Machine and Data Mining in Business
Week 1 Tutorial Exercises
The following exercises are review exercises about differentiation, probability, and the con- cepts of the bias and variance and variance of an estimator. We will need to take derivatives in Week 5 when we discuss optimisation, and in other lectures from there onwards. We use rules from probability theory and rely on concepts from statistical inference throughout the unit.
Question 1
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Find the derivative f′(x) of the following functions. Identify the rule for differentiation used in your solution. We will use these results in the next three exercises.
(a) f(x) = exp(x). (b) f(x) = exp(−x).
(c) f(x)=1+exp(−x). (d) f(x)=x1.
(e) f(x) = log(x).
(f) f(x)=log(1+x).
(g) f(x) = x2. Question 2
Consider the logistic sigmoid function
σ(x) = 1 ,
1 + exp(−x)
which appears in the logistic regression model and other classification methods in this
(a) Take the derivative σ′(x).
(b) Show that σ′(x) = σ(x)(1 − σ(x)).
Question 3
Consider the softplus function
ζ(x) = log(1 + exp(x)),
which also plays a role in understanding the logistic regression model and other classi-
fication methods.
(a) Take the derivative ζ′(x).
(b) Show that ζ′(x) corresponds to σ(x) as defined in the previous exercise. Question 4
Consider the function
J(β0,β1)=(y−β0 −β1x)2 (a) What are the partial derivatives of the function?
(b) What is the gradient ∇J? Question 5
Let θ be an estimator and θ the quantity to be estimated (called the estimand). You can think of θ as being a population parameter for this exercise, but the quantity to be estimated can be any quantity of interest. As an example, θ can be the sample mean and θ the population mean.
(a) What is an estimator (in words)?
(b) What is the mathematical definition of the bias of the estimator θ? Interpret the equation.
(c) We define the (population) mean squared error of the estimator as
MSE(θ) = Bias2(θ) + V(θ). Identify the property used at each step.
MSE(θ)=E θ−θ , where θ is the actual value of the parameter.
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