统计代写|化学计量学作业代写chemometrics代考|Self-Organizing Maps

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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|Training SOMs

A SOM is trained by repeatedly presenting the individual samples to the map. At each iteration, the current sample is compared to the codebook vectors. The most similar codebook vector (the “winning unit”) is then shifted slightly in the direction of the mapped object. This is achieved by replacing it with a weighted average of the old values of the codebook vector, $c v_{i}$, and the values of the new object obj:
$$c v_{i+1}=(1-\alpha) c v_{i}+\alpha o b j$$
The weight, also called the learning rate $\alpha$, is a small value, typically in the order of $0.05$, and decreases during training so that the final adjustments are very small.
As we shall see in Sect. $6.2 .1$, the algorithm is very similar in spirit to the one used in $k$-means clustering, where cluster centers and memberships are alternatingly estimated in an iterative fashion. The crucial difference is that not only the winning unit is updated, but also the other units in the “neighborhood” of the winning unit. Initially, the neighborhood is fairly large, but during training it decreases so that finally only the winning unit is updated. The effect is that neighboring units in general are more similar than units far away. Or, to put it differently, moving through the map by jumping from one unit to its neighbor would see gradual and more or less smooth transitions in the values of the codebook vectors. This is clearly visible in the mapping of the autoscaled wine data to a 5-by-4 SOM, using the kohonen package:

The result is shown in Fig. 5.2. Units in this example are arranged in a hexagonal fashion and are numbered row-wise from left to right, starting from the bottom left. The first unit at the bottom left for instance, is characterized by relatively large values of alcohol, flavonoids and proanth; the second unit, to the right of the first, has lower values for these variables, but still is quite similar to unit number one.
The codebook vectors are usually initialized by a random set of objects from the data, but also random values in the range of the data can be employed. Sometimes a grid is used, based on the plane formed by the first two PCs. In practice, the initialization method will hardly ever matter, however, starting from other random initial values will lead to different maps. The conclusions drawn from the different maps, however, tend to be very similar.

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

Several different visualization methods are provided in the kohonen package: one can look at the codebook vectors, the mapping of the samples, and one can also use SOMs for prediction. Here, only a few examples are shown. For more information, consult the manual pages of the plot. kohonen function, or the software description (Wehrens and Buydens 2007; Wehrens and Kruisselbrink 2018).

For multivariate data, the locations of the codebook vectors can not be visualized as was done for the two-dimensional data in Fig. 5.1. In the kohonen package, the default is to show segment plots, such as in Fig. $5.2$ if the number of variables is smaller than 15, and a line plot otherwise. One can also zoom in and concentrate on the values of just one of the variables:
$>\operatorname{for}(1 \ln c(1,8,11,13))$

• plotiwines.som. “property”.

for $(1 \ln c(1,8,11,13))$
$+\quad$ plot $($ wines.som, “property”
$+\quad$ property = getcodes(wines.som, 1) $[, 1] .$
$+\quad$ main = colnames $($ wines $)[1])$
property = getcodes (wines.som, 1) $[, 1]$.
main = colnames (Wines $[1]}$

Clearly, in these plots, shown in Fig. 5.3, there are regions in the map where specific variables have high values, and other regions where they are low. Areas of high values and low values are much more easily recognized than in Fig.5.2. Note the use of the accessor function getcodes here.

Perhaps the most important visualization is to show which objects map in which units. In the kohonen package, this is achieved by supplying the the type = “mapping ” argument to the plotting function. It allows for using different plotting characters and colors (see Fig. 5.4):

plot (wines.som, type = “mapping” ,

$c o 1=a s .$ integer (vintages), pch $=$ as. integer (vintages) )

Again, one can see that the wines are well separated. Some class overlap remains, especially for the Grignolinos (pluses in the figure). These plots can be used to make predictions for new data points: when the majority of the objects in a unit are, e.g., of the Barbera class, one can hypothesize that this is also the most probably class for future wines that end up in that unit. Such predictions can play a role in determining authenticity, an economically very important application.

Since SOMs are often used to detect grouping in the data, it makes sense to look at the codebook vectors more closely, and investigate if there are obvious class boundaries in the map-areas where the differences between neighboring units are relatively large. Using a color code based on the average distance to neighbors one can get a quick and simple idea of where the class boundaries can be found. This

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

The main attraction of SOMs lies in the applicability to large data sets; even if the data are too large to be loaded in memory in one go, one can train the map sequentially on (random) subsets of the data. It is also possible to update the map when new data points become available. In this way, SOMs provide a intuitive and simple visualization of large data sets in a way that is complementary to PCA. An especially interesting feature is that these maps can show grouping of the data without explicitly performing a clustering. In large maps, sudden transitions between units, as visualized by, e.g., a U-matrix plot, enable one to view the major structure at a glance. In smaller maps, this often does not show clear differences between groups-see Fig. $5.5$ for an example. One way to find groups is to perform a clustering of the individual codebook vectors. The advantage of clustering the codebook vectors rather than the original data is that the number of units is usually orders of magnitude smaller than the number of objects.

The kohonen package used in this chapter, originally based on the class package (Venables and Ripley 2002), has several noteworthy features not discussed yet (Wehrens and Kruisselbrink 2018). It can use distance functions other than the usual Euclidean distance, which might be extremely useful for some data sets, often avoiding the need for prior data transformations. One example is the WCC function mentioned earlier: this can be used to group sets of X-ray powder diffractograms where the position rather than the position of peaks contains the primary information (Wehrens and Willighagen 2006; Wehrens and Kruisselbrink 2018). For numerical variables, the sum-of-squares distance is the default (slightly faster than the Euclidean distance); for factors, the Tanimoto distance. In the kohonen package it is possible to supply several different data layers, where the rows in each layer correspond to different bits of information on the same objects. Separate distance functions can be defined for each single layer, which are then combined into one overall distance measure using weights that can be defined by the user. Apart from the usual “online” training algorithm described in this chapter, a “batch” algorithm is implemented as well, where codebook vectors are not updated until all records have been presented to the map. One advantage of the batch algorithm is that it dispenses with one of the parameters of the SOM: the learning rate $\alpha$ is no longer needed. The main disadvantage is that it is sometimes less stable and more likely to end up in a local optimum. The batch algorithm also allows for parallel execution by distributing the comparisons of objects to all codebook vectors over several cores (Lawrence et al. 1999) which may lead to considerable savings with larger data sets (Wehrens and Kruisselbrink 2018).

统计代写|化学计量学作业代写chemometrics代考|Training SOMs

C在一世+1=(1−一种)C在一世+一种这bj

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

kohonen 包中提供了几种不同的可视化方法：一种可以查看码本向量、样本的映射，还可以使用 SOM 进行预测。这里只展示了几个例子。有关更多信息，请参阅该图的手册页。kohonen 函数或软件描述（Wehrens 和 Buydens 2007；Wehrens 和 Kruisselbrink 2018）。

>为了⁡(1ln⁡C(1,8,11,13))

• plotiwines.som。“财产”。

+阴谋(wines.som，“财产”
+属性 = getcodes(wines.som, 1)[,1].
+主要 = 列名(葡萄酒)[1])

main = colnames（葡萄酒[1]}[1]}

C这1=一种s.整数（年份），pch=作为。整数（年份））

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

SOM 的主要吸引力在于对大数据集的适用性；即使数据太大而无法一次性加载到内存中，也可以在数据的（随机）子集上按顺序训练地图。当新的数据点可用时，也可以更新地图。通过这种方式，SOM 以与 PCA 互补的方式提供了大型数据集的直观和简单的可视化。一个特别有趣的功能是这些地图可以显示数据的分组，而无需明确执行聚类。在大型地图中，单元之间的突然转换，例如通过 U 矩阵图可视化，使人们能够一目了然地查看主要结构。在较小的地图中，这通常不会显示组之间的明显差异——见图。5.5例如。找到组的一种方法是对各个码本向量进行聚类。对码本向量而不是原始数据进行聚类的优点是单元的数量通常比对象的数量小几个数量级。

有限元方法代写

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MATLAB代写

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