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# 数学代写|随机过程Stochastic Porcesses代考|STAT507 Forecasting short-term behavior

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## 数学代写|随机过程Stochastic Porcesses代考|Forecasting short-term behavior

Suppose that we wish to predict future values of the chain. For example, we can predict the next value of the chain, at time $n+1$ using
\begin{aligned} P\left(X_{n+1}=j \mid \mathbf{x}\right) & =\int P\left(X_{n+1}=j \mid \mathbf{x}, \boldsymbol{P}\right) f(\boldsymbol{P} \mid \mathbf{x}) \mathrm{d} \boldsymbol{P} \ & =\int p_{x_n j} f(\boldsymbol{P} \mid \mathbf{x}) \mathrm{d} \boldsymbol{P}=\frac{\alpha_{x_n j}+n_{x_n j}}{\alpha_{x_n}+n_{x_n}}, \end{aligned}
where $\alpha_i=\sum_{j=1}^K \alpha_{i j}$
Prediction of the state at $t>1$ steps is slightly more complex. For small $t$, we can use
$$P\left(X_{n+t}=j \mid \mathbf{x}\right)=\int\left(\boldsymbol{P}^t\right){x_n j} f(\boldsymbol{P} \mid \mathbf{x}) \mathrm{d} \boldsymbol{P},$$ which gives a sum of Dirichlet expectation terms. However, as $t$ increases, the evaluation of this expression becomes computationally infeasible. A simple alternative is to use a Monte Carlo algorithm based on simulating future values of the chain as follows: For $s=1, \ldots, S$ : Generate $\boldsymbol{P}^{(s)}$ from $f(\boldsymbol{P} \mid \mathbf{x})$. Generate $x{n+1}^{(s)}, \ldots, x_{n+t}^{(s)}$ from the Markov chain with $\boldsymbol{P}^{(s)}$ and initial state $x_n$.

## 数学代写|随机过程Stochastic Porcesses代考|Forecasting stationary behavior

Often interest lies in the stationary distribution of the chain. For a low-dimensional chain where the exact formula for the equilibrium probability distribution can be derived, this is straightforward.

Example 3.5: Suppose that $K=2$ and $\boldsymbol{P}=\left(\begin{array}{cc}p_{11} & 1-p_{11} \ 1-p_{22} & p_{22}\end{array}\right)$. Then the equilibrium probability of being in state 1 can easily be shown to be
$$\pi_1=\frac{1-p_{22}}{2-p_{11}-p_{22}}$$
and the predictive equilibrium distribution is
$$E\left[\pi_1 \mid \mathbf{x}\right]=\int_0^1 \int_0^1 \frac{1-p_{22}}{2-p_{11}-p_{22}} f\left(p_{11}, p_{22} \mid \mathbf{x}\right) \mathrm{d} x$$
which can be evaluated by simple numerical integration techniques.
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## 数学代写|随机过程Stochastic Porcesses代考|Forecasting shortterm behavior

$$P\left(X_{n+1}=j \mid \mathbf{x}\right)=\int P\left(X_{n+1}=j \mid \mathbf{x}, \boldsymbol{P}\right) f(\boldsymbol{P} \mid \mathbf{x}) \mathrm{d} \boldsymbol{P} \quad=\int p_{x_n j} f(\boldsymbol{P} \mid \mathbf{x}) \mathrm{d} \boldsymbol{P}=\frac{\alpha_{x_{n j}}+n_{x_n j}}{\alpha_{x_n}+n_{x_n}},$$

$$P\left(X_{n+t}=j \mid \mathbf{x}\right)=\int\left(\boldsymbol{P}^t\right) x_n j f(\boldsymbol{P} \mid \mathbf{x}) \mathrm{d} \boldsymbol{P},$$

## 数学代写|随机过程Stochastic Porcesses代考|Forecasting stationary behavior

$$\pi_1=\frac{1-p_{22}}{2-p_{11}-p_{22}}$$

$$E\left[\pi_1 \mid \mathbf{x}\right]=\int_0^1 \int_0^1 \frac{1-p_{22}}{2-p_{11}-p_{22}} f\left(p_{11}, p_{22} \mid \mathbf{x}\right) \mathrm{d} x$$

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## MATLAB代写

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