统计代写|时间序列分析作业代写time series analysis代考|Other Oscillations

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

统计代写|时间序列分析作业代写time series analysis代考|Other Oscillations

The Madden-Julian Oscillation (MJO) is another unusual phenomenon both because it is not firmly fixed geographically and because it presents an oscillatory system not related to tides or to a seasonal trend. A review of MJO can be found in Zhang (2005).
Strictly speaking, the MJO phenomenon is a vector process and its spectra should be estimated in agreement with the approach discussed in Thomson and Emery $(2014$, Chap. 5). However, having in mind the methodological goals of the book, the MJO components will be treated here as either two scalar time series (this chapter and Chap. 6) or as a bivariate process (Chap. 8).

The MJO data used here consist of daily MJO indices RMM1 and RMM2 from January 1, 1979 through April $30,2017(N=14000, \Delta t=1$ day). Thus, MJO is a bivariate random process. The source of the data is the Australian Bureau of Meteorology, site #16 in Appendix. The graph of the time series is shown in Fig. 5.6a. The hypothesis of stationarity can be accepted through visual assessment, but it is also confirmed by using the method described in Chap. 4. The spectral densities of the time series components are very similar and contain a single wide peak at the frequency close to $0.02$ cpd. The spectral estimates are shown in Fig. $5.6 \mathrm{~b}$ for the part of the frequency axis up to $0.05$ cpd; at higher frequencies, the spectrum is monotonically decreasing. The confidence limits are not shown because they almost coincide with the spectra due to the high reliability of estimates obtained with these long time series. The contribution of higher frequencies is negligibly small. Thus, the Madden-Julian Oscillation presents a good example of an oscillatory system. The statistical predictability criterion $r_{e}$ (1) given with Eq. (3.7) amounts to about $0.98$, meaning that both components possess high statistical predictability at the unit lead time, that is, at 1 day.

统计代写|时间序列分析作业代写time series analysis代考|General Remarks

Forecasting geophysical processes is probably the most desired goal in Earth and related solar sciences. Reliable predictions are needed at time scales from hours and days (meteorology, hydrology, etc.) to decades and centuries (climatology and related sciences). With one exception, all geophysical processes in the atmosphereocean-land-cryosphere system are random, which means that none of them can be predicted at any lead time without an error. The exception is tides-a deterministic process which exists in the oceans, atmosphere, and in the solid body of the planet. The knowledge of tides is especially important for the oceans, and tides in the open ocean can be predicted almost precisely. Along the shorelines where tides play an important role, sea level variations can generally be predicted with sufficient accuracy as well, but there may be some cases when random disturbances should also be taken into account (Munk and Cartwright 1966).

The behavior of another astronomically caused process – the seasonal trend-is so irregular that one cannot even say for sure whether the next summer (or any other season) will be warmer or cooler than the current one.

The atmospheric, oceanic, terrestrial, and cryospheric processes and their interactions can be described with fluid dynamics equations; however, the equations are complicated, numerous, and cannot be solved analytically. Getting reliable numerical solutions encounters serious physical and computational problems, which cannot be discussed in this book. However, there is at least one important example of successful numerical solution of prediction problems-the weather forecasting. The forecasts given by meteorologists are reliable and rarely contain serious errors at lead times at least up to about a week. These forecasts are obtained by uploading information about the current (initial) state of processes involved in weather generation into a numerical computational scheme having discrete temporal and spatial resolution and then running the scheme forward in time and space to obtain forecasts. As the knowledge of the initial conditions cannot be ideal, the forecasts contain errors. Besides, the computational grid is discrete so that the processes whose scales are smaller than the distance between the grid nodes and shorter than the unit time step cannot be directly taken into account. The errors in the initial and other conditions grow with the forecast lead time, and eventually, the variance of the forecast errors becomes equal to the variance of the process that is being forecasted. The forecast becomes unusable. It means that the process has a predictability limit; the limit should be defined quantitatively through the ratio of the forecast error variance as a function of lead time to the variance of the process. These issues have been discussed in a number of classical works by Lorenz $(1963,1975,1995)$.

For weather forecasting, the predictability limit at which the error variance approaches the variance of the process amounts to about a week or slightly longer. The numerical models of climate used, first of all, to assess the influence of anthropogenic factors upon future climate require the same equations and initial conditions as in weather forecasting; however, the temporal and spatial resolutions of numerical climate models are much less detailed and the models cannot predict the natural variability of climate. This may be the reason (or one of the reasons) why the results of climate simulations with numerical general circulation models that show the behavior of climate in the twenty-first century are called climate projections rather than climate predictions (IPCC 2013).

Thus, the numerical models cannot ensure predictions beyond the predictability limits defined in accordance with Lorenz’s ideas. Under this situation, it becomes quite reasonable to deal with the problem by trying to predict the state of Earth system’s elements at lead times of weeks, months, or even longer by using the probabilistic approach (also see Lorenz 2007). This is the subject discussed in this chapter: probabilistic (traditionally – statistical) extrapolation of geophysical time series based upon information about their behavior in the past. The term “extrapolation” is equivalent to prediction and forecasting; for example, forecasting a geophysical process by its behavior in the past and, possibly, by the past behavior of other predictors is nothing else but an extrapolation of a random process.

统计代写|时间序列分析作业代写time series analysis代考|Method of Extrapolation

In both scalar and multivariate cases, the extrapolation means a forecast of the time series on the basis of its behavior in the past. The method of extrapolation used in this book to predict the behavior of stationary geophysical time series is based upon the autoregressive modeling (Box et al. 2015). It is discussed in this chapter for the case of scalar time series $x_{t}$ known over a finite time interval from $t=\Delta t$ through $t=N \Delta t$. The sampling interval $\Delta t$ is the unit time step, which can be a minute, hour, month, year, or whatever the data prescribes. Here, $\Delta t=1$. The only

assumption made about the time series $x_{i}$ is that it presents a sample record of a stationary random process.

The first stage of extrapolation procedure is to approximate the scalar time series with an AR model of a properly selected order $p$. The result of approximation is
$$x_{t}=\varphi_{1} x_{t-1}+\cdots+\varphi_{p} x_{t-p}+a_{t}$$
where $\varphi_{j}, j=1, \ldots, p$ are the AR coefficients and $a_{t}$ is a zero mean innovation sequence (white noise) with the variance $\sigma_{a}^{2}$.

Equation (6.1) describes the time series as a function of its behavior in the past, that is, exactly what is required for time series extrapolation. The unknown true value of the time series at lead time $\tau$ is
$$x_{t+\tau}=\varphi_{1} x_{t+\tau-1}+\cdots+\varphi_{p} x_{t+\tau-p}+a_{t+\tau}$$
so that at the lead time $\tau=1$
$$x_{t+1}=\varphi_{1} x_{t}+\cdots+\varphi_{p} x_{t-p+1}+a_{t+1}$$
At time $t$, all terms in the right-hand side of this equation, with the exception of $a_{t+1}$, are known because they belong to the observed initial time series. Therefore, the extrapolated (predicted, forecasted) value of the time series at the unit lead time is
$$\hat{x}{t}(1)=\varphi{1} x_{t}+\cdots+\varphi_{p} x_{t-p+1} .$$
As the extrapolation error at the unit lead time is $a_{t+1}$, its variance is $\sigma_{a}^{2}$. For $\tau=2$, one has
$$\hat{x}{t}(2)=\varphi{1} \hat{x}{t}(1)+\cdots+\varphi{p} x_{t-p+2}$$
so that the extrapolation error will be the sum of $\sigma_{a}^{2}$ with the error at $\tau=1$ (that is, $\sigma_{a}^{2}$ ) multiplied by the autoregression coefficient $\varphi_{1}$. The general solution for the extrapolation of an AR ( $p$ ) sequence at the lead time $\tau$ is
$$\hat{x}{t}(\tau)=\varphi{1} \hat{x}{t}(\tau-1)+\cdots+\varphi{p} \hat{x}{t}(\tau-p)$$ where $\hat{x}{t}(\tau-k)=x_{t+\tau-k}$ are the known time series elements if $\tau \leq k$.
Let
$$\varepsilon_{t}(\tau)=x_{t+\tau}-\hat{x}{t}(\tau)$$ be the error of extrapolation from time $t$ at lead time $\tau$; its variance $\sigma{\varepsilon}^{2}(\tau)$ can be defined in the following manner. In the operator form, Eq. (6.1) is

$$x_{t}=\left(1-\varphi_{1} B-\cdots-\varphi_{p} B^{p}\right)^{-1} a_{t}$$
or
$$x_{t}=\boldsymbol{\Phi}^{-1}(B) a_{t} .$$

统计代写|时间序列分析作业代写time series analysis代考|Other Oscillations

Madden-Julian 振荡 (MJO) 是另一种不寻常的现象，既因为它在地理上没有牢固固定，也因为它呈现出与潮汐或季节性趋势无关的振荡系统。可以在 Zhang (2005) 中找到对 MJO 的评论。

统计代写|时间序列分析作业代写time series analysis代考|Method of Extrapolation

X吨=披1X吨−1+⋯+披pX吨−p+一种吨

X吨+τ=披1X吨+τ−1+⋯+披pX吨+τ−p+一种吨+τ

X吨+1=披1X吨+⋯+披pX吨−p+1+一种吨+1

X^吨(1)=披1X吨+⋯+披pX吨−p+1.

X^吨(2)=披1X^吨(1)+⋯+披pX吨−p+2

X^吨(τ)=披1X^吨(τ−1)+⋯+披pX^吨(τ−p)在哪里X^吨(τ−ķ)=X吨+τ−ķ是已知的时间序列元素，如果τ≤ķ.

e吨(τ)=X吨+τ−X^吨(τ)是时间外推的误差吨在交货时间τ; 它的方差σe2(τ)可以通过以下方式定义。在运算符形式中，方程式。(6.1) 是X吨=(1−披1乙−⋯−披p乙p)−1一种吨

X吨=披−1(乙)一种吨.

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