## 统计代写|随机控制代写Stochastic Control代考|Conclusions and Directions for Future Research

In this paper we construct the minimum risk estimators of state of stochastic systems. The method used is that of the invariant embedding of sample statistics in a loss function in order to form pivotal quantities, which make it possible to eliminate unknown parameters from the problem. This method is a special case of more general considerations applicable whenever the statistical problem is invariant under a group of transformations, which acts transitively on the parameter space.

For a class of state estimation problems where observations on system state vectors are constrained, i.e., when it is not feasible to make observations at every moment, the question of how many observations to take must be answered. This paper models such a class of problems by assigning a fixed cost to each observation taken. The total number of observations is determined as a function of the observation cost.
Extension to the case where the observation cost is an explicit function of the number of observations taken is straightforward. A different way to model the observation constraints should be investigated.
More work is needed, however, to obtain improved decision rules for the problems of unconstrained and constrained optimization under parameter uncertainty when: (i) the observations are from general continuous exponential families of distributions, (ii) the observations are from discrete exponential families of distributions, (iii) some of the observations are from continuous exponential families of distributions and some from discrete exponential families of distributions, (iv) the observations are from multiparameter or multidimensional distributions, (v) the observations are from truncated distributions, (vi) the observations are censored, (vii) the censored observations are from truncated distributions.

## 统计代写|随机控制代写Stochastic Control代考|Takagi-Sugeno Fuzzy Model

The TS fuzzy inference system is composed by a set of IF-THEN rules which partitions the input space, so-called universe of discourse, into fuzzy regions described by the rule antecedents in which consequent functions are valid. The consequent of each rule $i$ is a functional expression $y_i=f_i(x)$ (King (1999); Papadakis \& Theocaris (2002)). The $i$-th TS fuzzy rule has the following form:
$R^{i \mid i=1,2, \ldots, l}:$ IF $x_1$ is $F_1^i$ AND $\cdots$ AND $x_q$ is $F_q^i$ THEN $y_i=f_i(\mathbf{x})$
where $l$ is the number of rules. The vector $x \in \Re^q$ contains the antecedent linguistic variables, which has its own universe of discourse partitioned into fuzzy regions by the fuzzy sets representing the linguistic terms. The variable $x_j$ belongs to a fuzzy set $F_j^i$ with a truth value given by a membership function $\mu_{F_j}^i: \Re \rightarrow[0,1]$. The truth value $h_i$ for the complete rule $i$ is computed using the aggregation operator, or t-norm, AND, denoted by $\otimes:[0,1] \times[0,1] \rightarrow[0,1]$,
$$h_i(x)=\mu_1^i\left(x_1\right) \otimes \mu_2^i\left(x_2\right) \otimes \ldots \mu_q^i\left(x_q\right)$$
Among the different t-norms available, in this work the algebraic product will be used, and
$$h_i(\mathbf{x})-\prod_{j=1}^q \mu_j^i\left(x_j\right)$$

The degree of activation for rule $i$ is then normalized as
$$\gamma_i(\mathbf{x})=\frac{h_i(\mathbf{x})}{\sum_{r=1}^l h_r(\mathbf{x})}$$
This normalization implies that
$$\sum_{i=1}^l \gamma_i(\mathbf{x})-1$$
The response of the TS fuzzy model is a weighted sum of the consequent functions, i.e., a convex combination of the local functions (models) $f_i$,
$$y=\sum_{i=1}^l \gamma_i(\mathbf{x}) f_i(\mathbf{x})$$
which can be seen as a linear parameter varying (LPV) system. In this sense, a TS fuzzy model can be considered as a mapping from the antecedent (input) space to a convex region (politope) in the space of the local submodels defined by the consequent parameters, as shown in Fig. 1 (Bergsten (2001)).

## 统计代写|随机控制代写随机控制代考|Takagi-Sugeno模糊模型

TS模糊推理系统是由一组IF-THEN规则组成的，这些规则将输入空间(所谓的话语宇宙)划分为由规则前因变量描述的模糊区域，在这些前因变量中，结果函数是有效的。每条规则的结果 $i$ 是一个函数表达式 $y_i=f_i(x)$ (King (1999);Papadakis ＆Theocaris(2002))。。 $i$-th TS模糊规则有以下形式:
$R^{i \mid i=1,2, \ldots, l}:$ 如果 $x_1$ 是 $F_1^i$ 和 $\cdots$ 和 $x_q$ 是 $F_q^i$ 然后 $y_i=f_i(\mathbf{x})$
where $l$ 是规则的数量。矢量 $x \in \Re^q$ 包含先行语言变量，它有自己的话语宇宙，被表示语言术语的模糊集划分为模糊区域。变量 $x_j$ 属于一个模糊集合 $F_j^i$ 由隶属函数给出的真值 $\mu_{F_j}^i: \Re \rightarrow[0,1]$。真值 $h_i$ 对于完整的规则 $i$ 是使用聚合运算符或t范数AND计算的，用 $\otimes:[0,1] \times[0,1] \rightarrow[0,1]$，
$$h_i(x)=\mu_1^i\left(x_1\right) \otimes \mu_2^i\left(x_2\right) \otimes \ldots \mu_q^i\left(x_q\right)$$在可用的不同t范数中，本文将使用代数乘积，
$$h_i(\mathbf{x})-\prod_{j=1}^q \mu_j^i\left(x_j\right)$$

$$\gamma_i(\mathbf{x})=\frac{h_i(\mathbf{x})}{\sum_{r=1}^l h_r(\mathbf{x})}$$

$$\sum_{i=1}^l \gamma_i(\mathbf{x})-1$$
TS模糊模型的响应是后续函数的加权和，即，局部函数(模型)$f_i$，
$$y=\sum_{i=1}^l \gamma_i(\mathbf{x}) f_i(\mathbf{x})$$

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