Is a mistake in the slide 9?

[vpozdnyakov]

@inakleinbottle thanks for your open talk on Signature Methods for Time Series Data in PyData, London, 2022. I've bit confused in the slide 9 "I don't understand, show me an example". I've looked your explanation many times but I'm still not sure about the last line, which you call a "tensor product" of exp(I1) and exp(I2). Let me show my understanding how it was derived.

• term 1 is the multiplication of the 1st elements in level 0, thats is 1 * 1 = 1
• term 2 is the summation of the 1st elements in level 1, that is 1 + 3
• term 3 is the summation of the 2nd elements in level 1, that is 2 + 4
• term 4 is the summation of the 1st elements of level 2 and multiplication of two elements of level 1 (1st from 1st vector and 1st from 2nd vector), that is 0.5 + 1 * 3 + 4.5 = 0.5 + 3 + 4.5
• term 5 is the summation of the 2nd elements of level 2 and multiplication of two elements of level 1 (1st from 1st vector and 2nd from 2nd vector), that is 1 + 1 * 4 + 6 = 1 + 4 + 6
• term 6 is the summation of the 3rd elements of level 2 and multiplication of two elements of level 1 (2nd from 1st vector and 1st from 2nd vector), that is 1 + 2 * 3 + 6 = 1 + 6 + 6
• term 7 is the summation of the 4th elements of level 2 and multiplication of two elements of level 1 (2nd from 1st vector and 2nd from 2nd vector), that is 2 + 2 * 4 + 8 = 2 + 8 + 8

Am I right? If so, there is a mistake in the slide.

[inakleinbottle]

I'm so glad you found my talk interesting. I shall try and explain here, but for reference there is some much better documentation that you can refer to. It is entirely possible that I made a mistake in my slides, so let's see.

If I write x_i for the ith element of x, so the free tensor is

(x_0, x_1, x_2, x_3, x_4, x_5, x_6, ...)

Then the 5th term in xy (the second element of the level 2 terms) is the sum

x_0y_5 + x_1y_2 +  x_5y_0

In the example on the slide, x_0 = 1, x_1 = 1, and x_5 = 1, and y_0=1, y_2 = 4, y_5=6, so we can compute (xy)_5 as

(xy)_5 = 1*6 + 1*4 + 1*1 = 6 + 4 + 1 = 11

So in fact you are correct, the 0.5 should be 1. I apologise for the confusion - I did write these slides the morning of my talk.