 # 数学代写|ENEE627 Information Theory

## ENEE627课程简介

The dynamics of complex systems are often described in terms of how they process information and self-organise; for example regarding how genes store and utilise information, how information is transferred between neurons in undertaking cognitive tasks, and how swarms process information in order to collectively change direction in response to predators. The language of information also underpins many of the central concepts of complex adaptive systems, including order and randomness, self-organisation and emergence. Shannon information theory, which was originally founded to solve problems of data compression and communication, has found contemporary application in how to formalise such notions of information in the world around us and how these notions can be used to understand and guide the dynamics of complex systems. This unit of study introduces information theory in this context of analysis of complex systems, foregrounding empirical analysis using modern software toolkits, and applications in time-series analysis, nonlinear dynamical systems and data science. Students will be introduced to the fundamental measures of entropy and mutual information, as well as dynamical measures for time series analysis and information flow such as transfer entropy, building to higher-level applications such as feature selection in machine learning and network inference. They will gain experience in empirical analysis of complex systems using comprehensive software toolkits, and learn to construct their own analyses to dissect and design the dynamics of self-organisation in applications such as neural imaging analysis, natural and robotic swarm behaviour, characterisation of risk factors for and diagnosis of diseases, and financial market dynamics.

## Prerequisites

At the completion of this unit, you should be able to:

• LO1. critically evaluate investigations of self-organisation and relationships in complex systems using information theory, and the insights provided
• LO2. develop scientific programming skills which can be applied in complex system analysis and design
• LO3. apply and make informed decisions in selecting and using information-theoretic measures, and software tools to analyse complex systems
• LO4. create information-theoretic analyses of real-world data sets, in particular in a student’s domain area of expertise
• LO5. understand basic information-theoretic measures, and advanced measures for time-series, and how to use these to analyse and dissect the nature, structure, function, and evolution of complex systems
• LO6. understand the design of, and to extend the design of a piece of software using techniques from class, and your own readings.

## ENEE627 Information Theory HELP（EXAM HELP， ONLINE TUTOR）

(CARD Shuffling) Let $X$ be an arbitrary random variable taking its values from the set ${1,2, \ldots, 52}$. Let $T$ be a random permutation of the numbers $1,2, \ldots, 52$, i.e., $T(1), T(2), \ldots, T(52)$ is a random re-ordering of the set ${1,2, \ldots, 52}$. We assume that $T$ is independent of $X$. Show that
$$H(T(X)) \geq H(X) .$$

Let $\mathbf{X}=X_1, X_2, \ldots$ be a binary memoryless stationary source with $\mathbf{P}\left{X_1=1\right}=10^{-6}$. Determine a variable-length code of $\mathbf{X}$ whose per letter expected codeword length is smaller than 1/10.

(Run Length Coding) Let $X_1, \ldots, X_n$ be binary random variables. Let $\mathbf{R}=\left(R_1, R_2, \ldots\right)$ denote the run lengths of the symbols in $X_1, \ldots, X_n$. That is, for example, the run lengths of the sequence 1110010001111 is $\mathbf{R}=(3,2,1,3,4)$. Determine the relation between $H\left(X_1, \ldots, X_n\right), H(\mathbf{R})$ and $H\left(\mathbf{R}, X_n\right)$ ?

(The SeCond LAW of thermodynamics) Let $\mathbf{X}=X_1, X_2, \ldots$ be a stationary Markov chain. Show that $H\left(X_n \mid X_1\right)$ is monotone increasing (in spite of the fact that $H\left(X_n\right)$ does not change with $n$ by stationarity).

(Properties of the binary entropy Function) Define the function $h(x)=-x \log x-$ $(1-x) \log (1-x)$, for $x \in(0,1)$, and $h(0)=h(1)=0$. Show that $h$ satisfies the following properties:

• symmetric around $1 / 2$;
• continuous in every point of $[0,1]$;
• strictly monotone increasing in $[0,1 / 2]$;
• strictly concave.