Super-resolved anomalous diffusion: deciphering the joint distribution of anomalous exponent and diffusion coefficient

TL;DR

The paper presents a universal, copula-based framework to reconstruct the joint distribution $(\alpha,D)$ of anomalous diffusion from TAMSD-based estimators $(\hat{\alpha},\hat{D})$, accounting for finite-trajectory noise and intrinsic parameter variability. By deriving moments of TAMSD-based statistics via a quadratic-form analysis and employing a Gaussian copula (with non-Gaussian marginals when needed), it yields practical transfer functions $p(\hat{\alpha},\hat{D}|\alpha,D)$ and enables both discrete mixtures and continuous joint distributions to be inferred. The work explains the observed $\hat{D}\propto \exp(\hat{\alpha} c_1+c_2)$ and provides exact expressions for the joint density under the Gaussian framework, with robust non-Gaussian extensions using Beta and Pearson IV marginals. It validates the approach on fractional Brownian motion simulations, analyzes trajectory-length effects via corrected Hellinger distances, and provides a MATLAB code for fitting the joint distribution from experimental-like data, enabling super-resolved inference of diffusion heterogeneity in complex media.

Abstract

The molecular motion in heterogeneous media displays anomalous diffusion by the mean-squared displacement $\langle X^2(t) \rangle = 2 D t^α$. Motivated by experiments reporting populations of the anomalous diffusion parameters $α$ and $D$, we aim to disentangle their respective contributions to the observed variability when this last is due to a true population of these parameters and when it arises due to finite-duration recordings. We introduce estimators of the anomalous diffusion parameters on the basis of the time-averaged mean squared displacement and study their statistical properties. By using a copula approach, we derive a formula for the joint density function of their estimations conditioned on their actual values. The methodology introduced is indeed universal, it is valid for any Gaussian process and can be applied to any quadratic time-averaged statistics. We also explain the experimentally reported relation $D\propto\exp(αc_1+c_2)$ for which we provide the exact expression. We finally compare our findings to numerical simulations of the fractional Brownian motion and quantify their accuracy by using the Hellinger distance.

Figures (7)

• Figure 1: (1): Histograms of $\hat{\alpha}$ versus theoretical predictions for Gaussian (blue line) and Beta distribution (yellow line); (2): Histograms of $\hat{Y}$ versus theoretical prediction for Gaussian (blue line) and Pearson IV (yellow line). (3): histogram $p(\hat{\alpha},\hat{D})$ versus theory (magenta mesh) using Gaussian approximation of $\hat{\alpha}$ and $\hat{Y}$ from Eq. (\ref{['eq15']}). (4): histogram $p(\hat{\alpha},\hat{D})$ versus theory (yellow mesh) using copula-based approach Eq. (\ref{['eq:joint_pdf_alpha_D_copula']}). All the empirical histograms were obtained from $M=10^5$ simulated fBm trajectories of $N=100$ steps with $\tau_{\max}=5$ and $\alpha=\lbrace 0.2,1,1.6\rbrace$ for rows A, B, C. In each case, we choose $D=1$ and $\Delta t=1$. For the yellow lines and mesh columns (1), (2), (4), the skewness and kurtosis used in marginal PDFs were obtained empirically.
• Figure 2: (A) Top: Histogram of the empirical $p(\hat{\alpha},\hat{Y})$ overlayed with the fitted distribution (A) Bottom: Histogram of the ground truth $p(\alpha,D)$ overlayed with the joint PDF inferred from the fitting of empirical $p(\hat{\alpha},\hat{Y})$. (B) Scatter plot of $\lbrace \hat{\alpha},\hat{D}\rbrace$ (black dots) and $\lbrace \alpha,D\rbrace$ (purple crosses) overlayed with teal isoline for the fitted joint PDF $p(\hat{\alpha},\hat{D})$ and pink isoline for the inferred joint PDF $p(\alpha,D)$. On top, histogram of ground truth $\alpha$ (grey bars) and $\hat{\alpha}$ (yellow bars), teal line is $p(\hat{\alpha})$ obtained from fitting and pink line is $p(\alpha)$ inferred from the fitting. On the right, histogram of ground truth $D$ (grey bars) and $\hat{D}$ (yellow bars), teal line is $p(\hat{D})$ obtained from fitting and pink line is $p(D)$ inferred from the fitting. The figure is based on a single simulated data set of $M=10^3$ trajectories of fBm of $N=100$ steps with jointly randomly distributed $\alpha$ and $D$ according to Eq. (\ref{['eq:joint_PDF_true_alpha_D']}) with parameters $\langle\alpha\rangle=1.4$, $\textrm{std}(\alpha)=0.2$, $\langle D\rangle=1$, $\textrm{std}(D)=0.5$ and $\rho_{\alpha,D}=-0.7$.
• Figure 3: Histogram $p(\hat{\alpha},\hat{D})$ versus theory (magenta mesh) using Gaussian approximation of $\hat{\alpha}$ and $\hat{Y}$ from Eq. (17) from main text where $p(\alpha,D)$ is distributed according to Eq. (19) from main text with three components $\alpha_1=0.1$, $D_1=0.1$, $\alpha_2=1$, $D_2=0.1$, $\alpha_3=1$, $D_3=0.15$ with probability $c_1=c_2=c_3=1/3$. Empirical histogram was obtained from $M=10^5$ trajectories of $N=100$ steps with $dt=0.03$ and fitted using $\tau_{\max}=5$.
• Figure 4: Theoretical curve of the maximum anomalous exponent $\alpha_{max}$ for which the Gaussian approximation holds as a function of trajectory length $N$ given a maximum lag-time $\tau_{max}=5$ for the fitting of TAMSD applied to fractional Brownian motion.
• Figure 5: Corrected Hellinger distance between theoretical joint PDF $p(\hat{\alpha},\hat{D})$ (using Gaussian approximation from Eq. (17) in the main text) and the empirical joint PDF of $\hat{\alpha}$ and $\hat{D}$. Results are presented as a function of the trajectories' length $N$ received from $M=10^5$ fBm-particles for different values of the anomalous exponent $\alpha$ with fixed $D=1$, $\Delta t=1$, and $\tau_{max}=5$.