# 统计代写|抽样理论作业代写sampling theory代考|STAT7124

#### Doug I. Jones

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## 统计代写|抽样理论作业代写sampling theory代考|Heterogeneity of a Zero-Dimensional Lot

In this category, we include lots that are essentially statistical populations, and we already know that they are affected by two kinds of heterogeneities:

1. the Constitution Heterogeneity $\mathrm{CH}_L$
2. the Distribution Heterogeneity $\mathrm{DH}_{\mathrm{L}}$.
The Distribution Heterogeneity is a direct consequence of the Constitution Heterogeneity. Without Constitution Heterogeneity, Distribution Heterogeneity cannot exist. The kind of Distribution Heterogeneity we are dealing with in a zero-dimensional lot is what could be defined as a small-scale Distribution Heterogeneity, which is the logical consequence of random fluctuations between the constitution of neighboring fragments. Such random fluctuations generate the Fundamental Sampling Error FSE already introduced, but they also give an opportunity for gravitational forces to proceed with a
3. rearrangement of the order between fragments, segregating families of fragments on the basis of their respective constitution. The more the difference in constitution (e.g., composition, shape, size, density, etc.), the stronger the possible segregation. In due course, we will determine the exact relationship between Constitution Heterogeneity and Distribution Heterogeneity and we will see that Distribution Heterogeneity is always smaller or equal to the Constitution Heterogeneity, which is intuitive. Two factors will be responsible for the sampling error introduced by Distribution Heterogeneity:
1. a segregation factor, which is a measure of spatial rearrangements
2. a grouping factor, which is a measure of the random selectivity.
Intuitively, we find that the Fundamental Sampling Error FSE is the minimum error generated when collecting a sample of a given weight. The minimum is reached only under one statistical condition: fragments making up the sample shall be collected strictly at random, and one by one. Of course, it does not happen that way in practice, which greatly bothers inexperienced empirical practitioners who wonder why the variability between their replicate samples in their kind of heterogeneity test does not agree with Gy’s formula. When collecting an increment to make up a sample, this increment is likely to be made of many fragments. Then, statistically speaking, one sample is not made of strictly random fragments, but only of random groups of fragments; the difference is huge and often completely misunderstood. Consequently, we introduce an additional error and the larger the groups, the larger the error, and there are no ways around it; it is the way it is. This invariably leads to endless and counterproductive debates, and irrelevant theoretical developments that plant total chaos and confusion in the world of TOS, unfortunately. We define the error introduced by Distribution Heterogeneity as the Grouping and Segregation error, GSE, a vastly misunderstood error in sampling, even by some wellknown sampling experts, which is worrisome. Such fundamental misunderstanding is magnified when a poor understanding of the true source of analytical errors takes place, which makes empirical approaches extremely vulnerable.

## 统计代写|抽样理论作业代写sampling theory代考|Heterogeneity of a One-Dimensional Lot

Industrial activities are characterized by a constant need to transport materials (e.g., ores, concentrates, coal, cereals, chemicals, etc.) from one location to another. The practical implementation of such activities necessarily generates long piles, running materials on conveyor belts, and streams with suspended solids, that are all defined as one-dimensional lots. The good news is that it is always possible to implement a correct sampling operation on one-dimensional lots.

What we said of a zero-dimensional lot is still true for a one-dimensional lot which will be affected by a certain Constitution Heterogeneity coupled with a transient term which is Distribution Heterogeneity. However, a one-dimensional lot is almost always generated by chronological operations. Consequently, it will be affected by fluctuations that mainly reflect human activities at the mine, at the mill, at the processing or chemical plant, and so on. These are not the intrinsic properties of the material making up the lot; instead they are trends and they lead to a new concept of heterogeneity that can be subdivided into two terms:

1. heterogeneity $h_2$ introduced by long-range trends, which could be defined as a large-scale segregation
2. heterogeneity $h_3$ introduced by cyclic phenomena, which are extremely frequent in a processing plant.

Consequently, we will define two new errors introduced by these types of heterogeneity:

1. the long-range Heterogeneity Fluctuation Error $H_F E_2$
2. the periodic Heterogeneity Fluctuation Error $\mathrm{HFE}_3$.
These two new errors are defined as the continuous components of the Overall Estimation Error, $O E E$.

Therefore, in the case of a chronological series of units, we can define the heterogeneity introduced by random Constitution Heterogeneity as the small-scale heterogeneity $h_1$, which introduces an error defined as the short-range Heterogeneity Fluctuation Error $H F E_1$. We will show that we may write the following relationships, making the assumption the Increment Weighting Error IWE is negligible:
$$\begin{gathered} H F E_1=F S E+G S E \ h=h_1+h_2+h_3 \ H F E=H F E_1+H F E_2+H F E_3 \end{gathered}$$
where $h$ is the total heterogeneity supported by a lot of any kind, HFE is total continuous Heterogeneity Fluctuation Error which could also be called the integration error. We may see now that $H F E_1$ will serve as a link between the continuous model and the discrete model. If a one-dimensional lot was considered as a zero-dimensional lot, $h_3$ would cancel, and $h_2$ would become part of $h_1$, which is obvious as the lot would be considered as a random population. In Part VIII we will address potential problems with the Increment Weighting Error IWE; at this stage we make the assumption that it is negligible, which is not always the case if preventive precautions are not taken.

# 抽样理论代考

## 统计代写|抽样理论作业代写采样理论代考|零维Lot的异质性

1. 体质异质性$\mathrm{CH}_L$
2. 分布异质性$\mathrm{DH}_{\mathrm{L}}$ .
分布异质性是体质异质性的直接结果。没有构成异质性，分布异质性就不可能存在。我们在零维批次中处理的分布异质性可以定义为小规模分布异质性，这是相邻碎片组成之间随机波动的逻辑结果。这种随机波动产生了已经介绍过的基本采样误差FSE，但它们也为引力提供了一个机会，使碎片之间的顺序发生
3. 的重新排列，根据它们各自的组成将碎片族分离出来。成分(如成分、形状、大小、密度等)的差异越大，可能的分离就越强。在适当的时候，我们将确定构成异质性和分布异质性之间的确切关系，我们将看到分布异质性总是小于或等于构成异质性，这是直观的。分布异质性引入的抽样误差有两个因素:
1. 是分离因子，是空间重排的度量。
2. 是分组因子，是随机选择性的度量。直观地，我们发现基本抽样误差FSE是在收集给定权重的样本时产生的最小误差。只有在一个统计条件下才能达到最小值:组成样本的碎片必须严格随机收集，并且一个接一个。当然，实际情况并非如此，这让经验不足的经验从业人员非常困惑，他们想知道为什么在他们的异质性检验中复制样本之间的可变性与Gy公式不一致。当收集一个增量来组成一个样本时，这个增量很可能由许多片段组成。然后，从统计学上讲，一个样本不是由严格随机的片段组成的，而是由随机的片段组组成的;这种差异是巨大的，而且经常被完全误解。因此，我们引入了一个额外的错误，组越大，错误就越大，没有办法绕过它;事情就是这样。不幸的是，这总是会导致无休止的和适得其反的争论，以及不相关的理论发展，给TOS的世界带来了完全的混乱和困惑。我们将分布异质性引入的误差定义为分组和分离误差，GSE，这是一个在抽样中被广泛误解的误差，甚至被一些著名的抽样专家所误解，这是令人担忧的。当对分析错误的真正来源缺乏理解时，这种根本的误解就会被放大，这使得经验方法极其脆弱
统计代写|抽样理论作业代写采样理论代考|一维Lot的异质性 .工业活动的特点是不断需要将材料(例如矿石、精矿、煤、谷物、化学品等)从一个地点运输到另一个地点。这些活动的实际实施必然会产生长桩、传送带上的物料以及带有悬浮固体的流，这些都被定义为一维地段。好消息是，在一维批次上始终可以实现正确的采样操作我们所说的零维地段对于一维地段仍然是正确的，一维地段将受到一定的构成异质性加上一个暂态项即分布异质性的影响。然而，一维地段几乎总是由时间顺序操作生成的。因此，它将受到波动的影响，这些波动主要反映了在矿山、磨坊、加工厂或化工厂等地的人类活动。这些并不是构成物质的内在属性;相反，它们是趋势，它们导致了一个新的异质性概念，可以细分为两个术语:
1. 异质性$h_2$由长期趋势引入，可定义为由循环现象引入的大规模分离
2. 异质性$h_3$，循环现象在加工厂中极为频繁因此，我们将定义由这些类型的异质性引入的两个新错误:
1. 远距离异质性波动误差$H_F E_2$
2. 周期性异质性波动误差$\mathrm{HFE}_3$ .
这两个新误差被定义为总体估计误差的连续分量$O E E$ .

因此，在时间序列单元的情况下，我们可以将随机构成异质性引入的异质性定义为小尺度异质性$h_1$，引入一个被定义为短范围异质性波动误差$H F E_1$的误差。我们将表明，我们可以写出以下关系，假设增量加权误差IWE可以忽略:
$$\begin{gathered} H F E_1=F S E+G S E \ h=h_1+h_2+h_3 \ H F E=H F E_1+H F E_2+H F E_3 \end{gathered}$$
，其中$h$是由许多任何类型支持的总异质性，HFE是总连续异质性波动误差，也可以称为积分误差。我们现在可以看到$H F E_1$将作为连续模型和离散模型之间的链接。如果一维地段被认为是零维地段，$h_3$将被取消，$h_2$将成为$h_1$的一部分，这是显而易见的，因为地段将被认为是一个随机总体。在第八部分中，我们将讨论增量加权误差IWE的潜在问题;在这一阶段，我们假设它可以忽略不计，但如果不采取预防措施，情况并不总是如此

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