For each of the following first construct an example and then show that it has the correct properties: (a) (Xt) with constant mean but has a variance that is a function of time. (b) (Wt) white noise process that is not strongly stationary. (c) (Zt) is nonstationary process with an autocovariance function such that γ(t, t) = σ 2 for all t. (d) (Vt) is nonstationary with an autocovariance function such that γ(t, t+ h) = 0 for all |h| > 1.
For each of the following first construct an example and then show that it has the...
Let Wt de a (Gaussian) white noise with variance σ 2 . Then, let
Xt = WtWt−1 + µ, where µ is a real constant. Determine the mean and
autocovariance of (Xt)? Is this process stationary?
Let W, de a (Gaussian) white noise with variance σ2. Then, let of where μ is a real constant. Determine the mean and (X)? Is this process stationary?
2.6 Consider a process consisting of a linear trend with an additive noise term consisting of independent random variables wt with zero means and variances o that is where Bo, B1 are fixed constants (a) Prove t is nonstationary. (b) Prove that the first difference series Vxt finding its mean and autocovariance function Xt t-s stationary by
2. Let (et) be a zero mean white noise process with variance 1. Suppose that the observed process is h ft + Xt where β is an unknown constant, and Xt-et- Explain why {X.) is stationary. Find its mean function μχ and autocorrelation function p for lk0,1,.. a. b. Show that {Yt3 is not stationary. C. Explain why w. = ▽h = h-K-1 is stationary. d. Calculate Var(Yt) Vt and Var(W) Vt . (Recall: Var(X+c)-Var(X) when c is a constant.)...
2. (a) Consider the following process: where {Z) is a white noise process with unit variance. [1 mark] ii. Find the infinite moving average representation of X,i.e., find the scquence [6 marks] i. Explain why the process is stationary. (6) such that Xt = Σ b,2-j. iii. Calculate the mean and the autocovariance "Yo, γι and 72 of the process. 7 marks iv. Given 40 = 0.1 and Xo = 1.8, find the 2-step ahead forecast of the time series...
11.8 A linear system has a transfer function given by H(W) + 15w+50 Determine the power spectral density of the output when the input function is a. a stationary random process X(t) with an autocorrelation function, Rxx(t)=10e ! b. white noise that has a mean-square value of 1.2 V/Hz
Q1) Let X(t) be a zero-mean WSS process with X(t) is input to an LTI system with Let Y(t) be the output. a) Find the mean of Y(t) b) Find the PSD of the output SY(f) c) Find RY(0) ------------------------------------------------------------------------------------------------------------------------- Q2) The random process X(t) is called a white Gaussian noise process if X(t) is a stationary Gaussian random process with zero mean, and flat power spectral density, Let X(t) be a white Gaussian noise process that is input to...
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4. Which of the following sets of characteristics would usually best describe an autoregressive process of order 3 (i.e. an AR(3)? (a) A slowly decaying acf, and a pacf with 3 significant spikes (b) A slowly decaying pacf and an acf with 3 significant spikes (c) A slowly decaying acf and pacf (d) An acf and a pacf with 3...
2.4 Let (e) be a zero mean white noise process. Suppose that the observed process is Y = e, + 0,-1, where is either 3 or 1/3. (a) Find the autocorrelation function for {Y} both when 0 = 3 and when 0 = 1/3. (b) You should have discovered that the time series is stationary regardless of the value of and that the autocorrelation functions are the same for 0 = 3 and 0 = 1/3. For simplicity, suppose that...
5. For ridge regression, we choose parameter estimators b which minimise j-0 where A is a constant penalty parameter. (a) Show that these estimators are given by (b) Calculate the ridge regression estimates for the data from Q2 with penalty parameter λ-0.5 In order to avoid penalising some parameters unfairly, we must first scale every predictor variable so that t is standardised (mean 0, variance 1), and centre the response variable (mean 0), in which case an intercept parameter...
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A random Process XCt) has an auto Correlation function Rxx (T) = 9+2e 121 a) find the mean of Xct) b) If X(+) is the input to a system having an impulse response h(t)= e Btult) (Where is positive), find the mean value of the output process