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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Consider the same house rent prediction problem where you are supposed to predict price
of a house based on just its area. Suppose you have n samples with their respective areas,
x(¹), x(²),...,x(n), their true house rents y(¹), y(2),..., y(n). Let's say, you train a linear regres-
sor that predicts f(x)) = 0 + 0₁x). The parameters, and 0₁ are scalars and are learned
by minimizing mean-squared-error loss with L1-regularization through gradient descent with
a learning rate a and the regularization strength constant A. Answer the following questions.
1. Express the loss function(L) in terms of x(i),y(i), n, 00, 01, X.
2. Compute L
200
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3. Compute 20₁
4. Write update rules for 0o and 0₁
Hint:
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Use R to answer the following question
According to the central limit theorem, the sum of n independent identically distributed random variables will start to resemble a normal distribution as n grows large. The mean of the resulting distribution will be n times the mean of the summands, and the variance n times the variance of the summands. Demonstrate this property using Monte Carlo simulation. Over 10,000 trials, take the sum of 100 uniform random variables (with min=0 and max=1). Note: the variance of the uniform distribution with min 0 and max 1 is 1/12. Include:
1. A histogram of the results of the MC simulation
2. A density plot of a normal distribution with the appropriate mean and standard deviation
3. The mean and standard deviation of the MC simulation.
ps(plz do not use chatgpt)
Consider the same house rent prediction problem where you are supposed to predict price
of a house based on just its area. Suppose you have n samples with their respective areas,
x(1), x(2), ... , x(n), their true house rents y(1), y(2),..., y(n). Let's say, you train a linear regres-
sor that predicts f(x()) = 00 + 01x(e). The parameters 6o and 0, are scalars and are learned
by minimizing mean-squared-error loss with L2-regularization through gradient descent with
a learning rate a and the regularization strength constant A. Answer the following questions.
1. Express the loss function(L) in terms of x), y@), n, 0, 01, A.
2. Compute L
3. Compute
4. Write update rules for 6, and O1
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- Explain how to use a histogram to estimate the size of a selection of the form σA≤v(r).arrow_forwardthe logit function(given as l(x)) is the log of odds function. what could be the range of logit function in the domain x=[0,1]?arrow_forwardPick one million sets of 12 uniform random numbers between 0 and 1. Sum up the 12 numbers in each set. Make a histogram with these one million sums, picking some reasonable binning. You will find that the mean is (obviously?) 12 times 0.5 = 6. Perhaps more surprising, you will find that the distribution of these sums looks very much Gaussian (a "Bell Curve"). This is an example of the "Central Limit Theorem", which says that the distribution of the sum of many random variables approaches the Gaussian distribution even when the individual variables are not gaussianly distributed. mean Superimpose on the histogram an appropriately normalized Gaussian distribution of 6 and standard deviation o = 1. (Look at the solutions from the week 5 discussion session for some help, if you need it). You will find that this Gaussian works pretty well. Not for credit but for thinking: why o = 1 in this case? (An explanation will come once the solutions are posted).arrow_forward
- 1. The impulse response of a causal system is: h(t) = A cos(wt) e¯¹/¹u(t) where u(t) is the Heaviside step function. The response is measured experimentally with a sampling interval of T. a. Write an expression for the sampled impulse response h[n]. b. Calculate the z transform of h[n] and write an expression for H[z]. Use the tables provided below as necessary. c. Does the system have an infinite impulse response (IIR) or finite impulse response (FIR)? Justify your answer. d. What is the DC gain of H[z]? e. Write a difference equation that describes the output y[n] in terms of input x[n].arrow_forwardLet pn(x) be the probability of selling the house to the highest bidder when there are n people, and you adopt the Look-Then-Leap algorithm by rejecting the first x people. For all positive integers x and n with x < n, the probability is equal to p(n(x))= x/n (1/x + 1/(x+1) + 1/(x+2) + … + 1/(n-1)) If n = 100, use the formula above to determine the integer x that maximizes the probability n = 100 that p100(x). For this optimal value of x, calculate the probability p100(x). Briefly discuss the significance of this result, explaining why the Optimal Stopping algorithm produces a result whose probability is far more than 1/n = 1/100 = 1%.arrow_forwardCorrect answer will be upvoted else Multiple Downvoted. Computer science. There are n+2 towns situated on an arrange line, numbered from 0 to n+1. The I-th town is situated at the point I. You fabricate a radio pinnacle in every one of the towns 1,2,… ,n with likelihood 12 (these occasions are autonomous). From that point forward, you need to set the sign power on each pinnacle to some integer from 1 to n (signal powers are not really the equivalent, yet in addition not really unique). The sign from a pinnacle situated in a town I with signal power p arrives at each city c to such an extent that |c−i|<p. Subsequent to building the pinnacles, you need to pick signal powers so that: towns 0 and n+1 don't get any transmission from the radio pinnacles; towns 1,2,… ,n get signal from precisely one radio pinnacle each. For instance, if n=5, and you have assembled the pinnacles in towns 2, 4 and 5, you might set the sign force of the pinnacle around 2 to 2, and the sign…arrow_forward
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