In the problems on this page, we will work through the definitions and minimization described above for a simple data configuration.
Given 3 data points \mathbf{x}^{(1)},\mathbf{x}^{(2)},\mathbf{x}^{(3)} that are equidistant (at distance A) from each other in p-dimensional space, compute the probability distribution \mathbf{P} (given by p_{12}, p_{23}, p_{13}), and enter the value of p_{12} and p_{13} below.
p_{12}=\quad
unanswered
p_{13}=\quad
To compute the probability distribution \mathbf{P}, we need to consider the distances between each pair of data points.
Given that the data points \mathbf{x}^{(1)},\mathbf{x}^{(2)},\mathbf{x}^{(3)} are equidistant (at distance A) from each other in p-dimensional space, we can calculate the probabilities as follows:
p_{12} is the probability that \mathbf{x}^{(1)} and \mathbf{x}^{(2)} are nearest neighbors. Since all three data points are equidistant, there are no other data points closer to either \mathbf{x}^{(1)} or \mathbf{x}^{(2)}. Therefore, we can conclude that p_{12} = 1.
p_{13} is the probability that \mathbf{x}^{(1)} and \mathbf{x}^{(3)} are nearest neighbors. Again, since all three data points are equidistant, there are no other data points closer to either \mathbf{x}^{(1)} or \mathbf{x}^{(3)}. Therefore, p_{13} = 1.
Therefore, we have:
p_{12} = 1
p_{13} = 1
To compute the probability distribution \(\mathbf{P}\) for the given data points \(\mathbf{x}^{(1)}, \mathbf{x}^{(2)}, \mathbf{x}^{(3)}\), we need to calculate the values \(p_{12}\) and \(p_{13}\).
Since the data points are equidistant at a distance \(A\) from each other, the probability distribution can be computed as follows:
1. Calculate the Euclidean distance between \(\mathbf{x}^{(1)}\) and \(\mathbf{x}^{(2)}\). Let's call this distance \(d_{12}\).
2. Calculate the Euclidean distance between \(\mathbf{x}^{(1)}\) and \(\mathbf{x}^{(3)}\). Let's call this distance \(d_{13}\).
The probability distribution can be computed using the formula:
\(p_{ij} = \frac{1}{Z} \exp\left(-\frac{d_{ij}}{\epsilon}\right)\)
Where \(Z\) is a normalization constant and \(\epsilon\) is a scaling parameter.
Please provide the values of \(d_{12}\) and \(d_{13}\) so that we can calculate \(p_{12}\) and \(p_{13}\).
To compute the probability distribution \(\mathbf{P}\) for the given equidistant data points \(\mathbf{x}^{(1)},\mathbf{x}^{(2)},\mathbf{x}^{(3)}\) in p-dimensional space, we need to calculate \(p_{12}\) and \(p_{13}\).
Since the data points are equidistant from each other, with a distance of A, we can use the formula for computing probability distribution in the above problem set.
To calculate \(p_{12}\), we consider the probability of a random point falling in the region between \(\mathbf{x}^{(1)}\) and \(\mathbf{x}^{(2)}\). In this case, there are two regions (one on each side) of equal size, each with a width of A/2. Therefore, \(p_{12}\) can be calculated as:
\(p_{12} = \frac{{\text{{width of region between }} \mathbf{x}^{(1)} \text{{ and }} \mathbf{x}^{(2)}}}}{{\text{{total width of the space}}}} = \frac{{A/2}}{{A}} = \frac{1}{2}\)
To calculate \(p_{13}\), we consider the probability of a random point falling in the region between \(\mathbf{x}^{(1)}\) and \(\mathbf{x}^{(3)}\). In this case, there is only one region (on one side) with a width of A. Therefore, \(p_{13}\) can be calculated as:
\(p_{13} = \frac{{\text{{width of region between }} \mathbf{x}^{(1)} \text{{ and }} \mathbf{x}^{(3)}}}}{{\text{{total width of the space}}}} = \frac{{A}}{{A}} = 1\)
So, the values of \(p_{12}\) and \(p_{13}\) are:
\(p_{12} = \frac{1}{2}\)
\(p_{13} = 1\)