1. Vector notation

"x(i) is a vector indexed by time i, x_{j}(i) is the j-th component of x(i)" Could anyone please explain to me what this notation really means? What does it mean for a vector to be "indexed by time"? In the same context there is also this: "x^n = (x(1), x(2), ..., x(n))"

2. Re: Vector notation

Originally Posted by mathlearner100
"x(i) is a vector indexed by time i, x_{j}(i) is the j-th component of x(i)" Could anyone please explain to me what this notation really means? What does it mean for a vector to be "indexed by time"? In the same context there is also this: "x^n = (x(1), x(2), ..., x(n))"
Think of the position of a satellite or a baseball or anything moving through space.

At any given point in time t0 it has 3 spatial coordinates {x, y, z}

These change as the thing moves. It will be at {x0, y0, z0} at t0, and {x1, y1, z1} at t1, etc.

You can treat this as a vector of 3 spatial components, that is further indexed by a time component.

Now just make your space n dimensional rather than 3. Sure a baseball doesn't move through n dimensional space but you know what I mean.

You can index these n dimensional spatial vectors by time just as you did with the 3 dimensional ones.

These spatial vectors don't have to be "space" space either. The can be the configuration space of some huge dynamical system.

They are just vectors that change with time and you are capturing snapshots of them at times t0, t1, t2.. tk, etc.

3. Re: Vector notation

Originally Posted by romsek
Think of the position of a satellite or a baseball or anything moving through space.

At any given point in time t0 it has 3 spatial coordinates {x, y, z}

These change as the thing moves. It will be at {x0, y0, z0} at t0, and {x1, y1, z1} at t1, etc.

You can treat this as a vector of 3 spatial components, that is further indexed by a time component.

Now just make your space n dimensional rather than 3. Sure a baseball doesn't move through n dimensional space but you know what I mean.

You can index these n dimensional spatial vectors by time just as you did with the 3 dimensional ones.

These spatial vectors don't have to be "space" space either. The can be the configuration space of some huge dynamical system.

They are just vectors that change with time and you are capturing snapshots of them at times t0, t1, t2.. tk, etc.
So x^n = (x(1), x(2), ... , x(n)) means the vector (or set?) of all the vectors, iterated over the allowed time domain (all possible n)?

4. Re: Vector notation

Originally Posted by mathlearner100
So x^n = (x(1), x(2), ... , x(n)) means the vector (or set?) of all the vectors, iterated over the allowed time domain (all possible n)?
no. There's no reason your time index and space index have any dependence on each other. You can take M samples of an N dimensional vector.

A better notation would be

let x = {x1, x2, .... , xn}

your time indexed spatial vector x is now xj or x[j] with j as your time index.

If you need to get at individual pieces of x you can use (xj)i, or

xi[j], i=1,N, j=0, #samples

There's nothing written in stone about notation. Just try to be clear.

5. Re: Vector notation

Originally Posted by romsek
no. There's no reason your time index and space index have any dependence on each other. You can take M samples of an N dimensional vector.

A better notation would be

let x = {x1, x2, .... , xn}

your time indexed spatial vector x is now xj or x[j] with j as your time index.

If you need to get at individual pieces of x you can use (xj)i, or

xi[j], i=1,N, j=0, #samples

There's nothing written in stone about notation. Just try to be clear.
So the notation is unclear? I can agree with that, this notation is from a set of lecture notes. I would be happy to use a different notation or read it in a different way, but I still don't understand what this notation really means.

It is stated that x refers to values of random column vectors with specified dimensions. Is n in x^n the dimension? Or is it something else?

In x^n = (x(1), x(2), ..., x(n)) what is e.g. x(1)?

From the earlier definition I mentioned "x(i) is a vector indexed by time i, x_{j}(i) is the j-th component of x(i)" it seems that x(1) is a vector indexed by time 1. Hence this vector (x(1), x(2), ..., x(n)) is a vector of all the vectors, indexed from 1 to n. If it was a set, wouldn't they have used curly brackets { } to denote the set?

6. Re: Vector notation

You're losing me. Where does randomness come into this? I don't understand what you mean by x^n.

You're making this much harder than it is. If X is an n dimensional vector that changes in time so you call call it X(t), and I take snapshots of it at t0, t1, tk, etc.

I can then refer to those vector snapshots as X[tk] or Xk where it's understood that X is still an n dimensional vector.

If you need to get at the components of X at a given time tk, again there is no notation carved in stone, just be clear

{X[tk]}i works where i the vector component you want.

X[tk][i] also works. Just be clear and consistent.

7. Re: Vector notation

Originally Posted by romsek
You're losing me. Where does randomness come into this? I don't understand what you mean by x^n.

You're making this much harder than it is. If X is an n dimensional vector that changes in time so you call call it X(t), and I take snapshots of it at t0, t1, tk, etc.

I can then refer to those vector snapshots as X[tk] or Xk where it's understood that X is still an n dimensional vector.

If you need to get at the components of X at a given time tk, again there is no notation carved in stone, just be clear

{X[tk]}i works where i the vector component you want.

X[tk][i] also works. Just be clear and consistent.
These lecture notes use lower case letters, x, y, z to denote random variables. However, the same lecture notes (and this is specified in the notes) also denote random column vectors with lower case variables, like x, y, z, with a specified dimension.