**MY-ASSIGNMENTEXPERT™**可以为您提供sydney. **CSYS5030 Information Theory**信息论课程的代写代考和辅导服务！

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#### 这是悉尼大学信息论课程的代写成功案例。

**CSYS5030**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.

**CSYS5030 Information Theory** HELP（EXAM HELP， ONLINE TUTOR）

The Shannon-Fano CODE IS ALmost optimal Let $l(x)=\left\lceil\log \frac{1}{p(x)}\right\rceil$ be the length of the codeword corresponding to $x \in \mathcal{X}$ in a binary Shannon-Fano code, where $p(x)=\mathbf{P}{X=x}$. Let $l^{\prime}(x)$ be the codeword length of $x$ in an arbitrary uniquely decodable code. Prove that for any $c>1$,

$$

\mathbf{P}\left{l(X) \geq l^{\prime}(X)+c\right} \leq \frac{1}{2^{c-1}} .

$$

(Log-Sum inequality) Assume that the numbers $a_i, b_i \geq 0, i=1, \ldots, n$ satisfy

$$

\sum_{i=1}^n a_i=a \quad \text { and } \quad \sum_{i=1}^n b_i=b .

$$

Prove that

$$

\sum_{i=1}^n a_i \log \frac{b_i}{a_i} \leq a \log \frac{b}{a},

$$

with equality if and only if for all $i, b_i / a_i=$ constant.

The entropy of a more uniform Distribution is Larger Show that the entropy of the distribution

$$

\left(p_1, \ldots, p_i, \ldots, p_j \ldots, p_n\right)

$$

cannot be larger then the entropy of the distribution

$$

\left(p_1, \ldots, \frac{p_i+p_j}{2}, \ldots, \frac{p_i+p_j}{2} \ldots, p_n\right)

$$

The optimal code of a more uniform distribution is worse Consider the distributions

$$

\mathbf{p}=\left(p_1, \ldots, p_i, \ldots, p_j \ldots, p_n\right) \quad \text { and } \mathbf{q}=\left(p_1, \ldots, \frac{p_i+p_j}{2}, \ldots, \frac{p_i+p_j}{2} \ldots, p_n\right) .

$$

Show that the expected codeword length of the optimal code of $\mathbf{q}$ (i.e., that of minimal expected codeword length) cannot be smaller than that of $\mathbf{p}$. (Where the expected values are taken with respect to the corresponding distribution.)

The optimal code of a more uniform distribution is worse Consider the distributions

$$

\mathbf{p}=\left(p_1, \ldots, p_i, \ldots, p_j \ldots, p_n\right) \quad \text { and } \mathbf{q}=\left(p_1, \ldots, \frac{p_i+p_j}{2}, \ldots, \frac{p_i+p_j}{2} \ldots, p_n\right)

$$

Show that the expected codeword length of the optimal code of $\mathbf{q}$ (i.e., that of minimal expected codeword length) cannot be smaller than that of $\mathbf{p}$. (Where the expected values are taken with respect to the corresponding distribution.)

**MY-ASSIGNMENTEXPERT™**可以为您提供SYDNEY. **CSYS5030 INFORMATION THEORY**信息论课程的代写代考和辅导服务！