An Introduction to Statistical Learning: with Applications in R (Springer Texts in Statistics)
13th Edition
ISBN: 9781461471370
Author: Gareth James
Publisher: SPRINGER NATURE CUSTOMER SERVICE
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Chapter 4, Problem 3E
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Density function
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Consider a real random variable X with zero mean and variance σ2X . Suppose that we cannot directly observe X, but instead we can observe Yt := X + Wt, t ∈ [0, T ], where T > 0 and {Wt : t ∈ R} is a WSS process with zero mean and correlation function RW , uncorrelated with X.Further suppose that we use the following linear estimator to estimate X based on {Yt : t ∈ [0, T ]}:ˆXT =Z T0h(T − θ)Yθ dθ,i.e., we pass the process {Yt} through a causal LTI filter with impulse response h and sample theoutput at time T . We wish to design h to minimize the mean-squared error of the estimate.a. Use the orthogonality principle to write down a necessary and sufficient condition for theoptimal h. (The condition involves h, T , X, {Yt : t ∈ [0, T ]}, ˆXT , etc.)b. Use part a to derive a condition involving the optimal h that has the following form: for allτ ∈ [0, T ],a =Z T0h(θ)(b + c(τ − θ)) dθ,where a and b are constants and c is some function. (You must find a, b, and c in terms ofthe…
Imagine a regression model on a single feature, defined by the function f (x) = wx + b where
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4. E: (wa; +b - t;)
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4
Computer Science
Suppose we have 3 independent classifiers, each of which can correctly predict the label of a data point with 80% accuracry. Using the hard voting approach, prove that the ensemble of these classifiers can correctly predict with at least 89% accuracy.
Chapter 4 Solutions
An Introduction to Statistical Learning: with Applications in R (Springer Texts in Statistics)
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