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The following Toeplitz auto-correlation matrix R = E[x(nJx" (n)] , with x(n) = [x(n) I(n - N +1]" and where x(n) are observed time series samples from sta...

Question

The following Toeplitz auto-correlation matrix R = E[x(nJx" (n)] , with x(n) = [x(n) I(n - N +1]" and where x(n) are observed time series samples from stationary ergodic random process; has the following decomposition with N = 20.99 R = 0.991.99 0.01 44 72 7 A. Find the KLT transformation leading to x'(n) with a diagonal auto-correlation matrix What is the resulting diagonal matrix? Find the transformation leading with a identity auto-correlation matrix: C. How could the KLT trans

The following Toeplitz auto-correlation matrix R = E[x(nJx" (n)] , with x(n) = [x(n) I(n - N +1]" and where x(n) are observed time series samples from stationary ergodic random process; has the following decomposition with N = 2 0.99 R = 0.99 1.99 0.01 44 72 7 A. Find the KLT transformation leading to x'(n) with a diagonal auto-correlation matrix What is the resulting diagonal matrix? Find the transformation leading with a identity auto-correlation matrix: C. How could the KLT transformation be used to perform compression with Principal Component Analysis (PCA)?



Answers

A subsample of a random sequence $X[n]$ is obtained by observing every $k$ th element of
the sequence, for some integer $k>1 .$ The resulting random sequence, $Y[n],$ is given by $Y[n]=X[k n] .$
(a) Find the mean and autocorrelation functions of $Y[n]$ in terms of those of $X[n]$ .
(b) If $X[n]$ is WSS, is the subsample $Y[n]$ also WSS?

Yeah, Part one. The first order. Auto correlation. Issa. Correlation of a time series variable with its first bag. We have four Siri's to examine the first order auto relation here. The first one is the lock of I M V P C. It's row one. That's how we did. Note the first order auto correlation coefficient. It's about 0.6 39 for the same Siri's, but the trended row one hat become smaller. It is 0.485 mhm, or the log of price we get. Row one had equals 10.9 for nine. And after we d train this Siri's through one hat is still very high 0.8 to 2. So for the Siri's I M v p c, we can roll out Ah, unit root in this variable. But for the other variable price, we cannot confidently say that this this Siri's is not a unit route. And if you recall, you need root Refers to you a time Siri's that is highly persistent for a unit root. Siri's Row one hat is roughly one right here. This is our estimated equation. Okay, The coefficient on the first difference of lock price implies that a 1% point increased in the growth in price leads to you a 3.88 percent increase in housing investment and because we have a temp trend in the right hand side of the equation. So this increase is compared to the trend. This is an increase above the trend of housing investment Park pre. If we regress the de Trended Law I m. V p. C. On the first difference of law, price and the time trend, we wouldn't get an estimation result with our square of 0.303 This is much lower than what we got previously when we do not d train lock of I M v p. C. In this part, the first difference in law price or the growth in law price, explains Onley. About 30% of the variation in the log of I M v P C above its trend at four. This is the estimated equation. The coefficient on the difference of lock price has reduced substantially. It is no longer significant at the 5% level against a positive one sided alternative. The out square is also much smaller, indicating that the change in love price explains very little variation in the change of lock I M V P. C. Different thing eliminates linear time trends. That's why the estimate on the trend thanks one is very small and very statistically insignificant.


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