## Exercise 1

From the textbook “Nonparametric Statistical Methods” by Hollander, Wolfe, and Chicken (Stanford link) (Problem 7, p. 654):

Use the dwt command to obtain the wavelet coefficients of the first 512 components of the sunspots data (you can find the data in package datasets). Use n.levels=9.

Use the dwt command to obtain the wavelet coefficients of the first 512 components of the sunspots data. Use n.levels=9.

1. Use the unlist command on this object to create a single vector of coefficients. Make two histograms: one of the coefficient vector and one of the untransformed data. How do these histogram shapes illustrate the sparsity of the wavelet representation of the data?

2. Create a histogram using only the highest level of detail coefficients (use [[1]]] or \$d1 to access these coefficients from the output of dwt). Do these coefficients appear to be symmetric about 0? Normal?

3. Threshold the wavelet coefficients for this data using both SureShrink and VisuShrink (Tip: Use package waveslim). Describe the differences in the reconstructions.

## Exercise 2

Analyze one of the networks from here using Matrix Estimation by Universal Singular Value Thresholding (USVT).

1. Implement USVT. Use $$\eta = 0.01$$. Hint: You can rewrite this Matlab script in R.
2. Rearrange nodes according to the empirical degree. Plot reconstructed matrix with rearranged columns and rows.
3. Try to interpret the results.