Convergent Mapping Research Articles

Causality analysis in process control based on denoising and periodicity-removing CCM

Purpose1. To improve the causality analysis performance, a novel causality detector based on time-delayed convergent cross mapping (TD-CCM) is proposed in this work. 2. Identify the root cause of plant-wide oscillations in process control system.Design/methodology/approachA novel causality analysis framework is proposed based on denoising and periodicity-removing TD-CCM (time-delayed convergent cross mapping). We first point out that noise and periodicity have adverse effects on causality detection. Then, the empirical mode decomposition (EMD) and detrended fluctuation analysis (FDA) are combined to achieve denoising. The periodicities are effectively removed through singular spectrum analysis (SSA). Following, the TD-CCM can accurately capture the causalities and locate the root cause by analyzing the filtered signals.Findings1. A novel causality detector based on denoising and periodicity-removing time-delayed convergent cross mapping (TD-CCM) is proposed. 2. Simulation studies show that the proposed method is able to improve the causality analysis performance. 3. Industrial case study shows the proposed method can be used to analyze the root cause of plant-wide oscillations in process control system.Originality/value1. A novel causality detector based on denoising and periodicity-removing time-delayed convergent cross mapping (TD-CCM) is proposed. 2. The influences of noise and periodicity on causality analysis are investigated. 3. Simulations and industrial case shows that the proposed method can improve the causality analysis performance and can be used to identify the root cause of plant-wide oscillations in process control system.

Sparse Bayesian mass-mapping with uncertainties: Full sky observations on the celestial sphere

ABSTRACT To date weak gravitational lensing surveys have typically been restricted to small fields of view, such that the flat-sky approximation has been sufficiently satisfied. However, with Stage IV surveys (e.g. LSST and Euclid) imminent, extending mass-mapping techniques to the sphere is a fundamental necessity. As such, we extend the sparse hierarchical Bayesian mass-mapping formalism presented in previous work to the spherical sky. For the first time, this allows us to construct maximum a posteriori spherical weak lensing dark-matter mass-maps, with principled Bayesian uncertainties, without imposing or assuming Gaussianty. We solve the spherical mass-mapping inverse problem in the analysis setting adopting a sparsity promoting Laplace-type wavelet prior, though this theoretical framework supports all log-concave posteriors. Our spherical mass-mapping formalism facilitates principled statistical interpretation of reconstructions. We apply our framework to convergence reconstruction on high resolution N-body simulations with pseudo-Euclid masking, polluted with a variety of realistic noise levels, and show a significant increase in reconstruction fidelity compared to standard approaches. Furthermore, we perform the largest joint reconstruction to date of the majority of publicly available shear observational data sets (combining DESY1, KiDS450, and CFHTLens) and find that our formalism recovers a convergence map with significantly enhanced small-scale detail. Within our Bayesian framework we validate, in a statistically rigorous manner, the community’s intuition regarding the need to smooth spherical Kaiser-Squires estimates to provide physically meaningful convergence maps. Such approaches cannot reveal the small-scale physical structures that we recover within our framework.

Inferring species interactions using Granger causality and convergent cross mapping

Identifying directed interactions between species from time series of their population densities has many uses in ecology. This key statistical task is equivalent to causal time series inference, which connects to the Granger causality (GC) concept: $x$ causes $y$ if $x$ improves the prediction of $y$ in a dynamic model. However, the entangled nature of nonlinear ecological systems has led to question the appropriateness of Granger causality, especially in its classical linear Multivariate AutoRegressive (MAR) model form. Convergent-cross mapping (CCM), a nonparametric method developed for deterministic dynamical systems, has been suggested as an alternative. Here, we show that linear GC and CCM are able to uncover interactions with surprisingly similar performance, for predator-prey cycles, 2-species deterministic (chaotic) or stochastic competition, as well as 10- and 20-species interaction networks. There is no correspondence between the degree of nonlinearity of the dynamics and which method performs best. Our results therefore imply that Granger causality, even in its linear MAR($p$) formulation, is a valid method for inferring interactions in nonlinear ecological networks; using GC or CCM (or both) can instead be decided based on the aims and specifics of the analysis.

Exploring causes of streamflow alteration in the Medjerda river, Algeria

Study regionThe Medjerda is a transboundary catchment located in North-Eastern Algeria and shared with Tunisia. Study focusIn this study, we explore the causes of hydrological alteration of streamflow in a sub-catchment of the Medjerda in Algeria. The hydrological alteration was explored through the application of Mann-Kendall test based on possible explanatory factors, namely, precipitation, evapotranspiration, temperature, irrigation, and Normalized Difference of Vegetation Index (NDVI). Furthermore, the causal factors of streamflow variation were addressed using Convergent Cross Mapping (CCM) method. New hydrological insights for the regionResults of the trend analysis yield that the streamflow is altered during the period 1981−2012. This is consistent with the trends of the possible explanatory factors of this alteration. The Convergent Cross Mapping (CCM) method showed that streamflow alteration is unidirectionally caused by changes in patterns of precipitation, temperature, evapotranspiration, irrigation, and NDVI, and that there is little feedback from streamflow alteration to these causing factors. Overall, our assessment showed that the method used to identify the causal relationships in dynamical systems based on the CCM algorithm is suitable for exploring the drivers of the hydrologic alteration in multivariate time series, in particular when nonlinear dynamics determine the system.

Optimal void finders in weak lensing maps

ABSTRACT Cosmic voids are a key component of the large-scale structure that contain a plethora of cosmological information. Typically, voids are identified from the underlying galaxy distribution, which is a biased tracer of the total matter field. Previous works have shown that 2D voids identified in weak lensing (WL) maps – WL voids – correspond better to true underdense regions along the line of sight. In this work, we study how the properties of WL voids depend on the choice of void finder, by adapting several popular void finders. We present and discuss the differences between identifying voids directly in the convergence maps, and in the distribution of WL peaks. Particular effort has been made to test how these results are affected by galaxy shape noise (GSN), which is a dominant source of noise in WL observations. By studying the signal-to-noise ratios (S/N) for the tangential shear profile of each void finder, we find that voids identified directly in the convergence maps have the highest S/N but are also the ones most affected by GSN. Troughs are least affected by noise, but also have the lowest S/N. The tunnel algorithm, which identifies voids in the distribution of WL peaks, represents a good compromise between finding a large tangential shear S/N and mitigating the effect of GSN.

Ambient temperature and stroke hospitalization: causal analysis with empirical dynamic modelling

Background: In the climate change context, the role of ambient temperature in stroke attack has drawn much research interest. However, prior epidemiologic evidence from mechanistic model-based regressions remains mixed. Inferring causation on these findings is a big challenge. Methods: Here we applied a novel approach, empirical dynamic modeling (EDM), to explore the causality between air temperature and stroke onset in Hong Kong from 2004-2011. EDM, a data-driven analysis scheme using convergent cross mapping (CCM) to detect causal pairs and multivariate S-map to quantify effect strength, is based on state space reconstruction for nonlinear dynamical systems where climate-health relations may accommodate.Results: By comparing with surrogate data to exclude seasonality, CCM determined that daily mean temperature and its deviation from previous week were potential drivers for hemorrhagic stroke (HS) hospitalization. Yet, no significant causal signal was identified for ischemic stroke (IS). S-map further found that daily mean temperatures ranging from 8.8°C to 31.8°C drove HS counts in a negative manner with the protective effect being stronger on warmer days. By contrast, temperature swings within a week showed a positive forcing in HS and a cooling trend may be more adverse than warming scenario. Also, we traced that mean temperature drove HS with delayed effects of 2 to 17 lagged days, while the impact of short-run temperature swings on HS could persist over the lagged period from 6 to 15 days. The strength of temperature-HS link peaked on the closest lag day of exposure. The effect strengths varied through time and by exposure levels, implying the nonlinearities internal to the system.Conclusion and Significance: Our analyses support that HS attack may be more sensitive to temperature condition than IS. We envision this EDM method could offer insights for epidemiologic studies to move beyond correlation when assessing health effects of climatic exposure.

Dark Energy Survey Year 1 results: the lensing imprint of cosmic voids on the cosmic microwave background

ABSTRACT Cosmic voids gravitationally lens the cosmic microwave background (CMB) radiation, resulting in a distinct imprint on degree scales. We use the simulated CMB lensing convergence map from the Marenostrum Institut de Ciencias de l’Espai (MICE) N-body simulation to calibrate our detection strategy for a given void definition and galaxy tracer density. We then identify cosmic voids in Dark Energy Survey (DES) Year 1 data and stack the Planck 2015 lensing convergence map on their locations, probing the consistency of simulated and observed void lensing signals. When fixing the shape of the stacked convergence profile to that calibrated from simulations, we find imprints at the 3σ significance level for various analysis choices. The best measurement strategies based on the MICE calibration process yield S/N ≈ 4 for DES Y1, and the best-fitting amplitude recovered from the data is consistent with expectations from MICE (A ≈ 1). Given these results as well as the agreement between them and N-body simulations, we conclude that the previously reported excess integrated Sachs–Wolfe (ISW) signal associated with cosmic voids in DES Y1 has no counterpart in the Planck CMB lensing map.

Using empirical dynamic modeling to assess relationships between atmospheric trace gases and eukaryotic phytoplankton populations in coastal Southern California

Many different atmospheric trace gases have been directly and indirectly linked to biological sources and sinks. Here we assess how atmospheric mixing ratios of a range of halocarbons (CH3Br, CH2Br2, CHBr3, CH3Cl, CHCl3, and CH3I) and COS are causally connected to naturally occurring marine eukaryotic phytoplankton in coastal Southern California. We use a self-organizing map to characterize the abiotic environment and empirical dynamic modeling with convergent cross mapping to identify causal interactions between multiple in situ 8-year time-series, sampled at the Ellen Browning Scripps Pier at Scripps Institution of Oceanography. Our work supports previous findings that halocarbon production is found in a variety of marine phytoplankton taxa and suggests that local phytoplankton may have the ability to affect changes in the mixing ratios of halocarbons in nearshore environments. There were notable links between changes in CH3I and several different diatom taxa and between changes in CHCl3 and a group of phytoplankton during specific ecosystem states. Our results suggest that both seasonal and non-seasonal shifts in eukaryotic phytoplankton structure contribute to small fluctuations in atmospheric halocarbon mixing ratios that exhibit strong seasonality and may occasionally play a larger role in atmospheric mixing ratios of halocarbons that display reduced seasonality.

Long-term warming destabilizes aquatic ecosystems through weakening biodiversity-mediated causal networks.

Understanding how ecosystems will respond to climate changes requires unravelling the network of functional responses and feedbacks among biodiversity, physicochemical environments, and productivity. These ecosystem components not only change over time but also interact with each other. Therefore, investigation of individual relationships may give limited insights into their interdependencies and limit ability to predict future ecosystem states. We address this problem by analyzing long-term (16-39years) time series data from 10 aquatic ecosystems and using convergent cross mapping (CCM) to quantify the causal networks linking phytoplankton species richness, biomass, and physicochemical factors. We determined that individual quantities (e.g., total species richness or nutrients) were not significant predictors of ecosystem stability (quantified as long-term fluctuation of phytoplankton biomass); rather, the integrated causal pathway in the ecosystem network, composed of the interactions among species richness, nutrient cycling, and phytoplankton biomass, was the best predictor of stability. Furthermore, systems that experienced stronger warming over time had both weakened causal interactions and larger fluctuations. Thus, rather than thinking in terms of separate factors, a more holistic network view, that causally links species richness and the other ecosystem components, is required to understand and predict climate impacts on the temporal stability of aquatic ecosystems.

Reconstructing coupled time series in climate systems using three kinds of machine-learning methods

Abstract. Despite the great success of machine learning, its application in climate dynamics has not been well developed. One concern might be how well the trained neural networks could learn a dynamical system and what will be the potential application of this kind of learning. In this paper, three machine-learning methods are used: reservoir computer (RC), backpropagation-based (BP) artificial neural network, and long short-term memory (LSTM) neural network. It shows that the coupling relations or dynamics among variables in linear or nonlinear systems can be inferred by RC and LSTM, which can be further applied to reconstruct one time series from the other. Specifically, we analyzed the climatic toy models to address two questions: (i) what factors significantly influence machine-learning reconstruction and (ii) how do we select suitable explanatory variables for machine-learning reconstruction. The results reveal that both linear and nonlinear coupling relations between variables do influence the reconstruction quality of machine learning. If there is a strong linear coupling between two variables, then the reconstruction can be bidirectional, and both of these two variables can be an explanatory variable for reconstructing the other. When the linear coupling among variables is absent but with the significant nonlinear coupling, the machine-learning reconstruction between two variables is direction dependent, and it may be only unidirectional. Then the convergent cross mapping (CCM) causality index is proposed to determine which variable can be taken as the reconstructed one and which as the explanatory variable. In a real-world example, the Pearson correlation between the average tropical surface air temperature (TSAT) and the average Northern Hemisphere SAT (NHSAT) is weak (0.08), but the CCM index of NHSAT cross mapped with TSAT is large (0.70). And this indicates that TSAT can be well reconstructed from NHSAT through machine learning. All results shown in this study could provide insights into machine-learning approaches for paleoclimate reconstruction, parameterization scheme, and prediction in related climate research.Highlights: i The coupling dynamics learned by machine learning can be used to reconstruct time series. ii Reconstruction quality is direction dependent and variable dependent for nonlinear systems. iii The CCM index is a potential indicator to choose reconstructed and explanatory variables. iv The tropical average SAT can be well reconstructed from the average Northern Hemisphere SAT.

Accounting for linefish dependency in management of the South African small pelagic fishery

To further implement an ecosystem approach to management of the small pelagic fishery in South Africa, we attempted to develop functional relationships between spatialised time-series of the biomass of three small pelagic fish species and the catch per unit effort (CPUE) of four linefish species (i.e. species caught on hook and line, excluding longlines) that could be used to account for the forage needs of dependent predators. Granger causality tests indicated significant effects of anchovy Engraulis encrasicolus biomass on silver kob Argyrosomus inodorus and yellowtail Seriola lalandi CPUE, significant effects of sardine Sardinops sagax biomass on geelbek Atractoscion aequidens CPUE, but no significant effects of biomass of west coast redeye round herring Etrumeus whiteheadi and east coast redeye round herring Etruemus wongratanai on any linefish CPUE. Sequential t-test analysis of regime shift (STARS) indicated temporal matching in regimes between anchovy biomass east of Cape Agulhas (EoCA) and yellowtail west coast (west of Cape Point) CPUE, and between sardine biomass EoCA and geelbek southwest coast (Cape Point to Cape Infanta) CPUE. Convergent cross mapping found no causality for anchovy biomass EoCA on yellowtail west coast CPUE but causality in the reverse direction. A high degree of causality in both directions for sardine biomass EoCA and geelbek southwest coast CPUE was observed, indicating strong predator–prey interaction, and a highly significant linear relationship between sardine biomass EoCA in year n–1 and geelbek southwest coast CPUE in yearn was obtained. This functional relationship should be considered for inclusion as a proxy for ecosystem state in management procedures that adopt the ecosystem approach to fisheries for the South African small pelagic fishery.

Higher order spectra of weak lensing convergence maps in parametrized theories of modified gravity

ABSTRACT We compute the low-ℓ limit of the family of higher order spectra for projected (2D) weak lensing convergence maps. In this limit these spectra are computed to an arbitrary order using tree-level perturbative calculations. We use the flat-sky approximation and Eulerian perturbative results based on a generating function approach. We test these results for the lower order members of this family, i.e. the skew- and kurt-spectra against state-of-the-art simulated all-sky weak lensing convergence maps and find our results to be in very good agreement. We also show how these spectra can be computed in the presence of a realistic sky-mask and Gaussian noise. We generalize these results to 3D and compute the equal-time higher order spectra. These results will be valuable in analysing higher order statistics from future all-sky weak lensing surveys such as the Euclid survey at low-ℓ modes. As illustrative examples, we compute these statistics in the context of the Horndeski and beyond Horndeski theories of modified gravity. They will be especially useful in constraining theories such as the Gleyzes–Langlois–Piazza–Vernizzi (GLPV) theories and degenerate higher order scalar-tensor theories as well as the commonly used normal-branch of Dvali–Gabadadze–Porrati model, clustering quintessence models and scenarios with massive neutrinos.

Dark Energy Survey Year 3 results: cosmology with moments of weak lensing mass maps – validation on simulations

ABSTRACT We present a simulated cosmology analysis using the second and third moments of the weak lensing mass (convergence) maps. The second moment, or variances, of the convergence as a function of smoothing scale contains information similar to standard shear two-point statistics. The third moment, or the skewness, contains additional non-Gaussian information. The analysis is geared towards the third year (Y3) data from the Dark Energy Survey (DES), but the methodology can be applied to other weak lensing data sets. We present the formalism for obtaining the convergence maps from the measured shear and for obtaining the second and third moments of these maps given partial sky coverage. We estimate the covariance matrix from a large suite of numerical simulations. We test our pipeline through a simulated likelihood analyses varying 5 cosmological parameters and 10 nuisance parameters and identify the scales where systematic or modelling uncertainties are not expected to affect the cosmological analysis. Our simulated likelihood analysis shows that the combination of second and third moments provides a 1.5 per cent constraint on S8 ≡ σ8(Ωm/0.3)0.5 for DES Year 3 data. This is 20 per cent better than an analysis using a simulated DES Y3 shear two-point statistics, owing to the non-Gaussian information captured by the inclusion of higher order statistics. This paper validates our methodology for constraining cosmology with DES Year 3 data, which will be presented in a subsequent paper.

Systematic identification of causal relations in high-dimensional chaotic systems: application to stratosphere-troposphere coupling

Obtaining reliable causal inference is crucial for understanding the climate system. Convergent Cross Mapping (CCM), a recently developed method to infer causal relationships from time series has been shown to be superior to previous methods which are based on linearity assumptions. However, CCM has so far been only tested on low-dimensional or bivariate models, while real-world systems, like the climate system, are high-dimensional and have many more interacting variables. Here, we demonstrate that standard CCM cannot reliably infer causal relations in high-dimensional chaotic systems. However, by using a hierarchy of conceptual models and observational data we show that time-lagged CCM reliably identifies causal relationships in contrast to standard CCM and Pearson correlation. Furthermore, we systematically demonstrate that time-lagged CCM is able to identify long-distance causal interactions. Moreover, we apply time-lagged CCM to detect causal relations in stratosphere-troposphere coupling, and demonstrate the downward causal chain induced by polar vortex activity.

Some stronger forms of topological transitivity and sensitivity for a sequence of uniformly convergent continuous maps

Let (H,d) be a metric space, F be a Furstenberg family, and (gm)m∈Z+ be a sequence of continuous map on H, which converges uniformly to a map g on H. In this paper, under the condition limm→∞⁡d∞(gmm,gm)=0, a necessary and sufficient condition for g to be F-mixing is established. Moreover, let T⊂Z+ be an infinite set andlimm∈T:m→∞⁡d∞(gmm,gm)=0. Then, some necessary and sufficient conditions, or sufficient conditions, for g to be some stronger forms of topological transitivity, sensitivity, ergodic, or mixing are obtained.

Influence of Individual Heart Rate on Nonlinear Brain-Heart Interactions Estimated by Convergent Cross Mapping in Schizophrenic Patients and Healthy Controls.

Quantification of directed (nonlinear) brain-heart interactions has turned to be an emerging topic of research and is important for the better understanding of central autonomic processing during specific diseases such as schizophrenia. Convergent Cross Mapping (CCM) was able to provide directed, frequency-selective and topographic views on existent interaction pattern of those patients. Investigations of the influence of individual heart rate (HR) on CCM estimations may further contribute to this topic. Relationship of mean HR and CCM was analyzed in a group of schizophrenic patients (N=17) and healthy controls (N=21). Influence of individual HR values was most pronounced for patients, for interactions from brain to heart and for the subgroup of patients with highest mean HR values.

Combining Deep Learning with Handcrafted Features for Cell Nuclei Segmentation.

Segmentation of cell nuclei in fluorescence microscopy images provides valuable information about the shape and size of the nuclei, its chromatin texture and DNA content. It has many applications such as cell tracking, counting and classification. In this work, we extended our recently proposed approach for nuclei segmentation based on deep learning, by adding to its input handcrafted features. Our handcrafted features introduce additional domain knowledge that nuclei are expected to have an approximately round shape. For round shapes the gradient vector of points at the border point to the center. To convey this information, we compute a map of gradient convergence to be used by the CNN as a new channel, in addition to the fluorescence microscopy image. We applied our method to a dataset of microscopy images of cells stained with DAPI. Our results show that with this approach we are able to decrease the number of missdetections and, therefore, increase the F1-Score when compared to our previously proposed approach. Moreover, the results show that faster convergence is obtained when handcrafted features are combined with deep learning.

Evaluation of Methods for Causal Discovery in Hydrometeorological Systems

AbstractUnderstanding causal relations is of utmost importance in hydrology and climate research for systems identification, prediction, and understanding systems behavior in a changing climate. Traditionally, researchers in hydrometeorology attempted to study causal questions by conducting controlled experiments using numerical models. This approach, however, in most cases of interest provides uncertain results because the models are approximate representation of the natural system. An alternative approach that has recently drawn significant attention in several fields is to infer causal relations from purely observational data. It possesses several traits to its utility particularly in hydrometeorology due to the rapid accumulation of in situ and remotely sensed data records. The first objective of this study is to present a brief description of four causal discovery methods (Granger causality, Transfer Entropy, graph‐based algorithms, and Convergent Cross Mapping) with special emphasis on the assumptions on which they are built. Second, using synthetic data generated from a hydrological model, we assess their performance in retrieving causal information taking into account sensitivity to sample size and presence of noise. Last, we use causal analysis to examine and formulate hypotheses on causal drivers of evapotranspiration in a shrubland region during summer and winter seasons. An interpretation of the hypotheses based on canopy seasonal dynamics and evapotranspiration processes is presented. It is hoped that the results presented here can be useful in guiding researchers studying hydrometeorological systems as to which causal method is most appropriate to the characteristics of the system under study.

Testing the reliability of fast methods for weak lensing simulations: wl-moka on pinocchio

ABSTRACT The generation of simulated convergence maps is of key importance in fully exploiting weak lensing by large-scale structure (LSS) from which cosmological parameters can be derived. In this paper, we present an extension of the pinocchio code that produces catalogues of dark matter haloes so that it is capable of simulating weak lensing by Modify LSS into Large Scale Structures (LSS). Like wl-moka, the method starts with a random realization of cosmological initial conditions, creates a halo catalogue and projects it on to the past light-cone, and paints in haloes assuming parametric models for the mass density distribution within them. Large-scale modes that are not accounted for by the haloes are constructed using linear theory. We discuss the systematic errors affecting the convergence power spectra when Lagrangian perturbation theory at increasing order is used to displace the haloes within pinocchio, and how they depend on the grid resolution. Our approximate method is shown to be very fast when compared to full ray-tracing simulations from an N-body run and able to recover the weak lensing signal, at different redshifts, with a few percent accuracy. It also allows for quickly constructing weak lensing covariance matrices, complementing pinocchio’s ability of generating the cluster mass function and galaxy clustering covariances and thus paving the way for calculating cross-covariances between the different probes. This work advances these approximate methods as tools for simulating and analysing survey data for cosmological purposes.

Euclid: Reconstruction of weak-lensing mass maps for non-Gaussianity studies

Weak lensing, which is the deflection of light by matter along the line of sight, has proven to be an efficient method for constraining models of structure formation and reveal the nature of dark energy. So far, most weak-lensing studies have focused on the shear field that can be measured directly from the ellipticity of background galaxies. However, within the context of forthcoming full-sky weak-lensing surveys such as Euclid, convergence maps (mass maps) offer an important advantage over shear fields in terms of cosmological exploitation. While it carry the same information, the lensing signal is more compressed in the convergence maps than in the shear field. This simplifies otherwise computationally expensive analyses, for instance, non-Gaussianity studies. However, the inversion of the non-local shear field requires accurate control of systematic effects caused by holes in the data field, field borders, shape noise, and the fact that the shear is not a direct observable (reduced shear). We present the two mass-inversion methods that are included in the official Euclid data-processing pipeline: the standard Kaiser & Squires method (KS), and a new mass-inversion method (KS+) that aims to reduce the information loss during the mass inversion. This new method is based on the KS method and includes corrections for mass-mapping systematic effects. The results of the KS+ method are compared to the original implementation of the KS method in its simplest form, using the Euclid Flagship mock galaxy catalogue. In particular, we estimate the quality of the reconstruction by comparing the two-point correlation functions and third- and fourth-order moments obtained from shear and convergence maps, and we analyse each systematic effect independently and simultaneously. We show that the KS+ method substantially reduces the errors on the two-point correlation function and moments compared to the KS method. In particular, we show that the errors introduced by the mass inversion on the two-point correlation of the convergence maps are reduced by a factor of about 5, while the errors on the third- and fourth-order moments are reduced by factors of about 2 and 10, respectively.