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AlekM

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- Meaning of each member being a unit vector, and how the products of each tensor can be averaged.

**Summary:**Meaning of each member being a unit vector, and how the products of each tensor can be averaged.

Hello!

I am struggling with understanding the meaning of "each member is a unit vector":

Let n^{(1)}, n^{(2)}, and n^{(N)}be observed directional data, where each member is a unit vector. The most fundamental quantities are various averages of them. Since we are trying to seek tensor quantities to characterize the data distribution, we first consider the average of their tensor product, or the “moment”, and put

N_{i1i2...in}= ⟨n_{i1}n_{i2}...n_{in}⟩

I can see that N would represent the number of samples, and the pointy bracket represents an average:

Average of i^{th}component of a = ⟨a_{i}⟩ = (a_{i}^{1}+a_{i}^{2}+a_{i}^{N})/N

However the notation of the averaged tensors in the text is n

_{i2}...n

_{in}, which leads me to wonder what i and n represent. It states that each member (of n?) is a unit vector, does that mean that each value of n is a normalized vector representing a direction? Or could a single value of n contain multiple vectors of directional data?

The paper later states that they adopt the summation convention over tensor indices, would this mean that something like i represents the axes that make up each vector (such as x,y,z in Cartesian) and 1 through n represents the axes that make up each vector? In that case it would seem that the directional vectors from each sample are averaged together. I'm not sure if I am on the right track or entirely incorrect.

Thanks in advance for any help,

Alek