And I'll write here that h of x1, x2 is equal to so now

Beyond it is another dimension a dimension of sound... a dimension of sight... a dimension of mind.

Produce simple xy plots of scientific data

So now I'm going to show you a number of projects that we've built, from one dimensional, two dimensional, three dimensional and even four dimensional systems.

I now have a more challenging quiz for you in which I give you multivariant data in this case 2 dimensional data.

So we remap the data onto the space over here, with x1 over here and x2 over here.

In linear regression, we are given data and data has still has more than one dimension.

This is what string theory says about the extra dimensions.

The dimensionality reduction looks a little bit silly when you go from 2 dimensions to 1 dimension.

In this case X is one dimensional which is N 1.

There's a set of linear algebra equations that implement the Kalman filter and higher dimensions.

A deeper, higher dimension

The orthogonal dimension in this direction carries alomst information, so it suffices, in most cases, to project the data onto this 1 dimensional space.

But it's just a story.

They have some deficiencies.

The best pilot they ever had, next to me.

Let me explain k means by an example.

Let's say that a hypersphere is the 4D equivalent of a 3D sphere.

I can draw, really easily, a nice 1 dimensional space that follows these data points.

We're going to start with a little quiz, in which I will check your intuition.

cubes game

3D Rubik's cube game

Consider the following data set where plus indicates a positive traning example and minus a negative training example in this 2 dimensional space.

And notice I'm going from a space that has two dimensions to a space it has three dimensions, or three components.

The one dimension applies to the x axis and the other to the y axis whereas histograms only apply to 1D data where the y axis becomes the count of that data.

My codomain by definition is R3.

sort of... a portal between this reality and the next.

Even though moving one over here and one over there would give a better solution.

And that's where I work.

It doesn't work in a universe with four dimensions of space, nor five, nor six.

We have to understand that we are interacting with the forth dimensional reality when we are just activating our third eyes.

So let's dive straight in so let's talk about lines and let's talk about the technology to fit lines to data called linear regression.

They both use vertical bars, and they both aggregate data.

The answer is here.

They don't really scale to high dimensional spaces.

Now when we talk about a three dimensional surface it's a three dimensional surface of a four dimensional sphere that means that any of the three dimensions that are over here on the surface, I can only draw two but that means, if this is true, if the Universe is a three dimensional surface of a four dimensional sphere, that means if you go up and if you just keep going up, you'll eventually come back from the bottom.

Galaxies seem to fit all over this particular family of profiles.

If that angle becomes 0 we end up with a degenerate triangle

a 3D city simulator game

The flux capacitor.

Nearest neighbor works really well for small input spaces like three or four dimensions.

Further, if you re represent each data vector using those eigenvectors, you'll find a 3 dimensional space where original data falls into a varity of different places.

Let me draw them for you just so you can visualize it.

It might be possible to have a fairly small uncertainty in 1 dimension, but a huge uncertainty in the other.

A gaussian with a small amount of uncertainty might look like this.