Frame theory in directional statistics

Introduction

Observations that inherit a direction occur in many scientific disciplines. For example, directional data arise naturally in the biomedical field for protein structure, cell-cycle, and circadian clock experiments . Further examples occur in statistical mechanics, where experiments containing only rod-shaped particles can develop complex directional ordering . A simple pattern change for rod shaped objects is the density-dependent, order-disorder phase transition shown for macroscopic granular rod experiments in Fig. 1(a) and 1(b). Quantification of this transition relies on a principle component analysis (PCA) type measure that is linked with statistical mechanical theories . When applied to experimental samples whose rod orientations shift from uniform to unidirectional, this measure finds the dominant direction (director) and strength of rod ordering (order parameter).

In reality, however, experimental rod orientations are rarely unidirectional, and spatial distortions in the director field frequently occur. These distortions may result from fluctuations and/or other competing forces, like those exerted by the container boundaries in Fig. 1(c) and 1(d). Accurately quantifying rod orientations can, in fact, yield information about the collective behavior of rods, for example elastic properties . While accurate orientation measurements of molecular-sized rods require special techniques , recent advances in single molecule detection may make such measurements more widely accessible . For example, “labeled” rods can be inserted into various environments and serve as local directional sensors by aligning with the rod-shaped material around them. This technique can in principle be used in environments as complicated as a cell and potentially uncover intricate patterns that require more sophisticated measures of directional order.

The resulting complex patterning may reduce the value of the order parameter, sometimes to the point that the sample is inaccurately classified as disordered, Fig. 1(d). To predict which multidirectional patterns cause such misclassifications, we merge directional statistics with frame theory. Frames have proven useful in fields like spherical codes, compressed sensing, signal processing, and wavelet analysis . A frame is a basis-like system that spans a vector space but allows for linear dependency, which can be used to reduce noise, find sparse representations, or obtain other desirable features unavailable with orthonormal bases. Tight frames even provide a parseval type formula similar to orthonormal bases. Moreover, the frame concept has recently been generalized to probability distributions on the unit sphere .

To analyze granular rod patterning, we consider statistical testing for directional uniformity, focusing on the Bingham test. We characterize non-uniform sample distributions that lead to failure of rejection and find that these distributions are probabilistic tight frames. Since these frames are well-understood in terms of algebraic/geometric conditions , further synergistic effects may develop between directional statistics and frame theory.

Directional statistics

where the polar representation $\bar{x}=\bar{r}\bar{x}_{0}$ splits the mean into a mean direction $\bar{x}_{0}\in S^{d-1}$ and a mean resultant length $\bar{r}=\|\bar{x}\|$. The Rayleigh test rejects the hypothesis of uniformity if $\bar{r}$ is large. More precisely, the asymptotic large-sample distribution of $dn\bar{r}^{2}$ under uniformity is $\chi^{2}_{d}$ distributed with an error $\mathcal{O}(n^{-1})$, while the modified Rayleigh statistic $(1-\frac{1}{2n})dn\bar{r}^{2}+\frac{1}{2n(d+2)}d^{2}n^{2}\bar{r}^{4}$ is $\chi^{2}_{d}$ distributed with an error $\mathcal{O}(n^{-2})$ .

To describe the Bingham test, let $\sigma$ denote the uniform probability measure on the sphere with respect to the Borel sigma algebra $\mathcal{B}$. We first observe that the second moments of $\sigma$ satisfy

of a sample $\{x_{i}\}_{i=1}^{n}\subset S^{d-1}$ equals the matrix of second moments of the underlying counting measure. Recalling that the matrix of second moments of the uniform measure equals $\frac{1}{d}\mathcal{I}_{d}$, the Bingham test rejects the hypothesis of directional uniformity of a sample if its Fisher matrix $T_{\{x_{i}\}_{i=1}^{n}}$ is far from $\frac{1}{d}\mathcal{I}_{d}$. In fact, the Bingham statistic $\frac{d(d+2)}{2}n(\operatorname{trace}(T_{\{x_{i}\}_{i=1}^{n}}^{2})-\frac{1}{d})$ under uniformity is $\chi^{2}_{(d-1)(d+2)/2}$ distributed with an error $\mathcal{O}(n^{-1})$, cf. .

We say that the Bingham test is inconsistent when rejection of uniformity fails for a particular non-uniform sample distribution. Here, we focus on those distributions that are multi-modal, where a mode is a local maximum of the distribution’s density. Other analysis tools have been customized to spherical and more general manifold data in .

Frames

Every finite tight frame gives rise to the expansion

Joining frame theory and directional statistics

As introduced in Section 2, the Rayleigh test rejects uniformity if the mean resultant length is far from , while the Bingham test rejects uniformity if the sample’s Fisher matrix is far from $\frac{1}{d}\mathcal{I}_{d}$. We therefore call a probability measure $\mu$ on the sphere a Rayleigh-alternative if its mean $\bar{\mu}=\int_{S^{d-1}}xd\mu(x)$ is and a Bingham-alternative if $M_{i,j}(\mu)=\frac{1}{d}\delta_{i,j}$ for all $i,j=1\ldots,d$. We first characterize Rayleigh- and Bingham-alternatives in terms of maximizers and minimizers, respectively, of certain potentials. Subsequently, we provide a connection to probabilistic frames.

Bjoerck verifies in that, among all probability measures $\mu\in\mathcal{M}(\mathcal{B},S^{d-1})$, the maximizers of the probabilistic Riesz-$2$-potential

are exactly the zero mean probability measures. Therefore, the maximizers of (4) are the Rayleigh-alternatives.

To characterize Bingham-alternatives, we introduce the directional force $F$ between two points $a$ and $b$ on the sphere $S^{d-1}$ as $F(a,b)=2|\langle a,b\rangle|(a-b).$ The physical potential between $a$ and $b$ is $|\langle a,b\rangle|^{2}$ . The minimizers of the probabilistic frame potential

Results in and the present work imply that the Bingham-alternatives with zero mean are the minimizers of the fractional frame-Riesz-$2$-potential

Hence, probabilistic unit norm tight frames with zero mean are both, Rayleigh- and Bingham-alternatives.

Patterning of granular rods

A collection of rod shaped particles can undergo an order-disorder phase transition that, in the simplest model, is controlled by entropy. At low rod densities, the maximal total entropy occurs when both rotational and translational entropies are independently maximized. Beyond a critical density, however, rotational entropy is sacrificed for significant gains in translational entropy, resulting in a phase transition from randomly (uniformly) rotated rods, Fig. 1(a), to directionally oriented rods, Fig. 1(b). A PCA-type method measures the average rod direction (director) and strength of rod alignment (order parameter) that results from rotational entropy loss and is described in the following: Let $x_{i}\in S^{d-1}$ denote the direction of the $i$-th rod out of $n$ total rods. For simplicity, let us assume that the alignment is measured in a plane, hence $d=2$. The covariance type matrix

is therefore used to determine the director. Since $Q_{2}$ is symmetric, the eigenvectors form an orthogonal basis, where the nonnegative eigenvalue $\lambda$ corresponds to the order parameter and the associated eigenvector corresponds to the director, cf. . In fact, $0\leq\lambda\leq 1$ and the second eigenvalue of $Q_{2}$ equals $-\lambda$.

This PCA-type method only measures unidirectional rod ordering. In experiments, however, fluctuations and/or competing forces like container boundaries, Figs. 1(c) and 1(d), can influence rod alignment. Therefore, the director field can vary spatially, potentially resulting in complex multidirectional patterning, Fig. 1(d), that sometimes cannot be distinguished from a disordered state when analyzed by the traditional order parameter.

2 New model by means of directional statistics

We propose a complementary analysis of rod ordering that can more accurately quantify multidirectional alignment. We fit a probability model to experimentally observed rod patterning by first identifying a set of directions, cf. Fig. 1(b)-(d). Since a rod rotated by $180$ degrees is indistinguishable from an unrotated rod, we do not differentiate between $x$ and $-x$. Consistent with this criteria, let $\sigma\in\mathcal{M}(\mathcal{B},S^{d-1})$ be the uniform measure on the sphere, $z_{0}\in S^{d-1}$, and $\kappa\neq 0$, the Watson measure $\mu$ is then given by

where $c_{d}(\kappa)=\frac{\Gamma(d/2)}{2\pi^{d/2}F(1/2,d/2,\kappa)}$, $\Gamma$ the usual Gamma function, and $F$ a confluent hypergeometric function . For $\kappa>0$, the density tends to concentrate around $\pm z_{0}$, whereas for $\kappa<0$, the density concentrates around the great circle orthogonal to $z_{0}$. And as $|\kappa|$ increases, the density peaks tighten.

Next, we model each sample with a mixture of Watson distributions, i.e., for a collection of directors $\{z_{i}\}_{i=1}^{N}\subset S^{1}$, we consider

Finally, this approach extracts two parameter sets from the sample, directional modes and associated widths , where the widths represent a measure for directional ordering. These parameters will be used in a forthcoming paper to quantify the differences between experimental rod patterns and the expected behavior from theories and simulations. Furthermore, these tools may provide a method to identify more subtle rod patterning transitions, like the one described in .

Acknowledgements

The authors are supported, in part, by intramural research funds from the National Institute of Child Health and Human Development, National Institutes of Health. The Fritz Haber research center is supported by the Minerva Foundation, Munich, Germany. In addition, ME is supported by the NIH/DFG Research Career Transition Awards Program (EH 405/1-1/575910).

References