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, . . . , T ? 1. The measurements are made at some sampling rate, say once every 10 ms. The engineer then wants to fit a model of the form yk = X M j=1 ajyk?j + X N j=0 bjxk?j + k, (1) for coefficients aj 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 yk and xk in it. (c) (Graduate students only) Show that, for T N and T M, the coefficients of (1/T)ATA and (1/T)ATy can be approximately computed from the so-called autocorrelation functions Rxy(`) = 1 T T X?1 k=0 xkyk+` , Ryy(`) = 1 T T X?1 k=0 ykyk+` , Rxx(`) = 1 T T X?1 k=0 xkxk+` , In the sum, we take xk = 0 or yk = 0 whenever k

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 xx and velocity of the car yx at time instants k = 0, 1, .... T - 1. 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 yk = Lajyr-it _bick-it ek, (1) j=1 j=0 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 S 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 = AB + E. Your matrix A will have entries of yx and Tx in it. (c) (Graduate students only) Show that, for T > N and T >> M, the coefficients of (1/T)A A and (1/T)A y can be approximately computed from the so-called auto- correlation functions "-1 T-1 T-1 Rry (() = I kykte, Ryy(() = T ykykte, RII(!) = ) =0 *=0 *=0 In the sum, we take rx = 0 or yx = 0 whenever k T