### 数学代写|随机过程统计代写Stochastic process statistics代考|STAT4061

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|随机过程统计代写Stochastic process statistics代考|Mathematical Modeling and Optimization

Although optimization may occur by trial-and-error procedures, such strategy can be quite expensive or even dangerous. In other cases, the number of possible solutions is considerably high; therefore, it is unpractical

to test each of them. When those situations occur, rigorous optimization techniques are necessary. Moreover, such methods require, in several cases, counting with a mathematical model, which properly represents the phenomena or system of interest. A mathematical model is an abstract representation of the system under study, and it relates the important variables through mathematical expressions. Such mathematical equations can be expressed as equalities $(A=B)$, inequalities $(A \leq B$ or $A \geq B)$, or logical expressions $(A \rightarrow B)$. Furthermore, relationships between the variables can be merely algebraic, which happens for static systems, or can be differential or integro-differential, which is observed in dynamic systems. Despite the type of mathematical equations and relationships conforms to the model, it should be used for better understanding the system under study, and obtaining information about the relationship between the different components of the system. Furthermore, the model will be beneficial for examining the effects of manipulating the input variables on the entire performance of the case of study. Moreover, it allows avoiding the high costs of multiple experiments and the risks of manipulating a not well understood system. Certainly, experimentation is necessary to obtain the unknown information required for the model or to validate the results obtained, but the required number of tests will be small.

An important concept, which is the first link between mathematical modeling and optimization, is the number of degrees of freedom. Let us assume a mathematical model with $M$ independent equations and $N$ variables. The number of degrees of freedom, $F$, is then defined as follows:
$$F=N-M$$
Thus, the degrees of freedom can be defined as a set of variables in excess, which avoids the model to be solved in a direct way. To solve the model, an $\mathbf{M} \times \mathbf{M}$ matrix should be obtained. Consequently, additional equations are required, which can be obtained by fixing $F$ variables in a given value. Three situations can be observed when analyzing the number of degrees of freedom:
Case I. The number of equations is greater than the number of variables $(M>N)$, and thus, the number of degrees of freedom is negative. This situation commonly implies that there are some errors in the model, and it is said that the problem is overspecified. Another possibility for the existence of this situation is that there are some dependent equations in the model, which should not be considered for computing $M$.

## 数学代写|随机过程统计代写Stochastic process statistics代考|Classification of Optimization Problems

When a mathematical model is used for solving an optimization problem, it can be classified into different categories in terms of the number of degrees of freedom, including the type of mathematical relationships, equations, and variables. Depending on the type of optimization problem, the solution strategy will be different. In terms of the number of degrees of freedom, there are univariate problems, when there is only one degree of freedom; and multivariable problem, when there exist two or more degrees of freedom. For the univariate optimization problems, there are search methods, such as the golden section or the Fibonacci methods, which are considerably beneficial for solving that ty pe of problems (Jiménez Gutiérrez, 2003). For multivariable optimization, more robust methods are required.

Optimization problems can also be classified in terms of the type of mathematical relationships on the model, which can be algebraic or differential/ integro-differential. For both cases, uncertainties may or may not occur for the model components. If the model has only algebraic equations and there are no uncertainties, we discuss about a classical mathematical programming problem. If there are uncertainties, the case is known as a stochastic programming problem. When the model consists of differential/integro-differential relationships, but there are no uncertainties, we discuss about an optimal control problem. Finally, if there are uncertainties, a stochastic optimal control problem arises. Figure $1.1$ shows this classification in a graphical way.

## 数学代写|随机过程统计代写Stochastic process statistics代考|Objective Function

We have mentioned that optimizing implies selecting the best alternative among a set of possibilities. Nevertheless, the term “the best” is quite relative, and the selection of the best alternative strongly depends on the personal opinion of the decision maker. Thus, to avoid taking subjective decisions, a more trustworthy, numeric criteria should be established, which allows selecting the solution independent to the personal criteria of the one responsible of taking the decision. That criteria is called the objective function.

The objective function can be defined as a way for measuring the effectiveness of the system (Sarker and Newton, 2008), or a way for measuring the performance of the system (Pierre, 1986). In other words, it indicates whether a given solution is good in comparison to others, or if it can be considered as the best solution. In Figure $1.3 a$, a one-variable objective function is shown. In Figure 1.3a, the optimal solution is the one marked as $x^{}$. For that solution, the objective function takes a value of $f\left(x^{}\right)$. It can be observed that there is no other value of $f(x)$ smaller than $f\left(x^{}\right)$ for any other $x$. Thus, it is said that the solution is a global minimum. In the case of the objective function in Figure 1.3a, finding the optimal solution is quite simple, implying the use of the first derivative criteria. Nevertheless, when the number of decision variables is higher, the solution of the optimization problem is not that easy. Figure $1.3 \mathrm{~b}$ shows an objective function with two independent variables. It can be observed that there are two points, which can be classified as minimums, $\bar{x}{1}^{}$ and $\bar{x}{2}^{}$. For both solutions, the gradient is equal to zero; thus, they are both optimal solutions. Nevertheless, the value of the function evaluated for $\bar{x}{2}^{}$ is lower than the value of the function for $\bar{x}{1}^{}$. Moreover, $f\left(\bar{x}{2}^{}\right)$ is the lowest value the function can consider, and it is a global minimum. The solution given by $f\left(\vec{x}{1}\right)$ is a minimum, but it is the lowest value of the function only for the surroundings of $\bar{x}_{1}^{*}$. Thus, it is known as a local minimum.

An unconstrained optimization problem can be stated in a general form as follows:
optimize $Z=z(\bar{x})$
where the term “optimize” is replaced by “min” for minimization and “max” for maximization, depending on what type of solution is desired.

F=ñ−米

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