代写Statistical Programming、代做R语言程序、代做fragment、代写R编程

日期：2018-11-20 09:10

Statistical Programming Assignment 2

Gordon Ross

November 19, 2018

Submission: Only one submission attempt is allowed. The deadline

is 23:59 on Monday December 3rd. Late submissions will incur a 15%

penalty.

To submit, create an R script called matriculationnumberA2.R where

matriculationnumber refers to your matriculation number, and upload

to the Assessment 1 section of Learn. Failure to give your script

file the correct name will incur a 5% penalty. You have to submit a

single script file, i.e., matriculationnumberA1.R. Failure to comply

with this will incur a 5% penalty. Your answer for each question must

be included in a corresponding section of your R script file. For example,

your answer/code for question 1.1 must be included in a section

which looks like:

## ;;

## ---------------------------------------------

## Q1: -- add your code below

## ---------------------------------------------

## ;;

## 1.1

code goes here

## ---------------------------------------------

I will deduct 5% of marks for script files which are disorganised (e.g questions

are not answered in numerical order, or where it is not clear which question

a code fragment is answering) so please make sure your file has a sensible

structure.

Guidance - Assessment criteria.

1

–  A marking scheme is given. Additionally to the marking

scheme, your code will be assessed according to the following criteria:

 Style: follow https://google.github.io/styleguide/

Rguide.xml with care;

 Writing of functions: avoid common pitfalls of local

vs global assignments; wrap your code in a coherent set of

instructions and try to make it as generic as possible; Also,

functions that are meant to be optimized with optim must

be written accordingly, see ?optim.

Executability: your code must be executable and should

not require additional code in order to run. A common pitfall

is failure to load R packages required by your code.

Deadline: Monday December 3rd, 23:59.

Individual feedback will be given.

2

Please answer all three questions. The first question is a fairly straightorward

test of Monte–Carlo integration. The last 2 questions are more conceptually

challenging, and will apply computational techniques to less artificial

problems than we have seen in lectures.

Question 1

Use Monte Carlo integration with 100,000 random numbers to evaluate the

following integrals. In your script file, report both your code, the estimated

integral value, and the Monte Carlo error.

R 3

1

x

2

e

((x?2)2

)dx, using a N(2, 1) proposal distribution [5].

R 5

1

y

5

log(y)dy using a Uniform(1,5) proposal [5].

It is well known that π is the solution to the following integral:

Z 1

0

4

1 + x

2

dx

Use Monte Carlo to approximate this integral, for sample sizes (i.e. the

number of random numbers) N ∈ (10, 100, 1000). For each value, also compute

the Monte Carlo approximation error . Write the values and the errors

in your script files [5].

3

Question 2

The below plot shows the monthly percentage returns to a financial asset over

a 20 month period. The numbers generating the plot are shown beneath it.

5 10 15 20

0.04 0.02 0.00 0.02 0.04 0.06

month

percentage change

y <- c( 0.48, 0.50, -0.86, -0.83, -0.32, -1.30, -1.42,

1.74, -0.29, -1.31, -0.07, -1.22, 3.24, -1.97,

1.81, 4.00, 1.87, 1.50, 6.81, -4.14)

In finance, it is often useful to know whether there has been a change in

the variance over time, i.e. some point k such that the variance of observations

y1, . . . , yk is equal to σ

2

1

and the variance of observations yk+1, . . . , yn

is equal to σ

2

2

, where σ

2

1

6= σ

2

2

(note that by convention, yk belongs to the

pre-change segment).

1. Assuming that the observations are Gaussian, describe how an F-test

could be used to test whether a change has occurred at location k = 10.

Clearly state the null and alternative hypotheses. (Write your words

as a comment in your R script). [2]

2. Implement this test in R and make a conclusion based on your p-value

(remember that the var.test() function carries out the F test). [3]

4

3. In practice, we do not know which specific k to test for (i.e. we do

not know in advance where the change occurred). Instead, we wish to

estimate which value of k the change occurred at. One approach for this

is to perform the F test at every possible value of k ∈ (2, 3, . . . , n ? 2),

so 17 tests in total given 20 observations (note that we need at least 2

observations in each segment to compute the variance, hence why we

do not consider k ∈ {1, n ? 1, n}).

In other words, for each value of k ∈ (2, 3, . . . , n?2), split the observations

into the sets y1, . . . , yk and yk+1, . . . , yn, then perform an F-test

and record the p-value.

After carrying out these 17 tests, the best estimate of k will be the

value of k which gives the lowest p-value since this provides the most

evidence for a difference in variance. Perform this procedure in R and

hence determine which value of k is most likely to be the change point.

in the above data. [10]

4. Next, we need to determine whether the change point we found is statistically

significant, i.e. is there really evidence to suggest that there

is a change in variance at the value of k you found above? Unfortunately,

we cannot just check whether the p-value of the F test at this

point is less than 0.05 because we did not just perform one test, we

performed 17 tests and chose the lowest p-value. This multiple testing

issue means that a more sophisticated procedure is necessary.

Instead we can use a variant of permutation testing. Let the null

hypothesis be that there is no change point anywhere, i.e. that all

observations have the same distribution. Let the alternative be that

there is a change point at some unknown value of k. If the null hypothesis

is true, we can rearrange the 20 observations in any order we like.

For each rearrangement, compute the minimum p-value over all 17 F

tests, and hence approximate the distribution of the minimum p-value

under the null hypothesis. Plot this distribution, and hence conclude

whether there is evidence to suggest that a change has occurred in the

given sequence (i.e. check if your minimum p-value from part 3. above

is in the lower 5th quintile of p-values from this null distribution). [10].

Question 3

Statistical methods can be used to determine the (unknown) author of an

unidentified piece of writing. This is known as stylometry. Research has

5

"a", "all", "also", "an", "and", "any", "are", "as", "at", "be", "been",

"but", "by", "can", "do", "down", "even", "every", "for", "from",

"had", "has", "have", "her", "his", "if", "in", "into", "is", "it",

"its", "may", "more", "must", "my", "no", "not", "now", "of", "on",

"one", "only", "or", "our", "shall", "should", "so", "some", "such",

"than", "that", "the", "their", "then", "there", "things", "this", "to",

"up", "upon", "was", "were", "what", "when", "which", "who", "will",

"with", "would", "your"

Figure 1: The 70 grammatical words that characterise writing style

shown that people differ in how frequently they use basic grammatical English

words such as ‘a’ and ‘the’. These differences are quite small (perhaps

one person only uses the word ‘the’ 1% more often than another person does)

but they do exist, and can be identified given a large enough sample of a

person’s writing. As such, if we are given a large sample of a person’s writing

and a new text that has an unknown author, then it is possible to statistically

test whether the person wrote it simply by counting up the number of

times these basic grammatical words appear in the new text, and comparing

it to their writing sample

This question will explore an example of this technique. You will download

a text file which contains counts of how often each of 3 different authors

used the 70 grammatical word from Figure 1. You will then use this to determine

which of the 3 was the most likely author of a new piece of writing that

has an unknown author. The three authors in question are Agatha Christie

(British crime noveltist), Charles Dickens, and George R R Martin (author

of Song of Ice and Fire/Game of Thrones)

First download the ‘authorship.csv” file from the course webpage and load

it into R. This contains a 3x71 matrix of counts, where the rows correspond

to each of the above three authors in order. The columns are counts of how

many times each author used each of the words from Figure 1, summed up

over all of their published books. So for example, the first column of the

first row counts how many times Agatha Christie used the word ‘a’ in her

published novels, while the 2nd column of the third row counts how many

times George R R Martin used the word ‘all’. The 71st column counts the

number of non-grammatical words each author used (i.e. every word which

wasnt one of the 70 in this list). The total number of words used by each

6

authors are equal to the row sums, i.e. 3,808,305 for Christie, 3,825,393 for

Dickens, and 1,753,671 for George R R Martin.

1. Download the authorship.csv file and then load it into R, as a numerical

matrix of counts [3].

2. For each author i, we have a 71 element vector corresponding to

the number of times they used each word. We will model this as a

Multinomial(θi) distribution. The Multinomial distributionn is a generalisation

of the Binomial distribution, where a random variabel can

take one of K different outcomes (in this case, K = 71).

For a particular author i, the unknown parameter θi

is a 71 element

vector, θi = (θi,1, θi,2, . . . , θi,71). Suppose that yi = (yi,1, yi,2, . . . , yi,71)

is the vector of counts (i.e. y1,1 is how many times author 1 used the

word ‘a’ and so on). Then the maximum likelihood estimate of θi

is:

θi,k =

yi,k

P71

j=1 yi,j

(i.e. the MLE for the proportion of times each word is used by author

i is simply the empirical proportion of times they used that word).

Write R code to compute the maximum likelihood estimate of the 71-

element θi vector for each of the three authors and report each one as

a seperate commented line in your script file [5].

3. Download the ‘unknowntext.csv’ file from the course website and load

it into R [2].

4. The unknowntext.csv file contains an extract of 10,000 words taken

from a novel written by one of these three authors. As above, this is

a 71 element vector which counts how many times each of the above

grammatical words were used. We will try to determine which author

wrote it by testing which of the estimated ?θi parameters it is consistent

with. This can be done using hypothesis testing, i.e. for each θi we

will test:

H0 : p(z) = Multinomial(θi)

H1 : p(z) 6= Multinomial(θi)

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where z = (z1, . . . , z71) is the word counts for the unknown text. First,

normalise this vector so that it sums to 1 by dividing each element by

10,000. Next, we define the test statistic:

Ti =

X

71

k=1

(zkθi,k)

2

where zk is the normalised count for the k

th word. Compute this test

statistic for all 3 authors and write down the values in your script file.

[5]

5. Ti essentially measures the distance between the new text, and the

parameter for each author. As such for each author i, we will reject

the null hypothesis if Ti > γi and conclude that this author did not

write the text. We need to choose γi

in order to make the Type 1 error

equal to the usual 0.05 (so that we only mistakenly reject the null 5%

of the time, if the author really did write the text).

For each author, use Monte Carlo simulation to find the appropriate

value of γi

. You can do this by simulating sample data under the

assumption that the null hypothesis is true, computing the test statistic

for each simulated piece of data, and then defining γi to be the

95th quantile of these simulated test statistics. This means that if the

null hypothesis is true, only 5% of observations simulated from the

Multinomial(θi) distribution will be greater than this value of γi

.

In other words: for each i, simulate a large number (e.g. S = 100, 000)

of observations from the Multinomial(θi) with 71 categories and 10,000

words (i.e. equal to the number of words in the unknown text). Compute

the test statistic Ti above for each simulated observation. Then,

define γi to be the 0.95S largest of these values (similar to bootstrapping)

[10]

6. Based on the above, compute which of the three null hypotheses are

rejected and hence determine the most likely author of the unknown

text. [5]