当我尝试编译我的 main.tex 文件时,使用 mdframed 包创建的框架内的代码行为异常并给出以下警告:
“包 mdframed 警告:您遇到了一个错误,因为最后一个框是空的,您必须通过更改输入行 3775 上的文本、空格或其他内容来手动更改它。”
这是框架代码:
\begin{mdframed}
\textbf{Remarks:}
\begin{itemize}
\item In general $\hat{y}(t|t-1,\theta)$ depends on previous predictions $\rightarrow$ initialization is required $\rightarrow$ transient effect $\rightarrow$ the predictor is asymptotically optimal.
Let's explain why. For ARMAX models we have $G(z^{-1})=\frac{B(z^{-1})}{A(z^{-1})}$ and $H(z^{-1})=\frac{C(z^{-1})}{A(z^{-1})}$, so the predictor will be:
\begin{equation*}
\hat{y}(t|t-1,\theta)=\left(1-\frac{A(z^{-1})}{C(z^{-1})}\right)y(t)+\frac{B(z^{-1})}{C(z^{-1})}u(t)
\end{equation*}
In this case the optimal predictor is a dynamic system itself:
\begin{equation*}
C(z^{-1})\hat{y}(t|t-1,\theta)=\left(C(z^{-1})-A(z^{-1})\right)y(t)+B(z^{-1})u(t)
\end{equation*}
When we apply the operator $z^{-1}$ we obtain the previous prediction:
\begin{align*}
\hat{y}(t|t-1,\theta)&=-c_1 \hat{y}(t-1|t-2,\theta)-c_2 \hat{y}(t-2|t-3,\theta)-\dots-c_n \hat{y}(t-n|t-n-1,\theta)+\\
&+(c_1-a_1)y(t-1)+\dots+(c_n-a_n)y(t-n)+b_1 u(t-1)+\dots+b_n u(t-n)
\end{align*}
The problem is that if we are at time $t-1$ and we want to predict the value of $y(t)$ we need all previous input output samples, all the parameters $a_i,b_i,c_i$ (where $i=1,\dots,n$) and all the previous predictions. When we compute the prediction at the starting point we haven't all this information, so we must initialize the system and we will have a transient effect and for this reason the predictor will be optimal only asymptotically.
Since $C(z^{-1})$ is at the denominator in the expression of the predictor and it has to be a stationary process, so we need that all the roots must be within the unitary circle (those of $C(z^{-1})$ as well).
A rough way to start can be: start to consider the predictions from time $t=1$ (so after $n+1$ steps) and set to zero the previous predictions.
\item In general, the $\hat{\theta}$ minimizing $J(\theta)$ cannot be found analytically (no closed-form solutions available), so we are forced to apply iterative identification algorithms (e.g. Newton-Raphson algorithm).
\end{itemize}
\end{mdframed}
结果如下:
是否仍可以使用 mdframed 软件包来避免此行为?我搜索了类似的问题,但没有一个得到正确的答案(如果不是更换软件包的建议)。
在此先感谢您的帮助!
答案1
你也可以简单地把它放在
\begin{figure}[H]
...
\end{figure}
[H]
当然,这需要 float 包。