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Problem 2. Mean Absolute Error - Adding a Data Point Suppose we have a dataset y₁ ≤ Y2 < ... < Yn and want to minimize absolute loss on it for the prediction h. As we've seen before, the corresponding empirical risk is mean absolute error, n Rabs(h) = | Yi - h n Suppose that Rabs(a) = Rabs (α + 3) = a) b) M, where M is the minimum value of Rabs(h) and a is some constant. Suppose we add to the dataset a new data point yn+1 whose value is a + 1. For this new larger dataset: What is the minimum of Rabs(h)? At what value of h is this minimum achieved? Your should only involve the variables n, M, a, and constants. Let n be odd. Let ya and yɩ be two values in our dataset such that ya < yɩ and that the slope of Rabs(h) is the same between h Ya and h = y. Specifically, let d be the slope of Rabs (h) between Ya and yb. = 1 = Suppose we introduce a new value q to our dataset such that q> y. In our new dataset of n + 1 values, the slope of Rabs (h) is still the same between h == Ya and h Yb, but it's no longer equal to d. What is the slope of Rabs (h) between h = Ya and h = y in our new dataset? Your answer should depend on d, n, q, and/or one or more constants.

Problem 2. Mean Absolute Error - Adding a Data Point Suppose we have a dataset y₁ ≤ Y2 < ... < Yn and want to minimize absolute loss on it for the prediction h. As we've seen before, the corresponding empirical risk is mean absolute error, n Rabs(h) = | Yi - h n Suppose that Rabs(a) = Rabs (α + 3) = a) b) M, where M is the minimum value of Rabs(h) and a is some constant. Suppose we add to the dataset a new data point yn+1 whose value is a + 1. For this new larger dataset: What is the minimum of Rabs(h)? At what value of h is this minimum achieved? Your should only involve the variables n, M, a, and constants. Let n be odd. Let ya and yɩ be two values in our dataset such that ya < yɩ and that the slope of Rabs(h) is the same between h Ya and h = y. Specifically, let d be the slope of Rabs (h) between Ya and yb. = 1 = Suppose we introduce a new value q to our dataset such that q> y. In our new dataset of n + 1 values, the slope of Rabs (h) is still the same between h == Ya and h Yb, but it's no longer equal to d. What is the slope of Rabs (h) between h = Ya and h = y in our new dataset? Your answer should depend on d, n, q, and/or one or more constants.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter7: Distance And Approximation
Section7.3: Least Squares Approximation
Problem 31EQ
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Problem 2. Mean Absolute Error - Adding a Data Point Suppose we have a dataset y₁ ≤ Y2 < ... < Yn and want to minimize absolute loss on it for the prediction h. As we've seen before, the corresponding empirical risk is mean absolute error, n Rabs(h) = | Yi - h n Suppose that Rabs(a) = Rabs (α + 3) = a) b) M, where M is the minimum value of Rabs(h) and a is some constant. Suppose we add to the dataset a new data point yn+1 whose value is a + 1. For this new larger dataset: What is the minimum of Rabs(h)? At what value of h is this minimum achieved? Your should only involve the variables n, M, a, and constants. Let n be odd. Let ya and yɩ be two values in our dataset such that ya < yɩ and that the slope of Rabs(h) is the same between h Ya and h = y. Specifically, let d be the slope of Rabs (h) between Ya and yb. = 1 = Suppose we introduce a new value q to our dataset such that q> y. In our new dataset of n + 1 values, the slope of Rabs (h) is still the same between h == Ya and h Yb, but it's no longer equal to d. What is the slope of Rabs (h) between h = Ya and h = y in our new dataset? Your answer should depend on d, n, q, and/or one or more constants.
Transcribed Image Text:Problem 2. Mean Absolute Error - Adding a Data Point Suppose we have a dataset y₁ ≤ Y2 < ... < Yn and want to minimize absolute loss on it for the prediction h. As we've seen before, the corresponding empirical risk is mean absolute error, n Rabs(h) = | Yi - h n Suppose that Rabs(a) = Rabs (α + 3) = a) b) M, where M is the minimum value of Rabs(h) and a is some constant. Suppose we add to the dataset a new data point yn+1 whose value is a + 1. For this new larger dataset: What is the minimum of Rabs(h)? At what value of h is this minimum achieved? Your should only involve the variables n, M, a, and constants. Let n be odd. Let ya and yɩ be two values in our dataset such that ya < yɩ and that the slope of Rabs(h) is the same between h Ya and h = y. Specifically, let d be the slope of Rabs (h) between Ya and yb. = 1 = Suppose we introduce a new value q to our dataset such that q> y. In our new dataset of n + 1 values, the slope of Rabs (h) is still the same between h == Ya and h Yb, but it's no longer equal to d. What is the slope of Rabs (h) between h = Ya and h = y in our new dataset? Your answer should depend on d, n, q, and/or one or more constants.
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