### 统计代写|机器学习作业代写machine learning代考| Inter-Class Boundaries: Linear

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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|Essence

To begin, let us constrain ourselves to Boolean domains where each attribute is either true or false. To be able to use these attributes in algebraic functions, we will represent them by integers: true by 1 , and false by 0 .

Linear Classifier In Fig. 4.1, one example is labeled as positive and the remaining three as negative. In this particular case, the two classes are separated by the linear function defined as follows:

In the expression on the left-hand side, $x_{1}$ and $x_{2}$ represent attributes. If we substitute for $x_{1}$ and $x_{2}$ the concrete values of a given example $(0$ or 1$)$, the expression $-1.2+0.5 x_{1}+x_{2}$ will be either positive or negative. The sign then determines the example’s class. The table on the right shows how the four examples from the left are thus classified.

Equation $4.1$ is not the only one capable of doing the job. Other expressions, say, $-1.5+x_{1}+x_{2}$, will label the four examples in exactly the same way. As a matter of fact, the same can be accomplished by infinitely many classifiers of the following generic form:
$$w_{0}+w_{1} x_{1}+w_{2} x_{2}=0$$
The function is easy to generalize to domains with $n$ attributes:
$$w_{0}+w_{1} x_{1}+\ldots+w_{n} x_{n}=0$$
If $n=2$, Eq. $4.2$ defines a line; if $n=3$, a plane; and if $n>3$, a hyperplane. If we introduce a “zeroth” attribute, $x_{0}$, that is not used in example description and whose value is always fixed at $x_{0}=1$, the equation can be re-written in the following compact form:
$$\sum_{i=0}^{n} w_{i} x_{i}=0$$

## 统计代写|机器学习作业代写machine learning代考|Perceptron Learning

Having developed some basic understanding of how the linear classifier works, we are ready to take a look at how to induce it from training data.

Learning Task Let us assume that each training example, $\mathbf{x}$, is described by $n$ binary attributes whose values are either $x_{i}=1$ or $x_{i}=0$. A positive example is indicated by $c(\mathbf{x})=1$, and a negative by $c(\mathbf{x})=0$. To make sure we do not confuse the example’s real class with the one suggested by the classifier, we will denote the latter by $h(\mathbf{x})$ where the letter $h$ emphasizes that this is the classifier’s hypothesis. If $\sum_{i=0}^{n} w_{i} x_{i}>0$, the classifier “hypothesizes” that the example is positive and therefore returns $h(\mathbf{x})=1$. Conversely, if $\sum_{i=0}^{n} w_{i} x_{i} \leq 0$, the classifier returns $h(\mathbf{x})=0$. Figure $4.2$ reminds us that the classifier labels $\mathbf{x}$ as positive only if the cumulative evidence supporting this class exceeds 0 .

Finally, we will assume that examples with $c(\mathbf{x})=1$ are linearly separable from those with $c(\mathbf{x})=0$. This means that there exists a linear classifier that will label correctly all training examples so that $h(\mathbf{x})=c(\mathbf{x})$ for any $\mathbf{x}$. The task for machine learning is to find the weights, $w_{i}$, that make this happen.

Learning from Mistakes Here is the essence of the most common approach to induction of linear classifiers. Suppose we have a working version of the classifier, even if imperfect. When presented with a training example, $\mathbf{x}$, the classifier suggests a label, $h(\mathbf{x})$. If this differs from the true class, $h(\mathbf{x}) \neq c(\mathbf{x})$, the learner concludes that the weights should be modified in a way likely to correct this error.

Let the true class be $c(\mathbf{x})=1$. In this event, $h(\mathbf{x})=0$ will only happen if $\sum_{i=0}^{n} w_{i} x_{i}<0$, an indication that the weights are too small. If we increase them, the sum, $\sum_{i=0}^{n} w_{i} x_{i}$, may exceed zero, making the returned label positive, and therefore correct. Note that it is enough to increase only the weights of attributes with $x_{i}=1$; when $x_{i}=0$, then the value of $w_{i}$ does not matter because anything multiplied by zero is still zero: $0 \cdot w_{i}=0$.

Likewise, if $c(\mathbf{x})=0$ and $h(\mathbf{x})=1$, then the weights of all attributes with $x_{i}=1$ should be decreased so as to give the sum the chance to drop below zero, $\sum_{i=0}^{n} w_{i} x_{i}<0$, in which case the classifier will label $\mathbf{x}$ as negative.

## 统计代写|机器学习作业代写machine learning代考|Domains with More Than Two Classes

Having only two sides, a hyper-plane may separate the positive examples from the negative examples-and that is all. When it comes to multi-class domains, the tool seems helpless. Or is it?

Groups of Binary Classifiers What exceeds the powers of an individual can be solved by a team. One practical solution is shown in Fig. 4.4. The “team” consists of four binary classifiers, each specializing on one of the four classes, $C_{1}$ through $C_{4}$. Ideally, the presentation of an example from $C_{i}$ results in the $i$-th classifier returning $h_{i}(\mathbf{x})=1$, and all the other classifiers returning $h_{j}(\mathbf{x})=0$, assuming, again, that each class is linearly separable from the other classes.

Modifying the Training Data To exhibit this behavior, the individual classifiers need to be properly trained. This training can be accomplished by any of the two algorithms from the previous sections. The only additional trick is that the engineer needs to modify the training data.

Table $4.5$ illustrates the principle. On the left is the original training set, $T$, where each example is labeled with one of the four classes. On the right are four “derived” sets, $T_{1}$ through $T_{4}$, each consisting of the same six examples which have now been re-labeled so that an example that in the original set, $T$, represents class $C_{i}$ is labeled with $c(\mathbf{x})=1$ in $T_{i}$ and with $c(\mathbf{x})=0$ in all other sets.

Needing a Master Classifier The training sets, $T_{i}$, are presented to a program that induces from each of them a linear classifier dedicated to the corresponding class. This is not the end of the story, though. The training examples may poorly represent the classes, they may be corrupted by noise, and even the requirement of linear separability may be violated. As a result, the induced classifiers may overlap each other in the sense that two or more of them will respond to the same example, $\mathbf{x}$, with $h_{i}(\mathbf{x})=1$, leaving the incorrect impression that $\mathbf{x}$ simultaneously belongs to more than one class. This is why a master classifier is needed; its task is to choose from the returned classes the one most likely to be correct.

∑一世=0n在一世X一世=0

## 广义线性模型代考

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