Efficient, nonparametric removal of noise and recovery of probability distributions from time series using nonlinear-correlation functions: Additive noise.

Single-molecule and related experiments yield time series of an observable as it fluctuates due to thermal motion. In such data, it can be difficult to distinguish fluctuating signal from fluctuating noise. We present a method of separating signal from noise using nonlinear-correlation functions. The method is fully nonparametric: No a priori model for the system is required, no knowledge of whether the system is continuous or discrete is needed, the number of states is not fixed, and the system can be Markovian or not. The noise-corrected, nonlinear-correlation functions can be converted to the system's Green's function; the noise-corrected moments yield the system's equilibrium-probability distribution. As a demonstration, we analyze synthetic data from a three-state system. The correlation method is compared to another fully nonparametric approach-time binning to remove noise, and histogramming to obtain the distribution. The correlation method has substantially better resolution in time and in state space. We develop formulas for the limits on data quality needed for signal recovery from time series and test them on datasets of varying size and signal-to-noise ratio. The formulas show that the signal-to-noise ratio needs to be on the order of or greater than one-half before convergence scales at a practical rate. With experimental benchmark data, the positions and populations of the states and their exchange rates are recovered with an accuracy similar to parametric methods. The methods demonstrated here are essential components in building a complete analysis of time series using only high-order correlation functions.

Tinea Gladiatorum Detection With a Dermatophyte Test Strip.

OBJECTIVE: Determine sensitivity, specificity, positive predictive value (PPV), negative predictive value (NPV), and concordance of diafactory hs-TP (DTS) to detect tinea gladiatorum using direct potassium hydroxide (KOH) microscopy as the reference standard. DESIGN: Prospective, comparative study. SETTING: Seventeen Minnesota high schools during the winter wrestling season. PATIENTS: Seventy-one consecutive high school wrestlers identified with a suspicious rash during skin inspection. INTERVENTIONS: Samples were obtained from each rash for both DTS and direct KOH microscopy. MAIN OUTCOME MEASURES: Readings were recorded as positive or negative. RESULTS: Direct KOH microscopy identified tinea gladiatorum in 35 of the 71 samples (46%). DTS sensitivity was 80% (95% confidence interval 63%-92%), and specificity was 82% (66%-92%). PPV was 85% (68%-95%), and NPV was 86% (72 %-91%). The DTS result was 83% concordant (72%-91%) with direct KOH microscopy. CONCLUSIONS: Similar to rapid Covid antigen tests, DTS required brief, basic training to perform and gave onsite results in 5 to 30 minutes. Although DTS is not approved for use in the United States by the FDA, concordance compared with direct KOH microscopy in diagnosing tinea gladiatorum was similar to results reported for DTS-TU in tinea unguium and tinea pedis. Further study comparing DTS to a reference standard using PCR plus direct microscopy is warranted.

Nonlinear measurements of kinetics and generalized dynamical modes. I. Extracting the one-dimensional Green's function from a time series.

Often, a single correlation function is used to measure the kinetics of a complex system. In contrast, a large set of k-vector modes and their correlation functions are commonly defined for motion in free space. This set can be transformed to the van Hove correlation function, which is the Green's function for molecular diffusion. Here, these ideas are generalized to other observables. A set of correlation functions of nonlinear functions of an observable is used to extract the corresponding Green's function. Although this paper focuses on nonlinear correlation functions of an equilibrium time series, the results are directly connected to other types of nonlinear kinetics, including perturbation-response experiments with strong fields. Generalized modes are defined as the orthogonal polynomials associated with the equilibrium distribution. A matrix of mode-correlation functions can be transformed to the complete, single-time-interval (1D) Green's function. Diagonalizing this matrix finds the eigendecays. To understand the advantages and limitation of this approach, Green's functions are calculated for a number of models of complex dynamics within a Gaussian probability distribution. Examples of non-diffusive motion, rate heterogeneity, and range heterogeneity are examined. General arguments are made that a full set of nonlinear 1D measurements is necessary to extract all the information available in a time series. However, when a process is neither dynamically Gaussian nor Markovian, they are not sufficient. In those cases, additional multidimensional measurements are needed.

Nonlinear measurements of kinetics and generalized dynamical modes. II. Application to a simulation of solvation dynamics in an ionic liquid.

Solvation dynamics in ionic liquids show features that are often associated with supercooled liquids, including "stretched" nonexponential relaxation. To better understand the mechanism behind the stretching, the nonlinear mode-correlation methods proposed in Paper I [S. R. Hodge and M. A. Berg, J. Chem. Phys. 155, 024122 (2021)] are applied to a simulation of a prototypical ionic liquid. A full Green's function is recovered. In addition, specific tests for non-Gaussian dynamics are made. No deviations from Gaussian dynamics are found. This finding is incompatible with rate heterogeneity as a cause of the nonexponential relaxation and appears to be in conflict with an earlier multidimensional analysis of the same data. Although this conflict is not resolved here, this work does demonstrate the practicality of mode-correlation analysis in the face of finite datasets and calculations.

Jump-precursor state emerges below the crossover temperature in supercooled o-terphenyl.

In a supercooled liquid, the crossover temperature T_{c} separates a high-temperature region of diffusive dynamics from a low-temperature region of activated dynamics. A molecular-dynamics simulation of all-atom, flexible o-terphenyl [Eastwood et al., J. Phys. Chem. B 117, 12898 (2013)10.1021/jp402102w] is analyzed with advanced statistical methods to reveal the molecular features associated with this crossover. The simulations extend to an α-relaxation time of 14 µs (272.5 K), two orders of magnitude slower than at T_{c} (290 K). At T_{c} and below, a distinct state emerges that immediately precedes an orientational jump. Compared to the initial, tightly caged state, this jump-precursor state has a looser cage, with solid-angular excursions of 0.054-0.0125 × 4π sr. At T_{c} (290 K), rate heterogeneity is already the dominant cause of stretched relaxation. Exchange within the distribution of rates is faster than α relaxation at T_{c}, but becomes equal to it at the lowest temperature simulated (272.5 K). The results trend toward a recent experimental observation near the glass transition (243 K) [Kaur et al., Phys. Rev. E 98, 040603(R) (2018)10.1103/PhysRevE.98.040603], which saw exchange substantially slower than α relaxation. Overall, the dynamic crossover comprises multiple phenomena: the development of heterogeneity, an increasing jump size, an emerging jump-precursor state, and a lengthening exchange time. The crossover is neither sharp, nor a simple superposition of the high- and low-temperature regimes; it is a broad region that contains unique and complex phenomena.

Micelle Heterogeneity from the 2D Kinetics of Solute Rotation.

The chemical and physical properties of microstructured materials vary with position. The photophysics of solute molecules can measure these local properties, but they often show multiple rates (rate dispersion), which complicates the interpretation. In the case of micelles, rate dispersion in a solute's anisotropy decay has been assigned to either local anisotropy or heterogeneity in the local viscosity. To resolve this conflict, the rotation of PM597 molecules in SDS micelles has been measured by polarized MUPPETS (multiple population-period transient spectroscopy). This 2D technique shows that heterogeneity is strong and that local anisotropy is minimal. The results suggest that on a subnanosecond time scale, the solute sees only one strong fluctuation of the micelle structure. The anisotropic, average structure emerges on longer time scales.

Nonparametric analysis of nonexponential and multidimensional kinetics. I. Quantifying rate dispersion, rate heterogeneity, and exchange dynamics.

The quantification of nonexponential (dispersed) kinetics has relied on empirical functions, which yield parameters that are neither unique nor easily related to the underlying mechanism. Multidimensional kinetics provide more information on dispersed processes, but a good approach to their analysis is even less clear than for standard, one-dimensional kinetics. This paper is the first in a series that analyzes kinetic data in one or many dimensions with a scheme that is nonparametric: it quantifies nonexponential decays without relying on a specific functional form. The quantities obtained are directly related to properties of the mechanism causing the rate dispersion. Log-moments of decays, which parallel the standard moments of distributions (mean, standard deviation, etc.), are introduced for both one- and multi-dimensional decays. Kinetic spectra are defined to visualize the data. The utility of this approach is demonstrated on a simple, but general, model of dispersed kinetics-a nonexponential homogeneous decay combined with slowly exchanging rate heterogeneity. The first log-moments give a geometric-mean relaxation time. Second log-moments quantify the magnitude of rate dispersion, the fraction of the dispersion due to heterogeneity, and the dynamics of exchange between different rate subensembles. A suitable combination of these moments isolates exchange dynamics from three-dimensional kinetics without contamination by the rate-filtering effects that were identified in a recent paper [M. A. Berg and J. R. Darvin, J. Chem. Phys. 145, 054119 (2016)].

Measuring a hidden coordinate: Rate-exchange kinetics from 3D correlation functions.

Nonexponential kinetics imply the existence of at least one slow variable other than the observable, that is, the system has a "hidden" coordinate. We develop a simple, but general, model that allows multidimensional correlation functions to be calculated for these systems. Homogeneous and heterogeneous mechanisms are both included, and slow exchange of the rates is allowed. This model shows that 2D and 3D correlation functions of the observable measure the distribution and kinetics of the hidden coordinate controlling the rate exchange. Both the mean exchange time and the shape of the exchange relaxation are measurable. However, complications arise because higher correlation functions are sums of multiple "pathways," each of which measures different dynamics. Only one 3D pathway involves exchange dynamics. Care must be used to extract exchange dynamics without contamination from other processes.

Rate and Amplitude Heterogeneity in the Solvation Response of an Ionic Liquid.

In contrast with conventional liquids, ionic liquids have solvation dynamics with more rate dispersion and with average times that do not agree with dielectric measurements. A kinetic analog of multidimensional spectroscopy is introduced and used to look for heterogeneity in simulations of coumarin 153 in [Im12][BF4]. Strong heterogeneity is found in the diffusive solvation rate. An unanticipated heterogeneity in the amplitude of the inertial solvation is also seen. Both heterogeneities exchange at the same rate. This rate is similar to the mean diffusive solvation time, putting it in the intermediate-exchange region. Overall, there are multiple violations of the assumptions usually invoked in the theory of reaction dynamics.

When is a single molecule heterogeneous? A multidimensional answer and its application to dynamics near the glass transition.

Even for apparently simple condensed-phase processes, bulk measurements of relaxation often yield nonexponential decays; the rate appears to be dispersed over a range of values. Taking averages over individual molecules is an intuitive way to determine whether heterogeneity is responsible for such rate dispersion. However, this method is in fundamental conflict with ergodic behavior and often yields ambiguous results. This paper proposes a new definition of rate heterogeneity for ergodic systems based on multidimensional time correlation functions. Averages are taken over both time and molecules. Because the data set is not subdivided, the signal-to-noise ratio is improved. Moment-based quantities are introduced to quantify the concept of rate dispersion. As a result, quantitative statements about the fraction of the dispersion due to heterogeneity are possible, and the experimental noise is further averaged. The practicality of this approach is demonstrated on single-molecule, linear-dichroism trajectories for R6G in poly(cyclohexyl acrylate) near its glass transition. Single-molecule averaging of these data does not provide useful conclusions [C. Y. Lu and D. A. Vanden Bout, J. Chem. Phys. 125, 124701 (2006)]. However, full-ensemble, two- and three-dimensional averages of the same data give clear and quantitative results: the rate dispersion is 95% ± 5% due to heterogeneity, and the rate exchange is at least 11 times longer than the mean rotation time and possibly much longer. Based on these results, we suggest that the study of heterogeneous materials should not focus on "ensemble" versus "single-molecule" experiments, but on one-dimensional versus multidimensional measurements.

Two-Dimensional Anisotropy Measurements Showing Local Heterogeneity in a Polymer Melt.

In polymers, the rotation of a small solute is nonexponential. Either heterogeneity in the local friction or local anisotropy-a homogeneous process-may be responsible. A new, two-dimensional anisotropy experiment is demonstrated on this problem. In poly(dimethylsiloxane), the rotation of individual solute molecules is found to be exponential, and the observed rate dispersion is primarily due to variation in the local friction. This sample is far from its glass transition. Studies of rate heterogeneity associated with the glass transition must account for the contribution from this polymer-related mechanism.

Multiple population-period transient spectroscopy (MUPPETS) of CdSe/ZnS nanoparticles. I. Exciton and biexciton dynamics.

The nonradiative relaxation of both the exciton and biexciton in CdSe/ZnS core-shell nanoparticles have complicated, nonexponential kinetics. This paper presents data on this system from multiple population-period transient spectroscopy (MUPPETS), a method for two-dimensional kinetics. An initial report of a dispersed (nonexponential) biexciton decay [J. Am. Chem. Soc. 2013, 135, 1002] is confirmed in a more rigorous analysis. Additional transient-grating data allow a quantitative treatment of the full, complex MUPPETS data set. The MUPPETS signal has a strong fluence dependence. With extrapolation to the low fluence limit, the ratio of cross sections for ground-to-exciton and exciton-to-biexciton absorption is found to be close to the predictions of the uncorrelated-electron model. The full two-dimensional MUPPETS data set is reported for the first time and is analyzed to detect heterogeneity in the exciton decay. The exciton has a substantial (>40%) nonradiative decay, but it is not due to a subset of defective particles. A surface relaxation in response to formation of the exciton is suggested. This data set is the first capable of detecting correlations between the biexciton and exciton decay mechanism. None is found.

Multiple population-period transient spectroscopy (MUPPETS) of CdSe/ZnS nanoparticles. II. Effects of high fluence and solvent heating.

Multiple population-period transient spectroscopy (MUPPETS) is a six-pulse experiment with two time dimensions that is capable of adding information about systems with complicated kinetics. The core theory for MUPPETS focuses on the χ(5) response of the chromophores. This theory was used to analyze the dynamics of excitons and biexcitons in CdSe/ZnS core-shell nanoparticles in part I of this paper [J. Phys. Chem. B 2013, DOI:10.1021/jp405785a]. In real experiments, the potential role of additional processes must also be considered, in particular, the χ(7), "saturation" of the MUPPETS signal and nonresonant signals from heating of the solvent. A pathway method for calculating fluence effects in MUPPETS is developed. The fluence dependence of the biexciton signal and its sign reversal, as found in part I, are explained without invoking higher excitons or unexpected species. A method is presented for quantitatively predicting the magnitude of signals from solvent heating using an external standard. Thermal effects in this system are found to be too small to affect the conclusions in part I. Their small size, combined with small, systematic errors in the data, also makes it difficult to measure the yield of solvent heat in these experiments.

Multiple population-period transient spectroscopy (MUPPETS) in excitonic systems.

Time-resolved experiments with more than one period of incoherent time evolution are becoming increasingly accessible. When applied to a two-level system, these experiments separate homogeneous and heterogeneous contributions to kinetic dispersion, i.e., to nonexponential relaxation. Here, the theory of two-dimensional (2D) multiple population-period transient spectroscopy (MUPPETS) is extended to multilevel, excitonic systems. A nonorthogonal basis set is introduced to simplify pathway calculations in multilevel systems. Because the exciton and biexciton signals have different signs, 2D MUPPETS cleanly separates the exciton and biexciton decays. In addition to separating homogeneous and heterogeneous dispersion of the exciton, correlations between the exciton and biexciton decays are measurable. Such correlations indicate shared features in the two relaxation mechanisms. Examples are calculated as both 2D time decays and as 2D rate spectra. The effect of solvent heating (i.e., thermal gratings) is also calculated in multidimensional experiments on multilevel systems.

Rate dispersion in the biexciton decay of CdSe/ZnS nanoparticles from multiple population-period transient spectroscopy.

Measurements of biexciton decays in semiconductor nanoparticles are easily contaminated by contributions from photoproducts or higher excitons. Theoretical work has shown that multiple population-period transient spectroscopy (MUPPETS) can measure biexciton decays free from these interferences. In this communication, the biexciton decay of CdSe/ZnS core-shell nanoparticles is measured with MUPPETS. The decay is strongly dispersed (nonexponential) with a more than 5-fold range of rates. This large dispersion must be accounted for in the decay mechanism and in the measurement of biexciton dynamics by more conventional methods. The success of MUPPETS in this context lays the foundation for using it to study exciton-exciton interactions in a variety of materials.

Heterogeneity of the electron-trapping kinetics in CdSe nanoparticles.

The kinetics of electron trapping in CdSe nanoparticles are examined from 0.5 ps to 1.8 ns. The ensemble kinetics fit a slow power law, but two-dimensional measurements show that the decay of each nanoparticle is exponential. A model is proposed in which defect sites provide a gateway for surface trapping and are randomly distributed on the surface. The electric field from the particle's dipole moment creates the observed heterogeneity in rates.

Heterogeneous reaction rates in an ionic liquid: quantitative results from two-dimensional multiple population-period transient spectroscopy.

The hypotheses that ionic liquids are structurally heterogeneous at the molecular level and, even further, that this heterogeneity can transfer to the rates of reactions run in ionic liquids is being actively debated. Here, this hypothesis is tested using multiple population-period transient spectroscopy (MUPPETS), an emerging type of multidimensional measurement that resolves the kinetics of subensembles within a heterogeneous sample. A previous MUPPETS study of the excited-state twisting and electronic relaxation of auramine indicated that an ionic-liquid solvent induces rate dispersion due to a combination of heterogeneous and homogeneous processes, but those data could not quantitatively separate these contributions [Khurmi, C.; Berg, M. A. J. Phys. Chem. Lett.2010, 1, 161]. New MUPPETS data that include phase resolution and subtraction of thermal gratings are presented here and are successfully modeled. The total range of reaction rates (10--90%) is a factor of 70. If the solvent effect is viewed as a set of local viscosities, the viscosity distribution is broad and highly asymmetric. However, if the solvent is viewed as changing a reaction barrier, the data correspond to a Gaussian distribution of barrier heights. The relaxation of each subensemble is nonexponential with an initial induction period, but the shape of the decay is invariant across the rate distribution. A small (2%), long-lived component is identified as a part of the homogeneous kinetic scheme and thus as a secondary channel for excited-state relaxation, not as an impurity or alternative ground-state form of auramine. On the basis of these results, we suggest that the primary cause of rate heterogeneity is a long-lived local electric field acting on the charge redistribution during the reaction.

Thermal gratings and phase in high-order, transient-grating spectroscopy.

Thermal gratings are a well known feature in one-dimensional (i.e., single excitation) transient-grating spectroscopy. This paper presents theory and experiments for thermal gratings in multiple dimensions (i.e., with many excitations). The theory of thermal gratings is extended to an arbitrary number of dimensions using an incoherent Hilbert-space formalism. Interference between Hilbert-space pathways makes it impossible for a thermal grating to propagate across multiple time intervals. The only surviving signal is a hybrid--a population grating between excitations and a thermal grating between the final excitation and the probe. This theory is tested on auramine O in methanol (1D) and in an ionic liquid (3-butyl-1-methylimidazolium hexafluorophosphate) (1D and 2D). In methanol, the ground-state recovery and thermal-grating signals are well separated in time; in the ionic liquid, they are not. Using the results of the theory, accurate subtraction of the thermal-grating signal is possible, extending the useful time range of the experiments. Both the comparison to the theory and the subtraction of the thermal-grating signal are dependent on accurate measurements of the time-dependent phase in these systems. Models are proposed to account for the time-dependent phase. Beer's law is generalized to multidimensional grating spectroscopy. This law provides conventions for consistently comparing the absolute phases and magnitudes between grating and nongrating experiments and between experiments of differing dimensionality.