# Kalman Filter

Kalman filter
References for the derivation
Lacey, Tony. “Tutorial: The kalman filter.” Georgia Institute of Technology. [pdf]
Unknown author. [pdf]

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Kalman filter
References for the derivation
Lacey, Tony. “Tutorial: The kalman filter.” Georgia Institute of Technology. [pdf]
Unknown author. [pdf]

You can directly download the cheat sheet here [pdf]. If you want to edit the pdf based on the current version, you can turn to my overleaf project oneline in this link [overleaf project]. I am sorry I only open the rights to view the projects.
Inner/outer product$e_ie_j^T\rightarrow E_{i,j}=1$, $R(A)=span(A)={Ax:x\in R^n}$
Vector input into a single hidden neuron: $f(x)=f_a.(w^Tx+b)$
Euclidean Norm (length) of vector: $||x||_2=\sqrt{x^Hx}$
If $Q$ is a unitary matrix then: $||Qx||_2=||x||_2$
$Trace$ call only applied to square matrix.
Left product a matrix is to transform the row while right product a matrix is to transform the column. Similarly, $e_i^TA$ is $i$-th row of $A$ while $Ae_i$ is the $i$-th column of $A$.
Determinant$$\det (A)= \ det (A^T),\det(AB)=\det(BA)$$Optional:$$\det \left[\begin{array}{cc} A & B \\ C & D\end{array} \right]=\det(A)\det(D-CA^{-1}B)$$ $$\det \left[\begin{array}{cc} A & B \\ C & D\end{array} \right]=\det(D)\det(A-BD^{-1}C)$$
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IntroductionSometimes we need to submit the homework with equations and code, then the Jupyter Notebook will be a good choice.
Jupyter enable you use both markdown and the code in the same environment, that is, it is quite convenient to give a brief explanation to your code. From this perspective, Jupyter Notebook is beyond all the other development environments.
Here is an example of the Jupyter Notebook.
In Jupyter Notebook, there are two kinds of cells: markdown and code. In the markdown cell, you can write the explanantion such as some important equations. You can simply insert the equations in markdown cell by typing the equations within the $...$ or $$...$$ environment. Refer to [Link] for typing the mathematical expressions.
Actually, I think that Jupyter Notebook is designed to do the research-related work instead of being used as a development environment. It is not that efficient for one to develop a large-scale project.
InstallMaybe there are many methods to install the Ju
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I completed this almost within one day…..so this can only be used as a reference. I cannot guarantee all of below are right. Email me if any question.
Notations:
$f(x)$: objective function
$X$: feasible region
$h_i(x)$: equality constraints
$g_i(x)$: inequality constraints
then for a optimization problem, the general formulation is:
$$z=\min_{x\in X}f(x)$$
subject to:
$$\left\{ \begin{aligned} h_i(x) = 0 \\ g_i(x)\le 0 \end{aligned} \right.$$
Formulate the Optimization ProblemThree main steps to formulate the optimization problem:
Translate the problem: A optimization are composed of objective, decision variables, parameters and constraints. Think what is the decision variables, objective and constraints before writting down the mathematical expression;
Mathematical expression of the problem;
Fine-tuning.
General principles when fine-tuning,
Minimize the number of decision variables and constraints.
Try to maintain linearity or convexity.
Avoid introducing binary or intege
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This is my personal blog. Maybe I will update my recent progress, share some useful tools which can help a researcher work more efficiently and also share my stories with my parents and my best friends. This is the first year of my study in US. It is quite important for me to develop a good habit during such a long journey so that I decide to write this blog to record my life and simply use it as my notebook. There is an old Chinese saysing, “工欲善其事，必先利其器”. I hope this blog will be my powerful weapon in my daily life and work.
Here I want to thank my undergraduate roommate Jacob Zhong [Link] for the help to establish this blog. Zhong inspired me a lot during my undergraduate study especially on the use of all kinds of fancy tools. It is so lucky of me to continue to be the neighbor of Zhong maybe for the whole graduate study in U of M. Now I can still learn from him.
The following instructions maybe can help me to quickly familiar with writing this blog.
Insert EquationThe following co
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