### 统计代写|机器学习作业代写machine learning代考|Polynomial Classifiers

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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|Polynomial Classifiers

Let us now abandon the strict requirement that positive examples be linearly separable from negative ones. Quite often, they are not. Not only can the linear separability be destroyed by noise; the very shape of the region occupied by one of the classes can render linear decision surface inadequate. Thus in the training set shown in Fig. 4.5, no linear classifier ever succeeds in separating the two squares from the circles. Such separation can only be accomplished by a non-linear curve such as the parabola shown in the picture.

Non-linear Classifiers The point having been made, we have to ask how to induce these non-linear classifiers from data. To begin with, we have to decide what type of function to employ. This is not difficult. Math teaches us that any $n$ dimensional curve can be approximated to arbitrary precision with some polynomial of a sufficiently high order. Let us therefore take a look at how to induce from data these polynomials. Later, we will discuss their practical utility.

Polynomials of the Second Order The good news is that the coefficients of polynomials can be induced by the same techniques that we have used for linear classifiers. Let us explain how.

For the sake of clarity, we will begin by constraining ourselves to simple domains with only two Boolean attributes, $x_{1}$ and $x_{2}$. The second-order polynomial is then defined as follows:
$$w_{0}+w_{1} x_{1}+w_{2} x_{2}+w_{3} x_{1}^{2}+w_{4} x_{1} x_{2}+w_{5} x_{2}^{2}=0$$
The expression on the left is a sum of terms that have one thing in common: a weight, $w_{i}$, multiplies a product $x_{1}^{k} x_{2}^{l}$. In the first term, we have $k+l=0$, because $w_{0} x_{1}^{0} x_{2}^{0}=w_{0}$; next come the terms with $k+l=1$, concretely, $w_{1} x_{1}^{1} x_{2}^{0}=w_{1} x_{1}$ and $w_{2} x_{1}^{0} x_{2}^{1}=w_{1} x_{2}$; and the sequence ends with three terms that have $k+l=2$ : specifically, $w_{3} x_{1}^{2}, w_{4} x_{1}^{1} x_{2}^{1}$, and $w_{5} x_{2}^{2}$. The thing to remember is that the expansion of the second-order polynomial stops when the sum of the exponents reaches 2 .
Of course, some of the weights can be $w_{i}=0$, rendering the corresponding terms “invisible” such as in $7+2 x_{1} x_{2}+3 x_{2}^{2}$ where the coefficients of $x_{1}, x_{2}$, and $x_{1}^{2}$ are zero.

## 统计代写|机器学习作业代写machine learning代考|Specific Aspects of Polynomial Classifiers

Now that we understand that the main strength of polynomials is their almost unlimited flexibility, it is time to turn our attention to their shortcomings and limitations.

Overfitting Polynomial classifiers tend to overfit noisy training data. Since the problem of overfitting is typical of many machine-learning paradigms, it is a good idea discuss its essence in some detail. Let us constrain ourselves to twodimensional continuous domains that are easy to visualize.

The eight training examples in Fig. $4.7$ fall into two groups. In one group, all examples are positive (empty circles); in the other, all save one are negative (filled circles). Two attempts at separating the two classes are shown. The one on the left uses a linear classifier, ignoring the fact that one training example is thus misclassified. The one on the right resorts to a polynomial classifier in an attempt to avoid any error on the training set.

Inevitable Trade-Off Which of the two is to be preferred? The answer is not straightforward because we do not know the underlying nature of the data. It may be that the two classes are linearly separable, and the only cause for one positive example to be found in the negative region is class-label noise. If this is the case, the single error made by the linear classifier on the training set is inconsequential, whereas the polynomial on the right, cutting deep into the negative area, will misclassify those future examples that find themselves on the wrong side of the

curve. Conversely, it is possible that the outlier does represent some legitimate, even if rare, aspect of the positive class. In this event, the use of the polynomial is justified. Practically speaking, however, the assumption that the single outlier is only noise is more likely to be correct than the “special-aspect” alternative.

A realistic training set will contain not one, but quite a few, perhaps many examples that appear to be in the wrong area of the instance space. And the interclass boundary that the classifier seeks to approximate may indeed be curved, though how much curved is anybody’s guess. The engineer may regard the linear classifier as too crude, and opt instead for the more flexible polynomial. This said, a highorder polynomial will separate the two classes even in a very noisy training set-and then fail miserably on future data. The ideal solution is usually somewhere between the extremes and has to be determined experimentally.

## 统计代写|机器学习作业代写machine learning代考|Support Vector Machines

Now that we understand that polynomial classifiers do not call for any new learning algorithms, we can return to linear classifiers, a topic we have not yet exhausted. Let us abandon the restriction to the Boolean attributes, and consider also the possibility of the attributes being continuous. Can we then still rely on the two training algorithms described above?

Perceptron Learning in Numeric Domains In the case of perceptron learning, the answer is easy: yes, the same weight-modification formula can be used. Practical experience shows, however, that it is a good idea to normalize all attribute values so that they fall into the unit interval, $x_{i} \in[0,1]$. We can use to this end the normalization technique described in the chapter dealing with nearest-neighbor classifiers, in Sect. 3.3.
$$w_{i}=w_{i}+\eta[c(\mathbf{x})-h(\mathbf{x})] x_{i}$$
Learning rate, $\eta$, and the difference between the real and hypothesized class labels, $[c(\mathbf{x})-h(\mathbf{x})]$, have the same meaning and impact as before. What has changed is the role of $x_{i}$. In the case of Boolean attributes, the value of $x_{i}=1$ or $x_{i}=0$ decided whether or not the corresponding weight should change. Here, however, the value of $x_{i}$ decides how much the weight should be affected: the change is greater if the attribute’s value is higher.

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