Solve the following

 1) Here, we further explore the cosine and correlation measures.1. What is the range of values possible for the cosine measure? 2. If two objects have a cosine measure of 1, are they identical? Explain. 3. What is the relationship of the cosine measure to correlation, if any? (Hint: Look at statistical measures such as mean and standard deviation in cases where cosine and correlation are the same and different.) 4. Figure 2.22(a) shows the relationship of the cosine measure to Euclidean distance for 100,000 randomly generated points that have been normalized to have an L2 length of 1. What general observation can you make about the relationship between Euclidean distance and cosine similarity when vectors have an L2 norm of 1?5. Figure 2.22(b) shows the relationship of correlation to Euclidean distance for 100,000 randomly generated points that have been standardized to have a mean of 0 and a standard deviation of 1. What general observation can you make about the relationship between Euclidean distance and correlation when the vectors have been standardized to have a mean of 0 and a standard deviation of 1? 6. Derive the mathematical relationship between cosine similarity and Euclidean distance when each data object has an L2 length of 1. 7. Derive the mathematical relationship between correlation and Euclidean distance when each data point has been standardized by subtracting its mean and dividing by its standard deviation.

2) Assume that we apply a square root transformation to a ratio attribute x to obtain the new attribute x*. As part of your analysis, you identify an interval (a, b) in which x* has a linear relationship to another attribute. 1.) What is the corresponding interval (A, B) in terms of x ? 2.) Give an equation that relates y to x.