Projected Rayleigh Statistic (PRS)

• Projected Rayleigh Statistic (PRS) is a modified Rayleigh test that maps axial data to circular data via angle doubling to detect preferential alignments.
• It computes the cosine sum of doubled relative angles and projects it onto the x-axis, with positive or negative values indicating parallel or perpendicular alignment respectively.
• PRS offers improved statistical power over histogram binning, making it valuable for analyzing magnetic field and gas structure orientations in molecular clouds.

The projected Rayleigh statistic (PRS) is a modification of the classic Rayleigh statistic for testing non-uniform relative orientation between two pseudo-vector fields, particularly when the relevant alternatives are preferential parallel or preferential perpendicular alignment. It was introduced by Jow et al. as a statistic tuned to detect a mean direction of zero after mapping axial relative-orientation data to circular data, and it has since been used in analyses of magnetic-field and gas-structure alignment in molecular clouds (Jow et al., 2017, Mazzei et al., 2023).

1. Definition and inferential target

Suppose two overlapping pseudo-vector fields on the sky are denoted $U$ and $V$. At each position $i$, one forms the relative angle

$$\phi_i = \arctan\!\left(\frac{|U_i \times V_i|}{U_i \cdot V_i}\right),$$

with $\phi_i \in [-\pi/2,\pi/2]$. The inferential question is whether the $\phi_i$ are uniformly distributed, corresponding to no preferred alignment, or whether they cluster around $0^\circ$ or $90^\circ$, corresponding respectively to parallel or perpendicular relative orientation (Jow et al., 2017).

The PRS, denoted $Z_x$, is specifically tuned to test preferential alignment at $\phi=0$. A strongly positive $V$0 indicates statistically significant parallel alignment, while a strongly negative $V$1 indicates perpendicular alignment. In this sense, the PRS is not merely a generic non-uniformity test; it is a directional test adapted to the geometry of pseudo-vectors, or spin-2 quantities, for which $V$2 and $V$3 are equivalent (Jow et al., 2017).

In molecular-cloud applications, the same construction is used with $V$4, the gradient of a gas-tracer Moment 0 map or intensity map, and $V$5, the unit vector of the polarization electric field. In that setting, $V$6 means magnetic field and gas structures are locally parallel, while $V$7 means perpendicular (Mazzei et al., 2023).

2. Derivation from the classic Rayleigh test

The starting point is the classic Rayleigh statistic for circular data $V$8: $V$9 Under the null hypothesis of uniform $i$0, $i$1 on average, whereas clustering away from uniformity drives $i$2 large (Jow et al., 2017).

For relative-orientation data between pseudo-vectors, the angles are axial rather than circular. Jow et al. therefore use angle doubling: $i$3 Under this mapping, parallel alignment at $i$4 maps to $i$5, while perpendicular alignment at $i$6 maps to $i$7. Rather than using the full two-dimensional Rayleigh vector sum, the PRS projects that sum onto the $i$8-axis only, because the target alternative is a mean direction of zero. The resulting statistic is

$i$9

Under the null hypothesis of uniform $$\phi_i = \arctan\!\left(\frac{|U_i \times V_i|}{U_i \cdot V_i}\right),$$0, one has $$\phi_i = \arctan\!\left(\frac{|U_i \times V_i|}{U_i \cdot V_i}\right),$$1 and $$\phi_i = \arctan\!\left(\frac{|U_i \times V_i|}{U_i \cdot V_i}\right),$$2. By the central-limit theorem for large $$\phi_i = \arctan\!\left(\frac{|U_i \times V_i|}{U_i \cdot V_i}\right),$$3, $$\phi_i = \arctan\!\left(\frac{|U_i \times V_i|}{U_i \cdot V_i}\right),$$4. Jow et al. also give the measurement-error-driven variance of a single $$\phi_i = \arctan\!\left(\frac{|U_i \times V_i|}{U_i \cdot V_i}\right),$$5 as

$$\phi_i = \arctan\!\left(\frac{|U_i \times V_i|}{U_i \cdot V_i}\right),$$6

This normalization is the basis for interpreting $$\phi_i = \arctan\!\left(\frac{|U_i \times V_i|}{U_i \cdot V_i}\right),$$7 directly in units of Gaussian significance (Jow et al., 2017).

3. Computation, weighting, and significance

In practical use, the computation is straightforward. One first measures two pseudo-vectors at each pixel, such as a column-density gradient direction and a polarization pseudo-vector. One then computes the relative angle $$\phi_i = \arctan\!\left(\frac{|U_i \times V_i|}{U_i \cdot V_i}\right),$$8, doubles it to obtain $$\phi_i = \arctan\!\left(\frac{|U_i \times V_i|}{U_i \cdot V_i}\right),$$9, and evaluates the cosine sum. If no weighting is used, the denominator is $\phi_i \in [-\pi/2,\pi/2]$0, where $\phi_i \in [-\pi/2,\pi/2]$1 is the number of samples (Jow et al., 2017).

Jow et al. also define a weighted form,

$\phi_i \in [-\pi/2,\pi/2]$2

where the per-angle weights $\phi_i \in [-\pi/2,\pi/2]$3 may represent, for example, $\phi_i \in [-\pi/2,\pi/2]$4 of polarization. Under uniform $\phi_i \in [-\pi/2,\pi/2]$5, $\phi_i \in [-\pi/2,\pi/2]$6 also tends to $\phi_i \in [-\pi/2,\pi/2]$7. The weighted form is therefore a direct extension of the same null calibration while allowing heterogeneous uncertainties or data quality across the map (Jow et al., 2017).

In the formulation summarized by Mazzei et al., the computation begins from synthetic or observed Stokes $\phi_i \in [-\pi/2,\pi/2]$8 maps, from which one computes the polarization fraction $\phi_i \in [-\pi/2,\pi/2]$9, the polarization angle $\phi_i$0, and hence $\phi_i$1. The local gradient vector $\phi_i$2 is computed from the same $\phi_i$3 map or from a Moment 0 line map, and the relative angle is written as

$\phi_i$4

The PRS is then evaluated using the number of independent samples $\phi_i$5, with an oversampling correction implemented via the white-noise map protocol. In that application, $\phi_i$6 is interpreted as a global tendency toward parallel alignment and $\phi_i$7 as a tendency toward perpendicular alignment; $\phi_i$8 corresponds to a $\phi_i$9 detection of a preferred orientation (Mazzei et al., 2023).

4. Statistical power relative to histogram binning

A central motivation for the PRS is improved statistical efficiency relative to histogram-binning methods. The relevant comparator in Jow et al. is the histogram of relative orientations (HRO) shape statistic

$0^\circ$0

where $0^\circ$1 counts central angles with $0^\circ$2 and $0^\circ$3 counts edge angles with $0^\circ$4. Under the null hypothesis, $0^\circ$5 is approximately Gaussian $0^\circ$6, but it ignores $0^\circ$7 of the angular range (Jow et al., 2017).

Jow et al. compare the two procedures using von Mises alternatives $0^\circ$8. For $0^\circ$9 from $90^\circ$0 and detection thresholds of $90^\circ$1, $90^\circ$2, and $90^\circ$3, the PRS power curves lie systematically above those of $90^\circ$4. At intermediate $90^\circ$5, PRS power can exceed HRO power by $90^\circ$6. In a second experiment with fixed $90^\circ$7 and $90^\circ$8 varying from $90^\circ$9, the PRS reaches $Z_x$0 power at $Z_x$1 for a $Z_x$2 threshold, whereas HRO requires $Z_x$3 (Jow et al., 2017).

The stated conclusion is that the PRS uses every angle continuously and is the Neyman-Pearson optimal test for von Mises alternatives with known mean direction, whereas HRO loses information in binning. This suggests that the primary gain is not a change in the scientific question being asked, but a more efficient use of the same relative-orientation data (Jow et al., 2017).

5. Empirical performance in the Vela C molecular cloud

Jow et al. apply the PRS to the Vela C molecular cloud using BLASTPol polarization at $Z_x$4, $Z_x$5, and $Z_x$6 for $Z_x$7 and Herschel column-density gradients for $Z_x$8. The four sub-regions South-Nest, South-Ridge, Centre-Nest, and Centre-Ridge contain $Z_x$9–$\phi=0$0 independent $\phi=0$1 (Jow et al., 2017).

The explicit values reported for several region-band combinations are:

Region / Band	HRO $\phi=0$2 ($\phi=0$3)	PRS $\phi=0$4 ($\phi=0$5)
South-Nest, $\phi=0$6	$\phi=0$7	$\phi=0$8
South-Nest, $\phi=0$9	$V$00	$V$01
South-Nest, $V$02	$V$03	$V$04
South-Ridge, $V$05	$V$06	$V$07
South-Ridge, $V$08	$V$09	$V$10

In every band and region, $V$11, with the largest factor being $V$12 in South-Nest at $V$13. Jow et al. state that the PRS reveals the same transition from parallel to perpendicular orientation with increasing $V$14, but with tighter uncertainties and higher significance. The Vela C application therefore serves as an observational demonstration of the same efficiency gain already seen in the power calculations (Jow et al., 2017).

6. Broader uses, physical interpretation, and limitations

Jow et al. describe the PRS as applicable wherever two orientation fields are to be compared, especially when one or both are spin-2 quantities. The examples they give include the relative alignment of galaxy major axes within filaments or sheets in large-scale structure, the orientation of protostellar disks or binary orbital planes versus host-cluster shear, and small-scale polarization versus large-scale shear patterns in galaxy clusters. They also note that one may form a two-dimensional cross-correlation $V$15 by computing $V$16 between one field and a spatially shifted version of the other, in order to measure correlation length scales (Jow et al., 2017).

Mazzei et al. provide a detailed later application in synthetic observations of molecular clouds. In their trans-Alfvénic simulation, which they describe as more strongly magnetized, there is a transition to perpendicular alignment at densities above $V$17. This transition is recovered in most synthetic observations of optically thin molecular tracers, whereas for $V$18 the PRS remains in parallel alignment across the whole observer-space. They attribute this difference to optical depth, reporting that the $V$19 surface for $V$20 is largely in front of the cloud midplane, suggesting that $V$21 mainly probes low volume density gas. In their super-Alfvénic simulation, the magnetic field becomes significantly more tangled and all observed tracers tend toward no preference for perpendicular or parallel alignment (Mazzei et al., 2023).

That study also articulates several limitations. Beam convolution reduces $V$22 by washing out small-scale structure, although larger beams can partially counteract this by merging low-density sightlines into dense ones. There is a viewing-angle degeneracy: if the mean field is inclined toward the line of sight, even a strong field can yield $V$23, mimicking a weak-field case. Pixel-selection and sensitivity cuts matter because using only the top fraction of pixels can bias the inferred sign of $V$24. The radiative-transfer method, such as LTE versus LVG, changes molecular optical depth and can shift $V$25 accordingly. Mazzei et al. also note that assuming uniform tracer abundance is a limitation, since real chemistry may intensify the perpendicular signature by raising high-density abundances relative to low density (Mazzei et al., 2023).

Across these applications, the PRS functions as a single-number diagnostic of preferential alignment whose sign distinguishes parallel from perpendicular organization and whose magnitude is interpretable on an approximately Gaussian significance scale. Within the scope described by Jow et al. and Mazzei et al., its main advantages are continuous use of the full angular information, compatibility with weighting, and superior power relative to binning-based alternatives when the target alternative is known a priori (Jow et al., 2017).