### 统计代写|化学计量学作业代写chemometrics代考| Scaling

statistics-lab™ 为您的留学生涯保驾护航 在代写化学计量学chemometrics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写化学计量学chemometrics代写方面经验极为丰富，各种代写化学计量学chemometrics相关的作业也就用不着说。

• 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

## 统计代写|化学计量学作业代写chemometrics代考|Scaling

The scaling method that is employed can totally change the result of an analysis. One should therefore carefully consider what scaling method (if any) is appropriate. Scaling can serve several purposes. Many analytical methods provide data that are not on an absolute scale; the raw data in such a case cannot be used directly when comparing different samples. If some kind of internal standard is present, it can be used to calibrate the intensities. In NMR, for instance, the TMS (tetramethylsilane, added to indicate the position of the origin on the $x$-axis) peak can be used for this if its concentration is known. Peak heights can then be compared directly. However, even in that situation it may be necessary to further scale intensities, since samples may contain different concentrations. A good example is the analysis of a set of urine samples by NMR. These samples will show appreciable global differences in concentrations, perhaps due to the amount of liquid the individuals have been consuming. This usually is not of interest-rather, one seeks one or perhaps a couple of metabolites with concentrations that deviate from the general pattern. As an example, consider the first ten spectra of the prostate data:

The intensity differences within these first ten spectra are already a factor five for both statistics. If these differences are not related to the phenomenon we are interested in but are caused, e.g., by the nature of the measurements, then it is important to remove them. As stated earlier, also in cases where alignment is necessary, this type of differences between samples can hamper the analysis.

Several options exist to make peak intensities comparable over a series of spectra. The most often-used are range scaling, length scaling and variance scaling. In range scaling, one makes sure that the data have the same minimal and maximal values. Often, only the maximal value is considered important since for many forms of spectroscopy zero is the natural lower bound. Length scaling sets the length of each spectrum to one; variance scaling sets the variance to one. The implementation in $R$ is easy. Here, these three methods are shown for the first ten spectra of the prostate data. Range scaling can be performed by

The sweep function is very similar to apply-it performs an action for every row or column of a data matrix. The MARGIN argument states which dimension is affected. In this case the MARGIN $=1$ indicates the rows; column-wise sweeping would be achieved with MARGIN $=2$. The third argument is the statistic that is to be swept out, here the vector of the per-row maximal values. The final argument states how the sweeping is to be done. The default is to use subtraction; here we use division. Clearly, the differences between the spectra have decreased.

## 统计代写|化学计量学作业代写chemometrics代考|Missing Data

Missing data are measurements that for some reason have not led to a valid result. In spectroscopic measurements, missing data are not usually encountered, but in many other areas of science they occur frequently. The main question to be answered is: are the data missing at random? If yes, then we can probably get around the problem, provided there are not too many missing data. If not, then it means that there is some rationale behind the missingness of data points. If we would know what it was, we could use that to decide how to handle the missing values. Usually, we don’t, and that means trouble: if the missingness is related to the process we are studying our results will be biased and we can never be sure we are drawing correct conclusions.
Missing values in R are usually indicated by NA. Since many types of analysis do not accept data containing NAs, it is necessary to think of how to handle the missing values. If there are only a few, and they occur mostly in one or a few samples or one or a few variables, we might want to leave out these samples (or variables). Especially when the data set is rather large this seems a small price to pay for the luxury of having complete freedom in choosing any analysis method that is suited to our aim. Alternatively, one can try to find suitable replacements for the missing values, e.g. by estimating them from the other data points, a process that is known as imputation. Intermediate approaches are also possible, in which variables or samples with too many missing values are removed, and others, with a lower fraction of missing data, are retained. Sometimes imputation is not needed for statistical analysis: fitting linear models with $1 \mathrm{~m}$ for instance is possible also in the presence of missing values – these data points will simply be ignored in the fit process. Other functions such as var and cor have arguments that define several ways of dealing with missing values. In var, the argument is na. rm, allowing the user to either throw out missing values or accept missing values in the result, whereas cor has a more elaborate mechanism of defining strategies to deal with missing values. For instance, one can choose to consider only complete cases, or use only pairwise complete observations. Consult the manual pages for more information and examples. One example of dealing with missing values is shown in Sect. 11.1.

## 统计代写|化学计量学作业代写chemometrics代考|Principal Component Analysis

Principal Component Analysis or PCA (Jackson 1991; Jolliffe 1986) is a technique which, quite literally, takes a different viewpoint of multivariate data. It has many uses, perhaps the most important of which is the possibility to provide simple twodimensional plots of high-dimensional data. This way, one can easily assess the presence of grouping or outliers, and more generally obtain an idea of how samples and variables relate to each other. PCA defines new variables, consisting of linear combinations of the original ones, in such a way that the first axis is in the direction containing most variation. Every subsequent new variable is orthogonal to previous variables, but again in the direction containing most of the remaining variation. The new variables are examples of what is called latent variables (LVs) – in the context of PCA the term principal components (PCs) is used.

The central idea is that more often than not many of the variables in highdimensional data are superfluous. If we look at high-resolution spectra, for example, it is immediately obvious that neighboring wavelengths are highly correlated and contain similar information. Of course, one can try to pick only those wavelengths that appear to be informative, or at least differ from the other wavelengths in the selected set. This could, e.g., be based on clustering the variables, and selecting for each cluster one “representative”. However, this approach is quite elaborate and will lead to different results when using different clustering methods and cutting criteria. Another approach is to use variable selection, given some criterion-one example is to select a limited set of variables leading to a matrix with maximal rank. Variable selection is notoriously difficult, especially in high-dimensional cases. In practice, many more or less equivalent solutions exist, which makes the interpretation quite difficult. We will come back to variable selection methods in Chap. $10 .$

PCA is an alternative. It provides a direct mapping of high-dimensional data into a lower-dimensional space containing most of the information in the original data. The tacit assumption here is that variation equals information. This is not always true, since variation may also be totally meaningless, e.g., in the case of noise. ${ }^{1}$ The coordinates of the samples in the new space are called scores, often indicated with the symbol $T$. The new dimensions are linear combinations of the original variables, and are called loadings (symbol $\boldsymbol{P}$ ). The term Principal Component ( $\mathrm{PC}$ ) can refer to both scores and loadings; which is meant is usually clear from the context. Thus, one can speak of sample coordinates in the space spanned by PC 1 and 2 , but also of variables contributing greatly to $\mathrm{PC} 1$.

## 统计代写|化学计量学作业代写chemometrics代考|Missing Data

R 中的缺失值通常用 NA 表示。由于许多类型的分析不接受包含 NA 的数据，因此有必要考虑如何处理缺失值。如果只有几个，并且它们主要出现在一个或几个样本或一个或几个变量中，我们可能希望省略这些样本（或变量）。特别是当数据集相当大时，这似乎是一个很小的代价，可以完全自由地选择适合我们目标的任何分析方法。或者，可以尝试为缺失值找到合适的替换，例如通过从其他数据点估计它们，这一过程称为插补。中间方法也是可能的，其中删除具有太多缺失值的变量或样本，而保留缺失数据比例较低的其他变量或样本。1 米例如，在存在缺失值的情况下也是可能的——这些数据点在拟合过程中将被简单地忽略。var 和 cor 等其他函数的参数定义了处理缺失值的几种方法。在 var 中，参数是 na。rm，允许用户在结果中丢弃缺失值或接受缺失值，而 cor 具有更精细的机制来定义处理缺失值的策略。例如，可以选择只考虑完整的案例，或者只使用成对的完整观察。有关更多信息和示例，请参阅手册页。处理缺失值的一个例子在 Sect. 11.1。

## 统计代写|化学计量学作业代写chemometrics代考|Principal Component Analysis

PCA 是一种替代方案。它提供了将高维数据直接映射到包含原始数据中大部分信息的低维空间。这里的默认假设是变化等于信息。这并不总是正确的，因为变化也可能完全没有意义，例如在噪声的情况下。1新空间中样本的坐标称为分数，通常用符号表示吨. 新维度是原始变量的线性组合，称为载荷（符号磷）。术语主成分 (磷C) 可以指分数和载荷；这通常是从上下文中清楚的。因此，可以说 PC 1 和 2 跨越的空间中的样本坐标，但也可以说变量对磷C1.

## 有限元方法代写

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。