Radial visualization & spring forces

[jst]

Hi, I am taking a Data Mining course and have this problem in radial visualization. To complete this problem, we use some physics formulas, which I really do not understand.

Derive formulas for radial visualization of:

a. 3-dimensional samples

b. 8-dimensional samples

Homework Equations

Here's what our notes have:
http://glomawr.com/radialvisualization.jpg

The Attempt at a Solution

I thought it would be very similar to the example in the notes so for part A I did:

x = (K1 - K3 - K5 - K7)/(K1 + K2 + K3 + K4 + K5 + K6 + K7 + K8)
y = (K2 - K4 - K6 - K8)/(K1 + K2 + K3 + K4 + K5 + K6 + K7 + K8)

and part B, I did:

x = (K1 - K3)/(K1 + K2 + K3)
y = (K2)/(K1 + K2 + K3)

I am told: "sqrt (2)/2, or sqrt(3)/2 or 1/2 are in the final expressions because of trig functions: sin and cos for angles of 30, 45 or 60 degrees."

I'm really lost, our book doesn't have what I need it sort of assumes that this example was enough, but I'm lost. I really don't need the solution, as much as an explanation of how to do it myself.

Thanks a lot,

Jason

 

[Nate810]

The formulas for radial visualization of 3-dimensional and 8-dimensional samples can be derived using the trigonometric functions cosine (cos) and sine (sin).For 3-dimensional samples, the formula is:x = cos(α) * K1 - sin(α) * K3y = sin(α) * K1 + cos(α) * K3where α is the angle measured in radians and K1 and K3 are constants. To derive the formula for an 8-dimensional sample, we use the same principles but with different constants. The formula is:x = cos(α) * (K1 - K3 - K5 - K7) - sin(α) * (K2 - K4 - K6 - K8)y = sin(α) * (K1 - K3 - K5 - K7) + cos(α) * (K2 - K4 - K6 - K8)where again, α is the angle measured in radians and K1, K2, K3, K4, K5, K6, K7, and K8 are constants. The constants are determined by the sample data points.The constants sqrt (2)/2, or sqrt(3)/2 or 1/2 are in the final expressions because of the trigonometric functions. As the angle increases, the sine and cosine values approach these constants, which allows us to simplify our equations. For example, if the angle is 45 degrees (1.57 radians), then cos(1.57) = 0.707 and sin(1.57) = 0.707, so sqrt (2)/2 is used in the final expression.

 

[Moetasim]

Hello Jason,

Radial visualization is a technique used to represent high-dimensional data in a 2D or 3D space. It is based on the concept of spring forces, where data points are represented as particles connected by springs. The closer the data points are in the high-dimensional space, the stronger the spring force between them, and the closer they will be in the radial visualization.

To derive the formulas for radial visualization of 3-dimensional and 8-dimensional samples, we need to understand the concept of spring forces and how they are applied in this visualization technique.

In the given notes, the formula for the position of a data point in the radial visualization is given as:

x = (K1 - K3 - K5 - K7)/(K1 + K2 + K3 + K4 + K5 + K6 + K7 + K8)
y = (K2 - K4 - K6 - K8)/(K1 + K2 + K3 + K4 + K5 + K6 + K7 + K8)

Here, K1 to K8 represent the spring forces between the data point and its neighboring points in the high-dimensional space. These spring forces are calculated using the Euclidean distance between the data points in the high-dimensional space.

Now, let's look at the formula for the position of a data point in the radial visualization for 3-dimensional samples:

x = (K1 - K3)/(K1 + K2 + K3)
y = (K2)/(K1 + K2 + K3)

In this formula, the spring forces K1, K2, and K3 represent the forces between the data point and its neighboring points in the high-dimensional space. These forces are calculated using the Euclidean distance between the data points in the high-dimensional space. The denominator in the formula represents the sum of all the spring forces acting on the data point.

Similarly, for 8-dimensional samples, the formula for the position of a data point in the radial visualization is:

x = (K1 - K3 - K5 - K7)/(K1 + K2 + K3 + K4 + K5 + K6 + K7 + K8)
y = (K2 - K4 - K6 - K8)/(K1 + K2 + K3 + K4 + K5 + K6 + K7 + K8)

Here, the spring forces K1 to K