You are analysing data from a wind turbine with the purpose to detect and minimise defects in the internal mechanical system of the wind turbine. You observe outputs from several sensors: vibration (three sensors), acoustic emission (two sensors), strain (three sensors), torque (two sensors), and bending moment (three sensors). These measurements are known to be noisy due to measurement errors. A vector with the 13 values, x R13, is observed every hour aggregating the information in that period. (a) You have access to data from a single wind turbine in the previous 1000 days (i.e. a 24000 x 13 matrix). (1) Using SVD, explain in detail how PCA can be computed from the raw data and used to provide a 2D projection of the observed data. State any assumptions you make and indicate the dimension of all the vectors/matrices involved. [3] (ii) A 2D PCA has already been computed and the projected data is shown in Figure 1. Your boss suggests that the data can be summarised with two Normal distributions defined by their mean and covariance: 2 0 D1: 14 = [7,7", E = 1 D2: H2 =-5,- 5]", E2 2z={ i] You are informed that the two groups of data points refer primarily to "normal operating conditions" (D1) and "defect operating conditions" (D2). Discuss the pros and cons of your boss' approach to summarising the data and suggest alternatives where relevant (including detailed justification). [4] = 0 2 You are analysing data from a wind turbine with the purpose to detect and minimise defects in the internal mechanical system of the wind turbine. You observe outputs from several sensors: vibration (three sensors), acoustic emission (two sensors), strain (three sensors), torque (two sensors), and bending moment (three sensors). These measurements are known to be noisy due to measurement errors. A vector with the 13 values, x R13, is observed every hour aggregating the information in that period. (a) You have access to data from a single wind turbine in the previous 1000 days (i.e. a 24000 x 13 matrix). (1) Using SVD, explain in detail how PCA can be computed from the raw data and used to provide a 2D projection of the observed data. State any assumptions you make and indicate the dimension of all the vectors/matrices involved. [3] (ii) A 2D PCA has already been computed and the projected data is shown in Figure 1. Your boss suggests that the data can be summarised with two Normal distributions defined by their mean and covariance: 2 0 D1: 14 = [7,7", E = 1 D2: H2 =-5,- 5]", E2 2z={ i] You are informed that the two groups of data points refer primarily to "normal operating conditions" (D1) and "defect operating conditions" (D2). Discuss the pros and cons of your boss' approach to summarising the data and suggest alternatives where relevant (including detailed justification). [4] = 0 2
