R语言统计代写: Statistical computing MATH10093

Statistical computing MATH10093 Coursework A 2017/18 Solutions

Finn Lindgren 7/2/2018

Summary

Handout Wednesday 7/2/2018, electronic (PDF) handin on Learn with deadline by end of Wednesday 28/2/2018. You may discuss the assignment with others, but you should hand in your own individual solutions. The work is marked out of 50, and counts for 50% of the total grade. Carefully read the questions to ensure that you’re answering them fully.

General credits

A total of 10 marks is awarded for general acquired skills:
1. RMarkdown used to produce the handin
2. Basic mathematical formula typesetting with RMarkdown 3. Code readability, useful code comments

The following four questions are worth 10 marks each.

Question 1: A new score

Write a new function score interval that implements the Interval score: LetLF andUF belowerandupperboundsofpredictionintervalsforsomeerror probability α, e.g. α = 0.05.

SINT(F,y)=UF −LF + 2(LF −y)I(y<LF)+ 2(y−UF)I(y>LF) αα

On average, this score is minimised when (LF , UF ) is the equal tail probability interval (α/2 on each side) under the true distribution G, and it is a proper score.

When used on output from model predict, the new function should use the lwr and upr bounds. For top marks, the function should allow the user to specify α, and it should be included in the score comparisons in Question 3, with α = 1/3 (this will require providing α for model predict as well).

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Question 2: The 3D printer problem

Problem description

A 3D-printer for home use typically operates by heating the end of a coil of plastic or other material just enough to squeeze through a nozzle, and onto an assembly plate, or already printed material. The object to be printed is first designed in a CAD (Computer Aided Design) program, which also calculates the expected amount of material required to print the object. When the object has been printed, we can weigh it to learn how much material was actually used. Due to variable material properties, the CAD-predicted amount and the actual amount will not match perfectly. It is also expected that the variability is more noticeable for large objects. Hence, it is expected that the actual amount will depend on both the material type, identified by colour, and the nominally expected amount as calculated by the CAD program.

Question

Look at the data generating code in CWAcode.R and determine the 12-parameter model structure (ideally, describe it in mathematical notation as well as plain text).

Construct an appropriate model Z function, and estimate θ for the 3D- printer problem.

Before you start, read through Question 3, and the solutions to Lab 4!

Question 3: Assessing models for the 3D printer problem

In Lab 4, a model that used only cad to model the expectation and variance was used for the 3D-printer problem. Compare the new, colour aware, model with that simpler model by computing proper scores.

Briefly discuss the meaning of the results.

Hint: Since the two models need different model Z functions, you may use the following technique to switch between the two models:

Question 4: Finite differences

Choose a non-trivial function f(θ) with known derivatives, and a point θ. Compute the error of first and second order finite differences when using an optimal h step value. For the second order derivative, also compare with the error obtained if one uses the h-value that is optimal for a first-order difference. Refer to the notes for Lecture 2 and Lecture 3 for approximate optimality

criteria, and use that when choosing h values.

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