## 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`

.

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?

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?

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).

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

## Deadline

Please email your solutions in the form of a Rmd file to Nan Bi and Lexi Guan.

This homework is due on Friday, May 27th at 1:30 pm.

You are encouraged to work through this homework with your peers. But write your own code and explain the results in your own words.