Industrial Data Analytics for Diagnosis and Prognosis - Yong Chen

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      Based on (3.30) and (3.31), the following R codes calculate the posterior mean and covariance matrix for μ using the first five (n = 5) observations in the data set.

      The posterior mean and covariance matrix are obtained as

table row cell mu subscript 5 equals open parentheses table row 1930 row 1856 row 1854 end table close parentheses comma capital sigma subscript 5 equals open parentheses table row cell 83.37 end cell cell negative 2.61 end cell cell 2.83 end cell row cell negative 2.61 end cell cell 46.85 end cell cell 15.37 end cell row cell 2.83 end cell cell 15.37 end cell cell 48.235 end cell end table close parentheses. end cell end table

      Compared to the sample mean of the first five observations, which is (1943 1850 1838)T, the posterior mean has some deviations from both the sample mean and the prior mean μ0. Now we use the first 100 (n = 100) observations to find the posterior mean by changing n in the R codes from 5 to 100. The posterior mean and covariance matrix are

table row cell mu subscript 100 equals open parentheses table row 1940 row 1849 row 1865 end table close parentheses comma capital sigma subscript 100 equals open parentheses table row cell 20.28 end cell cell negative 0.87 end cell cell 1.03 end cell row cell negative 0.87 end cell cell 4.97 end cell cell 2.72 end cell row cell 1.03 end cell cell 2.72 end cell cell 5.235 end cell end table close parentheses. end cell end table

      Compared to the sample mean vector of the first 100 observations, which is (1944 1849 1865)T, the posterior mean with n = 100 observations is very close to the sample mean, while the influence of the prior mean is very small. In addition, the posterior variance for the mean temperature at each of the three locations is much smaller for n = 100 than for n = 5.

      Bibliographic Notes

      Exercises

      1 Consider two discrete random variables X and Y with joint probability mass function p(x, y) given in the following table:
x y p(x, y)
–1 –1 0.24
–1 1 0.06
0 –1 0.16
0 1 0.14
1 –1 0.40
1 1 0.00

      1 Find the mean vector, the covariance matrix, and the correlation matrix of the random vector (X Y )T.

      1 A random vector X = (X1 X2 X3 X4)T has mean vector and covariance matrix given as

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      1 Let and . Please findE(X1)E(AX1)cov(X1)var(AX1)E(X2)E(BX2)cov(X2)cov(BX2)

      1 Repeat Exercise 2, but with A and B replaced by

 table row cell bold A equals open parentheses table row 1 row cell negative 1 end cell end table close parentheses space and space bold B equals open parentheses table row 2 cell negative 1 end cell row 0 2 end table close parentheses. end cell end table

      1 Let X = (X1 X2 X3)T be a random vector with X ∼ N(μ, Σ) with

 table row cell bold mu equals open parentheses table row cell negative 2 end cell row 1 row 4 end table close parentheses comma bold capital sigma equals open parentheses table row 1 cell negative 2 end cell 0 row cell negative 2 end cell 5 0 row 0 0 4 end table close parentheses. end cell end table

      1 Which of the following random variables are independent? Please explain.X1 and X2X2 and X3X1 and X3(X1, X2) and X3 and X3

      1 Consider the random vector X in Exercise 2.Find the distribution of X2 + X3 + X4.Find the distribution of 3X2 − 2X3 + X4.Find the joint distribution of X2 + X3 + X4 and 3X2 − 2X3 + X4.Find the distribution of X1 − X2 + 2X3 + X4Find a 2×1 vector c such that X2 and are independent.Find a 2×1 vector c such that X2 and are independent.

      2 Consider the random vector in Exercise 4.Find the conditional distribution of X1, given that X3 = x3.Find the conditional distribution of X1, given that X2 = x2.

      3 Consider the random vector X in Exercise 2.Find the conditional distribution of X1, given that X2 = x2 and X3 = x3.Find the conditional distribution of X2, given that X3 = x3 and X4 = x4.Find the conditional distribution of X3, given that X2 = x2 and X4 = x4.Find the conditional distribution of (X2 X3)T, given that X4 = x4.

      4 Calculate by hand the maximum likelihood estimates of the mean vector μ and the covariance matrix Σ of (X2 X3)T based on the first five observations of the last two variables in Table 2.1, assuming the observations are from a bivariate normal population.

      5 Consider a random sample of size n = 3 from a bivariate normal population as shown in the following table.
x 1 x 2
5 8
9 5
7 2

      1 Evaluate the T2-statistic used to test H0 : μ = μ0 based on this data set, where μ0 (8 4)T. What is the distribution of the T2-statistic in this case?

      1 Consider the data from a bivariate normal population in the following table:

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