**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！

**MY-ASSIGNMENTEXPERT™**

#### 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）

In class, we showed that if $X$ and $Y$ are two independent jointly-distributed discrete random variables with probability mass function $(\mathrm{pmf}) p(x, y)$, then the joint entropy $H(X, Y)$ is given by

$$

H(X Y)=H(X)+H(Y)

$$

We now wish to generalize this result. Show that if $X$ and $Y$ are two jointly-distributed discrete random variables with $\operatorname{pmf} p(x, y)$, , that in general

$$

H(X Y) \leq H(X)+H(Y)

$$

with equality if and only if the outcomes from the marginal ensembles $X$ and $Y$ are statistically independent.

Prove the following three equalities of (average) mutual information that were stated in class:

(i) $I(X ; Y)=I(Y ; X)$,

(ii) $I(X ; Y)=H(Y)-H(Y \mid X)$,

(iii) $I(X ; Y)=H(X)+H(Y)-H(X Y)$

Let $\mathbf{X}$ be a discrete random variable taking on values $2,3,4, \ldots$, with probability

$$

P{\mathbf{X}=n}=\frac{1}{A n \log ^2 n}

$$

where $A$ is a constant. Show that $H(\mathbf{X})=+\infty$, that is, that the defining series for $H(\mathbf{X})$ does not converge.

Consider a binary symmetric channel with error crossover probability $\epsilon$ as shown below. Assume the input ensemble is $X={0,1}$ with $P_X(0)=P_X(1)=1 / 2$. Calculate $I(X ; Y)$ in this situation as a function of $\epsilon$.

Problem 1 (Alternative definition of unique decodability) An $f: \mathcal{X} \rightarrow \mathcal{Y}^*$ code is called uniquely decodable if for any messages $\mathbf{u}=u_1 \cdots u_k$ and $\mathbf{v}=v_1 \cdots v_k$ (where $u_1, v_i, \ldots, u_k, v_k \in \mathcal{X}$ ) with

$$

f\left(u_1\right) f\left(u_2\right) \cdots f\left(u_k\right)=f\left(v_1\right) f\left(v_2\right) \cdots f\left(v_k\right),

$$

we have $u_i=v_i$ for all $i$. That is, as opposed to the definition given in class, we require that the codes of any pair of messages with the same length are equal. Prove that the two definitions are equivalent.

**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！