0:30

So in looking at the higher dimensional drawings or

Â higher dimensional spaces, you can think of something like a square.

Â A square consists of two one-dimensional lines,

Â these two vertical lines, that are separated horizontally.

Â I've taken a one dimensional line, copied into a second vertical line and

Â then connected them up with lines between the vertices.

Â So I've now got a two-dimensional square.

Â If I want to make this two-dimensional square into a three-dimensional cube,

Â I just copy the square, and move it in a direction, and then I just link up

Â the corresponding vertices with lines, and that gives me a three-dimensional cube.

Â If I want a four-dimensional cube, I just take the three-dimensional cube and

Â I make another copy of it and offset it in another direction, and

Â then just draw lines between the corresponding vertices and it gives me

Â a tesseract, some two-dimensional projection of a four-dimensional cube.

Â And so you can do this, but it's not very helpful for

Â visualization, because you've got all these different directions that are being

Â projected into a two-dimensional image, and after three dimensions, we're just

Â not used to perceiving the world in four or higher dimensions, and so we just have

Â difficulty perceiving four-dimensional or higher data being projected that way.

Â If I have a scatter plot, for example, I can pretty readily see x and

Â y encoded in the positional Cartesian coordinates of x and y here.

Â If I had a z coordinate that's being projected,

Â z is being foreshortened as it's being projected here, and

Â I've gotta mentally remember that z is some combination of horizontal and

Â vertical that's different than the horizontal x or the vertical y axis.

Â So you can draw these hints that are basically similar to a shadow,

Â where I'm indicating, with these dashed lines,

Â what the coordinates of this green dot are in three dimensions,

Â because it lacks other visual cues of where it is in three dimensions.

Â If you try to do this in four dimensions,

Â then it becomes a nightmare to manage all those dimensions.

Â So there's a better technique for doing this called parallel coordinates that

Â Al Inselberg demonstrated to me back in the early 90s,

Â when they were first invented, and I was quite blown away by his demonstration, and

Â I'll try to reproduce it here.

Â 2:55

So on parallel coordinates, we're going to take the Cartesian coordinates.

Â We have a horizontal x-axis and a vertical y-axis, and

Â we're going to take these axes and we're going to

Â make them parallel instead of orthogonal, at right angles, as they usually are.

Â So I'm going to take the x-axis, I'm going to put it here, and

Â I'm going to take the y-axis and I'm going to put it here.

Â And so I'll label this axis x and this axis y.

Â So now the two axes are not orthogonal.

Â They're parallel to each other and they don't extend from the same origin.

Â So the origin here is horizontally at the bottom and

Â increasing x goes up this axis, increasing y goes up this axis.

Â So now I've got the data points, and I need to figure out where these data points

Â occur, so if I take the y-coordinates of each data point,

Â I can map each point onto its corresponding position on the y-axis.

Â I'm just basically dragging them across horizontally, because their position in y

Â in this chart corresponds to their height along this y-axis.

Â If I want to do the same for

Â the x-axis, I'm going to basically take the x position of each point, and then I'm

Â going to drag it to the corresponding x position on this vertical x-axis, so

Â the horizontal length here corresponds to the vertical length here, and

Â so blue and red are at the same horizontal x coordinate, and so

Â they overlap each other on the x-axis.

Â And in green is a little bit farther to the right, so

Â it's going to be a little bit higher on the parallel x-axis.

Â Yellow is a little bit farther to the right so

Â it's going to be a little bit higher.

Â And then this blue green color dot is the farthest right so

Â it'll be the highest on the x-axis.

Â So now I've got this one set of points that's now appearing as two

Â sets of points, and I've got a correspondence between color,

Â and color's a good perceptual indicator of category, but

Â it's very difficult to perceive what's going on here.

Â So what we're going to do is, instead of displaying these as points on the axis,

Â I'm going to connect these points with lines, and

Â I'm going to delete the original points, and

Â now we get this nice duality between points in the coordinate system,

Â here, and lines in the coordinate system, here.

Â So, this orange point, here, has an x-coordinate and a y-coordinate.

Â It corresponds to this line connecting its x-coordinate to its y-coordinate.

Â And so each one of these five points in the Cartesian x y coordinate

Â system corresponds to a line in the parallel x y coordinate system here.

Â 5:49

Some other features are collinearity, so if these three points lie along

Â the same line, then you get this nice convergence in parallel coordinates.

Â And so you can see some co-linearity that happens, because lines in parallel

Â coordinates, corresponding to points in Cartesian coordinates that are collinear,

Â will basically converge in a point in the parallel coordinate system.

Â So you can add extra dimensions.

Â If I add a z-axis here, I'd have to add all sorts of three-dimensional queues to

Â these data points to figure out where they are in three dimensions.

Â In parallel coordinates, I just add a z-axis here.

Â And then I can connect the z-coordinates to the y-coordinates,

Â and follow this line from its x-coordinate to its y-coordinate to its z-coordinate.

Â And if I add a fourth w-axis.

Â It's actually easy to add a fourth axis here.

Â And in fact, all of these points may be collinear in the z w coordinate system,

Â and you can see because these lines are kind of converging to the same point.

Â The point that they converge to doesn't necessarily have to be between the axes.

Â But there are some ways of helping a person to see these points.

Â And so these parallel coordinates are really useful for high-dimensional data.

Â 7:07

And you get this correspondence of being able to follow these lines through

Â the various coordinates and have them correspond to points here, and

Â it's easier to see some relationships.

Â There could be correspondences between the w-axis and the y-axis, and

Â we wouldn't see those unless we also had lines from these points on

Â the w-axis to the points on the y-axis, and so you get a bit of a combinatorial

Â nightmare when you try to connect every axis to every other axis.

Â So there's some decision making that needs to happen if you're going to show

Â correspondences between axes,

Â to choose to have those axes next to each other in the parallel coordinates system.

Â 7:49

So parallel coordinates take the orthogonal axes of the Cartesian

Â coordinate system and they lay them out in parallel, and you can have any number of

Â these parallel coordinate axes laid next to each other, and you can start to

Â perceive higher dimensional data using these parallel coordinates.

Â Some of the problems of parallel coordinates are the fact that you're only

Â seeing the pairwise relationship between the axes, but they can be very useful for

Â finding certain features in your data, for example, collinearity.

Â [MUSIC]

Â