11 方向、旋转和超球面
Directions, Rotations, and Hyperspheres
本节译者：张梓源
Directional statistics involve data and/or parameters that are constrained to be directions. The set of directions forms a sphere, the geometry of which is not smoothly mappable to that of a Euclidean space because you can move around a sphere and come back to where you started. This is why it is impossible to make a map of the globe on a flat piece of paper where all points that are close to each other on the globe are close to each other on the flat map. The fundamental problem is easy to visualize in two dimensions, because as you move around a circle, you wind up back where you started. In other words, 0 degrees and 360 degrees (equivalently, 0 and \(2 \pi\) radians) pick out the same point, and the distance between 359 degrees and 2 degrees is the same as the distance between 137 and 140 degrees.
方向统计涉及到的数据和参数被限制为方向。这些方向的集合形成了一个球体，这个球体的几何结构无法平滑地映射到欧几里得空间。这是因为在球面上不断移动可以返回到起点。因此，想要在一张平面纸上制作一个球体的地图，使球体上彼此接近的点在平面地图上也彼此接近，是行不通的。这个基本问题在二维空间中很容易可视化。想象你沿着一个圆周移动，最终会回到起点。换句话说，0度和360度（等同于弧度值中的0和 \(2 \pi\) 弧度）指的是同一个点，359度和2度之间的距离与137度和140度之间的距离是一样的。
Stan supports directional statistics by providing a unit-vector data type, the values of which determine points on a hypersphere (circle in two dimensions, sphere in three dimensions).
Stan 通过提供一种单位向量的数据类型来支持对方向数据的统计分析，这种数据类型的值确定超球面上的点（在二维空间里是圆周上的点，在三维空间里则是球面上的点）。
11.1 Unit vectors
单位向量
The length of a vector \(x \in \mathbb{R}^K\) is given by
向量 \(x \in \mathbb{R}^K\) 的长度，可通过以下公式获得
\[ \Vert x \Vert = \sqrt{x^{\top}\,x} = \sqrt{x_1^2 + x_2^2 + \cdots + x_K^2}. \] Unit vectors are defined to be vectors of unit length (i.e., length one).
单位向量是具有单位长度的向量（即长度为一）。
With a variable declaration such as
变量声明如下
unit_vector[K] x;the value of x will be constrained to be a vector of size K with unit length; the reference manual chapter on constrained parameter transforms provides precise definitions.
变量 x 是一个长度为1，维度为 K 的向量。具体的定义可在参考手册的受限参数变换章节中找到。
Warning: An extra term gets added to the log density to ensure the distribution on unit vectors is proper. This is not a problem in practice, but it may lead to misunderstandings of the target log density output (lp__ in some interfaces). The underlying source of the problem is that a unit vector of size \(K\) has only \(K - 1\) degrees of freedom. But there is no way to map those \(K - 1\) degrees of freedom continuously to \(\mathbb{R}^N\)—for example, the circle can’t be mapped continuously to a line so the limits work out, nor can a sphere be mapped to a plane. A workaround is needed instead. Stan’s unit vector transform uses \(K\) unconstrained variables, then projects down to the unit hypersphere. Even though the hypersphere is compact, the result would be an improper distribution. To ensure the unit vector distribution is proper, each unconstrained variable is given a “Jacobian” adjustment equal to an independent standard normal distribution. Effectively, each dimension is drawn standard normal, then they are together projected down to the hypersphere to produce a unit vector. The result is a proper uniform distribution over the hypersphere.
注意： 为确保单位向量的分布是正确的，会在对数密度中添加一个额外的项。这在实践中通常可行，但这可能导致目标对数密度输出（某些接口中为 lp__）出错。问题的根源在于，一个维度为 \(K\) 的单位向量实际上只有 \(K - 1\) 个自由度，但是没有办法将这 \(K - 1\) 个自由度连续地映射到欧几里得空间 \(\mathbb{R}^N\) 中。例如，圆不能连续地映射到直线上同时保证极限可求，球体也不能映射到平面上。因此需要一种变通方法。Stan 的单位向量转换使用了 \(K\) 个无约束变量，然后将其投影到单位超球面上。即便超球面本身是一个紧致的集合，这样的变换结果仍将导致一个非正规分布。为确保单位向量分布的正确性，对每个无约束变量都做了一个等同于独立标准正态分布的“雅可比”调整。实际上，每个维度都是按照标准正态分布抽样，然后将每个维度上抽取到的标准正态分布的值投影到超球面上，从而得到一个单位向量。最后可以在超球面上得到正确的均匀分布。
11.2 Circles, spheres, and hyperspheres
圆、球体和超球体
An \(n\)-sphere, written \(S^{n}\), is defined as the set of \((n + 1)\)-dimensional unit vectors,
记 \(n\) 维球体为 \(S^{n}\) ，它是一组 \((n + 1)\) 维的单位向量的集合
\[ S^{n} = \left\{ x \in \mathbb{R}^{n+1} \: : \: \Vert x \Vert = 1 \right\}. \]
Even though \(S^n\) is made up of points in \((n+1)\) dimensions, it is only an \(n\)-dimensional manifold. For example, \(S^2\) is defined as a set of points in \(\mathbb{R}^3\), but each such point may be described uniquely by a latitude and longitude. Geometrically, the surface defined by \(S^2\) in \(\mathbb{R}^3\) behaves locally like a plane, i.e., \(\mathbb{R}^2\). However, the overall shape of \(S^2\) is not like a plane in that it is compact (i.e., there is a maximum distance between points). If you set off around the globe in a “straight line” (i.e., a geodesic), you wind up back where you started eventually; that is why the geodesics on the sphere (\(S^2\)) are called “great circles,” and why we need to use some clever representations to do circular or spherical statistics.
尽管 \(S^n\) 由 \((n+1)\) 维的点组成，它仍然只是一个 \(n\) 维流形。例如，\(S^2\) 被定义为 \(\mathbb{R}^3\) 中的一组点，但是每个这样的点都可以通过一个纬度和一个经度唯一地表示。从几何学角度，\(\mathbb{R}^3\) 中由 \(S^2\) 定义的表面在局部表现类似于平面，即 \(\mathbb{R}^2\)。然而，\(S^2\) 的整体形状与平面不同，它是紧致的（即点与点之间存在着最大距离）。如果沿着球体表面的“直线”（即测地线也就是球面上的最短路径）出发，最终会回到起点。这就是为什么在球面（\(S^2\)）上的测地线被称为“大圆”。因此我们需要使用一些巧妙的表示方法来进行圆形或球形统计。
Even though \(S^{n-1}\) behaves locally like \(\mathbb{R}^{n-1}\), there is no way to smoothly map between them. For example, because latitude and longitude work on a modular basis (wrapping at \(2\pi\) radians in natural units), they do not produce a smooth map.
尽管 \(S^{n-1}\) 与 \(\mathbb{R}^{n-1}\) 局部类似，但它们之间却无法进行平滑映射。例如，经度和纬度基于模运算（在自然单位中以 \(2\pi\) 弧度为周期），因此它们之间无法进行平滑映射。
Like a bounded interval \((a, b)\), in geometric terms, a sphere is compact in that the distance between any two points is bounded.
就像一个有界区间 \((a, b)\) ，在几何学上，一个球体是紧致的，因为任意两点之间的距离是有界的。
11.3 Transforming to unconstrained parameters
无约束参数变换
Stan (inverse) transforms arbitrary points in \(\mathbb{R}^{K+1}\) to points in \(S^K\) using the auxiliary variable approach of Muller (1959). A point \(y \in \mathbb{R}^K\) is transformed to a point \(x \in S^{K-1}\) by
Stan 使用 Muller (1959) 提出的辅助变量法，将 \(\mathbb{R}^{K+1}\) 中的任意点（逆）转换为 \(S^K\) 中的点。通过以下方式，点 \(y \in \mathbb{R}^K\) 转换为点 \(x \in S^{K-1}\)
\[ x = \frac{y}{\sqrt{y^{\top} y}}. \]
The problem with this mapping is that it’s many to one; any point lying on a vector out of the origin is projected to the same point on the surface of the sphere. Muller (1959) introduced an auxiliary variable interpretation of this mapping that provides the desired properties of uniformity; the reference manual contains the precise definitions used in the chapter on constrained parameter transforms.
这种映射的问题在于它是多对一的。任何从原点出发的向量上的点，都会被投影到球面上的同一点。 Muller (1959) 对这一映射引入了辅助变量解释，它提供了期望具有的均匀性特性，参考手册的受限参数变换章节中有精确的定义。
Warning: undefined at zero!
在零点处无定义！
The above mapping from \(\mathbb{R}^n\) to \(S^n\) is not defined at zero. While this point outcome has measure zero during sampling, and may thus be ignored, it is the default initialization point and thus unit vector parameters cannot be initialized at zero. A simple workaround is to initialize from a small interval around zero, which is an option built into all of the Stan interfaces.
上述从 \(\mathbb{R}^n\) 到 \(S^n\) 的映射在零点处无定义。虽然在抽样过程中这一点的测度为零，可以忽略不计，但它是默认的初始化点，因此单位向量参数不能在零点初始化。一个简单的解决方法是从零点周围的一个小区间进行初始化，这是 Stan 所有接口内置的一个选项。
11.4 Unit vectors and rotations
单位向量和旋转
Unit vectors correspond directly to angles and thus to rotations. This is easy to see in two dimensions, where a point on a circle determines a compass direction, or equivalently, an angle \(\theta\). Given an angle \(\theta\), a matrix can be defined, the pre-multiplication by which rotates a point by an angle of \(\theta\). For angle \(\theta\) (in two dimensions), the \(2 \times 2\) rotation matrix is defined by
单位向量直接与角度相对应，因此也对应旋转。这一点在二维空间中尤为直观，圆上的一个点对应罗盘上的一个方向（或等价于一个角度 \(\theta\)）。给定一个角度 \(\theta\)，可定义一个矩阵，通过该矩阵的左乘（前乘），可以将一个点旋转 \(\theta\) 角度。二维情况下，角度 \(\theta\) 的 \(2 \times 2\) 旋转矩阵定义如下：
\[ R_{\theta} = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}. \] Given a two-dimensional vector \(x\), \(R_{\theta} \, x\) is the rotation of \(x\) (around the origin) by \(\theta\) degrees.
给定一个二维向量 \(x\)，\(R_{\theta} \, x\) 表示将 \(x\)（绕原点）旋转 \(\theta\) 度后的结果。
Angles from unit vectors
根据单位向量计算角度
Angles can be calculated from unit vectors. For example, a random variable theta representing an angle in \((-\pi, \pi)\) radians can be declared as a two-dimensional unit vector then transformed to an angle.
可以通过单位向量计算得出角度。例如，可以声明一个表示 \((-\pi, \pi)\) 弧度范围内角度的随机变量 theta 为一个二维单位向量，然后将其转换为一个角度。
parameters {
unit_vector[2] xy;
}
transformed parameters {
real<lower=-pi(), upper=pi()> theta = atan2(xy[2], xy[1]);
}If the distribution of \((x, y)\) is uniform over a circle, then the distribution of \(\arctan \frac{y}{x}\) is uniform over \((-\pi, \pi)\).
如果 \((x, y)\) 在圆上是均匀分布的，那么 \(\arctan \frac{y}{x}\) 在 \((-\pi, \pi)\) 上也是均匀分布的。
It might be tempting to try to just declare theta directly as a parameter with the lower and upper bound constraint as given above. The drawback to this approach is that the values \(-\pi\) and \(\pi\) are at \(-\infty\) and \(\infty\) on the unconstrained scale, which can produce multimodal posterior distributions when the true distribution on the circle is unimodal.
直接将 theta 作为参数声明，并给它设置上述的上下界限制，这样做看似很诱人。但是它的缺点是，\(-\pi\) 和 \(\pi\) 的值在无约束尺度上分别为 \(-\infty\) 和 \(\infty\)，当圆上的真实分布是单峰分布时，可能会导致具有多峰性的后验分布。
With a little additional work on the trigonometric front, the same conversion back to angles may be accomplished in more dimensions.
再通过一些三角函数的处理，就可以在更高维空间中实现类似的角度转换。
11.5 Circular representations of days and years
日与年的循环表示法
A 24-hour clock naturally represents the progression of time through the day, moving from midnight to noon and back again in one rotation. A point on a circle divided into 24 hours is thus a natural representation for the time of day. Similarly, years cycle through the seasons and return to the season from which they started.
24小时制时钟自然地表示了一天中时间的流逝，从午夜到正午，再从正午到午夜。因此，被划分为24小时的圆上的某个点，自然而然地可以代表一天中某个时间。同样地，年份通过季节循环，最终又回到它们开始的季节。
In human affairs, temporal effects often arise by convention. These can be modeled directly with ad-hoc predictors for holidays and weekends, or with data normalization back to natural scales for daylight savings time.
在人类社会中，时间规律往往是约定俗成的。这些可以通过为假期和周末设置特设的预测因子直接建模，或者为了适应夏令时制，对数据进行标准化调整，使其回归到正常的时间标准。