STA437H1S/2005

Sebastian Maier博士（统计科学系）金融随机方法II第3周模块概要（第1部分）
引言（第1-2章）数学背景（第3-4章）投资时机（第5章）操作时间（Ch 6）进入、退出、搁置和报废（第7章）Sebastian Maier博士（统计科学系）金融随机方法II第3周讲座提纲基本型号通过动态编程的解决方案通过或有索赔分析解决方案最优投资的特点
替代随机过程Sebastian Maier博士（统计科学系）金融随机方法II第3周3/28基本模型：最佳时机假设项目价值V根据GBM发展，即。，dV=αVdt+σVdz，可以在I的沉没成本下获得投资的最佳时机是什么时候？Sebastian Maier博士（统计科学系）金融随机方法II第3周Assignment #2 STA437H1S/2005H1Sdue Friday February 17, 2023

1.Instructions

Solutions to problems 1–3 are to be submitted on Quercus (PDF files only).1. Andrews curves (conceived the University of Toronto’s own David Andrews) represent aninteresting approach to multivariate visualization. The idea is to represent each multivariateobservation (xi1, · · · , xip) (which is possibly normalized) by a sinusoidal function on [0, 1]:gi(t) =+ xi2 sin(2pit) + xi3 cos(2pit) + xi4 sin(4pit) + xi5 cos(4pit) + · · ·Observations that are similar will have similar Andrews curves while outlying observationswill often have curves that are distinctively different.On Quercus, there is a file andrews.txt, which contains a function andrews that computesAndrews curves for a data matrix whose columns are variables and rows are observations;for example, source("andrews.txt") # read the function into R> x <cbind(rnorm(100),rnorm(100),rnorm(100),rnorm(100),rnorm(100))> r <- andrews(x,scale=T) # scales columns to have mean 0 and variance 1The file testdata.txt contains 100? k observations from a 10-variate normal distributionand k outliers generated from another distribution (where k ≤ 15).(a) Look at the data using Andrews curves. How many clear outliers do there seem to be?(b) Using the information from the Andrews curves as well as pairwise scatterplots, principalcomponents etc, give an estimate of how many outliers are in the data.2. (a) If {gi(t)} are the Andrews curves defined in question 1, show that2xi, what is the Andrews curve of xˉ?(c) Suppose that xk lies on a line between xi and xj, that is, xk = λxi + (1? λ)xj for some0 < λ < 1. What can you say about the Andrews curve of xk relative to those of xi and xj?3. In Assignment #1, you looked at two dimensional scatterplots of data on two species ofrock crabs; here, you will do a principal components analysis of these data.As before, the data are in a file crabs.txt on Quercus; the columns of the file are species (Bor O), sex (M or F), index (1-50 within each species-sex combination), width of the frontallip (LP), the rear width of the shell (RW), length along the midline of the shell (CL), themaximum width of the shell (CW), and the body depth (BD).The data can be read into R using the following code:> x <- scan("crabs.txt",skip=1,what=list("c","c",0,0,0,0,0,0))> colour1 <- ifelse(x[[1]]=="B","blue","orange") # species colours> colour2 <- ifelse(x[[2]]=="M","black","red") # sex colours> sex <- x[[2]]

> FL <- x[[4]]
> RW <- x[[5]]
> CL <- x[[6]]
> CW <- x[[7]]
> BD <- x[[8]]

(a) Using the correlation matrix, do a principal component analysis of the 5 variables.> r <- princomp(~FL+RW+CL+CW+BD,cor=T)> summary(r,loadings=T)Give an interpretation of the first two principal components based on their loadings.(b) Look at pairwise scatterplots of the 5 principal components using colour1 to distinguishthe two species: pairs(r$scores,col=colour1)Which pairs of principal components seem to separate the two species?(c) Now look at pairwise scatterplots of the 5 principal components using colour2 to dis-tinguish the two sexes: pairs(r$scores,col=colour2)Which pairs of principal components seem to separate the two sexes?(d) Suppose you are given the following measurements for the 5 variables: FL = 18.7,RW = 15.0, CL = 35.0, CW = 40.3, BD = 16.6. What is your prediction of the species andsex of this crab?
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