of Cantor's bijection: Bounds for the Hölder exponent at almost every irrational point
Nicolay, Samuel ULg; Simons, Laurent

in Fractals (2016), 24(2), 1650014-1-1650014-9

In this note, we investigate the regularity of Cantor’s one-to-one mapping between the irrational numbers of the unit interval and the irrational numbers of the unit square. In particular, we explore the ... [more ▼]

In this note, we investigate the regularity of Cantor’s one-to-one mapping between the irrational numbers of the unit interval and the irrational numbers of the unit square. In particular, we explore the fractal nature of this map by showing that its Hölder regularity lies between 0.35 and 0.72 almost everywhere (with respect to the Lebesgue measure). [less ▲]

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See detailA Refined Method for Estimating the Global Hölder Exponent
Kleyntssens, Thomas ULg; Kreit, Damien; Nicolay, Samuel ULg

Conference (2016, April 12)

We give a wavelet characterization of the generalized Hölder spaces and show how this result can be applied to detect logarithmic corrections appearing in Brownian processes.

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See detailA Refined Method for Estimating the Global Hölder Exponent
Nicolay, Samuel ULg; Kreit, Damien

in Latifi, Shahram (Ed.) Information Technology: New Generations (2016, April)

In this paper, we recall basic results we have obtained about generalized Hölder spaces and present a wavelet characterization that holds under more general hypothesis than previously stated. This ... [more ▼]

In this paper, we recall basic results we have obtained about generalized Hölder spaces and present a wavelet characterization that holds under more general hypothesis than previously stated. This theoretical tool gives rise to a method for estimating the global Hölder exponent which seems to be more precise than other wavelet-based approaches. This work should prove helpful for estimating long range correlations. [less ▲]

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See detailPoster EGU 2016: Mars Topography Investigated Through the Wavelet Leaders Method
Deliège, Adrien ULg; Kleyntssens, Thomas ULg; Nicolay, Samuel ULg

Poster (2016, April)

This work examines the scaling properties of Mars topography through a wavelet-based formalism. We conduct exhaustive one-dimensional (both longitudinal and latitudinal) and two-dimensional studies based ... [more ▼]

This work examines the scaling properties of Mars topography through a wavelet-based formalism. We conduct exhaustive one-dimensional (both longitudinal and latitudinal) and two-dimensional studies based on Mars Orbiter Laser Altimeter (MOLA) data using the multifractal formalism called Wavelet Leaders Method (WLM). This approach shows that a scale break occurs at approximately 15 km, giving two scaling regimes in both 1D and 2D cases. At small scales, these topographic profiles mostly display a monofractal behavior while a switch to multifractality is observed in several areas at larger scales. The scaling exponents extracted from this framework tend to be greater at small scales. In the 1D context, these observations are in agreement with previous works and thus suggest that the WLM is well-suited for examining scaling properties of topographic fields. Moreover, the 2D analysis is the first such complete study to our knowledge. It gives both a local and global insight on the scaling regimes of the surface of Mars and allows to exhibit the link between the scaling exponents and several famous features of the Martian topography. These results may be used as a solid basis for further investigations of the scaling laws of the Red planet and show that the WLM could be used to perform systematic analyses of the surface roughness of other celestial bodies. [less ▲]

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See detail[Beamer] A New Wavelet-Based Mode Decomposition for Oscillating Signals and Comparison with the Empirical Mode Decomposition
Deliège, Adrien ULg; Nicolay, Samuel ULg

Conference (2016, April)

We introduce a new method based on wavelets (EWMD) for decomposing a signal into quasi-periodic oscillating components with smooth time-varying amplitudes. This method is inspired by both the “classic” ... [more ▼]

We introduce a new method based on wavelets (EWMD) for decomposing a signal into quasi-periodic oscillating components with smooth time-varying amplitudes. This method is inspired by both the “classic” wavelet-based decomposition and the empirical mode decomposition (EMD). We compare the reconstruction skills and the period detection ability of the method with the well-established EMD on toys examples and the ENSO climate index. It appears that the EWMD accurately decomposes and reconstructs a given signal (with the same efficiency as the EMD), it is better at detecting prescribed periods and is less sensitive to noise. This work provides the first version of the EWMD. Even though there is still room for improvement, it turns out that preliminary results are highly promising. [less ▲]

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See detailA New Wavelet-Based Mode Decomposition for Oscillating Signals and Comparison with the Empirical Mode Decomposition
Deliège, Adrien ULg; Nicolay, Samuel ULg

in Information Technology: New Generations (2016, April)

We introduce a new method based on wavelets (EWMD) for decomposing a signal into quasi-periodic oscillating components with smooth time-varying amplitudes. This method is inspired by both the “classic” ... [more ▼]

We introduce a new method based on wavelets (EWMD) for decomposing a signal into quasi-periodic oscillating components with smooth time-varying amplitudes. This method is inspired by both the “classic” wavelet-based decomposition and the empirical mode decomposition (EMD). We compare the reconstruction skills and the period detection ability of the method with the well-established EMD on toys examples and the ENSO climate index. It appears that the EWMD accurately decomposes and reconstructs a given signal (with the same efficiency as the EMD), it is better at detecting prescribed periods and is less sensitive to noise. This work provides the first version of the EWMD. Even though there is still room for improvement, it turns out that preliminary results are highly promising. [less ▲]

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See detailThe Fractal Nature of Mars Topography Analyzed via the Wavelet Leaders Method
Kleyntssens, Thomas ULg; Deliège, Adrien ULg; Nicolay, Samuel ULg

in Information Technology: New Generations (2016, April)

This paper studies the scaling properties of Mars topography based on Mars Orbiter Laser Altimeter (MOLA) data through the wavelet leaders method (WLM). This approach shows a scale break at 15 km. At ... [more ▼]

This paper studies the scaling properties of Mars topography based on Mars Orbiter Laser Altimeter (MOLA) data through the wavelet leaders method (WLM). This approach shows a scale break at 15 km. At small scales, these topographic profiles display a monofractal behavior while a multifractal nature is observed at large scales. The scaling exponents are greater at small scales. They also seem to be influenced by latitude and may indicate a slight anisotropy in topography. [less ▲]

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See detailWavelet-based Methods to Study the Regularity of a Signal: from Theory to Practice
Kleyntssens, Thomas ULg; Nicolay, Samuel ULg

Conference (2016, March 23)

In this talk, I use the notion of wavelet to design multifractal formalisms. I present the theoritical results obtained on the generalized Snu spaces and I show the utility of these generalization ... [more ▼]

In this talk, I use the notion of wavelet to design multifractal formalisms. I present the theoritical results obtained on the generalized Snu spaces and I show the utility of these generalization. Besides, I also apply these formalisms on a practical example: the Mars topography. [less ▲]

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See detailSome characterizations about Generalized Hölder-Zygmund Spaces Λ_{σ,N}^{α}(R^d)
Kreit, Damien ULg; Nicolay, Samuel ULg

E-print/Working paper (2016)

Generalized Hölder-Zygmund spaces $\Lambda_{\sigma, N}^{\alpha}(\R^{d})$ were recently introduced and are based on a generalization of Besov spaces. Under some conditions, generalized Hölder-Zygmund and ... [more ▼]

Generalized Hölder-Zygmund spaces $\Lambda_{\sigma, N}^{\alpha}(\R^{d})$ were recently introduced and are based on a generalization of Besov spaces. Under some conditions, generalized Hölder-Zygmund and Besov spaces are equal. It has been proved that most properties of classical Hölder-Zygmund spaces are held for spaces $\Lambda^{\sigma,\alpha}(\R^{d})$, which constitute a particular case of spaces $\Lambda_{\sigma, N}^{\alpha}(\R^{d})$ with $N_{j}=2^{j}$. The goal of the present document is to prove that most of these properties are kept for $\Lambda_{\sigma, N}^{\alpha}(\R^{d})$ spaces. [less ▲]

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See detailUne construction des nombres hyperréels
Nicolay, Samuel ULg

in Quadrature : Magazine de Mathématiques Pures et Appliquées (2016), 102

Nous proposons ici une brève construction classique des nombres hyperréels à partir des nombres réels. Dans un soucis de pédagogie, nous omettons les détails les plus techniques. La présentation est ... [more ▼]

Nous proposons ici une brève construction classique des nombres hyperréels à partir des nombres réels. Dans un soucis de pédagogie, nous omettons les détails les plus techniques. La présentation est divisée en deux parties. La première fait presqu' exclusivement appel à l'intuition, tandis que la seconde reprend les mêmes linéaments mais se veut plus rigoureuse. [less ▲]

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