Industrial Data Analytics for Diagnosis and Prognosis. Yong Chen

straight x subscript ij minus straight x with bar on top subscript straight j right parenthesis left parenthesis straight x subscript ik minus straight x subscript straight k right parenthesis over denominator straight n minus 1 end fraction."/> (2.5)

The diagonal elements of S, sjj, j = 1,…,p are the sample variance of the jth variable. It is easy to see that when k = j, the sample covariance in (2.5) is equal to sj², the sample variance of the jth variable. So both notations sjj and sj² represent the sample variance of xj. It is also obvious from (2.5) that skj. So the sample covariance matrix S is a symmetric matrix. The sample covariance matrix S can also be written by the observation vector xi as

       (2.6)

      Similarly, we define the sample correlation matrix as

      The (j, k)th element of R is the sample correlation of the jth and kth variables:

      The sample correlation between a variable and itself is equal to 1. So the diagonal elements of a sample correlation matrix are all equal to 1. The sample correlation matrix R is obviously symmetric since rjk = rkj.

      Example 2.4 Consider the data set in Table 2.1. In Example 2.2, we found that x̄₁ = 2479.5 and x̄₂ = 170.35. Similarly, we can obtain x̄₃ = 65.41. So the mean vector of x = (x₁ x₂ x₃)T is given by

      In Example 2.2, we calculated the sample variances, sample covariance, and sample correlation of x₁ and x₂. Similarly, we can obtain the sample variance of x₃ and its sample covariance and correlation with the other two variables as

      Note that while s₂₃ is much smaller than s₁₃, r₂₃ is greater than r₁₃, which indicates that the linear association between x₂ and x₃ is stronger than that of x₁ and x₃. This clearly shows that the magnitude of the covariance itself is not meaningful in characterizing how strong the relationship of two variables is. Combining all the sample variance, covariance, and correlation information, the sample covariance matrix and sample correlation matrix of x = (x₁ x₂ x₃)T can be written as

2.2.3 Linear Combination of Variables

      We are often interested in some linear combinations of the variables x₁, x₂,…, xp. For example, for the auto_spec data set, two of the variables are city.mpg and highway.mpg. If you expect that 60% of the mileage for a car is on highway and 40% is on local roads, then the average MPG for a car can be estimated as 0.6 × highway.mpg + 0.4 × city.mpg, which is a linear combination of city.mpg and highway.mpg. In general, let c₁, c₂,…, cp be constants and consider the linear combination of the variables x₁, x₂,…, xp given by

      For each observation of the data set, the corresponding value of the variable z can be found by

      where cT = (c₁ c₂ … cp). It can be seen that the sample mean of z is

(2.7)

      The sample variance of z can be found as

(2.8)

      Because sample variance is always non-negative, for any c ∈ ℛp we have c^T Sc ≥ 0 from (2.8). Therefore, the sample covariance matrix S is always a positive semidefinite matrix.

      In general, if we have q linear combinations of x₁, x₂,…, xp defined