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- https://stats.libretexts.org/Bookshelves/Advanced_Statistics/Time_Series_Analysis_(Aue)/4%3A_Spectral_Analysis/4.4%3A_Linear_FilteringTo investigate this in further detail, let \(\gamma_X(h)\) and \(\gamma_Y(h)\) denote the ACVF of the input process \((X_t\colon t\in\mathbb{Z})\) and the output process \((Y_t\colon t\in\mathbb{Z})\)...To investigate this in further detail, let \(\gamma_X(h)\) and \(\gamma_Y(h)\) denote the ACVF of the input process \((X_t\colon t\in\mathbb{Z})\) and the output process \((Y_t\colon t\in\mathbb{Z})\), respectively, and denote by \(f_X(\omega)\) and \(f_Y(\omega)\) the corresponding spectral densities. \[ \Psi(\omega)=\sum_{s=0}^\infty\psi_s\exp(-2\pi i\omega s)=\psi(e^{-2\pi i\omega})=\dfrac{\theta(e^{-2\pi i\omega})}{\phi(e^{-2\pi i\omega})}. \nonumber \]
- https://stats.libretexts.org/Bookshelves/Advanced_Statistics/Time_Series_Analysis_(Aue)/3%3A_ARMA_Processes/3.6%3A_Model_SelectionThe main focus is on the selection of \(p\) and \(q\) in the likely case that these parameters are unknown. Here, \(L(\phi,\theta,\sigma^2)\) denotes the Gaussian likelihood defined in (3.5.4) and \(S...The main focus is on the selection of \(p\) and \(q\) in the likely case that these parameters are unknown. Here, \(L(\phi,\theta,\sigma^2)\) denotes the Gaussian likelihood defined in (3.5.4) and \(S(\phi,\theta)\) is the weighted sum of squares in (3.5.5). The introduction of the penalty term on the right-hand side of (3.6.1) reduces the risk of overfitting. The last step in the analysis is concerned with diagnostic checking by applying the goodness of fit tests of Section 1.5.
- https://stats.libretexts.org/Bookshelves/Advanced_Statistics/Time_Series_Analysis_(Aue)/3%3A_ARMA_Processes/3.5%3A_Parameter_EstimationThe mean estimate can be obtained from rec.yw$x.mean as \(\hat{\mu}=62.26\), while the autoregressive parameter estimates and their standard errors are accessed with the commands rec.yw$ar and sqrt(re...The mean estimate can be obtained from rec.yw$x.mean as \(\hat{\mu}=62.26\), while the autoregressive parameter estimates and their standard errors are accessed with the commands rec.yw$ar and sqrt(rec.yw$asy.var.coef as \(\hat{\phi}_1=1.3316(.0422)\) and \(\hat{\phi}_2=-.4445(.0422)\).
- https://stats.libretexts.org/Bookshelves/Advanced_Statistics/Time_Series_Analysis_(Aue)/2%3A_The_Estimation_of_Mean_and_Covariances/2.1%3A_Estimation_of_the_MeanNotice that there is no difference in the computations between the standard case of independent and identically distributed random variables and the more general weakly stationary process considered h...Notice that there is no difference in the computations between the standard case of independent and identically distributed random variables and the more general weakly stationary process considered here. The first equality sign in the latter equation array follows from the fact that \(\mathrm{Var}(X)=\mathrm{Cov}(X,X)\) for any random variable \(X\), the second equality sign uses that the covariance function is linear in both arguments.
- https://stats.libretexts.org/Bookshelves/Advanced_Statistics/Time_Series_Analysis_(Aue)/4%3A_Spectral_Analysis/4.3%3A_Large_Sample_PropertiesBefore confidence intervals are computed for the dominant frequency of the recruitment data return for a moment to the computation of the FFT which is the basis for the periodogram usage. For our purp...Before confidence intervals are computed for the dominant frequency of the recruitment data return for a moment to the computation of the FFT which is the basis for the periodogram usage. For our purposes, we always use the specifications given above for the raw periodogram (taper allows you, for example, to exclusively look at a particular frequency band, log allows you to plot the log-periodogram and is the R standard).
- https://stats.libretexts.org/Bookshelves/Advanced_Statistics/Time_Series_Analysis_(Aue)/4%3A_Spectral_Analysis/4.1%3A_Introduction_to_Spectral_AnalysisIn the next section, it is established that the time domain approach (based on properties of the ACVF, that is, regression on past values of the time series) and the frequency domain approach (using a...In the next section, it is established that the time domain approach (based on properties of the ACVF, that is, regression on past values of the time series) and the frequency domain approach (using a periodic function approach via fundamental frequencies, that is, regression on sine and cosine functions) are equivalent.
- https://stats.libretexts.org/Bookshelves/Advanced_Statistics/Time_Series_Analysis_(Aue)/3%3A_ARMA_Processes/3.1%3A_Introduction_to_Autoregressive_Moving_Average_(ARMA)_ProcessesIn this chapter autoregressive moving average processes are discussed. They play a crucial role in specifying time series models for applications. As the solutions of stochastic difference equations w...In this chapter autoregressive moving average processes are discussed. They play a crucial role in specifying time series models for applications. As the solutions of stochastic difference equations with constant coefficients and these processes possess a linear structure.
- https://stats.libretexts.org/Bookshelves/Advanced_Statistics/Time_Series_Analysis_(Aue)/3%3A_ARMA_ProcessesIn this chapter autoregressive moving average processes are discussed. They play a crucial role in specifying time series models for applications. As the solutions of stochastic difference equations w...In this chapter autoregressive moving average processes are discussed. They play a crucial role in specifying time series models for applications. As the solutions of stochastic difference equations with constant coefficients and these processes possess a linear structure.
- https://stats.libretexts.org/Bookshelves/Advanced_Statistics/Time_Series_Analysis_(Aue)/2%3A_The_Estimation_of_Mean_and_CovariancesIn this brief second chapter, some results concerning asymptotic properties of the sample mean and the sample ACVF are collected. Throughout, \((X_t\colon t\in\mathbb{Z})\) denotes a weakly stationary...In this brief second chapter, some results concerning asymptotic properties of the sample mean and the sample ACVF are collected. Throughout, \((X_t\colon t\in\mathbb{Z})\) denotes a weakly stationary stochastic process with mean \(\mu\) and ACVF \(\gamma\). The mean \(\mu\) was estimated by the sample mean \(\bar{x}\), and the ACVF \(\gamma\) by the sample ACVF \(\hat{\gamma}\) defined in (1.2.1). In the following, some properties of these estimators are discussed in more detail.
- https://stats.libretexts.org/Bookshelves/Advanced_Statistics/Time_Series_Analysis_(Aue)/4%3A_Spectral_Analysis/4.5%3A_SummaryThese are based on a regression of the given data on cosine and sine functions varying at the Fourier frequencies. On the sample side, the periodogram has been shown to be an estimator for the unknown...These are based on a regression of the given data on cosine and sine functions varying at the Fourier frequencies. On the sample side, the periodogram has been shown to be an estimator for the unknown spectral density. Finally, linear filters were introduced which can, for example, be used to compute spectral densities of causal ARMA processes and to derive parametric spectral density estimators other than the periodogram.
- https://stats.libretexts.org/Bookshelves/Advanced_Statistics/Time_Series_Analysis_(Aue)/1%3A_Basic_Concepts_in_Time_Series/1.5%3A_Assessing_the_ResidualsMethod 1 (The sample ACF) It could be seen in Example 1.2.4 that, for \(j\not=0\), the estimators \(\hat{\rho}(j)\) of the ACF \(\rho(j)\) are asymptotically independent and normally distributed with ...Method 1 (The sample ACF) It could be seen in Example 1.2.4 that, for \(j\not=0\), the estimators \(\hat{\rho}(j)\) of the ACF \(\rho(j)\) are asymptotically independent and normally distributed with mean zero and variance \(n^{-1}\), provided the underlying residuals are independent and identically distributed with a finite variance.
