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

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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

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

PCA is not alone in its aim to find low-dimensional representations of highdimensional data sets. Several other methods try to do the same thing, but rather than finding the projection that maximizes the explained variance, they choose other criteria. In Principal Coordinate Analysis (PCoA) and the related Multidimensional Scaling (MDS) methods, the aim is to find a low-dimensional projection that reproduces the experimentally found distances between the data points. When these distances are Euclidean, the results are the same or very similar to PCA results; however, other distances can be used as well. Independent Component Analysis maximizes deviations from normality rather than variance, and Factor Analysis concentrates on reproducing covariances. We will briefly review these methods in the next paragraphs.

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

In some cases, applying PCA to the raw data matrix is not appropriate, for example in situations where regular Euclidean distances do not apply-similarities between chemical structures, e.g., can be expressed easily in several different ways, but it is not at all clear how to represent molecules into fixed-length structure descriptors (Baumann 1999), something that is required by distance measures such as the Euclidean distance. Even when comparing spectra or chromatograms, the Euclidean distance can be inappropriate, for instance in the presence of peak shifts (Bloemberg et al. 2010 ; de Gelder et al. 2001). In other cases, raw data are simply not available and the only information one has consists of similarities. Based on the sample similarities, the goal of methods like Multidimensional Scaling (MDS, (Borg and Groenen 2005; Cox and Cox 2001)) is to reconstruct a low-dimensional map of samples that leads to the same similarity matrix as the original data (or a very close approximation).

Since visualization usually is one of the main aims, the number of dimensions usually is set to two, but in principle one could find an optimal configuration with other dimensionalities as well.

The problem is something like making a topographical map, given only the distances between the cities in the country. In this case, an exact solution is possible in two dimensions since the original distance matrix was calculated from twodimensional coordinates. Note that although distances can be reproduced exactly, the map still has rotational and translational freedom-in practice this does not pose any problems, however. An amusing example is given by maps not based on kilometers but rather on travel time-the main cities will be moved to the center of the plot since they usually are connected by high-speed trains, whereas smaller villages will appear to be further away. In such a case, and in virtually all practical applications, a two-dimensional plot will not be able to reproduce all similarities exactly.

In MDS, there are several ways to indicate the agreement between the two distance matrices, and these lead to different methods. The simplest approach is to perform $\mathrm{PCA}$ on the double-centered distance matrix, ${ }^{4}$ an approach that is known as Principal Coordinate Analysis, or Classical MDS (Gower 1966). The criterion to be minimized is called the stress, and is given by
$$S=\sum_{j<i}\left(\left|x_{i}-x_{j}\right|-e_{i j}\right)^{2}=\sum_{j<i}\left(d_{i j}-e_{i j}\right)^{2}$$
where $e_{i j}$ corresponds with the true, given, distances, and $d_{i j}$ are the distances between objects $x_{i}$ and $x_{j}$ in the low-dimensional space.

## 统计代写|化学计量学作业代写chemometrics代考|Independent Component Analysis and Projection Pursuit

Variation in many cases equals information, one of the reasons behind the widespread application of PCA. Or, to put it the other way around, a variable that has a constant value does not provide much information. However, there are many examples where the relevant information is hidden in small differences, and is easily overwhelmed by other sources of variation that are of no interest. The technique of Projection Pursuit (Friedman 1987; Friedman and Tukey 1974; Huber 1985) is a generalization of PCA where a number of different criteria can be optimized. One can for instance choose a viewpoint that maximizes some grouping in the data. In general, however, there is no analytical solution for any of these criteria, except for the variance criterion used in PCA. A special case of Projection Pursuit is Independent Component Analysis (ICA, Hyvärinen et al. 2001), where the view is taken to maximize deviation from multivariate normality, given by the negentropy $J$. This is the difference of the entropy of a normally distributed random variable $H\left(x_{\mathrm{G}}\right)$ and the entropy of the variable under consideration $H(x)$
$$J(x)=H\left(x_{\mathrm{G}}\right)-H(x)$$
where the entropy itself is given by
$$H(x)=-\int f(x) \log f(x) d x$$
Since the entropy of a normally distributed variable is maximal, the negentropy is always positive (Cover and Thomas 1991). Unfortunately, this quantity is hard to calculate, and in practice approximations, such as kurtosis and the fourth moment are used.

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

PCA 并不是唯一一个致力于寻找高维数据集的低维表示的人。其他几种方法尝试做同样的事情，但不是找到最大化解释方差的投影，而是选择其他标准。在主坐标分析 (PCoA) 和相关的多维缩放 (MDS) 方法中，目标是找到一个低维投影，该投影再现实验发现的数据点之间的距离。当这些距离为欧式时，结果与 PCA 结果相同或非常相似；然而，也可以使用其他距离。独立成分分析最大限度地偏离正态性而不是方差，因子分析专注于再现协方差。我们将在接下来的段落中简要回顾这些方法。

## 统计代写|化学计量学作业代写chemometrics代考|Independent Component Analysis and Projection Pursuit

Ĵ(X)=H(XG)−H(X)

H(X)=−∫F(X)日志⁡F(X)dX

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## MATLAB代写

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