代写Statistical Programming作业、代做R语言程序作业、代做fragment作业、代写R编程作业
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)
7
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
.http://www.daixie0.com/contents/18/2079.html
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]
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