### 统计代写|时间序列分析作业代写time series analysis代考|Practical Analysis of Time Series

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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|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.

Detailed recommendations for setting the Nyquist frequency when dealing with recording and/or preparing time series for further analysis are given in Chap. 10 of the Bendat and Piersol book $(2010)$ and in the book by Thomson and Emery (2014).

## 统计代写|时间序列分析作业代写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 $\bar{N}$ is the effective number of independent observations in the time series:
$$\tilde{N}=N / \sum_{k=-\infty}^{k=\infty} r(k)$$
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
$$\hat{N}=N / \sum_{k=-\infty}^{\infty} r^{2}(k)$$
For more details, see Yaglom (1987). The formulae with the total number of observations $N$ used instead of $\tilde{N}$ and $\hat{N}$ are correct only if $x_{n}$ is a white noise sample. Generally, such an assumption is wrong and is not applicable to time series.
Thus, the test for stationarity with respect to the mean value and variance includes the following steps:

• fit a proper AR model to the entire series and to its halves.
• estimate mean values and variances for each of the three time series.

## 统计代写|时间序列分析作业代写time series analysis代考|Linear Filtering

Time series are often subjected to filtering designed isolate variations within specific frequency bands. Generally, the spectra of climate and many other geophysical data are smooth, and the only frequency band that dominates the spectrum is at the lower end of the frequency axis. This means that if one wants to “protect” some specific band within the spectrum from variations belonging to a different band, filtering

is not required. Moreover, with autoregressive (maximum entropy) spectral estimation, there is no interaction between different frequencies, which makes the filtering operation unnecessary or even harmful.

If the goal of filtering is to study variations within a specific frequency band in the time domain, one should remember that statistical properties of the initial and filtered time series are very different. In particular, if a filter which suppresses highfrequency components is applied, the numbers of mutually uncorrelated observations will be smaller and the reliability of all estimates will be worse than it was before the filter has been applied to the time series. The filtered time series has to be analyzed to determine variances of estimates obtained from it. In short, time series should not be filtered unless one has strong physical and/or probabilistic arguments in support of the filtering operation.
There are three types of filters:

• the low-pass filter removes high-frequency (fast) fluctuations.
• the high-pass filter removes low-frequency (slow) fluctuations.
• the band-pass filter removes fastest of the slow fluctuations and slowest of the fast fluctuations.

When passing a time series through a filter, one should have in mind the following factors:

• no physically realizable filter can remove fluctuations in a given frequency band without affecting all other frequencies.
• a longer weighting function of the filter means a narrower frequency band to which the filter is tuned.

The filtering operation is done in the time domain in accordance with the formula
$$\tilde{x}{t}=\sum{k=-K}^{K} \lambda_{k} x_{t+k}$$
where $\lambda_{k}$ is the weighting function of the filter. It makes the time series (4.6) shorter than the initial time series by $2 K$ terms. This latter effect can be avoided by building an AR model of the time series prior to its filtering and then simulating it at both ends of the time series for sufficiently long intervals. Then, the filter is applied to the resulting longer time series, which can now be studied within the entire time interval for which the observation data were available initially.

The properties of the filter in the frequency domain are defined by the filter’s frequency response function (FRF) which presents a Fourier transform of the weighting function:
$$H(f)=\sum_{k=-K}^{K} \lambda_{k} \mathrm{e}^{-i 2 \pi k f \Delta t}$$
where $i=\sqrt{-1}$. The spectral density of the time series that passed through a filter is transformed from $s(f)$ to
$$\bar{s}(f)=|H(f)|^{2} s(f)$$

## 统计代写|时间序列分析作业代写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)

• 将适当的 AR 模型拟合到整个系列及其一半。
• 估计三个时间序列中每一个的平均值和方差。

## 统计代写|时间序列分析作业代写time series analysis代考|Linear Filtering

• 低通滤波器去除高频（快速）波动。
• 高通滤波器去除低频（慢）波动。
• 带通滤波器去除最快的缓慢波动和最慢的快速波动。

• 没有物理上可实现的滤波器可以在不影响所有其他频率的情况下消除给定频带中的波动。
• 滤波器的加权函数越长意味着滤波器调谐到的频带越窄。

$$\tilde{x} {t}=\sum {k=-K}^{K} \lambda_{k} x_{t+k}$$

H(F)=∑ķ=−ķķλķ和−一世2圆周率ķFΔ吨

s¯(F)=|H(F)|2s(F)

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## MATLAB代写

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