4. An automobile engineer wants to model the relation between the accelerator control and the velocity...

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4. An automobile engineer wants to model the relation between the accelerator control and the velocity of the car. The relation may not be simple since there is a lag in depressing the accelerator and the car actually accelerating. To determine the relation, the engineers measures the acceleration control input xk and velocity of the car yk at time instants k = 0,1,..., T1. The measurements are made at some sampling rate, say once every 10 ms. The engineer then wants to fit a model of the form M N = aj Yk-j + bjxk-j+k, Yk = j=1 j=0 (1) for coefficients a; and bj. In engineering this relation is called a linear filter and it statistics it is called an auto-regressive moving average (ARMA) model. (a) Describe a vector with the unknown parameters. How many unknown parameters are there? (b) Describe the matrix A and target vector y so that we can rewrite the model (1) in matrix form, y = A+ . Your matrix A will have entries of y and xk in it. (c) (Graduate students only) Show that, for TN and T M, the coefficients of (1/T) ATA and (1/T)Ay can be approximately computed from the so-called auto- correlation functions T-1 1 Ray (l) = xkYk+, Ryy (l): T T-1 T-1 1 = T YkYk+l, Rxx(l) = xkxk+l, k=0 k=0 k=0 In the sum, we take k = 0 or Yk == = 0 whenever k < 0 or k > T. 4. An automobile engineer wants to model the relation between the accelerator control and the velocity of the car. The relation may not be simple since there is a lag in depressing the accelerator and the car actually accelerating. To determine the relation, the engineers measures the acceleration control input xk and velocity of the car yk at time instants k = 0,1,..., T1. The measurements are made at some sampling rate, say once every 10 ms. The engineer then wants to fit a model of the form M N = aj Yk-j + bjxk-j+k, Yk = j=1 j=0 (1) for coefficients a; and bj. In engineering this relation is called a linear filter and it statistics it is called an auto-regressive moving average (ARMA) model. (a) Describe a vector with the unknown parameters. How many unknown parameters are there? (b) Describe the matrix A and target vector y so that we can rewrite the model (1) in matrix form, y = A+ . Your matrix A will have entries of y and xk in it. (c) (Graduate students only) Show that, for TN and T M, the coefficients of (1/T) ATA and (1/T)Ay can be approximately computed from the so-called auto- correlation functions T-1 1 Ray (l) = xkYk+, Ryy (l): T T-1 T-1 1 = T YkYk+l, Rxx(l) = xkxk+l, k=0 k=0 k=0 In the sum, we take k = 0 or Yk == = 0 whenever k < 0 or k > T.