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# 数学代写|随机过程Stochastic Porcesses代考|STAT6540 Deterministic and indeterministic processes

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## 数学代写|随机过程Stochastic Porcesses代考|Deterministic and indeterministic processes

Deterministic process A time series $\left{X_t\right}$ will be called deterministic if there exists a function $g(t)$ of past and present values of $X_t$
such that \begin{aligned} g(t) & =g\left(X_{t-j}, j=0,1,2, \ldots\right) \ E\left(X_{t+1}-g(t)\right)^2 & =0\end{aligned}
If a function $g(t)$ is a linear function of $X_{t-j}, j \geq 0$, then $\left{X_t\right}$ will be called linear deterministic; e.g. consider the series $X_t=a e^{b t}$ so that $X_{t+1}=e^b X_t$. If $b$ is not known, it can be perfectly estimated from the past of the series, but the estimate will be a nonlinear function of the (sequence) series; for instance $b=1 / 2\left[\log X_t^2-\log X_{t-1}^2\right]$ allowing for the fact that $a$ may be negative. Other linear deterministic functions are the periodic functions $X_t=a \cos (\omega t+\theta)$ provided $a$ is known and there exists an integer $k$ such that $2 \pi k / \omega$ is an integer. For example, if $X_t=a \cos (2 \pi t / 12+\theta)$, then $X_{t+1}=X_{t-11}$ so that $X_t$ is clearly linearly deterministic. If $\omega$ has to be estimated from past data, the series becomes nonlinear deterministic. Let $X_t=\sum_{k=0}^n d_k t^k$ so that the sequence is a polynomial in $t$. Then $\Delta^n X_t=n ! d_n$ and $\Delta^{n+1} X_t=0$, i.e. $X_t$ series obeys a homogeneous linear difference equation and hence is linearly deterministic. To use this, procedure one has to know the value of $n$ or $n^{\prime}>n$.

A typical time series consists of a deterministic part and a random part, e.g.
Deterministic white noise
Indeterministic Process Consider a discrete time stationary time series $\left{X_t\right}$, $E\left(X_t\right)=0$ and $\operatorname{Var}\left(X_t\right)=\sigma^2$. Suppose the residual variance obtained by regressing $X_t$ on $\left(X_{t-q}, X_{t-q-1}, \ldots\right)$ is $\sigma_q^2\left(\leq \sigma^2\right)$. Certainly $\sigma_q^2$ is a non-decreasing bounded sequence, i.e.
$$\lim {\eta \rightarrow \infty} \sigma\eta^2=\sigma_0^2=\left{\begin{array}{c} \sigma^2 \ 0 \end{array}\right. \text { (considering two extreme situations) }$$
(i) If residual variance $\sigma^2>0$ then it is useless to consider the regression of $X_t$ on $\left{X_{t-q}, X_{t-q-1}, \ldots\right}$. Such a process $\left{X_t\right}$ is then called indeterministic, since the process cannot be used for forecasting purposes (MA, AR, ARMA, satisfy this property). Then linear regression on the remote past is useless for prediction purpose.
(ii) If residual variance is 0 then it can be used for forecasting purposes and hence is deterministic.

## 数学代写|随机过程Stochastic Porcesses代考|Analysis in frequency domain

Inference based on the conceptual tool known as spectral density function (spectrum) is called analysis in frequency domain. The spectral density is the natural tool for considering the frequency properties of a time series.

Theorem 9.3 Wiener-Khintchine’s Theorem Suppose $\left{X_t\right}$ is a covariance stationary time series with autocovariance function $\gamma(k)$. Then there is a function $F(\lambda) \uparrow$ in $\lambda$ such that
$$\gamma(k)=\int_0^\pi \cos (\lambda k) d F(\lambda)$$

This is called the spectral representation of autocovariance function, where $\lambda=$ frequency of the series $y_t=\cos (\lambda t)$ and $\lambda$ can be written as $\lambda=\frac{2 \pi}{\theta}$ where $\theta$ is the period of oscillation. Note that $\lambda \in(0, \pi)$.

Following is the direct physical interpretation of spectral representation (9.4.2):
If $k=0, \gamma(0)=\int_0^\pi d F(\lambda)=F(\pi)=\sigma^2$ (from equation (9.4.2))
As $F(\lambda) \uparrow$ in $\lambda, \max _{0 \leq \lambda \leq \pi} F(\lambda)=F(\pi)=\sigma^2=\operatorname{Var}\left(X_t\right)$.
Therefore $F(\lambda)$ is the contribution to the variance of the series which is accounted for frequency $\lambda \in(0, \pi) . F(\lambda)$ is an absolutely continuous function for any discrete stationary time series $\left{X_t\right}$ satisfying $\sum|\gamma(k)|<\infty$. Therefore $f(\lambda)=$ spectral density $=\frac{d F(\lambda)}{d \lambda}$ exists in this case and $(9.4 .2)$ becomes $\gamma(k)=$ $\int_0^\pi \cos (\lambda k) f(\lambda) d \lambda$

The autocovariance function and the spectral density function are equivalent in representing underlying time series. In some situations, the autocovariance function is seen to be useful in determining the underlying structure of the time series while in some situations the spectral density function $f(\lambda)$ is seen to be more useful. Fourier Analysis of spectral density $f(\lambda)$ plays an important role.

## 数学代写|随机过程Stochastic Porcesses代考|Deterministic and indeterministic processes

\gamma(k)=\int_0^\pi \cos (\lambda k) d F(\lambda)


\left 缺少或无法识别的分隔符 $\quad$ 令人满意 $\sum|\gamma(k)|<\infty$. 所以 $f(\lambda)=$ 谱密度 $=\frac{d F(\lambda)}{d \lambda}$ 在这种情况下存在并 且 $(9.4 .2)$ 成为 $\gamma(k)=\int_0^\pi \cos (\lambda k) f(\lambda) d \lambda$

## MATLAB代写

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