It is well-known that ridge regression tends to give similar coefficient values to correlated variables, whereas lasso may give quite different coefficient values to correlated variables. We will now explore this property in a very simple setting. Assume n are the number of training samples, p the number of dimensions, x in the input and y the output. Suppose that n = 2, p = 2, x11 = x12, and x21 = x22. Furthermore, suppose that y1 + y2 = 0 and x11 + x21 = 0 and x12 + x22 = 0, so that the estimate for the intercept in a least squares, ridge regression, or lasso model is zero: βˆ 0 = 0. (a) Write out the ridge regression optimization problem in this setting. (b) Argue that in this setting, the ridge coefficient estimates satisfy βˆ 1 = βˆ 2. (c) Write out the lasso optimization problem in this setting. (d) Argue that in this setting, the lasso coefficients βˆ 1 and βˆ 2 are not unique – in other words, there are many possible solutions to the optimization problem. Describe these solutions

Brittany has designed four different birdcages, each with the same volume: a square pyramid, a cylinder, a cone, and a sphere. the zoo wants to use one of brittany’s designs but plans to change the volume of the cage. find the design change that would result in the birdcage with the greatest volume. increase the area of the base of the square pyramid by a factor of ten. increase the height of the cylinder by a factor of ten. increase the radius of the cone by a factor of ten. increase the radius of a sphere by a factor of ten. juan wants to build a greenhouse in his back yard. he needs to maximize the ratio of floor area to structure volume in order to provide the most room for plants while minimizing the area to be heated. find the description of the structure that best meets juan’s criteria. a cube with a side length of 12 feet a cone with a diameter of 12 feet and a height of 10 feet a square pyramid with a base side length of 12 feet and a height of 9 feet a hemisphere with a diameter of 12 feet marion has a bicycle that she is planning to take for a ride. the rim of the rear wheel lies 41 cm from the center of the wheel. the tire itself has a thickness of 3 cm.in order to marion to ride her bicycle 96,712 cm, approximately how many rotations will the rear wheel make? 350 15.9 376 5133 an isosceles trapezoid has perimeter of 42 inches. each of the congruent nonparallel sides is 5 inches long, and the trapezoid is 3 inches tall. how long are the two parallel sides? 10 inches and 16 inches 12 inches and 20 inches 16 inches and 16 inches 10 inches and 22 inches