1 Introduction
Data in the form of three-dimensional rotations have applications in many scientific areas, such as bio-medical engineering, computer vision, and geological and materials sciences where such data represent the positions of objects within a three-dimensional reference frame. For example, (Humbert et al. 1996), (Bingham et al. 2009) and (Bachmann et al. 2010) apply rotation data to study the orientation of cubic crystals on the surfaces of metal. (Rancourt et al. 2000) use rotations to represent variations in human movement while performing a task.
A common goal shared in the analysis of rotation data across all fields is to estimate the main or central orientation for a sample of rotations. More formally, let \(SO(3)\) denote the rotation group, which consists of all real-valued \(3\times 3\) matrices \(\mathbf R\) with determinant equal to +1. Then observations \(\mathbf{R}_1,\ldots,\mathbf{R}_n \in SO(3)\) can be conceptualized as a random sample from a location model
\[\label{eq:locmodel} \mathbf{R}_i = \mathbf{S} \mathbf{E}_i, \quad i=1,\ldots,n, \tag{1} \]
where \(\mathbf S \in SO(3)\) is the fixed parameter of interest indicating the central orientation, and \(\mathbf{E}_1,\ldots,\mathbf{E}_n \in SO(3)\) denote i.i.d. random rotations which symmetrically perturb \(\mathbf{S}\). Model (1) is a rotation-matrix analog of a location model for scalar data \(Y_i = \mu + e_i\), where \(\mu \in \mathbb{R}\) denotes a mean and \(e_i \in \mathbb{R}\) denotes an additive error symmetrically distributed around zero.
Assuming the perturbations \(\mathbf E_i\) symmetrically perturb \(\mathbf S\) implies that the observations \(\mathbf R_i\) have no preferred direction relative to \(\mathbf S\) and that \(E\left(\mathbf R_i\right)=c\mathbf S\) for some \(c\in\mathbb{R}^+\) for all \(i\). Also note that under the symmetry assumption, (1) could be equivalently specified as \(\mathbf R_i=\mathbf E_i\mathbf S\), though the form given in (1) is the most common form in the literature (see Bingham et al. (2009) for details).
While there is a multitude of packages and functions available in R to estimate the mean in a location model, the toolbox for rotational data is limited. The orientlib (Murdoch 2003) package includes the definition of an orientation class along with a few methods to summarize and visualize rotation data. A strength of the orientlib package is its thorough exploration of rotation representations, but the estimation and visualization techniques are lacking and no methods for inference are available. The onion (Hankin 2011) package includes functions for rotation algebra but only the quaternion form is available and data analysis is not possible. The uarsbayes (Qiu 2013) package includes functions for data generation and Bayes inference but this package is currently not publicly available. Packages for circular and spherical data, e.g. circular (Agostinelli and Lund 2013) and SpherWave (Oh and Kim 2013), can possibly be used but their extension to rotation data is not straightforward.
The rotations
(Stanfill et al. 2014a) package fills this void by providing users with the
tools necessary to simulate rotations from (1) with four
distribution choices for the perturbation matrices \(\mathbf E_i\).
Estimation and inference for \(\mathbf{S}\) in (1) is
available along with two visualization techniques. The remainder of this
manuscript introduces rotation data more fully and discusses the ways
they are handled by the rotations package. For the latest on this
package as well as a full list of available functions, see
help(package = "rotations").
2 Rotation parameterizations
Several parameterizations of rotations exist. We consider two of the most commonly used: orthogonal \(3\times 3\) matrices with determinant one and four-dimensional unit vectors called quaternions. The rotations package allows for both parameterizations as input as well as transforming one into the other. We will briefly discuss each:
Matrix form
Rotations in three-dimensions can be represented by \(3\times3\) orthogonal matrices with determinant one. Matrices with these characteristics form a group called the special orthogonal group, or rotation group, denoted \(SO(3)\). Every element in \(SO(3)\) is associated with a skew-symmetric matrix \(\mathbf\Phi\left(\mathbf W\right)\) where \[\mathbf{\Phi}\left(\mathbf{W}\right) = \left(\begin{array}{ccc}0 & -w_3 & w_2 \\ w_3 & 0 & -w_1 \\-w_2 & w_1 & 0\end{array}\right)\] and \(\mathbf W\in\mathbb{R}^3\). Applying the exponential operator to the matrix \(\mathbf\Phi\left(\mathbf W\right)\) results in the rotation \(\mathbf R\)
\[\label{eq:expW} \mathbf R=\exp\left[\mathbf{\Phi}\left(\mathbf{W}\right)\right] = \sum\limits_{k=0}^\infty \frac{\left[\mathbf{\Phi}\left(\mathbf{W}\right)\right]^k}{k!}. \tag{2} \]
Since \(\mathbf\Phi\left(\mathbf W\right)\) is skew-symmetric, it can be shown that (2) reduces to
\[\label{eq:angleAxis} \mathbf R =\cos(r)\mathbf{I}_{3\times3} + \sin(r) \mathbf{\Phi}\left(\mathbf{U}\right) + [1-\cos (r)] \mathbf{U} \mathbf{U}^\top, \tag{3} \]
where \(r=\|\mathbf{W}\|\), \(\mathbf{U} =\mathbf{W}/\|\mathbf{W}\|\). In the material sciences literature \(r\) and \(\mathbf U\in\mathbb{R}^3\) are termed the misorientation angle and misorientation axis, respectively.
Given a rotation matrix \(\mathbf R\) one can find the associated skew-symmetric matrix \(\mathbf{\Phi}\left(\mathbf W\right)\) by applying the logarithm operator defined by
\[\label{eq:matLog} \text{Log}\left(\mathbf R\right)=\begin{cases} \mathbf 0 &\text{if } \theta=0\\ \frac{r}{2\sin r}\left(\mathbf R-\mathbf R^\top\right)&\text{otherwise,} \end{cases} \tag{4} \]
where \(r\in[-\pi,\pi)\) satisfies \(\text{tr}\left(\mathbf R\right)=1+2\cos r\) and \(\mathbf{tr}(\cdot)\) denotes the trace of a matrix. For more on the correspondence between \(SO(3)\) and skew-symmetric matrices see (Stanfill et al. 2013).
The rotations package defines the S3 class "SO3", which internally
stores a sample of \(n\) rotations as a \(n\times9\) matrix. If \(n=1\) then
an object of class "SO3" is printed as a \(3\times3\) matrix but for
\(n>1\) the \(n\times9\) matrix is printed. Objects can be coerced into, or
tested for the class "SO3" with the as.SO3 and is.SO3 functions,
respectively. Any object passed to is.SO3 is tested for three
characteristics: dimensionality, orthogonality and determinant one.
The as.SO3 function coerces the input into the class "SO3". There
are three types of input supported by the as.SO3 function. Given a
singe angle \(r\) and axis \(\mathbf U\), as.SO3 will form a rotation
matrix according to (3). Equivalently one could supply a
three-dimensional vector \(\mathbf W\), then the length of that vector
will be taken to be the angle of rotation \(r=\|\mathbf W\|\) and the axis
is taken to be the unit-vector in the direction of \(\mathbf W\),
i.e. \(\mathbf U=\mathbf W/\|\mathbf W\|\). One can also supply a rotation
\(\mathbf Q\) in the quaternion representation. The as.SO3 function will
return the matrix equivalent of \(\mathbf Q\). For all input types the
function as.SO3 returns an \(n\times9\) matrix of class "SO3" where
each row corresponds to a rotation matrix. Below we illustrate the use
of the as.SO3 function by constructing the \(3\times3\) matrix
associated with a \(90^\circ\) rotation about the \(y\)-axis, i.e. \(r=\pi/2\)
and \(\mathbf U=(0,1,0)\). In this example and all that follow, we have
rounded the output to three digits for compactness.
> r <- pi/2
> U <- c(0, 1, 0)
> W <- U * r
> R <- as.SO3(W)
> R
[,1] [,2] [,3]
[1,] 0 0 1
[2,] 0 1 0
[3,] -1 0 0
> identical(R, as.SO3(U, r))
[1] TRUEGiven a rotation matrix \(\mathbf R\), the functions mis.angle and
mis.axis will determine the misorientation angle and axis of an object
with class "SO3" as illustrated in the next example.
> mis.angle(R) * 2/pi
[1] 1
> mis.axis(R)
[,1] [,2] [,3]
[1,] 0 1 0Quaternion form
Quaternions are unit vectors in \(\mathbb{R}^4\) that are commonly written as
\[\label{eq:quat} Q = x_1 + x_2 i + x_3 j + x_4 k, \tag{5} \]
where \(x_l\in[-1,1]\) for \(l=1,2,3,4\) and \(i^2 = j^2= k^2 =ijk= -1\). We can write \(\mathbf Q=\left(s,\mathbf V\right)\) as tuple of the scalar \(s\) for coefficient \(\mathbf 1\) and vector \(\mathbf V\) for the remaining coefficients, i.e. \(s=x_1\) and \(\mathbf V= (x_2, x_3, x_4)\).
A rotation around axis \(\mathbf U\) by angle \(r\) translates to \(\mathbf Q=\left(s,\mathbf V\right)\) with \[s = \cos{(r/2)}, \ \ \mathbf V = \mathbf U \sin {(r/2)}.\] Note that rotations in quaternion form are over-parametrized: \(\mathbf Q\) and \(-\mathbf Q\) represent equivalent rotations. This ambiguity has no impact on the distributional models, parameter estimation or inference methods to follow. Hence, for consistency, the rotations package only generates quaternions satisfying \(x_1\geq0\). Data provided by the user does not need to satisfy this condition however.
The S3 class "Q4" is defined for the quaternion representation of
rotations. All the functionality of the "SO3" class also exists for
the "Q4" class, e.g. is.Q4 and as.Q4 will test for and coerce to
class "Q4", respectively. Internally, a sample of \(n\) quaternions is
stored in the form of a \(n\times4\) matrix with each row a unit vector.
Single quaternions are printed according to the representation in
(5) (see example below) while a sample of size \(n\) is printed
as a \(n\times 4\) matrix with column names Real, i, j and k to
distinguish between the four components.
The following code creates the same rotation from the previous section
in the form of a quaternion with the as.Q4 function. This function
works much the same way as the as.SO3 function in terms of possible
inputs but returns a vector of length four of the class "Q4".
> as.Q4(U, r)
0.707 + 0 * i + 0.707 * j + 0 * k
> as.Q4(as.SO3(U, r))
0.707 + 0 * i + 0.707 * j + 0 * k3 Data generation
If the rotation \(\mathbf{E}_i\in SO(3)\) from (1) has an axis \(\mathbf U\) that is uniformly distributed on the unit sphere and an angle \(r\) that is independently distributed about zero according to some symmetric distribution function then \(\mathbf E_i\) is said to belong to the uniform-axis random spin, or UARS, class of distributions. From (Bingham et al. 2009) the density for \(\mathbf E_i\) is given by
\[\label{eq:uarsden} f\left(\mathbf E_i|\kappa\right)=\frac{4\pi}{3-\text{tr}\left(\mathbf E_i\right)}C\left(\left.\text{acos}\left\{\frac{\text{tr}(\mathbf E_i)-1}{2}\right\}\right|\kappa\right), \tag{6} \]
where \(C(\cdot|\kappa)\) is the distribution function associated with the angle of rotation \(r\) with concentration parameter \(\kappa\). Members of the UARS family of distributions are differentiated based on the angular distribution \(C(\cdot|\kappa)\).
The rotations package gives the user access to four members of the
UARS class. Each member is differentiated by the distribution function
for \(r\): the uniform, the matrix Fisher
(Langevin 1905; Downs 1972; Khatri and Mardia 1977; Jupp and Mardia 1979), the Cayley
(Schaeben 1997; León et al. 2006) and the circular-von Mises distribution
(Bingham et al. 2009). Note: probability distribution functions on \(SO(3)\) such
as (6) are defined with respect to the Haar measure, which
we denote by \(\lambda\). That is, the expectation of a random rotation
\(\mathbf R\in SO(3)\) with corresponding misorientation angle \(r\) is
given by
\(E\left(\mathbf R\right)=\int_\Omega\mathbf Rf\left(\mathbf R|\kappa\right)\mathrm{d}\lambda\)
where \(\Omega=SO(3)\),
\(\mathrm{d} \lambda=[1-\cos(r)]\mathrm{d} r/(2\pi)\) and \(\mathrm{d} r\)
is the derivative of \(r\) with respect to the Lebesgue measure. Because
the Haar measure acts as the uniform measure on \(SO(3)\) and
\(\lambda\left(\Omega\right)=1\), then the angular distribution
\(C(r)=[1-\cos(r)]/(2\pi)\) is referred to as the uniform distribution for
misorientation angles \(r\) and has been included in the rotations
package under the name .haar (see Table 1).
The spread of the Cayley, matrix Fisher and circular-von Mises distributions is controlled by the concentration parameter \(\kappa\). Concentration is a distribution specific quantity and is not comparable across different distributions. To make comparisons across distributions possible we also allow for specification of the circular variance, which is defined as \(\nu=1-E[\cos(r)]\) where \(E[\cos(r)]\) is often referred to as the mean resultant length (Fisher 1996). The form of each angular distribution along with the circular variance as a function of the concentration parameter is given in Table 1.
| Name | Density \(C(r |\kappa)\) | Circular variance \(\nu\) | Function | |||
|---|---|---|---|---|---|---|
| Uniform | \(\frac{1-\cos(r)}{2\pi}\) | \(\frac{3}{2}\) | .haar |
|||
| Cayley | \(\frac{\Gamma(\kappa+2)(1+\cos r)^\kappa(1-\cos r)}{2^{(\kappa+1)}\sqrt{\pi}\Gamma(\kappa+1/2)}\) | \(\frac{3} {\kappa+2}\) | .cayley |
|||
| matrix Fisher | \(\frac{[1-\cos(r)]\exp[2\kappa \cos(r)]}{2\pi[\mathrm{I_0}(2\kappa)-\mathrm{I_1}(2\kappa)]}\) | \(\frac{3\mathrm{I}_0(2\kappa)-4\mathrm{I}_1(2\kappa)+\mathrm{I}_2(2\kappa)} {2[\mathrm{I}_0(2\kappa)-\mathrm{I}_1(2\kappa)]}\) | .fisher |
|||
| circular-von Mises | \(\frac{\exp[\kappa\cos(r)]}{2\pi \mathrm{I_0}(\kappa)}\) | \(\frac{\mathrm{I_0}(\kappa)-\mathrm{I_1}(\kappa)}{\mathrm{I_0}(\kappa)}\) | .vmises |
For a given concentration d, p and r take the same meaning as for
the more familiar distributions such as dnorm. To simulate a sample of
\(SO(3)\) data, the ruars function takes arguments n, rangle, and
kappa to specify the sample size, angular distribution and
concentration as shown below. Alternatively, one can specify the
circular variance \(\nu\). Circular variance is used in the event that
both circular variance and concentration are provided. The space
argument determines the parameterization to form. When a sample of
rotations is printed then a \(n\times 9\) matrix is printed with column
titles that specify which element of the matrix each column corresponds
to. For example, the \(\mathbf R_{\{1,1\}}\) element of a rotation matrix
is printed under the column heading R11 as illustrated below.
> Rs <- ruars(n = 20, rangle = rcayley, kappa = 1, space = "SO3")
> Qs <- ruars(n = 20, rangle = rcayley, kappa = 1, space = "Q4")
> Rs <- ruars(n = 20, rangle = rcayley, nu = 1, space = "SO3")
> Qs <- ruars(n = 20, rangle = rcayley, nu = 1, space = "Q4")
> head(Rs, 3)
R11 R21 R31 R12 R22 R32 R13 R23 R33
[1,] -0.425 -0.850 0.310 0.475 -0.501 -0.723 0.770 -0.160 0.617
[2,] -0.564 -0.733 0.379 0.745 -0.256 0.615 -0.354 0.630 0.691
[3,] 0.087 -0.716 0.692 0.117 0.698 0.707 -0.989 0.019 0.1454 Data analysis
In this section we present functions in the rotations package to compute point estimates and confidence regions for the central orientation \(\mathbf S\).
Estimation of central orientation
Given a sample of \(n\) observations \(\mathbf R_1,\dots,\mathbf R_{n}\) generated according to (1), the rotations package offers four built-in ways to estimate the central orientation \(\mathbf S\). These estimators are either Riemannian- or Euclidean-based in geometry and use either the \(L_1\)- or \(L_2\)- norm, i.e. they are median- or mean-type. We briefly discuss how the choice of geometry affects estimation of \(\mathbf S\).
The choice of geometry results in two different metrics to measure the distance between rotation matrices \(\mathbf{R}_1\) and \(\mathbf{R}_2 \in SO(3)\). The Euclidean distance, \(d_E\), between two rotations is defined by \[d_E\left(\mathbf{R}_1,\mathbf{R}_2\right)=\left\|\mathbf{R}_1-\mathbf{R}_2\right\|_F,\] where \(\|\mathbf{A}\|_F = \sqrt{\mathbf{tr}({\mathbf A^\top \mathbf A})}\) denotes the Frobenius norm. The Euclidean distance between two rotation matrices corresponds to the length of the shortest path in \(\mathbb{R}^{3\times3}\) that connects them and is therefore an extrinsic distance metric.
Estimators based on the Euclidean distance form the class of projected
estimators. The name is derived from the method used to compute these
estimators. That is, each estimator in this class is the projection of
the the generic \(3\times 3\) matrix that minimizes the loss function into
\(SO(3)\). For an object with class "SO3" the median or mean
function with argument type = "projected" will return a \(3\times 3\)
matrix in \(SO(3)\) that minimizes the first- or second-order loss
function, respectively.
By staying in the Riemannian space \(SO(3)\) the natural distance metric becomes the Riemannian (or geodesic) distance, \(d_R\), which for two rotations \(\mathbf{R}_1,\mathbf{R}_2\in SO(3)\) is defined as \[d_R\left(\mathbf{R}_1,\mathbf{R}_2\right)= \frac{1}{\sqrt{2}}\left\|\text{Log}\left(\mathbf{R}_1^\top\mathbf{R}_2\right)\right\|_F = |r|,\] where \(\text{Log}(\mathbf{R})\) denotes the logarithm of \(\mathbf{R}\) defined in (4) and \(r\in[-\pi,\pi)\) is the misorientation angle of \(\mathbf{R}_1^\top \mathbf{R}_2\). The Riemannian distance corresponds to the length of the shortest path that connects \(\mathbf{R}_1\) and \(\mathbf{R}_2\) within the space \(SO(3)\) and is therefore an intrinsic distance metric. For this reason, the Riemannian distance is often considered the more natural metric on \(SO(3)\). As demonstrated in (Stanfill et al. 2013), the Euclidean and Riemannian distances are related by \(d_E(\mathbf R_1,\mathbf R_2)=2\sqrt{2}\sin\left[d_R(\mathbf R_1,\mathbf R_2)/2\right]\).
Estimators based on the Riemannian distance metric are called geometric
estimators because they preserve the geometry of \(SO(3)\). These can be
computed using the mean and median functions with the argument
type = "geometric". Table 2 summarizes the four
estimators including their formal definition and how they can be
computed.
The estimators in Table 2 find estimates based on
minimization of \(L_1\)- and \(L_2\)-norms in the chosen geometry. The
function gradient.search provides the option to optimize for any other
arbitrary minimization criterion. As the name suggests, the minimization
is done along the gradient of the minimization function in the rotation
space. Starting from an initial, user-specified rotation, the algorithm
finds a (local) minimum by stepping iteratively in the direction of the
steepest descent. Step size is regulated internally by adjusting for
curvature of the minimization function.
We highlight this process in the example below. The function L1.error
is defined to minimize the intrinsic \(L_1\)-norm, the result from the
optimization should therefore agree with the geometric median of the
sample. In fact, the difference between the two results is at the same
level as the minimal difference (minerr) used for convergence of the
gradient search. What is gained in flexibility of the optimization is,
of course, paid for in terms of speed: the built-in median function is
faster by far than the gradient search.
Also illustrated in the example below is the rot.dist function, which
computes the distance between two objects of class "SO3", e.g. R1
and R2. The argument method specifies which type of distance to
compute: the "extrinsic" option will return the Euclidean distance and
the "intrinsic" option will return the Riemannian distance. If R1 is
an \(n\times9\) matrix representing a sample of rotations, then rot.dist
will return a vector of length \(n\) where the \(i\)th element represents
the specified distance between R2 and the \(i\)th row of R1.
> # error function definition
> L1.error <- function(sample, Shat) {
+ sum(rot.dist(sample, Shat, method = "intrinsic", p = 1))
+ }
> cayley.sample <- ruars(n = 50, rangle = rcayley, nu = 1, space = "SO3")
> # gradient based optimization
> system.time(SL1 <- gradient.search(cayley.sample, L1.error))
user system elapsed
3.464 0.007 3.473
> # in-built function
> system.time(S <- median(cayley.sample, type = "geometric"))
user system elapsed
0.004 0.000 0.005
> rot.dist(S, SL1$Shat)
[1] 1.492e-05| Estimator name | Definition | Code | ||||
|---|---|---|---|---|---|---|
| Projected Mean | \(\widehat{\mathbf{ S}}_E=\underset{\mathbf S\in SO(3)}{\text{argmin}}\sum\limits_{i=1}^n d_E^2(\mathbf S,\mathbf R_i)\) | mean(Rs, type = "projected") |
||||
| Projected Median | \(\widetilde{\mathbf{ S}}_E=\underset{\mathbf S\in SO(3)}{\text{argmin}}\sum\limits_{i=1}^nd_E(\mathbf S,\mathbf R_i)\) | median(Rs, type = "projected") |
||||
| Geometric Mean | \(\widehat{\mathbf{ S}}_R=\underset{\mathbf S\in SO(3)}{\text{argmin}}\sum\limits_{i=1}^n d_R^2(\mathbf S,\mathbf R_i)\) | mean(Rs, type = "geometric") |
||||
| Geometric Median | \(\widetilde{\mathbf{ S}}_R=\underset{\mathbf S\in SO(3)}{\text{argmin}}\sum\limits_{i=1}^nd_R(\mathbf S,\mathbf R_i)\) | median(Rs, type = "geometric") |
Confidence regions
Asymptotic results for the distribution of the projected mean \(\widehat{\mathbf{ S}}_E\) and median \(\widetilde{\mathbf{ S}}_E\) can be used to construct confidence regions for the central orientation \(\mathbf S\). In the literature two approaches are available to justify the limiting distribution of the vector in \(\mathbb{R}^3\) associated with the centered estimator through (2). More specifically, the vector \(\sqrt{n}\widehat{\mathbf h}\) has been shown to have a trivariate normal distribution where \(\widehat{\mathbf h}\in\mathbb{R}^3\) satisfies \[\exp\left[\mathbf{\Phi}\left(\widehat{\mathbf h}\right)\right]=\mathbf S^\top\widehat{\mathbf{ S}}_E.\] The first approach transforms a result from directional statistics while the second uses \(M\)-estimation theory in \(SO(3)\) directly. A summary of these methods is given next.
In the context of directional statistics, (Prentice 1984) used results found in (Tyler 1981) and the fact that \(\widehat{\mathbf{ S}}_E\) is a function of the spectral decomposition of \(\overline{\mathbf R}=\sum_{i=1}^n\mathbf{R}_i/n\) in order to justify a multivariate normal limiting distribution for the scaled vector \(\sqrt{n}\,\widehat{\mathbf h}\). Unsatisfied with the coverage rate achieved by (Prentice 1986), (Fisher et al. 1996) proposed a pivotal bootstrap procedure that results in coverage rates closer to the nominal level for small samples. A transformation from unit vectors in \(\mathbb{R}^d\) to rotation matrices is required in order to apply the results of (Prentice 1984) and (Fisher et al. 1996) to \(SO(3)\), therefore they are called transformation-based. The projected median \(\widetilde{\mathbf{ S}}_E\) cannot be expressed as a function of the sample spectral decomposition, therefore this approach cannot be used to create confidence regions based on \(\widetilde{\mathbf{ S}}_E\).
It has also been shown that both estimators \(\widehat{\mathbf{ S}}_E\) and \(\widetilde{\mathbf{ S}}_E\) are \(M\)-estimators, which motivates a direct approach to confidence region estimation in \(SO(3)\) (Chang and Rivest 2001). In (Stanfill et al. 2014b), a pivotal bootstrap method based on the direct approach was implemented to improve coverage rates in small samples. Because the results in (Chang and Rivest 2001) and (Stanfill et al. 2014b) deal with \(SO(3)\) data directly, this approach is called direct.
The six possible confidence regions that result from these two methods
are available through the wrapper function region. They are
differentiated based on the method, type and estimator arguments.
Set estimator = "mean" or estimator = "median" to estimate a region
based on \(\widehat{\mathbf{ S}}_E\) or \(\widetilde{\mathbf{ S}}_E\),
respectively. For \(\widehat{\mathbf{ S}}_E\) one can choose
method = "transformation" for the transformation-based methods or
method = "direct" for the direct method. Since the
transformation-based methods cannot be applied to
\(\widetilde{\mathbf{ S}}_E\) an error is returned if
estimator = "median" and method = "transformation" are combined. A
bootstrap version of the specified method is implemented if
type = "bootstrap" or the normal limiting distribution can be chosen
with type = "asymptotic". If a bootstrap type region is specified one
can additionally specify the bootstrap sample size with the m
argument, which is set to 300 by default. Regardless of the method and
type chosen a single value is returned on the interval \((0,\pi]\). This
value corresponds to the radius of the confidence region centered at
each of the axes of the specified estimator.
In the example code below a sample of \(n=50\) rotations are drawn from the Cayley-UARS(\(\mathbf I_{3\times 3}, \kappa=10\)) distribution then the four types of confidence regions based on the direct approach are constructed. For a graphical representation of this dataset along with an interpretation of the confidence regions see Figure 1.
> Rs <- ruars(50, rcayley, kappa = 10)
> region(Rs, method = "direct", type = "asymptotic",
+ estimator = "mean", alp = 0.05)
[1] 0.189
> region(Rs, method = "direct", type = "bootstrap", estimator = "mean",
+ alp = 0.05, m = 300)
[1] 0.201
> region(Rs, method = "direct", type = "asymptotic",
+ estimator = "median", alp = 0.05)
[1] 0.201
> region(Rs, method = "direct", type = "bootstrap", estimator = "median",
+ alp = 0.05, m = 300)
[1] 0.2495 Visualizations
The rotations package offers two methods to visualize rotation data in three-dimensions. Because rotation matrices are orthogonal, each column of a rotation matrix has length one and is perpendicular to the other axes. Therefore each column of a rotation matrix can be illustrated as a point on the surface of a unit sphere, which represents the position of the \(x\)-, \(y\)- or \(z\)-axis for that rotation matrix. Since each sphere represents one of the three axes, three spheres are required to fully visualize a sample of rotations. Though the use of separate spheres to represent each axis can be seen as a disadvantage, the proposed visualization method makes the idea of a central orientation and a confidence region interpretable.
An existing function that can be used to illustrate rotation data is the
boat3d function included in the orientlib package. Given a sample of
rotations, the boat3d function produces either a static or interactive
three-dimensional boat to represent the provided data. If only one
rotation is of interest, the boat3d function is superior to the
proposed method because it conveniently illustrates rotational data in a
single image. If multiple rotations are provided, however, the boat3d
function will produce separate side-by-side boats, which can be hard to
interpret. In addition, the illustration of a estimated central
orientation or a confidence region in \(SO(3)\) with the boat3d function
is not presently possible.
The rotations package can be used to produce high-quality static plots within the framework of the ggplot2 package (Wickham 2009). Static plots are specifically designed for datasets that are highly concentrated and for use in presentations or publications. Alternatively, the rotations package can produce interactive plots using functions included in the sphereplot package (Robotham 2013). Interactive plots are designed so that the user can explore a dataset and visualize a diffuse sample.
Calling the plot function with a "SO3" or "Q4" object will result
in an interactive or static sphere, differentiated by setting the
argument interactive to TRUE or FALSE, respectively. The center
argument defines the center of the plot and is usually set to the
identity rotation id.SO3 or an estimate of the central orientation,
e.g. mean(Rs). The user can specify which columns to visualize with
the col argument with options \(1\), \(2\) and \(3\) representing the \(x\)-,
\(y\)- and \(z\)- axes, respectively. For static plots, multiple axes can be
displayed simultaneously by supplying a vector to col; only one column
will be displayed at a time for interactive plots. Also available to
static plots is the argument to\(\_\)range, which when set to TRUE
will display the portion of the sphere where the observations are
present.
All four estimates of the central orientation can be plotted along with
a sample of rotations. Setting the argument
estimates\(\_\)show = "all" will display all four simultaneously. If
only a few estimates are of interest then any combination of
"proj.mean", "proj.median", "geom.mean" or "geom.median" are
valid inputs. The estimators are indicated by color and a legend is
provided, see Figure 1. Finally, the mean\(\_\)regions
and median\(\_\)regions options allow the user to draw a circle on the
surface of the sphere representing the confidence region for that axis,
centered at \(\widehat{\mathbf{ S}}_E\) and \(\widetilde{\mathbf{ S}}_E\)
respectively. If estimators are plotted along with the different regions
in static plots then shapes represent the estimators and colors
represent the region methods, see Figure 1, while
regions and estimators are always distinguished by colors for the
interactive plots. Given the sample of rotations generated in a previous
example, the example below illustrates how to produce static plots using
the plot function for objects of class "SO3" and Figure
1 illustrates the results of these commands.
> plot(Rs, center = mean(Rs), col = 1, show_estimates = "all",
+ interactive = FALSE)
> plot(Rs, center = mean(Rs), col = 1, show_estimates = "proj.mean",
+ mean_regions = "all", alp = .05, interactive = FALSE)
|
|
| (a)Point estimates, static. | (b)Confidence region estimates, static. |
6 Datasets
Datasets drill and nickel are included in the rotations package to
illustrate how the two representations of orientation data discussed
here are used in practice. The drill dataset was collected to assess
variation in human movement while performing a task (Rancourt 1995).
Eight subjects drilled into a metal plate while being monitored by
infrared cameras. Quaternions are used to represent the orientation of
each subjects’ wrist, elbow and shoulder in one of six positions. For
some subjects several replicates are available. See (Rancourt et al. 2000) for
one approach to analyzing these data. In the example below we load the
drill dataset, coerce the observations for subject one’s wrist into a
form usable by the rotations package via as.Q4, then estimate the
central orientation with the projected mean.
> data(drill)
> head(drill)
Subject Joint Position Replicate Q1 Q2 Q3 Q4
1 1 Wrist 1 1 0.944 -0.192 -0.156 0.217
2 1 Wrist 1 2 0.974 -0.120 -0.111 0.158
3 1 Wrist 1 3 0.965 -0.133 -0.141 0.177
4 1 Wrist 1 4 0.956 -0.134 -0.115 0.233
5 1 Wrist 1 5 0.953 -0.199 -0.061 0.222
6 1 Wrist 2 1 0.963 -0.159 -0.127 0.177
> Subj1Wrist<-subset(drill, Subject == '1' & Joint == 'Wrist')
> Subj1Wdata <- as.Q4(Subj1Wrist[, 5:8])
> mean(Subj1Wdata)
0.987 - 0.070 * i - 0.134 * j + 0.049 * kIn the nickel dataset, rotation matrices are used to represent the
orientation of cubic crystals on the surface of a nickel sample measured
with Electron Backscatter Diffraction. Each location on the surface of
the nickel is identified by the xpos and ypos columns while the
rep column identifies which of the fourteen replicate scans that
measurement corresponds to. The last nine columns, denoted v1-v9,
represent the elements of the rotation matrix at that location in vector
form. See (Bingham et al. 2009; Bingham et al. 2010) and (Stanfill et al. 2013) for more
details. In the example below we estimate the central orientation at
location one.
> data(nickel)
> head(nickel[, 1:6])
xpos ypos location rep V1 V2
1 0 0.346 1 1 -0.648 0.686
2 0 0.346 1 2 -0.645 0.688
3 0 0.346 1 3 -0.645 0.688
4 0 0.346 1 4 -0.646 0.688
5 0 0.346 1 5 -0.646 0.686
6 0 0.346 1 6 -0.644 0.690
> Location1<-subset(nickel, location == 1)
> Loc1data<-as.SO3(Location1[, 5:13])
> mean(Loc1data)
[,1] [,2] [,3]
[1,] -0.645 -0.286 -0.708
[2,] 0.687 -0.623 -0.374
[3,] -0.334 -0.728 0.5997 Summary
In this manuscript we introduced the rotations package and
demonstrated how it can be used to generate, analyze and visualize
rotation data. The rotations package is compatible with the quaternion
specific onion package by applying its as.quaternion function to a
transposed "Q4" object. Connecting to the onion package gives the
user access to a wide range of algebraic functions unique to
quaternions. Also compatible with the rotations package is the
orientlib package, which includes additional parameterizations of
rotations. To translate rotation matrices generated by the rotations
package into a form usable by the orientlib package, first coerce a
"SO3" object into a matrix of the same dimension, i.e. \(n\times9\),
then apply the rotvector function provided by the orientlib package.
Quaternions are defined in the orientlib package by
\(\mathbf Q = x_1 i + x_2 j + x_3 k + x_4\), cf. (5), which may
lead to confusion when translating quaternions between the orientlib
package and either of the onion or rotations packages. Below is a
demonstration of how quaternions and rotation matrices generated by the
rotations package can be translated into a form usable by the onion
and orientlib packages, respectively. See help(package = "onion")
and help(package = "orientlib") for more on these packages.
> Qs <- ruars(20, rcayley, space = 'Q4')
> Rs <- as.SO3(Qs)
> suppressMessages(require(onion))
> onionQs <- as.quaternion(t(Qs))
> suppressMessages(require(orientlib))
> orientRs <- rotvector(matrix(Rs, ncol = 9))Computational speed of the rotations package has been enhanced through use of the Rcpp and RcppArmadillo packages (Eddelbuettel 2013; Eddelbuettel and Sanderson 2014). In future versions of the package we plan to extend the parameterization and estimator sections to include robust estimators currently being developed by the authors.
8 Acknowledgements
We would like to thank the reviewers for their comments and suggestions. The rotations package and this article have benefited greatly from their time and effort.
CRAN packages used
orientlib, onion, circular, SpherWave, rotations, ggplot2, sphereplot, Rcpp, RcppArmadillo
CRAN Task Views implied by cited packages
Distributions, Environmetrics, HighPerformanceComputing, NumericalMathematics, Phylogenetics, Spatial, TeachingStatistics
Note
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