### 机器学习代写|聚类分析作业代写clustering analysis代考| Spectral representation 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

## 机器学习代写|聚类分析作业代写clustering analysis代考|Spectral representation of time series

Associated with every stationary stochastic process $\left{X_{t}\right}$ is the spectral density function which is a tool for considering the frequency properties of a stationary time series. The spectral density function, also referred to as the power spectral density function or the spectrum, is the derivative of the spectral distribution function $F(\omega)$, where $\omega$ is the frequency, which is defined as the number of radians per unit time. This is a continuous function that is monotone and bounded in the interval $[0, \pi]$. This derivative is denoted by $f(\omega)$, so that
$$f(\omega)=\frac{d F(\omega)}{d \omega} .$$
When $f(\omega)$ exists, Eq. $2.2$ can be expressed as
$$\gamma_{k}=\int_{0}^{\pi} \cos \omega k f(\omega) d \omega$$
When $k=0$, Eq. $2.3$ becomes
$$\gamma_{0}=\sigma_{X}^{2}=\int_{0}^{\pi} f(\omega) d \omega=F(\pi)$$

The interpretation of the spectrum is that $f(\omega) d \omega$ represents the contribution to variance of components of frequencies in the range $(\omega, \omega+d \omega)$. Eq. 2.4 indicates that the total area under the curve of the spectrum is equal to the variance of the process. A peak in the spectrum indicates an important contribution of variance at frequencies near the values that correspond to the peak.

It should be noted that the autocovariance function and the spectral density function are equivalent ways of describing a stationary stochastic process.
From Eq. 2.3, the corresponding inverse can be obtained, namely
$$f(\omega)=\frac{1}{\pi} \sum_{k=-\infty}^{\infty} \gamma_{k} e^{-i \omega k} .$$
This implies that the spectral density function is the Fourier transform of the autocovariance function. Refer to Chatfield (2004) for details on the Fourier transform. Since $\gamma_{k}$ is an even function of $k$, Eq. $2.5$ can be expressed as
$$f(\omega)=\frac{1}{\pi}\left[\gamma_{0}+2 \sum_{k=1}^{\infty} \gamma_{k} \cos \omega k\right] .$$
The normalized form of the spectral density function is given by
$$f^{}(\omega)=\frac{f(\omega)}{\sigma_{X_{t}}^{2}}=\frac{d F^{}(\omega)}{d \omega} .$$
This is the derivative of the normalized spectral distribution function. Hence, $f^{}(\omega)$ is the Fourier transform of the autocorrelation function, namely, $$f^{}(\omega)=\frac{1}{\pi}\left[1+2 \sum_{k=1}^{\infty} \rho_{k} \cos \omega k\right] .$$
This implies that $f^{*}(\omega) d \omega$ is the proportion of variance in the interval $(\omega, \omega+$ $d \omega$.

## 机器学习代写|聚类分析作业代写clustering analysis代考|Periodogram

An estimator of the spectral density function is the periodogram $I(\omega)$ where at an ordinate $p$ it is expressed as:
$$I\left(\omega_{p}\right)=\frac{1}{\pi}\left(c_{0}+2 \sum_{k=1}^{T-1} c_{k} \cos \left(\omega_{p} k\right)\right),$$
where $c_{k}$ is the sample autocovariance coefficient at lag $k, T$ is the length of the observed time series, and $p=1,2, \ldots,(T / 2)-1$. The periodogram is asymptotically unbiased, that is,

$$\lim _{T \rightarrow \infty} E[I(\omega)]=f(\omega)$$
However, $I(\omega)$ is not a consistent estimator of $f(\omega)$. It can be shown that neighbouring periodogram ordinates are asymptotically independent. Refer to Chatfield (2004) for more details on periodogram analysis including the distribution associated with the periodogram ordinates.

Example 2.2 Consider the observed time series of weekly sales of a consumer product from Example 2.1. Fig. 2.4 shows the periodogram of this series. The peak occurs around a normalized frequency between 0 and $0.05$, indicating that most of the largest contribution to the variance of the series is within this frequency range. The frequency range around which a peak occurs gives an indication of the frequency at which that cyclic component may exist.
Note that just as the spectral density function can be normalized, the periodogram can be normalized. In particular, the normalized periodogram is
$$I^{*}(\omega)=\frac{I(\omega)}{\operatorname{Var}\left(x_{t}\right)}$$

## 机器学习代写|聚类分析作业代写clustering analysis代考|Smoothed periodogram

While the periodogram is useful in assessing whether there are one or more strong cyclic components in a time series, the sampling error associated with its ordinates is quite large and confidence intervals set up around the ordinates would therefore be very wide. Therefore, the periodogram is not a very good estimator of the spectral density function especially when the signal to noise ratio of time series is low, that is, when the time series is very noisy. The periodogram can be smoothed to overcome these problems and there is a vast literature on windows that are used to smooth the periodogram. Refer to Chatfield (2004) for details about smoothing the periodogram.

Fig. $2.5$ shows a smoothed periodogram, from which it is clear that the fluctuations in the periodogram have been smoothed out to better assess which frequency bands account for most of the variance in the time series.

## 机器学习代写|聚类分析作业代写clustering analysis代考|Spectral representation of time series

F(ω)=dF(ω)dω.

Cķ=∫0圆周率因⁡ωķF(ω)dω

C0=σX2=∫0圆周率F(ω)dω=F(圆周率)

F(ω)=1圆周率∑ķ=−∞∞Cķ和−一世ωķ.

F(ω)=1圆周率[C0+2∑ķ=1∞Cķ因⁡ωķ].

F(ω)=F(ω)σX吨2=dF(ω)dω.

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