Nonlinear Correlation Functions Research Articles

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

A preceding paper [M. Dhar, J. A. Dickinson, and M. A. Berg, J. Chem. Phys. 159, 054110 (2023)] shows how to remove additive noise from an experimental time series, allowing both the equilibrium distribution of the system and its Green's function to be recovered. The approach is based on nonlinear-correlation functions and is fully nonparametric: no initial model of the system or of the noise is needed. However, single-molecule spectroscopy often produces time series with either photon or photon-counting noise. Unlike additive noise, photon noise is signal-size correlated and quantized. Photon counting adds the potential for bias. This paper extends noise-corrected-correlation methods to these cases and tests them on synthetic datasets. Neither signal-size correlation nor quantization is a significant complication. Analysis of the sampling error yields guidelines for the data quality needed to recover the properties of a system with a given complexity. We show that bias in photon-counting data can be corrected, even at the high count rates needed to optimize the time resolution. Using all these results, we discuss the factors that limit the time resolution of single-molecule spectroscopy and the conditions that would be needed to push measurements into the submicrosecond region.

Homogeneous Dephasing in Photosynthetic Bacterial Reaction Centers: Time Correlation Function Approach.

A new tractable linear electronic transition dipole moment time correlation function (ETDMTCF) that accurately accounts for electronic dephasing, asymmetry, and width of 1-phonon profile, which the zero-phonon line (ZPL) contributes to it, in Rhodopseudomonas viridis bacterial reaction center is derived. This time correlation function proves to be superior to other frequency-domain expressions in case of strong electron-phonon coupling (which is often the case in bacterial RCs and pigment-protein complexes), many vibrational modes involved, and high temperature, whereby more vibronic and electronic (sequence) transitions would arise. The Fourier transform of this ETDMTCF leads to asymmetric multiphonon profiles composed of Lorentzian distribution and Gaussian distribution on the high- and low-energy sides, respectively, whereby the overtone widths fold themselves with that of the one-phonon profile. This ETDMTCF also features expedient computation in large systems using asymmetric phonon profiles to account correctly for dephasing and pigment-protein interaction (electron-phonon coupling). The derived ETDMTCF allows computing all nonlinear optical signals in both time and frequency domains, through the nonlinear dipole moment time correlation functions (as guided by nonlinear optical response theory) in line with the eight Liouville space pathways. The linear transition dipole moment time correlation function is of a central value as the nonlinear transition dipole moment time correlation function is expressed in terms of the linear transition dipole moment time correlation function, derived herein. One of the great advantages of presenting this ETDMTCF is its applicability to nonlinear transition dipole moment time correlation functions in line with the eight Liouville space pathways needed in computing nonlinear signals. As such, there is more to the utility and applicability of the presented ETDMTCF besides computational expediency and efficiency. Results show good agreement with the reported literature. The intimate connection between a one-phonon profile and the corresponding bath spectral density in photosynthetic complexes is discussed.

Global correlation analysis of strongly nonlinear frequency responses using the arclength-based separation and the Correlation-Map

Global correlation analysis is an important technique to quantify both the shape and amplitude differences between two response vectors. In linear dynamic systems, differences between two Frequency Response Functions (FRFs) are quantified as scalar number curves of the Global Shape Criterion (GSC) and the Global Amplitude Criterion (GAC), to represent FRF similarities at different frequencies. From linear to nonlinear, responses are usually obtained at different frequencies to form the Frequency Response Curve (FRC), replacing the FRF for dynamic analysis. Extending the concept of global correlation analysis from linear FRFs to nonlinear FRCs could quantify shape and amplitude similarities between nonlinear models. However, global correlation analysis for multivalued FRCs with a strong nonlinearity is hard to conduct, as strongly nonlinear correlation functions have complex multivalued phenomena with real/fake characteristics. In this paper, the Global Shape Curve Criterion (GSCC) and Global Amplitude Curve Criterion (GACC) are proposed for the correlation analysis of strongly nonlinear FRCs, which can quantify the similarity between two FRCs with different and complex multivalued phenomena. Through the arclength-based separation, multivalued FRCs are separated to single-valued response branches, in order to compute single-valued correlation functions that form the multivalued correlation function. The computed correlations contain the GSCC and GACC, which separately represent shape and amplitude differences between two FRCs at each frequency. The multivalued correlation function is represented as a Correlation-Map (C-MAP) to extract real correlation characteristics, for accurate correlation analysis. The multivalued correlation analysis is first verified on a 3 DOF model with a strong nonlinearity. Differences between the reference and initial multivalued FRCs are successfully quantified as scalar curves and the GACC may be more sensitive than the GSCC on models with a local nonlinearity. Then, the proposed method is further validated on an experimental 3 DOF model. Very complex 15-valued correlation functions between FRCs with different multivalued phenomena are established. Even so, the real correlations are still successfully extracted by the C-MAP. These show the validity and superiority of the proposed method.

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.

Seismic site characterization using MASW and correlation study between shear wave velocity and SPT[sbnd

A seismic site characterization study is carried out in the south of the foothills of tectonically active Himalayan region. During an earthquake, the dynamic characteristics of the shallow subsurface strata can influence seismic shaking in the region, causing damage to civil infrastructures. The seismic site characterization study with shear wave velocity (Vs) is performed using the Multi-channel Analysis of Surface Wave (MASW) technique, which is particularly challenging to implement in the densely inhabited areas. However, the standard penetration test (SPTN) commonly measured by civil engineers can be utilized to compute Vs of such areas by establishing a correlation between Vs and the SPT-N values.In the study area, MASW data were acquired at twelve different locations, whereas SPT-N tests were available from 26 boreholes. Twelve borehole locations closer to the MASW survey lines were paired to establish relationships between SPT-N and Vs. The developed linear and non-linear correlation functions were found to be statistically similar. Furthermore, the linear correlation function is used for estimating Vs from SPT-N in remaining 14 boreholes. The new correlation functions were further compared with previously reported empirical relationships from other areas. The current relationship can be used for geoengineering purposes at different sites with similar geological conditions.Shear wave velocity derived from the MASW surveys and estimated from the new empirical relationship was utilized for seismic site characterization based on average shear wave velocity up to 30 m depth (Vs30). The Vs30 map of the study site is prepared and classified in accordance with the National Earthquake Hazard Reduction Program (NEHRP-2020). The calculated Vs30 value of the study site ranges from 285 to 375 m/s. The major portion of the study area lies under Site Class CD, with a minor portion under Site Class D. In the event of an earthquake, some zones can potentially have stronger ground amplification, particularly in Site Class D and may have damaging effects. A series of average shear wave velocity maps for different depths are prepared to help geotechnical engineers in identifying and mitigating geotechnical risks. Furthermore, there are potential geohazard risks as certain depths show low shear velocity.

Non-linear correlation functions and zero-point energy flow in mixed quantum-classical semiclassical dynamics.

Mixed quantum classical (MQC)-initial value representation (IVR) is a recently introduced semiclassical framework that allows for selective quantization of the modes of a complex system. In the quantum limit, MQC reproduces the semiclassical Double Herman-Kluk IVR results, accurately capturing nuclear quantum coherences and conserving zero-point energy. However, in the classical limit, although MQC mimics the Husimi-IVR for real-time correlation functions with linear operators, it is significantly less accurate for non-linear correlation functions with errors even at time zero. Here, we identify the origin of this discrepancy in the MQC formulation and propose a modification. We analytically show that the modified MQC approach is exact for all correlation functions at time zero, and in a study of zero-point energy (ZPE) flow, we numerically demonstrate that it correctly obtains the quantum and classical limits as a function of time. Interestingly, although classical-limit MQC simulations show the expected, unphysical ZPE leakage, we find that it is possible to predict and even modify the direction of ZPE flow through selective quantization of the system, with the quantum-limit modes accepting energy but preserving the minimum quantum mechanically required energy.

Quantum trajectory theory and simulations of nonlinear spectra and multiphoton effects in waveguide-QED systems with a time-delayed coherent feedback

We study the nonlinear spectra and multi-photon correlation functions for the waveguide output of a two-level system (including realistic dissipation channels) with a time-delayed coherent feedback. We compute these observables by extending a recent quantum trajectory discretized-waveguide (QTDW) approach which exploits quantum trajectory simulations and a collisional model for the waveguide to tractably simulate the dynamics. Following a description of the general technique, we show how to calculate the first and second order quantum correlation functions, in the presence of a coherent pumping field. With a short delay time, we show how feedback can be used to filter out the central peak of the Mollow triplet or switch the output between bunched and anti-bunched photons by proper choice of round trip phase. We further show how the loop length and round trip phase effects the zero-time second order quantum correlation function, an indicator of bunching or anti-bunching. New resonances introduced through the feedback loop are also shown through their appearance in the incoherent output spectrum from the waveguide. We explain these results in the context of the waiting time distributions of the system output and individual trajectories, uniquely stochastic observables that are easily accessible with the QTDW model.

Triggering Earthquakes Fluctuations of The Planetary Gravitational Field and Nonlinear Interactions with Matter

This research investigates the extent to which a non-linear correlation function can describe the influence of the planetary gravitational field on the triggering of earthquakes. This method differs from the methods I have known so far, which have only investigated the position of the Sun and Moon. A correlation function is developed that can describe changes in the probability of stable and unstable states in matter. It can be shown that about 6% of the investigated earthquakes can be triggered by the planetary gravitational field. The method could also be suitable as an element for artificial intelligence.

Constraints on f(R) and normal-branch Dvali-Gabadadze-Porrati modified gravity model parameters with cluster abundances and galaxy clustering

We present forecasted cosmological constraints from combined measurements of galaxy cluster abundances from the Simons Observatory and galaxy clustering from a DESI-like experiment on two well-studied modified gravity models, the chameleon-screened $f(R)$ Hu-Sawicki model and the nDGP braneworld Vainshtein model. A Fisher analysis is conducted using $\sigma_8$ constraints derived from thermal Sunyaev-Zel'dovich (tSZ) selected galaxy clusters, as well as linear and mildly non-linear redshift-space 2-point galaxy correlation functions. We find that the cluster abundances drive the constraints on the nDGP model while $f(R)$ constraints are led by galaxy clustering. The two tracers of the cosmological gravitational field are found to be complementary, and their combination significantly improves constraints on the $f(R)$ in particular in comparison to each individual tracer alone. For a fiducial model of $f(R)$ with $\text{log}_{10}(f_{R0})=-6$ and $n=1$ we find combined constraints of $\sigma(\text{log}_{10}(f_{R0}))=0.48$ and $\sigma(n)=2.3$, while for the nDGP model with $n_{\text{nDGP}}=1$ we find $\sigma(n_{\text{nDGP}})=0.087$. Around a fiducial General Relativity (GR) model, we find a $95\%$ confidence upper limit on $f(R)$ of $f_{R0}\leq5.68\times 10^{-7}$. Our results present the exciting potential to utilize upcoming galaxy and CMB survey data available in the near future to discern and/or constrain cosmic deviations from GR.

Electronic dephasing in mixed quantum–classical molecular systems using the spin-boson model

Pure electronic dephasing is investigated using the spin-boson Hamiltonian in mixed quantum–classical environment. The spin-boson model used here is a composite system made up of a quantum subsystem, an electronic 2-level subsystem linearly coupled to harmonic vibrations, interacting with a classical bath. Experimental results for a multitude of molecular systems indicate that the zero-phonon line (ZPL) profile is determined by electronic dephasing, which is not accounted for in the multimode Brownian oscillator (MBO) model due to the unphysical contribution from the MBO bath modes to the ZPL profile. Mixed quantum–classical dynamics formalism of non-equilibrium systems is employed to assess the contribution of the bath modes to pure electronic dephasing by probing the ZPL profile when coupled to a classical bath in the mixed quantum–classical condensed systems. Pure electronic dephasing is discussed in the context of mixed quantum–classical dynamics formalism which starts with mixed quantum–classical Liouville equation in a mixed quantum–classical environment. It is noteworthy, however, that the fundamental difference between the fully quantum MBO model and the mixed quantum–classical Brownian oscillator, is that the zero-phonon line calculated by the former shows unphysical asymmetry on the low-energy side as it has not been observed in real systems, whereas the ZPL reported herein eliminates this asymmetry. A systematic approach using matrix mechanics is developed to treat this phenomenon. To this end, a closed-form expression of linear and nonlinear optical electronic transition dipole moment time correlation functions in a dissipative media are derived. Linear absorption spectra and 4-wave mixing signals at various temperatures showing a sound thermal broadening, temporal decay, and accurate pure dephasing further ratify the applicability and correctness of the mixed quantum–classical dynamics approach to spectroscopy and dynamics are computed.

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.

A Focused Transport-based Kinetic Fractional Diffusion-advection Equation for Energetic Particle Trapping and Reconnection-related Acceleration by Small-scale Magnetic Flux Ropes in the Solar Wind

Abstract Analysis of energetic particle inner heliospheric spacecraft data increasingly suggests the existence of anomalous diffusion phenomena that should be addressed to achieve a better understanding of energetic particle transport and acceleration in the expanding solar wind medium. Related to this is fast-growing observational evidence supporting the long-standing prediction from magnetohydrodynamic (MHD) theory and simulations of the presence of an inner heliospheric, dominant quasi-two-dimensional MHD turbulence component that contains coherent contracting and merging (reconnecting) small-scale magnetic flux rope (SMFR) structures. This suggests that energetic particle trapping in SMFRs should play a role in anomalous diffusion in the solar wind that warrants further investigation. However, progress in studying such anomalous energetic particle transport phenomena in the solar wind is hampered by the lack of a fundamental derivation of a general fractional kinetic transport equation linking macroscopic energetic particle fractional transport to the microscopic physics of energetic particle interaction with SMFR structures. Here, we outline details of how one can derive a closed ensemble-averaged focused transport equation in the form of a general kinetic fractional diffusion-advection equation from first principles following the nonlinear Eulerian correlation function closure approach of Sanchez et al. With this equation one can model the anomalous diffusion of energetic particles in ordinary, momentum, and pitch-angle space in response to particle trapping in numerous SMFRs advected with the solar wind flow.

Nonquantum information gain from higher-order correlation functions

Nonlinear correlation functions are at the heart of quantum theory. The second-order correlation function $g^{(2)}(\tau)$ has been a cornerstone of quantum optics since over half a century and a myriad of quantum and classical applications has been discovered. In contrast, higher-order correlation functions have so far only been used to reveal the nonclassical character of the emitted fields. In this paper, we study the relation between the $k$th-order correlation function $g^{(k)}(0)$ and the projection of the underlying quantum state of light onto the subspace of Fock states with photon number less than $k$. We show, that when $g^{(k)}(0)$ falls below a critical value, lower bounds for the projection on this subspace can be concluded as well as on the ratio of the subspace with one upto $k-1$ photons and $k$ to infinity. These bounds are at face value only valid for nonclassical quantum states. However, when the quantum state includes a nonzero projection on the vacuum state, the value of $g^{(k)}(0)$ is artificially enhanced, potentially covering these projections. We derive an effective $k$th-order correlation function, which accounts for the effect of vacuum. We show that the information gained from the effective correlation function is not limited to nonclassical quantum states and thus constitute a quantum- and classical application of higher-order correlation functions.

Image correlation by one-dimensional signatures invariant to rotation, position, and scale using the radial Hilbert transform optimized.

This paper presents a new methodology for pattern recognition invariant to rotation, position, and scale. The method uses the correlation of signatures, where the signatures were created with a new equation called the radial Hilbert transform optimized (RHTO) for longer signatures. An analysis with eight non-homogeneous illumination patterns was performed with 2000 letter variants and 30 phytoplankton species. The higher confidence level was founded using the radial Hilbert optimized methodology. Also, it utilized a correlation called adaptive linear-nonlinear correlation, which gave a better discrimination performance than the nonlinear correlation function.

Quasiclassical Correlation Functions from the Wigner Density Using the Stability Matrix.

Accounting for zero-point energy in the initial conditions of classical trajectory calculations of time correlation functions requires sampling from a quantized phase space distribution, which is often chosen as the Weyl-Wigner transform of a thermalized operator. The numerical construction of the latter and its use as a sampling function can be challenging. We show that the operator dependence of the phase space distribution can be transferred to the dynamics, allowing sampling from the simpler Wigner phase space density. The method involves augmenting the classical equations of motion with additional differential equations for elements of the stability matrix. We also propose a local harmonic approximation for the dynamical derivatives, which significantly reduces the computational cost required to obtain correlation functions of nonlinear operators. We illustrate the method with application to linear and nonlinear correlation functions of model Hamiltonians. While the local harmonic approximation is not always successful in predicting nonlinear correlation functions of one degree of freedom, it quantitatively captures the full quasiclassical results for systems in contact with dissipative environments.

First results from the IllustrisTNG simulations: matter and galaxy clustering

Hydrodynamical simulations of galaxy formation have now reached sufficient volume to make precision predictions for clustering on cosmologically relevant scales. Here we use our new IllustrisTNG simulations to study the non-linear correlation functions and power spectra of baryons, dark matter, galaxies and haloes over an exceptionally large range of scales. We find that baryonic effects increase the clustering of dark matter on small scales and damp the total matter power spectrum on scales up to k ~ 10 h/Mpc by 20%. The non-linear two-point correlation function of the stellar mass is close to a power-law over a wide range of scales and approximately invariant in time from very high redshift to the present. The two-point correlation function of the simulated galaxies agrees well with SDSS at its mean redshift z ~ 0.1, both as a function of stellar mass and when split according to galaxy colour, apart from a mild excess in the clustering of red galaxies in the stellar mass range 10^9-10^10 Msun/h^2. Given this agreement, the TNG simulations can make valuable theoretical predictions for the clustering bias of different galaxy samples. We find that the clustering length of the galaxy auto-correlation function depends strongly on stellar mass and redshift. Its power-law slope gamma is nearly invariant with stellar mass, but declines from gamma ~ 1.8 at redshift z=0 to gamma ~ 1.6 at redshift z ~ 1, beyond which the slope steepens again. We detect significant scale-dependencies in the bias of different observational tracers of large-scale structure, extending well into the range of the baryonic acoustic oscillations and causing nominal (yet fortunately correctable) shifts of the acoustic peaks of around ~5%.

The clustering of dark matter haloes: scale-dependent bias on quasi-linear scales

We investigate the spatial clustering of dark matter halos, collapsing from $1-4 \sigma$ fluctuations, in the redshift range $0 - 5$ using N-body simulations. The halo bias of high redshift halos ($z \geq 2$) is found to be strongly non-linear and scale-dependent on quasi-linear scales that are larger than their virial radii ($0.5-10$ Mpc/h). However, at lower redshifts, the scale-dependence of non-linear bias is weaker and and is of the order of a few percent on quasi-linear scales at $z \sim 0$. We find that the redshift evolution of the scale dependent bias of dark matter halos can be expressed as a function of four physical parameters: the peak height of halos, the non-linear matter correlation function at the scale of interest, an effective power law index of the {\it rms} linear density fluctuations and the matter density of the universe at the given redshift. This suggests that the scale-dependence of halo bias is not a universal function of the dark matter power spectrum, which is commonly assumed. We provide a fitting function for the scale dependent halo bias as a function of these four parameters. Our fit reproduces the simulation results to an accuracy of better than 4 % over the redshift range $0\leq z \leq 5$. We also extend our model by expressing the non-linear bias as a function of the linear matter correlation function. It is important to incorporate our results into the clustering models of dark matter halos at any redshift, including those hosting early generations of stars and galaxies before reionization.

Synthesis, crystal structures, spectroscopic analysis and DFT calculations of 2-ethoxy-1-naphtaldehyde and (E)-N-((2-ethoxynaphthalen-1-yl)methylene)-3-methylaniline

(E)-N-((2-ethoxynaphthalen-1-yl)methylene)-3-methylaniline has been synthesized and characterized by using single-crystal X-ray diffraction, FT-IR, UV–Visible spectroscopy and computational methods. By using the same techniques, also for the first time, the 2-ethoxy-1-naphtaldehyde has been characterized. The molecular geometries, intra- and inter-molecular interactions of the compounds have been found by using X-ray crystallography. Characteristic infrared bands and the electronic bands have been discovered by experimental and theoretical IR and UV–Vis. spectroscopy. The geometry optimizations and the calculations of IR frequencies have been performed by using the Gaussian type orbitals at Gaussian 09W and the Slater type orbitals at ADF2009.01 software. In addition, the Fukui functions have been calculated to reveal active sites of the compounds. Furthermore, non-linear optical properties and thermodynamic correlation functions have been theoretically found for a further study of the titled compounds.

Cosmological constraints from the CFHTLenS shear measurements using a new, accurate, and flexible way of predicting non-linear mass clustering

We explore the cosmological constraints from cosmic shear using a new way of modelling the non-linear matter correlation functions. The new formalism extends the method of Angulo & White (2010), which manipulates outputs of $N$-body simulations to represent the three-dimensional nonlinear mass distribution in different cosmological scenarios. We show that predictions from our approach for shear two-point correlations at $1$ to $300$ arcmin separations are accurate at the $\sim10$\% level, even for extreme changes in cosmology. For moderate changes, with target cosmologies similar to that preferred by analyses of recent Planck data, the accuracy is close to $\sim5$\%. We combine this approach with a MonteCarlo Markov Chain sampler to explore constraints on a $\Lambda$CDM model from the shear correlation functions measured in the Canada-France Hawaii Telescope Lensing Survey (CFHTLenS). We obtain constraints on the parameter combination $\sigma_8 (\Omega_m/0.27)^{0.6} = 0.801 \pm 0.028$. Combined with results from CMB data, we obtain marginalised constraints on $\sigma_8 = 0.81 \pm 0.01$ and $\Omega_m = 0.29 \pm 0.01$. These results are fully compatible with previous analyses, which supports the validity of our approach. We discuss the advantages of our method and the potential it offers, including a path to incorporate in detail the effects of baryons, among others effects, in future high-precision cosmological analyses.

Delay differential analysis of time series.

Nonlinear dynamical system analysis based on embedding theory has been used for modeling and prediction, but it also has applications to signal detection and classification of time series. An embedding creates a multidimensional geometrical object from a single time series. Traditionally either delay or derivative embeddings have been used. The delay embedding is composed of delayed versions of the signal, and the derivative embedding is composed of successive derivatives of the signal. The delay embedding has been extended to nonuniform embeddings to take multiple timescales into account. Both embeddings provide information on the underlying dynamical system without having direct access to all the system variables. Delay differential analysis is based on functional embeddings, a combination of the derivative embedding with nonuniform delay embeddings. Small delay differential equation (DDE) models that best represent relevant dynamic features of time series data are selected from a pool of candidate models for detection or classification. We show that the properties of DDEs support spectral analysis in the time domain where nonlinear correlation functions are used to detect frequencies, frequency and phase couplings, and bispectra. These can be efficiently computed with short time windows and are robust to noise. For frequency analysis, this framework is a multivariate extension of discrete Fourier transform (DFT), and for higher-order spectra, it is a linear and multivariate alternative to multidimensional fast Fourier transform of multidimensional correlations. This method can be applied to short or sparse time series and can be extended to cross-trial and cross-channel spectra if multiple short data segments of the same experiment are available. Together, this time-domain toolbox provides higher temporal resolution, increased frequency and phase coupling information, and it allows an easy and straightforward implementation of higher-order spectra across time compared with frequency-based methods such as the DFT and cross-spectral analysis.