The k means in the algorithm Kmeansrepresents the number of samples to assign a sample to a class.

A university wants to identify groups of students based on their weekly study hours and exam performance. Use Kmeans = 2

Assume the initial centroids are:

• Centroid 1: (2,60)

• Centroid 2: (8,85)

1. Calculate the Euclidean distance between Student C (4,70) and each centroid. 

2. Assign Student C to the appropriate cluster. Explain your decision.

Using the initial centroids, suppose the first assignment produces the following clusters:

• Cluster 1: A, B, C

• Cluster 2: D, E, F

1. Calculate the new centroid of Cluster 1. 

2. Calculate the new centroid of Cluster 2. 

   What characteristics distinguish the two clusters?

K means is parametric algorithm. The goal is to find a decision boundary to separe clusters. 

What objective function does standard k-means minimize?

1. The number of points per cluster
2. The maximum distance between any two points in the same cluster
3. The distance between cluster centroids
4. The sum of squared distances between each point and its assigned centroid

Un sistema clasifica una medición x en una de dos clases. Las verosimilitudes de cada clase son Gaussianas con la misma varianza:

• Clase C₁: p(x | C₁) = N(0, 1)
• Clase C₂: p(x | C₂) = N(4, 1)

Las probabilidades a priori son P(C₁) = 0.8 y P(C₂) = 0.2.

a) Escribe la regla de decisión MAP para clasificar un nuevo punto x.

Se quiere minimizar la función f(w) = w² usando gradient descent. El valor inicial es w₀ = 3 y la tasa de aprendizaje es η = 0.1. ¿Cuál es el valor de w después de una iteración?

A person will be classified as high (class 1) or low risk (class2) of heart diseases using a single measurement and the maximum a posteriori decision rule.

• Assume the conditional probability densities are Gaussians with mean 7 for class 1 and  8 for class 2. The variance is equal to one for the two classes.
• Class 1 and class 2 are equally probable.

• Provide the decision rule for this problem
• Include all mathematical procedures

Hints:

P \left(x\mid C_{1}\right) = \frac{1}{\sqrt{2\pi \sigma}} \exp{\left(-\frac{1}{2}\right)}

A person will be classified as high (class 1) or low risk (class2) of heart diseases using a single measurement and the maximum a posteriori decision rule.

• Assume the conditional probability densities are Gaussians with mean 7 for class 1 and  8 for class 2. The variance is equal to one for the two classes.
• Class 1 and class 2 are equally probable.

• Provide the decision rule for this problem
• Include all mathematical procedures

Hints:

P(x∣C1)=1√2πσexp(−12)P(x∣C1)=12πσexp⁡(−12) P \left(x\mid C_{1}\right) = \frac{1}{\sqrt{2\pi \sigma}} \exp{\left(-\frac{1}{2}\right)}

P(X / C1) = \frac{1}{\sqrt{2\pi}} \left(\exp{-\frac{x^{2}}{2}}\right)

P(X / C2) = \frac{1}{\2/sqrt{2\pi}} \left(\exp{-\frac{x^{2}}{8}}\right)

Find the optimum decision rule in the sense of minimizing the total probability of error in terms of the likelihood ratio l(x).