a. How would you apply the CNN-based approach to a time series (e.g., represented as a one dimensional vector (array), as opposed to a 2D image)? For example, consider the following finite time series 2, 1, 4, 3, 7, 5. What would a filter be in this case and how can it be applied to this time series?

b. Apply the (0, 1, 0) filter to the time series from item (a) without any padding (P = 0) and show the output of this convolutional operation.

c. Computational complexity of the convolution operator. How many filtering operations did it take to apply the filter from Step (b) to the time series 2, 1, 4, 3, 7, 5 that is of length 6 (i.e., how many times was the filter (0, 1, 0) applied to compute the convolution)? If the length of the time series is n, how many times would the filter (0, 1, 0) be applied (without padding) to perform the convolution (e.g.,assume n = 10,000 or n = 100,000)? Give your answer in terms of n (write a formula in terms of n).

d. The same question as in Step (c) but for the 2-dimensional tensors (matrices): if the size of an image is nxn pixels, how many filtering operations (without padding) would it take to perform the convolution using a 3x3 filter (again, give your answer in terms of number n – write the formula in terms of n). The same question but for the 3-dimensional tensors (cubes) representing videos: if the size of a video is nxnxn, how many steps would it take to perform the convolution operation on that video (again, give your answer in terms of n)? 

• What conclusions can you draw from this analysis in terms of the ease of processing of the of the time series, imagers and 3D tensors, especially when they become large (i.e., n is “very large”)?