### 统计代写|时间序列分析代写Time-Series Analysis代考|STAT8002

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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|时间序列分析代写Time-Series Analysis代考|Practical Analysis of Time Series

Abstract After a preliminary processing, the time series should be tested for stationarity. The test may fail if the time series contains a trend or if its mean value, variance, or spectrum are found to be time-dependent. Deleting the trend can be justified if it is caused by external factors or if it interferes with the higher-frequency part of the spectral density of interest to the researcher. A test for stationarity is suggested through splitting the time series in halves and estimating the mean values, variances and, if possible, spectral densities of the entire time series and its halves. The confidence bounds for estimates of statistical moments depend upon the number of independent observations in the time series. These numbers depend upon the correlation structure of the time series, and they can be much smaller than the total number of terms in the time series. A linear filtering is generally not recommended. The autoregressive approach allows one to determine frequencies of even strictly periodic oscillations contained in the time series (tides) with exceptionally high accuracy providing that the time series is long. A detailed example of autoregressive analysis is given.
The two mandatory requirements in practical analysis of time series are

• using the proper methods of analysis and
• calculating confidence intervals for all estimated statistical characteristics.
This means, in particular, that the methods of time and frequency domain analysis should be mathematically suitable for the time series which is being analyzed. In this book, the fundamental approach to analysis is based upon autoregressive modeling, which has the ability to provide relatively reliable estimates of time series statistics even with short time series. Its other advantage is the explicit time domain model which is obtained at the initial stage of autoregressive analysis and which does not exist if a nonparametric approach is used. It will be shown later in this chapter (Sect. 4.5) that the autoregressive approach can be quite effective for estimating the spectral density of time series with a very complicated structure.

If the time series is short, its spectrum should not be estimated with nonparametric methods other than Thomson’s MTM. Moreover, all statistical estimates should be accompanied with respective reliability estimates. Any estimate of statistical characteristics is absolutely useless if it is not supplemented with confidence intervals or

some other quantitative indicator of its reliability in accordance with mathematical statistics. If confidence intervals or reliability estimates are available but not shown intentionally, their absence should be explained.

If the time series contains outliers, they should be handled before continuing the analysis. The next step is to build an autoregressive model of the time series, including respective spectral estimate, and to check for any unexpected features such as a significant low-frequency trend and/or unexplainable spectral peaks in the model selected by most order selection criteria. The latter problem is rare because normally the order selection criteria do not allow unreasonably high or low orders. The information obtained at that stage should be used to decide if any additional steps might be required.
The common problems of this type include

• an incorrect sampling interval,
• the presence of a linear or nonlinear trend in the time series,
• the presence of strong high-frequency and/or quasi-periodic fluctuations in the AR model’ spectrum, which need to be explained.

The trend can be a product of natural low-frequency components or it can be caused by some artificial external forcing (such as an anthropogenic effect). The decision whether to delete the trend or analyze the time series as is should not be taken without a justification.

## 统计代写|时间序列分析代写Time-Series Analysis代考|Selecting the Sampling Interval

The sampling interval $\Delta t$ should be set in agreement with the task of the analysis. If it is the climate variability at time scales longer than $2.5-3$ years, the interval $\Delta t=1$ year is generally sufficient. The resulting highest frequency in the time series spectrum is the Nyquist frequency $f_{N}=1 / 2 \Delta t$, that is, $0.5$ cpy when $\Delta t=1$ year. The frequencies that can be analyzed reliably begin from approximately $0.30-$ $0.35$ cpy. Simple interpolation between consecutive terms of the time series intended to get a faster sampling rate is useless. If the spectrum is expected to contain high energy at higher frequencies, the interval should be smaller; such cases are rather rare in Earth sciences. Setting $\Delta t=1$ month for studying climate variability is normally not reasonable, in particular, because it may transform a stationary time series into a sample of a periodically correlated (cyclostationary) random process. Also, an exceedingly small sampling interval creates redundant information and reduces the spectral resolution at lower frequencies. The general rule here is that the time series should contain at least several measurements per the smallest time scale of interest. For example, the choice of $\Delta t=1$ year could be too large for studying the QuasiBiennial Oscillation whose characteristic time scale is approximately $2.3$ year. These considerations are relevant for other conditions when the time series is not related to climate and when time is measured in seconds, hours, or any other units.

## 统计代写|时间序列分析代写Time-Series Analysis代考|Testing for Stationarity and Ergodicity

The only assumption, which is made about time series for its standard statistical analysis, including spectral estimation, is that it belongs to a stationary random process. To prove with probability one that the assumption is correct is not possible but one can verify whether it can be acceptable for a specific time series. This may be done in the following manner:

• split the time series in two equal parts and
• verify that the differences between the statistics of the entire time series and its halves do not lie outside the limits of sampling variability of respective estimates for the entire time series.

If the results of such verification are favorable, that is, if the differences can be ascribed to the sampling variability, there seems to be no ground to reject the initial assumption of stationarity. If, in addition, the probability distribution of the time series is Gaussian, the hypothesis of ergodicity that is usually accepted for stationary time series by default becomes reasonable as well. Comparisons should include at least the first two statistical moments, that is, the mean value and variance (or the root mean square value). The variances of estimated mean values and estimated variances should be calculated with account for the number of statistically independent observations in the time series.

For the time series $x_{t}, t=1, \ldots, N$, the root mean square (RMS) error of the estimated mean value $\bar{x}$ is
$$\sigma[\bar{x}] \approx \sigma_{x} / \sqrt{\bar{N}}$$
where $\sigma_{x}$ is the estimated standard deviation and $\tilde{N}$ is the effective number of independent observations in the time series:
$$\bar{N}=N / \sum_{k=-\infty}^{k=\infty} r(k)$$
where $r(k)$ is the correlation function of the time series.
The RMS error of the estimated variance $\sigma_{x}^{2}$ is
$$\sigma\left[\sigma_{x}^{2}\right]=\sigma_{x}^{2} / \hat{N}$$
where
$$\hat{N}=N / \sum_{k=-\infty}^{\infty} r^{2}(k)$$

## 统计代写|时间序列分析代写Time-Series Analysis代考|Practical Analysis of Time Series

• 使用适当的分析方法和
• 计算所有估计统计特征的置信区间。
这尤其意味着时域和频域分析的方法在数学上应该适用于正在分析的时间序列。在本书中，分析的基本方法是基于自回归模型，它能够提供相对可靠的时间序列统计估计，即使是短时间序列。它的另一个优点是在自回归分析的初始阶段获得的显式时域模型，如果使用非参数方法则不存在。本章稍后（第 4.5 节）将表明，自回归方法对于估计具有非常复杂结构的时间序列的谱密度非常有效。

• 不正确的采样间隔，
• 时间序列中存在线性或非线性趋势，
• AR 模型频谱中存在强烈的高频和/或准周期性波动，这需要解释。

## 统计代写|时间序列分析代写Time-Series Analysis代考|Testing for Stationarity and Ergodicity

• 将时间序列分成两个相等的部分，并且
• 验证整个时间序列及其一半的统计数据之间的差异不超出整个时间序列各自估计值的抽样变异性限制。

σ[X¯]≈σX/ñ¯

ñ¯=ñ/∑ķ=−∞ķ=∞r(ķ)

σ[σX2]=σX2/ñ^

ñ^=ñ/∑ķ=−∞∞r2(ķ)

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## MATLAB代写

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