机器学习代写|聚类分析作业代写clustering analysis代考| Time series models

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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|Stationary models

An autoregressive (AR) model is one for which the current value of the deviation of the process from the mean is expressed as a finite, linear combination of previous values of the process and a shock or error term. Given a stochastic process $\left{X_{t}\right}$, the AR model is expressed as:
$$\phi(B) Z_{t}=\varepsilon_{t}$$
where $Z_{t}=X_{t}-\mu$, is the deviation from the mean, $\varepsilon_{t}$ is a white noise or random process with mean 0 and variance $\sigma_{t}^{2}$,
$$\phi(B)=1-\phi_{1} B-\phi_{2} B^{2}-\ldots-\phi_{p} B^{p}$$
is the autoregressive operator, $p$ is the order of the AR model and $B$ is the backshift operator. In particular, we refer to it as an $\operatorname{AR}(p)$ model. This model is stationary and a necessary requirement for stationarity is that all roots of $\Phi(B)=0$ must lie outside the unit circle.

A moving average (MA) model is one where the current value of the deviation of the process from the mean is expressed as a linear combination of a finite number of previous error terms. The MA model is expressed as:
$$Z_{t}=\theta(B) \varepsilon_{t}$$
where
$$\theta(B)=1-\theta_{1} B-\theta_{2} B^{2}-\ldots-\theta_{q} B^{q},$$
and $q$ is the order of the MA model. In particular, we refer to it as an MA $(q)$ model. This model is also stationary with a similar stationarity condition as that of the AR model applying. While the AR and MA are useful representations of observed time series, it is sometimes useful to include both AR and MA terms in a model, resulting in an autoregressive, moving average (ARMA) model or an ARMA $(p, q)$ model which is expressed as:
$$\phi(B) Z_{t}=\theta(B) \varepsilon_{t}$$
In order for the $\operatorname{AR}(p), \operatorname{MA}(q)$ and $\operatorname{ARMA}(p, q)$ models to be fitted to an observed time series, it is assumed that the series is stationary, that is, it fluctuates about a fixed mean and its variance is constant.

In order to identify a suitable model that may be fitted to an observed stationary time series, we examine the ACF and PACF of this series to determine if it to some extent emulates the theoretical ACF and PACF associated with the model. For an AR(1) model, the ACF shows exponential decay while the PACF is zero beyond lag 1 . Hence, we can infer that an AR(1) model would be

an appropriate fit to an observed time series, when the ACF decays exponentially and when the PACF has a single significant spike at lag 1 . Given that this is the behaviour displayed by the ACF and PACF of the observed series in Example 2.1, we could infer that an AR(1) model is possibly an appropriate model to fit to this series. In general, for an $\operatorname{AR}(p)$ model, with $p \geq 2$, the ACF can show exponential decay or a damped sin wave pattern, whereas the $\mathrm{PACF}$ is zero beyond lag $q$.

For an MA(1) model, the PACF shows exponential decay while the $\mathrm{ACF}$ is zero beyond lag 1 . Hence, if the ACF of an observed stationary time series has a single significant spike at lag 1 , and the PACF decays exponentially, we can infer that an MA(1) model would be an appropriate fit to this series. In general, for an MA $(q)$ model, with $q \geq 2$, the ACF is zero beyond lag $p$, while the PACF can show exponential decay or a damped sin wave pattern. Refer to books such as Chatfield (2004), Makridakis et al. (1998) and Ord and Fildes (2013) for more details about the theoretical behaviour of ARMA models in general.

机器学习代写|聚类分析作业代写clustering analysis代考|Non-stationary models

Many time series encountered in various fields exhibit non-stationary behaviour and in particular they do not fluctuate about a fixed level. Although the level about which the series fluctuates may be different at different times, when differences in levels are taken, they may be similar. This is referred to as homogeneous non-stationary behaviour (Box et al., 1994 ) and the series can be represented by a model that requires the $d$-th difference of the process to be stationary. In practice, $d$ is usually no more than 2 . Hence, an ARMA model can be extended to what is known as an autoregressive, integrated moving average (ARIMA) model, or $\operatorname{ARIMA}(p, d, q)$ to represent a homogeneous non-stationary time series. This model is expressed as
$$\phi(B)(1-B)^{d} Z_{t}=\theta(B) \varepsilon_{t} .$$
In practice, time series may also have a seasonal component. Just as the consecutive data points of an observed time series may exhibit AR, MA or ARMA properties, so data separated by a whole season (for example, a year or a quarter) may exhibit similar properties. The ARIMA notation can be extended to incorporate seasonal aspects and in general we have an $\operatorname{ARIMA}(p, d, q)(P, D, Q){s}$ model which can be expressed as $$\phi(B) \Phi(B)(1-B)^{d}\left(1-B^{s}\right)^{D} Z{t}=\theta(B) \Theta(B) \varepsilon_{t}$$
where
\begin{aligned} &\Phi(B)=1-\Phi_{1} B^{s}-\Phi_{2} B^{2 s}-\ldots-\Phi_{P} B^{P_{s}} \ &\Theta(B)=1-\Theta_{1} B^{s}-\Theta_{2} B^{2 s}-\ldots-\Theta_{Q} B^{Q_{s}} \end{aligned} $D$ is the degree of seasonal differencing and $s$ is the number of periods per season. For example, $s=12$ for monthly time series and $s=4$ for quarterly time series. Refer to books such as Makridakis et al. (1994) and Ord and Fildes (2013) for more details about fitting non-stationary models that may be seasonal or not.

机器学习代写|聚类分析作业代写clustering analysis代考|Some other models

All these models discussed thus far are linear and are applicable to univariate time series. A popular extension to fitting models to stationary multivariate time series are vector autoregressive moving average models (VARMA). One of the large number of books in which details of these models can be found is Lutkepohl (1991). There are also several classes of non-linear models. A particular class is one that is concerned with modeling changes in variance or the volatility of a time series. These include autoregressive conditionally heteroscedastic (ARCH) and generalized autoregressive conditionally heteroscedastic (GARCH) models. One of the large number of books in which details of these models can be found is Tsay $(2010)$.

机器学习代写|聚类分析作业代写clustering analysis代考|Stationary models

φ(乙)从吨=e吨

φ(乙)=1−φ1乙−φ2乙2−…−φp乙p

θ(乙)=1−θ1乙−θ2乙2−…−θq乙q,

φ(乙)从吨=θ(乙)e吨

机器学习代写|聚类分析作业代写clustering analysis代考|Non-stationary models

φ(乙)(1−乙)d从吨=θ(乙)e吨.

有限元方法代写

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MATLAB代写

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