Support Vector Machines (SVMs) have become

A Simple Method For Estimating Conditional Probabilities For SVMs

Stefan R¨uping

CS Department,AI Unit

Dortmund University

44221Dortmund,Germany

rueping@ls8.cs.uni-dortmund.de

Abstract

Support Vector Machines(SVMs)have become

a popular learning algorithm,in particular for

large,high-dimensional classi?cation problems.

SVMs have been shown to give most accurate

classi?cation results in a variety of applications.

Several methods have been proposed to obtain

not only a classi?cation,but also an estimate of

the SVMs con?dence in the correctness of the

predicted label.In this paper,several algorithms

are compared which scale the SVM decision

function to obtain an estimate of the conditional

class probability.A new simple and fast method

is derived from theoretical arguments and empir-

ically compared to the existing approaches.

1Introduction

Support Vector Machines(SVMs)have become a pop-ular learning algorithm,in particular for large,high-dimensional classi?cation problems.SVMs have been shown to give most accurate classi?cation results in a vari-ety of applications.Several methods have been proposed to obtain not only a classi?cation,but also an estimate of the SVMs con?dence in the correctness of the predicted label. Usually,the performance of a classi?er is measured in terms of accuracy or some other performance measure based on the comparison of the classi?ers prediction of the true class.But in some cases,this does not give suf-?cient information.For example in credit card fraud de-tection,one has usually much more negative than positive examples,such that the optimal classi?er may be to the de-fault negative classi?er.But then,still one would like to ?nd out which transactions are most probably fraudulent, even if this probability is small.In other situations e.g.in-formation retrieval,one could be more interested in a rank-ing of the examples with respect to their interestingness in-stead of a simple yes/no-decision.Third,one may be inter-ested to integrate a classi?er into a bigger system,for ex-ample a multi-classi?er learner.To combine and compare the SVM prognosis with that of other learners,one would like a comparable,well-de?ned con?dence estimate.The best method to achieve a con?dence estimate that allows to rank the examples and gives well-de?ned,interpretable val-ues,is to estimate the conditional class probability. Obviously,this is a more complex problem than?nding a classi?cation,as it is possible to get a classi?cation function by comparing to the thresh-old,but not vice versa.

For numerical classi?ers,i.e.classi?ers of the type

with a numerical decision function, one usually tries to estimation the conditional class prob-ability from the decision function. This reduces the probability estimation from a multi-variate to a one-dimensional problem,where one has to?nd a scaling function such that. The idea behind this approach is that the classi?cation of examples that lie close to the decision boundary

can easily change when the examples are randomly perturbed by a small amount.This is very hard for examples with very high or very low(this argu-ment requires some sort of continuity or differentiability constraints on the function).Hence,the probability that the classi?er is correct should be higher for larger absolute values of.As was noted by Platt[10],this also means there is a strong prior for selecting a monotonic scaling function.

The rest of the paper is organized as follows:In the next section,we will shortly present the Support Vector Ma-chine and Kernel Logistic Regression algorithm,as far as it is necessary for this paper.In Section3,existing methods for probabilistic scaling of SVM outputs will be discussed and a new,simple scaling method will be presented.The effectiveness of this method will be empirically evaluated in Section4.

2Algorithms

2.1Support Vector Machines

Support Vector Machines are a classi?cation method based on Statistical Learning Theory[12].The goal is to?nd a function that minimizes the expected Risk

of the learner by minimizing the regularized risk reg, which is the weighted sum of the empirical risk with re-spect to the data and a complexity term reg

(2)

2.2The Kernel Trick

The inner product in Equation2can be replaced by a kernel function which corresponds to an

inner product in some space,called feature space.That is, there exists a mapping such that

.This allows the construction of non-linear classi?ers by an essentially linear algorithm.

The resulting decision function is given by

The actual SVM classi?cation is given by.It can be shown that the SVM solution depends only on its support vectors SV=.See[12;2]for a more detailed introduction on SVMs.

2.3Kernel Logistic Regression

Kernel Logistic Regression[13;5;14;11]is the kernelized version of the well-known logistic regression technique. The optimization problem is similar to the SVM problem in Equation1except that an exponential loss function is used instead of the L1loss:

In contrast to the SVM,Kernel Logistic Regression di-rectly models the conditional class probability,i.e.

can be estimated via which monotonously maps the decision functions value to the interval.The scaler assumes that for the decision function is of the type and hence for the classi?ers class decision is smallest such that is mapped to the conditional class probability.This allows to view softmax as a probability.However,this mapping is not very well founded,as the scaled values are not justi?ed from the data.

To justify the interpretation,it is better to use data to calibrate the scaling.One can use a subset of the data which has not been used for training(or use a cross-validation-like approach)and optimize the scal-ing function to minimize the error between the predicted class probability and the empirical class probabil-ity de?ned by the class values in the new data.There are two error measures which are usually used,cross-entropy and mean squared error.Cross-entropy is de?ned by

(where),which is the Kullback-Leibler dis-tance between the predicted and the empirical class proba-bility.For comparison of different data sets it is better to di-vide the cross-entropy by the number of examples and work with the mean cross-entropy mCRE.The mean squared er-ror is de?ned by

with to obtain a monotonically increasing function. The parameters and are found by minimization of the cross-entropy error over a test set with. For an ef?cient implementation,see[8].

Garczarek[4]proposes a method which scales classi?-cation values by

where is the Beta distribution function with param-eters and.The parameters and are se-lected such that over a test set

1.the average value of for each class is identical

to the classi?cation performance of the classi?er in this class and

2.the mean square error is minimized. Originally,the algorithm is designed for multiclass prob-lems and computes an individual scaler for each predicted class.For binary problems,it is better to modify this ap-proach such that only one scaler is generated.This avoids

-2.5

-2-1.5-1-0.500.511.522.53-3-2-1012345

(x,y)SVM KLR

Figure 1:One-dimensional comparison of SVM and KLR

predictions.Negatives examples are drawn from N(0,1)(dots at y=-1),positive examples from N(2,1)(dots at y=1).Both methods ?nd the class border at x=1,but the SVM prediction is essentially constant for y outside [-1,1].KLR correctly estimates higher con?dences for points nearer to class centers.

discontinuities in

when the prediction changes from one class to the other.

Binning has also been applied to this problem [3].The decision values are discretized into several bins and one can estimate the the conditional class probability by count-ing the class distribution in the single bins.Other,more complicated approaches also exists,see e.g.[7]or [12],Ch.11.11.

3.1Theoretical Limitations

Bartlett and Tewari [1]show that there is a tradeoff between sparseness of a classi?er and the ability to estimate condi-tional probabilities.Their result says,in short,that if one is able to estimate on some interval,sparse-ness is lost in that region.Hence,the question arises in how far the decision function of the SVM,which gener-ally produces sparse classi?ers,can approximate the true conditional density or the estimate of the non-sparse KLR,respectively.

The problem can be seen in Equation 1.To obtain a max-imally accurate classi?er,the SVM contains

in its objective function,i.e.the classi?er is punished if

(it becomes a support vector).In this case,

this forces an ordering on the values where the value is the higher,the more similar the example is to the rest of the examples in its class in feature space.Conse-quently,an estimation of

can be constructed from .When the example is classi?ed correctly with suf-?cient margin,i.e.,this example generates no loss and hence no speci?c order is enforced on these exam-ples.For the SVM,all the examples on the right side of the margin have the same probability .This behavior can be seen in Figure 1.

What can be said about the support vectors?In the pre-vious section we already saw that minimizing the mean

squared error between the estimation function

and gives a proper estimate of ,as for a ?xed the MSE is minimized for .How-ever,the error criterion in the SVM is the absolute er-ror,not the squared error,and one can show that for a ?xed the absolute error is minimized at iff and otherwise.What

comes to the rescue is that

is not determined for each independently,but for all together.Hence,if not over-?tting occurs,at least a value of

is an indicator of and it seems plausible that contains some useful information about .

3.2A Simple Estimation Method

From the previous discussion we know that decision func-tion value with are unreliable for estimating the

conditional class probability.Values with

di-rectly optimize the order of the examples with respect to

.Hence,the question arises if it is possible to esti-mate by the following trivial procedure

iff

covtype

495148diabetes 7688digits

77664ionosphere 35134liver

3456mushroom 8124126promoters 106228

SVM-Platt:SVM using Platt’s scaling.

SVM-Beta:SVM using Garczarek’s beta scaling.

SVM-Beta-2:SVM using binary beta scaling.

SVM-Bin:SVM and binning.

SVM-Softmax:SVM and softmax scaling.

SVM-01:SVM and output clipped between0and1. SVM-PP:SVM and output clipped between

and.

All reported results were10-fold cross-validated.For the linear SVM and KLR,the following results were obtained: Method MSE mCRE

SVM-010.09700.0317

SVM-PP0.09330.0296

With respect to the mean squared error,we get the fol-lowing ranking:SVM-Platt SVM-Beta-2SVM-PP SVM-01SVM-Softmax KLR SVM-Bin-10 SVM-Bin-50SVM-Beta.Sorting by mean cross-entropy,SVM-Beta-2and SVM-PP change places,as well as SVM-Softmax and Bin-10.

The RBF kernel gave the following results:

Method MSE mCRE

SVM-010.09160.0307

SVM-PP0.09040.0289

This gives the following ranking for MSE:KLR SVM-Platt SVM-Beta-2SVM-PP SVM-01SVM-Bin-10SVM-Softmax SVM-Bin-50SVM-Beta.

A close inspection reveals that these results do not give the full picture,as the error measures reach very different values for the individual data sets.E.g.,the MSE for Ker-nel Logistic Regression with radial basis kernel runs from (mushroom)to(liver).To allow for a better comparison,the methods were ranked according to their performance for each data set.The following table gives the average rank of each of the methods for the linear ker-nel:

avg.rank from

Method MSE mCRE

SVM-01 4.91 5.09

SVM-PP 3.45 3.55

The corresponding table for the radial basis kernel:

avg.rank from

Method MSE mCRE

SVM-01 5.36 5.64

SVM-PP 3.64 4.73

To validate the signi?cance of the results,a paired t-test ()was run over the cross-validation runs.The fol-lowing table shows the comparison of the cross-entropy for the linear kernel of the best?ve of the scaling algorithms. Each row of the table shows how often the hypothesis that the estimation in that row is better than the estimation in the corresponding column was rejected.E.g.,the6in the last row and?rst column shows that the hypothesis that soft-max scaling is better than KLR was rejected for6of the data sets.The contrary hypothesis was rejected on2data sets(?rst row,last column).

KLR Platt Beta2PP Bin10Soft KLR000000

Platt600100

Beta2760410

PP754020

Bin10833302

Soft997960 The corresponding tables for MSE show similar results. Summing up,we see that

Kernel Logistic Regression give the best estimation of the conditional class probability(with some outliers in the linear case).

The best scaling for the SVM is obtained by Platt’s method and binary Beta Scaling.

The trivial PP-scaling performs comparable to the much more complicated techniques.

Multiclass Beta scaling gives by far the worst results (which was expected from the non-continuicity of its method of scaling each predicted class on its own).

5Summary

The experiments in this paper showed that a trivial method of estimating the conditional class probability from the output of a SVM classi?er performs comparably to much more complicated estimation techniques.

Acknowledgments

The?nancial support of the Deutsche Forschungsgemein-schaft(SFB475,”Reduction of Complexity for Multivari-ate Data Structures”)is gratefully acknowledged.

References

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[3]Joseph Drish.Obtaining calibrated probability esti-

mates from support vector machines.Technical re-port,University of California,San Diego,June2001.

[4]Ursula Garczarek.Classi?cation Rules in Standard-

ized Partition Spaces.PhD thesis,Universit¨a t Dort-mund,2002.

[5]T.S.Jaakkola and D.Haussler.Probabilistic kernel

regression models.In Proceedings of the1999Con-ference on AI and Statistics,1999.

[6]S.S.Keerthi,K.Duan,S.K.Shevade,and A.N.Poo.

A fast dual algorithm for kernel logistic regression.

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port vector machine classi?ers.IEEE Transactions on Neural Networks,10(5):1018–1031,September1999.

[8]H.-T.Lin,C.-J.Lin,and R.C.Weng.A note on

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[9]P.M.Murphy and D.W.Aha.UCI repository of ma-

chine learning databases,1994.

[10]John Platt.Advances in Large Margin Classi?ers,

chapter Probabilistic Outputs for Support Vector Ma-chines and Comparisons to Regularized Likelihood Methods.MIT Press,1999.

[11]V olker Roth.Probabilistic discriminative kernel clas-

si?ers for multi-class problems.In B.Radig and S.Florczyk,editors,Pattern Recognition–DAGM’01, number2191in LNCS,pages246–253.Springer, 2001.

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and the import vector machine.In Neural Information Processing Systems,volume14,2001.

such_as_的各种不同用法

一、such as 的七个用法 1. 表示举例，意为“例如，诸如此类的，像……那样的”，相当于like或for example。如： There are few poets such as Keats and Shelly. 像济慈和雪莱这样的诗人现在很少了。 Adverbs are used to modify verbs, such as “quickly” in “she ran fast”. 副词用来修饰动词，例如“她跑得快”中的“快”。 Animals such as rabbits and deer continue to be active all winter，finding food wherever they can . 像兔和鹿这样的动物整个冬天都是很活跃的，它们到处寻找食物。 用于此义时的几点说明： (1) 这类结构既可表现为“名词+such as+例子”，也可表现为“such+名词+as+例子”。如： I enjoy songs such as this one.= I enjoy such songs as this one. 我喜欢像这首歌一样的歌。 (2) 若后接动词，通常用动名词，有时也可用动词原形。如： Don’t do anything silly such as marry him. 不要做什么蠢事，比如说去嫁给他。 Magicians often perform tricks such as pulling a rabbit out of a hat. 魔术师常常变从帽子里抓出兔子的戏法。 (3) 不要按汉语意思将such as用作such like。 (4) 其后不可列出前面所提过的所有东西。如： 正：I know four languages, such as Japanese and English. 我懂四种语言，如日语、英语。 误：I know four languages, such as Chinese, French, Japanese and English. 我懂四种语言，如汉语、法语、日语和英语。 (5) 在现代英语中，such as可与etc. 连用。如： They planted many flowers, such as roses, sunflowers，etc. 他们种了许多种花，如玫瑰花、向日葵等。 They export a 1ot of fruits，such as oranges，lemons，etc. 他们出口许多水果，如桔子、柠檬等。 2. 表示“像……这样的”，其中的as 用作关系代词，引导定语从句，as 在定语从句中用作主语或宾语。此外，不要按汉语意思把该结构中的as 换成like。如： He is not such a fool as he looks. 他并不像他看起来那么傻。

word短语用法小结

word短语用法小结 甘肃省民勤一中高雪萍邮编733300 一．中学阶段常用的含有word的短语主要有以下几个： 1．in other words换句话说，也就是说 He doesn’t like work. In other words, he is lazy. 他不爱工作，换句话说，他很懒。 2. in words 用文字 Can you describe it in words? 你能用文字描述它吗？ 3. in a / one word 总之，简言之 In a word, I think he’s a fool. 总之，我认为他是个傻瓜。 4. in word 口头上 The teacher asked his students to explain the law in word.老师让学生口头上解释一下这个定律。 5. have a word with sb.和某人说几句话 Can you spare me a few minutes? I’d like to have a word with you. 你能给我几分钟的时间吗？我有话跟你谈。 6. have words with sb.与某人争吵 She had words with his husbands about who should do the housework. 她和她丈夫就谁应该做家务吵了一架。 7. by word of mouth 口头上的 He received the news by word of mouth. 他得到的是口头上的消息。

8. a play on words 双关语 The advertising slogan was a play on words. 那条广告的口号是双关语。 9. word for word逐字地 He repeated what you said word for word.他一字不差地复述您说的话。 二．word常用的其他短语有： break one’s word失信； be as good as one’s word/keep one’s word守信；例如： You’ll find she is as good as her word. eat one’s words承认自己说错话； leave word留言；例如： Please leave word of your safe arrival/that you have arrived safely. big words大话； fair/good word恭敬话； go back on one’s word 食言； from the word go从一开始；例如： She knew from the word that it was going to be difficult.

For-example与such-as的用法及区别

For example与such as的用法及区别 1)for example和such as都可当作“例如”解。但such as用来列举事 物，插在被列举事物与前面的名词之间。例如： The farm grows various kinds of crops, such as wheat, corn, cotton and rice. 这个农场种植各种各样的庄稼，例如麦子，玉米，棉花和稻米。 2)for example意为用来举例说明，有时可作为独立语，插在句中，不影 响句子其他部分的语法关系。例如： A lot of people here, for example, Mr John, would rather have coffee. 这儿的许多人，例如约翰先生，宁愿喝咖啡。 【注意】 (a)such as一般不宜与and so on连用。 (b)对前面的复数名词部分起列举作用，一般不全部列出。故不可以说： He knows four languages, such as Chinese, English, French and German. 应将such as改成namely, 后面加逗号。即：He knows four languages, namely, Chinese, English, French and German. 这两个短语都可以表米“例如”，但含义及用法不同。 l）for example强调“举例”说明，而且一般只举同类人或物中的一个作为插入语，且用逗号隔开，可置于句首、句中或句末。如： Many people here, for example, John, would rather have coffee.这里有许多人，例如约翰很喜欢喝咖啡。 There are many kinds of pollution, for example, noise is a kind of pollution.有许多种污染方式，例如噪音就是一种污染。 2）such as用来“罗列”同类人或物中的几个例子，可置于被列举的事物与前面的名词之间，但其后边不能用逗号。如： Many of the English programmes are well received, such as Follow Me, Follow Me to Science . 其中有许多英语节目，如《跟我学》《跟我学科学》，就很受欢迎。 English is spoken in many countries, such as Australia, Canada and so on.许多国家说英语，如澳大利亚加拿大 1

ALEVEL数学术语

一般词汇 数学mathematics, maths(BrE), math(AmE) 公理axiom 定理theorem 计算calculation 运算operation 证明prove 假设hypothesis, hypotheses(pl.) 命题proposition 算术arithmetic 加plus(prep.), add(v.), addition(n.) 被加数augend, summand 加数addend 和sum 减minus(prep.), subtract(v.), subtraction(n.) 被减数minuend

减数subtrahend 差remainder 乘times(prep.), multiply(v.), multiplication(n.) 被乘数multiplicand, faciend 乘数multiplicator 积product 除divided by(prep.), divide(v.), division(n.) 被除数dividend 除数divisor 商quotient 等于equals, is equal to, is equivalent to 大于is greater than 小于is lesser than 大于等于is equal or greater than 小于等于is equal or lesser than 运算符operator 数字digit 数number

自然数natural number 整数integer 小数decimal 小数点decimal point 分数fraction 分子numerator 分母denominator 比ratio 正positive 负negative 零null, zero, nought, nil 十进制decimal system 二进制binary system 十六进制hexadecimal system 权weight, significance 进位carry 截尾truncation 四舍五入round

word的特殊用法

1. 问：WORD 里边怎样设置每页不同的页眉？如何使不同的章节显示的页眉不同？ 答：分节，每节可以设置不同的页眉。文件――页面设置――版式――页眉和页脚――首页不同。 2. 问：请问word 中怎样让每一章用不同的页眉？怎么我现在只能用一个页眉，一改就全部改了？ 答：在插入分隔符里，选插入分节符，可以选连续的那个，然后下一页改页眉前，按一下“同前”钮，再做的改动就不影响前面的了。简言之，分节符使得它们独立了。这个工具栏上的“同前”按钮就显示在工具栏上，不过是图标的形式，把光标移到上面就显示出”同前“两个字来。 3. 问：如何合并两个WORD 文档，不同的页眉需要先写两个文件，然后合并，如何做？ 答：页眉设置中，选择奇偶页不同/与前不同等选项。 4. 问：WORD 编辑页眉设置，如何实现奇偶页不同? 比如：单页浙江大学学位论文，这一个容易设；双页：（每章

标题），这一个有什么技巧啊？ 答：插入节分隔符，与前节设置相同去掉，再设置奇偶页不同。 5. 问：怎样使WORD 文档只有第一页没有页眉，页脚？ 答：页面设置－页眉和页脚，选首页不同，然后选中首页页眉中的小箭头，格式－边框和底纹，选择无，这个只要在“视图”――“页眉页脚”，其中的页面设置里，不要整个文档，就可以看到一个“同前”的标志，不选，前后的设置情况就不同了。 6. 问：如何从第三页起设置页眉？ 答：在第二页末插入分节符，在第三页的页眉格式中去掉同前节，如果第一、二页还有页眉，把它设置成正文就可以了 ●在新建文档中，菜单―视图―页脚―插入页码―页码格式―起始页码为0，确定；●菜单―文件―页面设置―版式―首页不同，确定；●将光标放到第一页末，菜单―文件―页面设置―版式―首页不同―应用于插入点之后，确定。第

Word2003使用技巧大全

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have got的详细用法回顾.doc

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alevel 数学介绍

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word的详细用法

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ALEVEL数学术语

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have_got的详细用法教学内容

h a v e_g o t的详细用 法

复习：have的用法及否定句、一般疑问句的变法。 ①have的意思是：________，它的单数形式是：_______。have是_______词。 例如：我有许多好朋友。_____ _____ many good friends.他有许多好朋友。_____ _____ many good friends. ②把下列两道题改为否定句： 1、I have many good friends：_____________________________ 总结：have 的句子改为否定句要 _______________________________________________________________ 2、He has a dog：_________________________ 总结：has的句子改为否定句要 _____________________________________ 同样的道理：请将下列两道题改为一般疑问句： I have many good friends：_____________________________ 总结： ______________________________________ He has a dog：_________________________ 总结： ________________________________________________________ 练习： 一、用have的正确形式填空： 1、He_____two brothers. 2、I_____a beautiful picture. 3、Betty_____ a lovely dog. 4、They_____some friends here. 二、请将下列的句子改为否定句和一般疑问句。 1-3题改为否定句：1、He has two brothers. ___________________________________________________ 2、I have a beautiful picture. ___________________________________________________ 3、Betty has some friends here. ___________________________________________________ 4-6题改为一般疑问句： 4、They have a good teacher. _________________________________肯定回答：_________________ 5、I have some cards. __________________________________________否定回答： ____________________ 6、Tony has a sister. __________________________________________否定回答： ____________________ 三、请用所给词的适当形式填空。 1、I _________ （have）a brother，but I_________ （not have）a sister. 2、He _________ （have）a beautiful pen. _________ you_________（have）a pen？ 3、Lingling _________ （have）an English dictionary. 4、_________ Tony_________（have）a car？ have got 的用法及否定句、一般疑问句的变法。

英国Alevel数学教材内容汇总.doc

Core Mathematics1（AS/A2）——核心数学1 1.Algebra and functions——代数和函数 2.Quadratic functions——二次函数 3.Equations and inequalities——等式和不等式 4.Sketching curves——画图（草图） 5.Coordinate geometry in the （x，y）plane——平面坐标系中的坐标几何 6.Sequences and series——数列 7.Differentiation——微分 8.Integration——积分 Core Mathematics2（AS/A2）——核心数学2 1.Algebra and functions——代数和函数 2.The sine and cosine rule——正弦和余弦定理 3.Exponentials and logarithm——指数和对数 4.Coordinate geometry in the （x，y）plane——平面坐标系中的坐标几何 5.The binomial expansion——二项展开式 6.Radian measure and its application——弧度制及其应用 7.Geometric sequences and series——等比数列 8.Graphs of trigonometric functions——三角函数的图形 9.Differentiation——微分 10.Trigonometric identities and simple equations——三角恒等式和简单的三角等式 11.Integration——积分 Core Mathematics3（AS/A2）——核心数学3 1.Algebra fractions——分式代数 2.Functions——函数 3.The exponential and log functions——指数函数和对数函数 4.Numerical method——数值法 5.Transforming graph of functions——函数的图形变换 6.Trigonometry——三角 7.Further trigonometric and their applications——高级三角恒等式及其应用 8.Differentiation——微分 Core Mathematics4（AS/A2）——核心数学4 1.Partial fractions——部分分式 2.Coordinate geometry in the （x，y）plane——平面坐标系中的坐标几何 3.The binomial expansion——二项展开式 4.Differentiation——微分 5.Vectors——向量 6.Integration——积分