### 统计代写|机器学习作业代写machine learning代考| Removing Redundant Examples

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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|机器学习作业代写machine learning代考|Removing Redundant Examples

Some training examples do not hurt classification, and yet we want to get rid of them because they are redundant: they add to computational costs without affecting the classifier’s classification performance.

Redundant Examples and Computational Costs In machine-learning practice, we may encounter domains with $10^{6}$ training examples described by some $10^{4}$ attributes. Moreover, one may need to classify thousands of objects as quickly as possible. To identify the nearest neighbor of a single object, the nearest classifier relying on Euclidean distance has to carry out $10^{6} \times 10^{4}=10^{10}$ arithmetic operations. Repeating this for thousands of objects results in $10^{10} \times 10^{3}=10^{13}$ arithmetic operations. This may be impractical.

Fortunately, training sets are often redundant in the sense that the $k=\mathrm{NN}$ classifier’s behavior will be unaffected by the deletion of many training examples. Sometimes, a great majority of the examples can thus be removed with impunity. This is the case of the domain shown in the upper-left corner of Fig. 3.9.

Consistent Subset Redundancy is reduced if we replace the training set, $T$, with its consistent subset, $S$. In the machine-learning context, $S$ is said to be a consistent subset of $T$ if replacing $T$ with $S$ does not affect the class labels returned by the $k$ NN classifier. This definition, however, is not very practical because we do not know how the $k$-NN classifier (whether using $T$ or $S$ ) will behave on future examples. Let

us therefore modify the criterion: $S$ will be regarded as a consistent subset of $T$ if any ex $\in T$ receives the same label from the classifier, no matter whether the $k$-NN classifier is applied to $T-{\mathbf{e x}}$ or to $S-{\mathbf{e x}}$.

Quite often, a realistic training set has many consistent subsets. How do we choose the best one? Intuitively, the smaller the subset, the better. But a perfectionist who insists on having the smallest consistent subset may come to grief because such ideal can usually be achieved only at the price of enormous computational costs. The practically minded engineer who does not believe exorbitant costs are justified will welcome a computationally efficient algorithm that “reasonably downsizes” the original set, unscientific though such formulation may appear to be.

Creating a Consistent Subset One such pragmatic technique is presented in Table 3.6. The algorithm starts by placing one random example from each class in set $S$. This set, $S$, is then used by the l-NN classifier to decide about the labels of all training examples. At this stage, it is likely that some training examples will thus be misclassified. These misclassified examples are added to $S$, and the whole procedure is repeated using this larger version of $S$. The procedure is then repeated all over again. At a certain moment, $S$ becomes sufficiently representative to allow the 1 -NN classifier to label all training examples correctly.

## 统计代写|机器学习作业代写machine learning代考|Limitations of Attribute-Vector Similarity

The successful practitioner of machine learning has to have a good understanding of the limitations of the diverse tools. Here are some ideas concerning classification based on geometric distances between attribute vectors.

Common Perception of Kangaroos Any child will tell you that a kangaroo is easily recognized by the poach on its belly. Among all the attributes describing the examples, the Boolean information about the presence or the absence of the “pocket” is the most prominent, and it is not an exaggeration to claim that its importance is greater than that of all the remaining attributes combined. Giraffe does not have it, nor does a mosquito or an earthworm.

One Limitation of Attribute Vectors Dividing attributes into relevant, irrelevant, and redundant is too crude. The “kangaroo” experience shows us that among the relevant ones, some are more important than others; a circumstance is not easily reflected in similarity measures, at least not in those discussed in this chapter.

Ideally, $k$-NN should perhaps weigh the relative importance of the individual attributes and adjust the similarity measures accordingly. This is rarely done, in this paradigm. In the next chapter, we will see that this requirement is more naturally addressed by linear classifiers.

Relations Between Attributes Another clearly observable feature in kangaroos is that their front legs are much shorter than the hind legs. This feature, however, is not immediately reflected by similarities derived from geometric distances between attribute vectors. Typically, examples of animals will be described by such attributes as the length of a front leg and the length of a hind leg (among many others), but relation between the different lengths is only implicit.

The reader will now agree that the classification may depend less on the original attributes than on the relations between individual attributes, such as $a_{1} / a_{2}$. One step further, a complex function of two or more attributes will be more informative than the individual attributes.

Low-Level Attributes In domains, the available attributes are of a very low informational level. Thus in computer vision, it is common to describe the given image by a matrix of integers, each given the intensity of one “pixel,” essentially a single dot in the image. Such matrix can easily comprise millions of such pixels.
Intuitively, though, it is not these dots, very low-level attributes, but rather the way that these dots are combined into higher-level features such as lines, edges, blobs of different texture, and so on.

Higher-Level Features Are Needed The ideas presented in the last few paragraphs all converge to one important conclusion. To wit, it would be good if some more advanced machine-learning paradigm were able to create from available attributes meaningful higher-level features that would be more capable of informing us about the given object’s class.

## 统计代写|机器学习作业代写machine learning代考|Summary and Historical Remarks

When classifying object $\mathbf{x}$, the $k$-NN classifier identifies in the training set $k$ examples most similar to $\mathbf{x}$ and then chooses the class label most common among these “nearest neighbors.”
The concrete behavior of the $k-\mathrm{NN}$ classifier depends to a great extent on how it evaluates similarities of attribute vectors. The simplest way to establish the similarity between $\mathbf{x}$ and $\mathbf{y}$ seems to be by calculating their geometric distance by the following formula:
$$d_{M}(\mathbf{x}, \mathbf{y})=\sqrt{\Sigma_{i=1}^{n} d\left(x_{i}, y_{i}\right)}$$
Usually, we use $d\left(x_{i}, y_{i}\right)=\left(x_{i}-y_{i}\right)^{2}$ for continuous-valued attributes. For discrete attributes, we put $d\left(x_{i}, y_{i}\right)=0$ if $x_{i}=y_{i}$ and $d\left(x_{i}, y_{i}\right)=1$ if $x_{i} \neq y_{i}$. However, more advanced methods are sometimes used.
The use of geometric distance in machine learning can be hampered by inappropriate scales of attribute values. This is why it is usual to normalize the domains of all attributes to the unit interval, $[0,1]$. The user should not forget to normalize the descriptions of future examples by the same normalization formula.

The performance of the $k-\mathrm{NN}$ classifier may disappoint if many of the attributes are irrelevant. Another difficulty is presented by the diverse domains (scales) of the attribute values. The latter problem can be mitigated by normalizing the attribute values to unit intervals.
Some examples are harmful in the sense that their presence in the training set increases error rate. Others are redundant in that they only add to computation costs without improving classification performance. Harmful and redundant examples should be removed.
In many applications, each of the nearest neighbors has the same vote. In others, the votes are weighted by distance.
Classical approaches to nearest-neighbor classification usually do not weigh the relative importance of individual attributes. Another limitation is caused by the fact that, in some domains, the available attributes are too detailed. A mechanism to construct from them higher-level features is then needed.

## 统计代写|机器学习作业代写machine learning代考|Summary and Historical Remarks

d米(X,是)=Σ一世=1nd(X一世,是一世)

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