An algorithm to implement a canonical representation of algebraic expressions and equations in AToM3

COMBINING HIDDEN MARKOV MODELS AND PHONETIC TRIGRAMSIN A THAI SOUNDEX SYSTEMTassawut Duangpanyasawang and Boonserm KijsirikulMachine Intelligence and Knowledge Discovery LaboratoryDepartment of Computer Engineering,Chulalongkorn University,Bangkok 10330, THAILANDEmail: {tassawut, boonserm}@mind.cp.eng.chula.ac.thABSTRACTMost of previous methods on Thai soundex encoding are based on Odell and Russell’s algorithm. These methods still have limitations in grouping words of similar sound. To improve the accuracy of Thai soundex encoding, we propose a new method that combines a Hidden Markov Model (HMM) and phonetic trigrams. Our soundex code consists of three features, i.e. initial consonant sound, vowel sound, and final consonant sound. The HMM and phonetic trigrams are used as a soundex converter. First, the HMM generates all possible soundex codes, and then the answers are re-ranked by combining probabilities from phonetic trigrams with probabilities from the HMM. The experimental results show that our algorithm surpasses the previous works and gives the best performance with 95% and 84% in precision and recall, respectively.INTRODUCTIONSoundex is a technique for identifying words that have similar pronunciation, and is used in many applications such as word retrieval in census database[Green], spelling correction [Arunwong-Na-Ayuthaya, 1991; Tangkhawanwanich, 1991], cross-language retrieval [Suwanvisat & Prasitjutrakul, 1999], etc. For English, Odell and Russell’s algorithm provides a good result of grouping words based on their sounds. The algorithm translates a string into a canonical form of at most four characters by ignoring vowels and ‘w’, ‘h’, ‘y’ and converting the rests into a phonetic code consisting of the first letter and three decimal digits.For Thai soundex systems, there are two board classes of approaches: (1) a soundex converter using a nondeterministic finite automaton (NFA) [Karoonboonyanan, et al., 1997], and (2) soundex schemes based on Odell and Russell’s algorithm. The NFA soundex converter generates all possible pronunciations but does not provide probabilities for ranking them. For this reason, the NFA soundex converter is impractical in most applications, because it cannot select the most probable answer.Most of Thai soundex systems are based on Odell and Russell’s algorithm [Lohjeerachoonhakul & Khuwinpan, 1982; Udompanich, 1983; Arunwong-Na-Ayuthaya, 1991; Suwanvisat & Prasitjutrakul, 1999]. The works of [Lohjeerachoonhakul & Khuwinpan, 1982] and [Udompanich, 1983] are almost the same as the original Odell and Russell’s algorithm, i.e. they preserve the first character of the input word and convert the rest intodecimal digits by ignoring vowels. The works of [Arunwong-Na-Ayuthaya, 1991; Suwanvisat & Prasitjutrakul, 1999] proposed improved versions of Odell and Russell’s algorithm for Thaisoundex systems by encoding both consonants and vowels into a phonetic code, but thesesystems still did not perform quite well. The following are specific characteristics of Thai that are overlooked in previous works.(1) As soundex can be though of as a phonetic code of an input string, the smallest unitin the soundex code should encode each syllable of the input string. However, some Thaisoundex systems did not take this into account, i.e. they ignored vowels, and thus mayproduce the same code for different words despite the fact that the words contain different syllables. For example, the word ‘ ¸ª¦’[WßLÚZoQ@ having two syllables and the word ‘ ¦’[WßoÚQ@ composed of only one syllable are encoded into the same soundex code [Lohjeerachoonhakul & Khuwinpan, 1982; Udompanich, 1983]. Thai soundex systems, which consider both consonants and vowels, still face the problem of Thai syllables [Arunwong-Na-Ayuthaya,1991; Suwanvisat & Prasitjutrakul, 1999]. This is because usually a Thai string can be segmented into many different sequences of syllables, but the previous Thai soundex systems use only one sequence of syllables in the original form.(2) There are special vowel structures which give the same pronunciation but are encoded into different soundex codes, such as vowels in the words ‘Á µ’[NDX] and ‘ µª’[NDX] [Lohjeerachoonhakul & Khuwinpan, 1982; Arunwong-Na-Ayuthaya, 1991]. Generally, the sounds of vowels ‘Á µ’[−DX] and ‘µ’[-DÚ-] are different, but the combination of the vowel ‘µ’[-DÚ-] with a final consonant ‘ª’[-X] provides the same sound as that of the vowel ‘Á µ’[−DX].(3) There are many ambiguous pronunciation rules in Thai that cause low precision in the previous works. Vowels ‘Á’[-HÚ-],‘Â’ [-(Ú-],‘Å’ [-DL],‘Ä’[-DL],‘Ã’[-RÚ-] can provide more than one pronunciation. For example, a word ‘Á ¬¤’ may be read as ‘Á ¬¤’[NHÚVRÚP] or ‘ ³ Á¬¤’[NDVHÚP], and in this case the correct one is ‘ ³ Á¬¤’[NDVHÚP]. However, a word ‘Á ¬¦’ which can also be read as ‘Á ¬¦’[NHÚVRÚQ] and ‘ ³ Á¬¦’[NDVHÚQ], but the correct one is ‘Á ¬¦’ [NHÚVRÚQ].(4) The reduced form of words is also an important cause of a limitation of Thai soundex systems. For instance, words such as ‘ª· ¥µ’[ZLWWDMDÚ@and ‘°´ ¦µ’[ D W WUDÚ@contain hidden pronunciation that need to be read as if they were ‘ª· ³¥µ’[ZLWWDMDÚ@and ‘°´ ¦ ¦µ’[ D W WUDÚ@, respectively.In this paper, we propose a new method for Thai soundex encoding that can solve theprevious limitations. First, to solve the problem of syllable segmentation, we segment an input string into all possible sequences of syllables. Next, we handle the problem of special vowel structures and the ambiguous in pronunciation rules by applying a Hidden Markov Model (HMM) and phonetic trigrams to generate all possible pronunciations and select the most probable one. Finally, to solve the problem of the reduced form of words, we employ a preprocessing method that adds the hidden pronunciation back into the words.To evaluate the effectiveness of our method, we construct a corpus consisting of twoparts: the first one is for training and the other is for testing. The experimental results showthat a combination of HMM and phonetic trigrams provides high precision and recall and ouralgorithm outperforms the previous works.COMBINING AN HMM AND PHONETIC TRIGRAMSBelow we describe our method for soundex encoding. We first give the definition ofsoundex, then explain the algorithm which employs an HMM and phonetic trigrams.THE ENCODING OF SOUNDEXWe employ the encoding scheme in [Karoonboonyanan, et al., 1997] which defines the properties of similarly-pronounced syllables as being composed of two main features, i.e.(1) common features consisting of initial consonant sounds, vowel sounds and final consonant sounds, and (2) variations consisting of clusters, vowel lengths and tones. Soundex encoding should preserve common features and eliminate variations. Therefore, a unit in the soundex code is composed of three letters representing an initial consonant sound, a vowel sound, anda final consonant sound. In Thai, there are 20 initial consonant sounds, 12 vowel sounds, and9 final consonant sounds as shown in Table 1,2,3 respectively.Table 1: Initial consonant encoding.Consonant(s) Phonetic Values Soundex Code¨ ¦ ª[k]¨ ¦ ª ª ¦ ¨[kh]® [1]¦[tß]¦ [tßh]­ ¬««¦«¨­¦­¨ ¦ ¦[s]¥°¥® ®¥[j] ¥¦ [d]¦ ¨ [t]¦ [th]® [n]¦ ¨[b]¦ ¨[p]¡¡¦¡¨ ¨£[ph] ¡¨ ¦¢¢¨¢¦[f] ¢¤®¤[m] ¤¨¦¯®¨®¦§[r],[l] ¨ª®ª[w] ª®±[h] ±°[ ] °* We group ‘¦’ (thrill [r]) and ‘¨’ (lateral [l]) together due to their close pronunciation.Table 2: Vowel encoding.Syllable Form Soundex Code°³> D@ °´ > D @ °¦¦ > D @ °> D@ °Îµ> DP@ Ä°> DL@ Å°> DL@ Å°¥> DL@ Á°µ> DX@ °µ > DÚ @ a°· > L @ °¸ > LÚ @ § > UL @ i °¹ > @ °º > Ú @ § > U @ v °» > X @ °¼ > XÚ @ uÁ°³> H@ Á°È > H @ Á° > HÚ @ e°³> (@ Â°È > ( @ ° > (Ú @ xð³> R@ ° > R @ ð > RÚ @ oÁ°µ³[-o] °È° > o @ °° > oÚ @ °> oÚ@ °¦> oÚQ@ cÁ°°³> \@ Á°° > \Ú @ Á°· > \Ú @ Á°¥> \ÚL@ § >U\Ú @ dÁ°¸¥³> L @ Á°¸¥ > L @ jÁ°º°³> @ Á°º° > Ú @ w°´ª³> X @ °´ª> XÚ @ °ª > XÚ @ $Table 3: Final consonant encoding.CodeSoundex Group Consonant(s)kok [-k] ¦ ¦kong [-1]kom [-m] ¤¤·¤kon [-n] ¦ ¨¯koew [-u] ªªkoey [-I] ¥¥kot [t] ­«¬¬ ¦¦ ¦ · » · · ¦kop [-p] ¡¢£Open syllable ±±With the above classification and all variations elimination applied, here are some encoding examples:a¥encodedÅ [nai] is°a a±encoded°´ ¦µ[ D W WUDÚ@ is¦¦¤³[thamma ] is encoded a¤¤a±OUR ALGORITHMWe now describe our algorithm for converting a string into its soundex code. The algorithm is shown in Table 4.Table 4: The algorithm for soundex encoding.1. Preprocess the input word by:1.1 eliminating tones and cancelled letters,1.2 recovering the hidden form, and1.3 creating new words for special vowels.2. Segment each word into all possible sequences of syllables.3. Apply a Hidden Markov Model (HMM) and phonetic trigrams to generate soundex codes,and output N-best answers according to the probabilities calculated from the HMM andphonetic trigrams.As shown in Table 4, the algorithm for generating soundex codes is composed of three steps.1. The first step is to preprocess the input word by eliminating tone markers and cancelled letters. We also solve the problem of the reduced forms of words by generating new words from the input word with hidden-pronunciation adding. For example, the preprocessing of ‘ª· ¥µ’ [ZLWWDMDÚ@will generate two new words: ‘ª· ¥µ’ [ZLWWDMDÚ@ itself and ‘ª· ³¥µ’ [ZLWWDMDÚ@ with the hidden-pronunciation ( ³) added. Furthermore, the ambiguous pronunciation of special vowels ‘Á’ [-HÚ-], ‘Â’ [-(Ú-], ‘Å’ [-DL ], ‘Ä’ [-DL ], ‘Ã’ [-RÚ-] is also solved in this step. For an input word containing these vowels, we generate all possible pronunciations for these vowels. For example, for an input word ‘Á ¬¤’ [NDVHÚP ], the algorithm will produce the word ‘Á ¬¤’ [NDVHÚP ] itself and another word ‘ Á¬¤’ [NDVHÚP ]. Therefore, the preprocessing of ‘Á ¬¤ª· ¥µ’ will provide four outputs: ‘ Á¬¤ª· ¥µ’, ‘ Á¬¤ª· ³¥µ’, ‘Á ¬¤ª· ¥µ’ and ‘Á ¬¤ª· ³¥µ’.2. The second step is to segment the words from Step 1 into all possible sequences of syllables. In other words, given a string of characters S = c 1c 2 ··· c n from Step 1, we want to find a set W = { o 1o 2 ··· o m | o 1, o 2,…, o m ∈ O and o 1o 2 ··· o m = S }, where O is a set of valid Thai syllable forms.3. The last step is to apply an HMM and phonetic trigrams to encode soundex for each sequence of syllables, and select the N-best answers according to the probabilities calculated from the HMM and phonetic trigrams. The HMM and phonetic trigrams are described below.HIDDEN MARKOV MODELAn HMM is a well-known and widely used statistical method, especially for speech recognition tasks [Rabiner & Juang, 1993]. Here, we apply an HMM as a soundex converter, by defining each state as a soundex code and outputs of each state as all syllable forms which have pronunciation matching with the state. A transition between two states is defined as the probability that two soundex codes are connected to each other. Figure 1 shows a part of our HMM.Figure 1: A part of our HMM.States ‘ a ±’ and ‘ o ’ in the figure represent soundex codes pronounced as ‘ ³[ka]’ and ‘ [nok]’, respectively. The outputs of the state ‘ a ±’ are the syllable forms (‘ ³’ and ‘ ’) of which pronunciation are ‘ ³[ka]’. Similarly, ‘ ’ and ‘à ’, which are outputs of the state ‘ o ’, have the same pronunciation ‘ [nok]’. The transition between the states ‘ a ±’ and ‘ o ’ is a probability that the state ‘ a ±’ is followed by the state ‘ o ’. o à a ± ³ P( s i | s i+1 ) P( s 1 | π ) P( o i | s i )i i ii Each soundex code contains three characters representing an initial consonant sound, a vowel sound, and a final consonant sound. Since there are 20 initial consonant sounds, 12 vowel sounds and 9 final consonant sounds, the total number of states is 20 x 12 x 9 = 2,160. Therefore, the total number of transitions is 2,160 × 2,160 = 4,665,600, in case of a fully-connected HMM. However, the transitions in our model are constructed from training data (not fully-connected). In our experiment, the model contains 10,896 transitions by the end of the training process. We create outputs of each state by combining all initial consonant forms, vowel forms, and final consonant forms shown in Table 1. For example, outputs of a state ‘ u ’ [1XÚ1] are ‘ ¼ ’ [1XÚ1], ‘® ¼ ’ [1XÚ1], ‘ » ’ [1XÚ1], and ‘® » ’ [1XÚ1], since there are 2 initial consonant forms ‘ ’ [1@and ‘® ’ [1@, 2 vowel forms t u >-X -] and t u >-XÚ-], and 1 final consonant forms ‘ ’ [1@.To apply the HMM for soundex encoding, we define the task as follows:Given:• a Hidden Markov Model H = {π, T, O} • a sequence of syllable forms X = o 1o 2 ··· o mwhere π is a set of states, T is a set of transitions, and O is a set of outputs of states, and o 1, o 2,…, o m ∈ O .Find:• the state sequence S = s 1s 2 ··· s m (the soundex code of the given X ) that maximizes P(S | X) by using the following equation:P(S | X) = P(s 1| π) Π P( s i | s i+1 ) P( o i | s i ) (1)where P(s 1 | π) is a probability that state s 1 is chosen as an initial state,P(s i | s i+1) is a transition probability between state s i and s i+1,P(o i | s i ) is a probability that output symbol o i will be emitted at state s i .Note that all these probabilities are calculated from training examples.TRIGRAM MODELWe employ phonetic trigrams for enhancing the performance of our method. Though various trigram models, e.g. word trigrams, character trigrams, have been successfully applied to many tasks [Golding & Schabes, 1996; Meknavin, et al., 1998], to our knowledge this is the first attempt to use a phonetic trigram model. To apply a phonetic trigrams to select the most probable answer, we define the task as follows. Given a state sequence S = s 1s 2 ··· s m from the HMM, the probability of this sequence can be calculated by using the phonetic trigrams defined as follows:P(S) = Π P(s i | s i+1, s i+2 ) (2)where s i , s i+1 and s i+2 are a state in state sequence S , the state next to s i and the state next to s i+1, respectively.We then define the combined probability (CP) that combines the phonetic trigrams with the HMM by using Equation (1) and Equation (2) as:CP = P(s 1| π) Π P(s i | s i+1 ) P(o i | s i ) × Π k P(s i | s i+1, s i+2 ) (3)where k is a constant that determines the relative values of trigrams ’s probabilities versus HMM ’s probabilities.EXPERIMENTAL RESULTSTo evaluate the effectiveness of our proposed method, we construct a corpus which contains about 26,000 words (100,000 syllables) from the databases of Telephone Organization of Thailand and Thai Royal Institute dictionary. About 70% of the whole corpus are used as a training set and the rest is used as a test set. We run the experiment to compare three algorithms: (1) an algorithm described in [Suwanvisat & Prasitjutrakul, 1999] which is based on Odell and Russell ’s Soundex algorithm (SP99), (2) the HMM alone (HMM), and (3) the combination of the HMM and phonetic trigrams (HMM+TRI). The results are shown in Table 4.The performances of the above methods are evaluated by standard precision (P) and recall (R). We also use F 1-measure (F1) that combines recall and precision with an equal weight and is defined as follows:Table 4. The comparison of the performances of three algorithms.SP99 HMM HMM+TRI P (%) R (%) F1 P (%) R (%) F1 P (%) R (%) F1 68.30 80.30 73.82 89.70*1 82.84*1 86.13*1 95.20*1 83.61*1 89.03*1 82.80*2 92.32*2 87.30*2 83.87*2 94.71*2 88.96*2 66.46*3 96.93*3 78.85*3 58.41*3 99.70*3 73.66*3 *1, *2 and *3 are the values obtained when the number of N best answers is set to 1, 2 and 3, respectivelyThe results show that our methods, both HMM and HMM+TRI, give better results than those of SP99, in both precision and recall. SP99 is a very fast method and usually finds the expected words, but it also gives a lot of false hits, resulting in high recall and low precision. Among these three algorithms, HMM+TRI performs best and significantly improves the use of HMM alone.HMM+TRI gives the best performance when using only the first answer, and in this case, HMM+TRI yields a satisfactory result of 95.20% and 83.61% in precision and recall respectively. The recall is very high (99.70%) when we use the best three soundex codes, but it causes very low precision (58.41%).CONCLUSIONSWe have proposed a method for improving Thai soundex system by using the soundex code that considers the initial consonant sound, vowel sound, and final consonant sound. We solve the problems of previous works by considering the syllable boundary and employing a preprocessing technique to handle the ambiguities in Thai pronunciation rules. We have applied the HMM with phonetic trigrams for Thai soundex encoding. The experimental results, comparing our algorithms and previous works, reveal that the combination of the HMM and phonetic trigrams gives the best result with 95% precision and 84% recall.2PR P+R F1 =REFERENCESArunwong-Na-Ayuthaya, N., (1991) Transformation on Thai strings using soundex principle (in Thai), Senior Project, Faculty of Engineering , Chulalongkorn University, Bangkok,Thailand.Green, J. H., Finding treasures in the U.S. federal census, /fv/uscensus.html.Golding, A. R., and Schabes, Y., (1996) Combining trigram-based and feature-based methods for context sensitive spelling correction, Proceedings of the 34th Annual Meeting of theAssociated for Computational Linguistics, pp. 71-78, Santa Cruz, California, USA. Karoonboonyanan, T., Somlertlamvanich, V. and Meknavin, S., (1997) A Thai soundex system for spelling correction, Proceeding of the National Language Processing Pacific Rim Symposium 1997, Phukhet, Thailand.Lohjeerachoonhakul, V., and Khuwinpan, C., (1982) Thai soundex algorithm and Thai syllable separation algorithm (in Thai), Research Report, Faculty of Applied Statistics,The National Institute of Development Administration, Bangkok, Thailand.Meknavin, S., Kijsirikul, B., Chotimonkol, A., and Nuttee, C., (1998) Combining trigram and winnow in Thai OCR error correction, Proceedings of COLING 1998.Rabiner, L., & Juang, B.H., (1993) Fundamentals of Speech Recognition, Prentice Hall, 1993. Suwanvisat, P., and Prasitjutrakul, S., (1999) Transliterated word encoding and retrieval algorithms for Thai-English cross-language retrieval, Proceeding of the NCSEC99,Thailand.Tangkhawanwanich, N., (1991) Spelling correction (in Thai), Senior Project, Faculty of Engineering, Chulalongkorn University, Bangkok, Thailand.Udompanich, W., (1983) Applying homonymity to searching Thai sound-alike strings (in Thai), Master Thesis, Chulalongkorn University, Bangkok, Thailand.。

算法是一把双刃剑作文材料英文回答：Algorithm is a powerful tool that has revolutionized the way we live and work. It has enabled us to automate tasks, make complex calculations, and process large amounts of data efficiently. However, it is also a double-edged sword, as it can be used for both good and bad purposes.On the positive side, algorithms have greatly improved our lives in many ways. For example, they have made it possible for us to search for information on the internet, navigate using GPS, and receive personalized recommendations for products and services. In the field of healthcare, algorithms are being used to analyze medical images, diagnose diseases, and develop new treatments. They have also made significant contributions to fields such as finance, transportation, and entertainment.However, the use of algorithms also raises concernsabout privacy, bias, and ethical issues. For example, algorithms can be used to track and monitor individuals without their consent, leading to violations of privacy. They can also perpetuate and even exacerbate existingbiases and discrimination, as they are often trained ondata that reflects societal inequalities. Moreover, the use of algorithms in decision-making processes, such as hiring and lending, can result in unfair outcomes for certain groups of people.In conclusion, while algorithms have the potential to bring about positive changes in our lives, it is importantto recognize that they also have the potential to cause harm. Therefore, it is crucial to carefully consider the ethical implications of their use and to developregulations and guidelines to ensure that they are used responsibly and for the greater good.中文回答：算法是一把强大的工具，彻底改变了我们生活和工作的方式。

Ambiguous operators need parentheses ----------- 不明确的运算需要用括号括起Ambiguous symbol ''xxx'' ---------------- 不明确的符号Argument list syntax error ---------------- 参数表语法错误Array bounds missing ------------------ 丢失数组界限符Array size toolarge ----------------- 数组尺寸太大Bad character in paramenters ------------------ 参数中有不适当的字符Bad file name format in include directive -------------------- 包含命令中文件名格式不正确Bad ifdef directive synatax ------------------------------ 编译预处理ifdef 有语法错Bad undef directive syntax --------------------------- 编译预处理undef 有语法错Bit field too large ---------------- 位字段太长Call of non-function ----------------- 调用未定义的函数Call to function with no prototype --------------- 调用函数时没有函数的说明Cannot modify a const object --------------- 不允许修改常量对象Case outside of switch ---------------- 漏掉了case 语句Case syntax error ------------------ Case 语法错误Code has no effect ----------------- 代码不可述不可能执行到Compound statement missing{ -------------------- 分程序漏掉"{"Conflicting type modifiers ------------------ 不明确的类型说明符Constant expression required ---------------- 要求常量表达式Constant out of range in comparison ----------------- 在比较中常量超出范围Conversion may lose significant digits ----------------- 转换时会丢失意义的数字Conversion of near pointer not allowed ----------------- 不允许转换近指针Could not find file ''xxx'' ----------------------- 找不到XXX 文件Declaration missing ; ---------------- 说明缺少" ；"Declaration syntax error ----------------- 说明中出现语法错误Default outside of switch ------------------ Default 出现在switch 语句之外Define directive needs an identifier ------------------ 定义编译预处理需要标识符Division by zero ------------------ 用零作除数Do statement must have while ------------------ Do-while 语句中缺少while 部分Enum syntax error --------------------- 枚举类型语法错误Enumeration constant syntax error ----------------- 枚举常数语法错误Error directive :xxx ------------------------ 错误的编译预处理命令Error writing output file --------------------- 写输出文件错误Expression syntax error ----------------------- 表达式语法错误Extra parameter in call ------------------------ 调用时出现多余错误File name too long ---------------- 文件名太长Function call missing ----------------- 函数调用缺少右括号Fuction definition out of place ------------------ 函数定义位置错误Fuction should return a value ------------------ 函数必需返回一个值Goto statement missing label ------------------ Goto 语句没有标号Hexadecimal or octal constant too large ------------------16 进制或8 进制常数太大Illegal character ''x'' ------------------ 非法字符xIllegal initialization ------------------ 非法的初始化Illegal octal digit ------------------ 非法的8 进制数字Illegal pointer subtraction ------------------ 非法的指针相减Illegal structure operation ------------------ 非法的结构体操作Illegal use of floating point ----------------- 非法的浮点运算Illegal use of pointer -------------------- 指针使用非法Improper use of a typedefsymbol ---------------- 类型定义符号使用不恰当In-line assembly not allowed ----------------- 不允许使用行间汇编Incompatible storage class ----------------- 存储类别不相容Incompatible type conversion -------------------- 不相容的类型转换Incorrect number format ----------------------- 错误的数据格式Incorrect use of default --------------------- Default 使用不当Invalid indirection --------------------- 无效的间接运算Invalid pointer addition ------------------ 指针相加无效Irreducible expression tree ----------------------- 无法执行的表达式运算Lvalue required --------------------------- 需要逻辑值0 或非0 值Macro argument syntax error ------------------- 宏参数语法错误Macro expansion too long ---------------------- 宏的扩展以后太长Mismatched number of parameters in definition --------------------- 定义中参数个数不匹配Misplaced break --------------------- 此处不应出现break 语句Misplaced continue ------------------------ 此处不应出现continue 语句Misplaced decimal point -------------------- 此处不应出现小数点Misplaced elif directive -------------------- 不应编译预处理elifMisplaced else ---------------------- 此处不应出现elseMisplaced else directive ------------------ 此处不应出现编译预处理elseMisplaced endif directive ------------------- 此处不应出现编译预处理endifMust be addressable ---------------------- 必须是可以编址的Must take address of memory location ------------------ 必须存储定位的地址No declaration for function ''xxx'' ------------------- 没有函数xxx 的说明No stack --------------- 缺少堆栈No type information ------------------ 没有类型信息Non-portable pointer assignment -------------------- 不可移动的指针（地址常数）赋值Non-portable pointer comparison -------------------- 不可移动的指针（地址常数）比较Non-portable pointer conversion ---------------------- 不可移动的指针（地址常数）转换Not a valid expression format type --------------------- 不合法的表达式格式Not an allowed type --------------------- 不允许使用的类型Numeric constant too large ------------------- 数值常太大Out of memory ------------------- 内存不够用Parameter ''xxx'' is never used ------------------ 能数xxx 没有用到Pointer required on left side of -> ----------------------- 符号-> 的左边必须是指针Possible use of ''xxx'' before definition ------------------- 在定义之前就使用了xxx （警告）Possibly incorrect assignment ---------------- 赋值可能不正确Redeclaration of ''xxx'' ------------------- 重复定义了xxxRedefinition of ''xxx'' is not identical ------------------- xxx 的两次定义不一致Register allocation failure ------------------ 寄存器定址失败Repeat cou-nt needs an lvalue ------------------ 重复计数需要逻辑值Size of structure or array not known ------------------ 结构体或数给大小不确定Statement missing ; ------------------ 语句后缺少" ；"Structure or union syntax error -------------- 结构体或联合体语法错误Structure size too large ---------------- 结构体尺寸太大Sub scripting missing ] ---------------- 下标缺少右方括号Superfluous & with function or array ------------------ 函数或数组中有多余的"&" Suspicious pointer conversion --------------------- 可疑的指针转换Symbol limit exceeded --------------- 符号超限Too few parameters in call ----------------- 函数调用时的实参少于函数的参数不Too many default cases ------------------- Default 太多(switch 语句中一个)Too many error or warning messages -------------------- 错误或警告信息太多Too many type in declaration ----------------- 说明中类型太多Too much auto memory in function ----------------- 函数用到的局部存储太多Too much global data defined in file ------------------ 文件中全局数据太多Two consecutive dots ----------------- 两个连续的句点Type mismatch in parameter xxx ---------------- 参数xxx 类型不匹配Type mismatch in redeclaration of ''xxx'' ---------------- xxx 重定义的类型不匹配Unable to create output file ''xxx'' ---------------- 无法建立输出文件xxxUnable to open include file ''xxx'' --------------- 无法打开被包含的文件xxxUnable to open input file ''xxx'' ---------------- 无法打开输入文件xxxUndefined label ''xxx'' ------------------- 没有定义的标号xxxUndefined structure ''xxx'' ----------------- 没有定义的结构xxxUndefined symbol ''xxx'' ----------------- 没有定义的符号xxxUnexpected end of file in comment started on line xxx ---------- 从xxx 行开始的注解尚未结束文件不能结束Unexpected end of file in conditional started on line xxx ---- 从xxx 开始的条件语句尚未结束文件不能结束Unknown assemble instruction ---------------- 未知的汇编结构Unknown option --------------- 未知的操作Unknown preprocessor directive: ''xxx'' ----------------- 不认识的预处理命令xxx Unreachable code ------------------ 无路可达的代码Unterminated string or character constant ----------------- 字符串缺少引号User break ---------------- 用户强行中断了程序V oid functions may not return a value ----------------- V oid 类型的函数不应有返回值Wrong number of arguments ----------------- 调用函数的参数数目错''xxx'' not an argument ----------------- xxx 不是参数''xxx'' not part of structure -------------------- xxx 不是结构体的一部分xxx statement missing ( -------------------- xxx 语句缺少左括号xxx statement missing ) ------------------ xxx 语句缺少右括号xxx statement missing ; -------------------- xxx 缺少分号xxx'' declared but never used ------------------- 说明了xxx 但没有使用xxx'' is assigned a value which is never used ---------------------- 给xxx 赋了值但未用过Zero length structure ------------------ 结构体的长度为零1."c" not an argument in function sum 该标识符不是函数的参数2.array bounds missing ] in function main 缺少数组界限符"]"3.Array size too large in function main 数组规模太大4.bad file name format in include directive 在包含指令中的文件名格式不正确.5.Call of non-function in function main 调用未经过定义的函数.6.cannot modify a const object in function main 对常量不能进行修改.7.character constant too long in function main 字符常量太大.8.constant expression required in funtion main 数组定义的时候,数组大小要求是常数pound statment missing } in function main 复合语句漏掉符号"{"10.declaration syntax error in function main 宣告语法错误11.expression syntax in function main 表达式语法错误12. extra parameter in call to sum in function 调用函数时使用了过多的参数13.illegal use of floating point in function main 浮点数的不合法使用14.illegal pionter subtraction in function main 不合法的指针相减15.invalid pointer addition in function main 无效的指针相加16.out of memory in function main 内存不足17.statement missing ; in function main 语句后面漏掉分号.警告报错18."k" is assigned a value which is never used 定义了一个变量,但程序从来没用过19.possibiy incorrect assignment in function main 这样的赋值可能不正确20.suspicious pointer conversion in function main 可疑的指针转换22.code has no effect in funtion main 代码对程序没效果23.Ambiguous operators need parentheses:不明确的运算需要用括号括起24.Ambiguous symbol 'xxx' :不明确的符号25.Argument list syntax error:参数表语法错误26.Array bounds missing :丢失数组界限符27.Array size toolarge :数组尺寸太大28.Bad character in paramenters :参数中有不适当的字符29.Bad file name format in include directive :包含命令中文件名格式不正确30.Bad ifdef directive synatax :编译预处理ifdef有语法错31.Bad undef directive syntax :编译预处理undef有语法错32.Bit field too large :位字段太长33.Call of non-function :调用未定义的函数34.Call to function with no prototype :调用函数时没有函数的说明35.Cannot modify a const object :不允许修改常量对象36.Case outside of switch :漏掉了case 语句37.Case syntax error :Case 语法错误38.Code has no effect :代码不可述不可能执行到pound statement missing{ :分程序漏掉"{"40.Conflicting type modifiers :不明确的类型说明符41.Constant expression required :要求常量表达式42.Constant out of range in comparison :在比较中常量超出范围43.Conversion may lose significant digits :转换时会丢失意义的数字44.Conversion of near pointer not allowed :不允许转换近指针45.Could not find file 'xxx' :找不到XXX文件46.Declaration missing ; :说明缺少"；"47.Declaration syntax error :说明中出现语法错误48.Default outside of switch efault 出现在switch语句之外49.Define directive needs an identifier :定义编译预处理需要标识符50.Division by zero :用零作除数51.Do statement must have while o-while语句中缺少while部分52.Enum syntax error :枚举类型语法错误53.Enumeration constant syntax error :枚举常数语法错误54.Error directive :xxx :错误的编译预处理命令55.Error writing output file :写输出文件错误56.Expression syntax error :表达式语法错误57.Extra parameter in call :调用时出现多余错误58.File name too long :文件名太长59.Function call missing ) :函数调用缺少右括号60.Fuction definition out of place :函数定义位置错误61.Fuction should return a value :函数必需返回一个值62.Goto statement missing label :Goto语句没有标号63.Hexadecimal or octal constant too large :16进制或8进制常数太大64.Illegal character 'x' :非法字符x65.Illegal initialization :非法的初始化66.Illegal octal digit :非法的8进制数字67.Illegal pointer subtraction :非法的指针相减68.Illegal structure operation :非法的结构体操作69.Illegal use of floating point :非法的浮点运算70.Illegal use of pointer :指针使用非法71.Improper use of a typedefsymbol :类型定义符号使用不恰当72.In-line assembly not allowed :不允许使用行间汇编73.Incompatible storage class :存储类别不相容74.Incompatible type conversion :不相容的类型转换75.Incorrect number format :错误的数据格式76.Incorrect use of default efault使用不当77.Invalid indirection 无效的间接运算78.Invalid pointer addition 指针相加无效79.Irreducible expression tree 无法执行的表达式运算80.Lvalue required 需要逻辑值0或非0值81.Macro argument syntax error 宏参数语法错误82.Macro expansion too long 宏的扩展以后太长83.Mismatched number of parameters in definition 定义中参数个数不匹配84.Misplaced break 此处不应出现break语句85.Misplaced continue 此处不应出现continue语句86.Misplaced decimal point 此处不应出现小数点87.Misplaced elif directive 不应编译预处理elif88.Misplaced else 此处不应出现else89.Misplaced else directive 此处不应出现编译预处理else90.Misplaced endif directive 此处不应出现编译预处理endif91.Must be addressable 必须是可以编址的92.Must take address of memory location 必须存储定位的地址93.No declaration for function 'xxx' 没有函数xxx的说明94.No stack 缺少堆栈95.No type information 没有类型信息96.Non-portable pointer assignment 不可移动的指针（地址常数）赋值97.Non-portable pointer comparison 不可移动的指针（地址常数）比较98.Non-portable pointer conversion 不可移动的指针（地址常数）转换99.Not a valid expression format type 不合法的表达式格式100.Not an allowed type 不允许使用的类型101.Numeric constant too large 数值常太大102.Out of memory 内存不够用103.Parameter 'xxx' is never used 能数xxx没有用到104.Pointer required on left side of -> 符号->的左边必须是指针105.Possible use of 'xxx' before definition 在定义之前就使用了xxx（警告）106.Possibly incorrect assignment 赋值可能不正确107.Redeclaration of 'xxx' 重复定义了xxx108.Redefinition of 'xxx' is not identical xx的两次定义不一致109.Register allocation failure 寄存器定址失败110.Repeat count needs an lvalue 重复计数需要逻辑值111.Size of structure or array not known 结构体或数给大小不确定112.Statement missing ; 语句后缺少"；"113.Structure or union syntax error X构体或联合体语法错误114.Structure size too large 结构体尺寸太大115.Sub scripting missing ] 下标缺少右方括号116.Superfluous & with function or array 函数或数组中有多余的"&" 117.Suspicious pointer conversion 可疑的指针转换118.Symbol limit exceeded 符号超限119.Too few parameters in call 函数调用时的实参少于函数的参数不120.Too many default cases Default太多(switch语句中一个)121.Too many error or warning messages 错误或警告信息太多122.Too many type in declaration 说明中类型太多123.Too much auto memory in function 函数用到的局部存储太多124.Too much global data defined in file 文件中全局数据太多125.Two consecutive dots 两个连续的句点126.Type mismatch in parameter xxx 数xxx类型不匹配127.Type mismatch in redeclaration of 'xxx' xx重定义的类型不匹配128.Unable to create output file 'xxx' 无法建立输出文件xxx129.Unable to open include file 'xxx' 无法打开被包含的文件xxx130.Unable to open input file 'xxx' 无法打开输入文件xxx131.Undefined label 'xxx' 没有定义的标号xxx132.Undefined structure 'xxx' 没有定义的结构xxx133.Undefined symbol 'xxx' 没有定义的符号xxx134.Unexpected end of file in comment started on line xxx 从xxx行开始的注解尚未结束文件不能结束135.Unexpected end of file in conditional started on line xxx 从xxx 开始的条件语句尚未结束文件不能结束136.Unknown assemble instruction 未知的汇编结构137.Unknown option 未知的操作138.Unknown preprocessor directive: 'xxx' 不认识的预处理命令xxx139.Unreachable code 无路可达的代码140.Unterminated string or character constant 字符串缺少引号er break 用户强行中断了程序142.V oid functions may not return a value V oid类型的函数不应有返回值143.Wrong number of arguments 调用函数的参数数目错144.'xxx' not an argument xxx不是参数145.'xxx' not part of structure xxx不是结构体的一部分146.'xxx' statement missing ( xx语句缺少左括号147.'xxx' statement missing ) xxx语句缺少右括号148.'xxx' statement missing ; xxx缺少分号149.'xxx' declared but never used 说明了xxx但没有使用150.'xxx' is assigned a value which is never used 给xxx赋了值但未用过151.Zero length structure 结构体的长度为零C++课程上机实验常见错误汇集1．在源码中遗失“；”调试器错误信息：syntax error : missing ';'2．缺少命名空间使用定义：即缺少“using namespace std;”调试器错误信息：例：error C2065: 'cout' : undeclared identifier例如cout/cin/endl/<</>>等在命名空间中定义的符号和标示符无法使用。

第28卷㊀第5期2023年10月㊀哈尔滨理工大学学报JOURNAL OF HARBIN UNIVERSITY OF SCIENCE AND TECHNOLOGY㊀Vol.28No.5Oct.2023㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀一种CCA -层次聚类的基因聚类算法林倩闽(厦门理工学院电气工程与自动化学院,福建厦门361024)摘㊀要:针对基因芯片技术带来的海量基因表达数据,为了充分挖掘其蕴含的生物信息和潜在的生物机制,提出一种基于CCA -层次聚类的基因聚类算法(CCA-Hc )㊂该算法在层次聚类的基础上引入典型相关分析,优化相似性矩阵计算方法㊂首先,利用典型相关分析方法结合基因的多个特征信息进行基因相关性度量,得到基因相似性矩阵㊂然后将该相似性矩阵作为层次聚类的邻近矩阵进行凝聚层次聚类㊂在Oryza sativa L.(水稻)的基因表达数据集上进行CCA-Hc 聚类效果测试实验,结果表明,与采用欧式距离的传统层次聚类算法(EUC-Hc )相比,CCA-Hc 的内部稳定性指标和生物功能性指标均优于EUC-Hc ,具有更佳的鲁棒性和聚类准确性,更有利于去发现基因间的共表达关系㊂关键词:基因表达数据;聚类算法;典型相关分析;层次聚类DOI :10.15938/j.jhust.2023.05.011中图分类号:TP391文献标志码:A文章编号:1007-2683(2023)05-0085-06A Gene Clustering Algorithm Based on the CCA-Hierarchical ClusteringLIN Qianmin(School of Electrical Engineering and Automation,Xiamen University of Technology,Xiamen 361024,China)Abstract :Aiming at the massive gene expression data brought by gene chip technology,in order to fully mine the biological information and potential biological mechanisms contained in it,this paper proposes a gene clustering algorithm based on CCA-hierarchical clustering (CCA-Hc).The algorithm introduces canonical correlation analysis on the basis of hierarchical clustering,and optimizes the calculation method of similarity matrix.First,the canonical correlation analysis method is used to measure the gene correlation by combining the multiple feature information of the gene,and the gene similarity matrix is obtained.Then the similarity matrix is used as the neighbor matrix of hierarchical clustering for agglomerative hierarchical clustering.The CCA-Hc clustering effect test experiment was performed on the gene expression dataset of Oryza sativa L.(rice).The results show that,compared with the traditional hierarchical clustering algorithm using Euclidean distance (EUC-Hc),CCA-Hc is superior to EUC-Hc in both internal stability index and biological functional index,and has better robustness and clustering accuracy.It is more conducive to discoveringthe co-expression relationship between genes.Keywords :gene expression data;clustering algorithm;canonical correlation analysis;hierarchical clustering㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀收稿日期:2022-06-08基金项目:福建省科技厅引导性项目(2019H0039);福建省中青年教师教育科研项目(JAT210341).通信作者:林倩闽(1992 ),女,硕士,助理实验师,E-mail:1023447133@.0㊀引㊀言随着高通量测序技术的不断快速发展,出现越来越多复杂度高㊁数据量大的生物数据㊂不同测序技术可以得到不同水平的生物数据,如通过基因组测序得到DNA 水平的生物数据,转录组测序得到RNA 水平的生物数据㊂基因表达数据是通过DNA微阵列技术(又称为基因芯片技术)检测得到,是不同细胞在不同条件下的基因动态表达水平[1]㊂基因是携带遗传物质的DNA片段,在不同细胞中会有不同的表达方向[2],从而可以控制不同的性状㊂为此基因表达数据蕴含着丰富且重要的生物机制,具有很大的研究价值㊂在基因表达数据分析中,聚类分析方法被广大研究者选用,用以发现具有相似表达行为的基因集,基因间的共表达㊁共调控关系等,对于推断未知的基因功能及在疾病诊断方面具有重要意义[2]㊂目前基因聚类算法根据聚类对象可以分为基于基因㊁基于样本聚类以及基于基因样本的双聚类[3-4]㊂根据聚类方式的不同,又可以分为以K-means算法[5]㊁K-MEDOIDS[6]为代表的基于分区的聚类算法,以BIRCH算法[7]㊁CURE算法[8]为代表的基于层次的聚类算法,以DBSCAN算法[9]㊁OPTICS算法[10]为代表的基于密度的聚类算法和以CLIQUE算法[11]为代表的基于网格的聚类算法㊂在对基因表达数据进行聚类分析时,主要是度量基因之间的相关性,把相关性程度高的基因聚在一起㊂很多基因聚类研究中把皮尔森相关系数㊁欧式距离㊁曼哈顿距离等作为相关性程度的度量方式[12]㊂这些度量方式是基于基因的整体表达水平进行的,即一个基因只由一个一维的数据矩阵表示㊂而在实际的的测序过程中,往往会在不同的细胞周期进行实验测量基因的表达水平,使得一个基因会有多组数据,每组数据代表该基因的一个特征㊂大部分的研究中采用求和的方式把基因多个特征的数据进行累加,进而分析基因之间的相关性㊂这种方法存在的问题是忽略了基因各个特征对表达水平的影响,从而对聚类结果造成影响㊂为了解决上述问题,本文把典型相关分析(Ca-nonical Correlation Analysis,CCA)引入到层次聚类中来,搭建出基于CCA-层次聚类的基因聚类算法(CCA-Hc)㊂典型相关分析是一种计算变量之间相关性的统计学分析方法,能结合变量的多个特征,得到变量的整体相关性[13]㊂利用典型相关分析度量基因之间的相关性,能充分考虑基因的多个特征信息,使得聚类结果中的基因集相似性程度更高㊂同时采用凝聚层次聚类,可以从聚类树状图中直观地分析聚类结果,从而整体上提高聚类效果㊂最后用GEO数据库上的基因数据集来验证CCA-Hc算法的有效性㊂1㊀CCA-Hc算法设计1.1㊀典型相关分析给定基因微阵列数据矩阵A nˑm=(G,T),n表示基因个数,m表示条件的种类数㊂每个基因可以看成是一个变量,使用典型相关分析方法分析变量相关性时,假设变量X有p个特征,变量Y有q个特征,pɤq,每个特征均对应m个不同条件的数据,则X=[x1, ,x p]T(1) Y=[y1, ,y q]T(2)变量X的数据矩阵为x11x12x13 x1mx21x22x23 x2mx31x32x33 x3m︙︙︙︙x p1x p2x p3 x pméëêêêêêêêùûúúúúúúú变量Y的数据矩阵为y11y12y13 y1my21y22y23 y2my31y32y33 y3m︙︙︙︙y q1y q2y q3 y qméëêêêêêêêùûúúúúúúú变量X和变量Y的协方差矩阵为ð=Cov(X,Y)=Var(X)Cov(X,Y)Cov(Y,X)Var(Y)()=ð11ð12ð21ð22()(3)变量X和变量Y的线性表达式记为U㊁V,表示为:U=a1x1+a2x2+ +a p x p=a T X(4) V=b1y1+b2y2+ +b q y q=b T Y(5)变量X和变量Y进行典型相关性分析时,可用这两个变量的线性表达式U㊁V之间相关系数的最大值来度量变量之间的相关性程度,即max a,b corr(U,V)=a Tð12b(a Tð11aˑb Tð22b)1/2(6)在求解上述最值表达式时,运用拉格朗日数乘法求解瑞利熵矩阵(ð-111ð12ð-122ð21)得到p个特征值,68哈㊀尔㊀滨㊀理㊀工㊀大㊀学㊀学㊀报㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀第28卷㊀记为λ1,λ2 λp ㊂这p 个特征值即变量X 和变量Y之间的典型相关系数㊂每一个相关系数再应用卡方检验进行显著性检验,得到p 个卡方检验p-value 值,记为p 1,p 2 p p ㊂为了更好地表示变量之间的典型相关程度,引入一个关于典型相关系数和p-value 值的权重函数W 来表示,定义为:W =ðp i =1λi I (log P i )ðp i =1I (log P i )(7)其中I (log P i )=0P >0.05-log PP ɤ0.05{这样每两个变量之间就能得到一个w 值来度量它们的相关性程度㊂对基因表达数据的n 个基因进行如上方法的典型相关分析后,最终得到一个n ˑn 的相似性矩阵㊂1.2㊀层次聚类目前常用的聚类算法有基于分区㊁基于层次㊁基于密度和基于网络4种类型[2],其中基于层次聚类的算法因原理通俗易懂㊁结果直观且精度高等优点而被广泛使用[14]㊂层次聚类分为自下而上的凝聚聚类和自上而下的分裂聚类两种[15],其中凝聚层次聚类运用最为广泛,同时凝聚层次聚类在无预先定义类别数的分类中具有明显优势[16]㊂故本文采用的是凝聚层次聚类,可以用树状图和嵌套簇图来表示,例如图1所示㊂图1㊀凝聚层次聚类的树状图和嵌套簇图Fig.1㊀Dendrogram and Nested Cluster Diagramfor Agglomerative Hierarchical Clustering下面介绍凝聚层次聚类的聚类过程:步骤1:视每一个数据点(如基因变量)为一个集群;步骤2:计算邻近矩阵,把类间距离最接近的两个集群进行合并;步骤3:重复步骤2,直到所有数据点合并完成㊂步骤2中的类间距离即两个集群之间的距离,传统的层次聚类类间距离计算方法有如下几种[17]:1)两个集群中距离最近的两个样本距离;2)两个集群中距离最远的两个样本距离;3)两个集群中所有样本之间的距离再求平均值;完成所有聚类步骤后会生产一个树状图(又叫聚类树)㊂采用不同的变量相关性程度度量方式和不同的类间距离计算方法都将对聚类结果造成影响㊂1.3㊀CCA-HC 算法传统的层次聚类算法其计算复杂度为O (n 3),由于在聚类过程中需要不断地重复计算类间距离㊁不断地更新邻近矩阵,从而消耗大量的时间与资源[18]㊂对于数据量庞大的基因微阵列数据,迫切需要对算法进行优化,降低复杂度㊂本文提出了一种基于CCA 和层次聚类的基因聚类算法(CCA-HC),优化相似性矩阵计算方法,把典型相关分析的输出作为层次聚类的输入,即把典型相关分析得到的相似性矩阵作为层次聚类的邻近矩阵㊂CCA-HC 在度量基因相关性程度时采用典型相关分析的方法,在层次聚类方式上选择自下而上的凝聚层次聚类㊂CCA-HC 充分利用了典型相关分析和层次聚类的优点,能够结合基因的多个特征来量化基因之间的相关性,使得聚类结果中的基因集相似性程度更高,也能自主选择集群数目以得到更佳的聚类效果[18]㊂2㊀实验与结果分析2.1㊀实验数据为了评价章节一中提出算法的聚类效果,在GEO 数据库上下载Oryza sativa L.(水稻)的基因表达数据集,得到的原始数据集共有45063个基因,样本数为41㊂由于原始数据集基因数庞大,对其计算分析时不论在存储空间还是计算程序上都提出了较高的要求,为此进行适当的数据预处理显得尤为重要㊂本文在数据预处理方面开展的主要工作有:把基因名未知的数据剔除;过滤掉样本表达量过低的基因;采用log2的对数函数对原始数据进行标准化处理等㊂经过如上处理后得到4564ˑ41的数据矩阵,用于后续的实验分析㊂预处理后的实验数据集78第5期林倩闽:一种CCA -层次聚类的基因聚类算法统计情况如表1所示㊂表1㊀预处理后的实验数据集统计情况表Tab.1㊀Statistical table of experimental dataset after preprocessing数据集基因数样本数基因功能类别Oryza sativa L.456441881.5㊀评价标准基因表达数据的聚类效果可以从聚类结果中同一集群的相关性程度以及聚类算法的稳定性等方面进行评价,用生物功能性指标和内部稳定性指标来描述㊂1.生物功能性指标生物同源性指标(biological homogeneity index, BHI)是用来评估聚类集群在生物功能意义上的同源性程度[19]㊂在基因本体(gene ontology,GO)数据库上下载水稻的基因功能类数据,可以得知每个水稻基因所对应的生物组织功能,用来分析同一聚类集群中的基因在功能上的相关性㊂BHI公式计算如下:BHI(K,B)=1KðK k=11nk(n k-1)ðiʂjɪC k I(B(i)=B(j))(8)式中:C为聚类结果中的任一集群;B为基因功能类集合,当基因i和基因j所对应的功能类存在交集,则I(B(i)=B(j))=1,否则为0㊂最终得到的BHI 是介于0~1的值,BHI值越大,表示基因聚类集群的生物功能相关性越大,聚类效果更佳[19]㊂2.内部稳定性指标内部稳定性指标在于评价聚类算法的鲁棒性,通过改变基因微阵列数据的某几列进行聚类,进而比较基于不同数据的聚类结果㊂优值系数(figure of merit,FOM)是内部稳定性指标中的一种,表示数据列改变后基因之间的平均群内方差[20]㊂FOM公式计算如下:FOM(l,K)=1NðK k=1ðiɪC k(l)dist(x i,l, x C k(l))(9)式中:FOM的取值范围是0到无穷大,FOM值越小表示该聚类算法的稳定性越好[20]㊂2.3㊀结果与分析为验证CCA-Hc的聚类效果,对比采用欧式距离的传统层次聚类算法(EUC-Hc),运用相同数据集进行实验㊂为了获得更加准确的聚类效果,本实验设置不同的聚类集群参数,确定聚类集群数目K 分别为2㊁4㊁6㊁7㊁9㊁11㊁12这7组实验,并通过BHI 和FOM指标对这7组实验的聚类结果进行评估, BHI和FOM指标值分别见表2和表3㊂表2㊀不同聚类集群数目下的BHI指标值Tab.2㊀BHI index values under different number of clusters 算法类型\集群数目CCA-Hc EUC-Hc差异率K=20.4660.233100.05%K=40.4630.34633.77%K=60.4670.37723.90%K=70.4670.41213.34%K=90.4650.4357.12%K=110.4640.4512.72%K=120.4630.456 1.48%表3㊀不同聚类集群数目下的FOM指标值Tab.3㊀FOM index values under different number of clusters算法类型\集群数目CCA-Hc EUC-Hc差异率K=22.6974.633-41.78%K=42.6974.298-37.26%K=62.6964.047-33.37%K=72.6963.995-32.52%K=92.6963.816-29.35%K=112.6953.693-27.03%K=12 2.695 3.636-25.89%㊀㊀表2中的差异率指的是CCA-Hc的BHI指标比EUC-Hc的BHI指标相差的百分比,同理可以计算表3中的差异率㊂根据表2和表3的实验指标数据发现,对于7组不同的聚类集群数目实验,本文提出的CCA-Hc 的BHI指标均高于EUC-Hc,FOM指标均低于EUC-Hc,这表明CCA-Hc的鲁棒性更好,聚类结果中同一集群的基因相关性更大,聚类效果更加显著㊂同时还发现,集群数目对CCA-Hc的影响较小,K选不同的值,BHI指标值稳定在0.463~0.467之间,FOM88哈㊀尔㊀滨㊀理㊀工㊀大㊀学㊀学㊀报㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀第28卷㊀指标值稳定在2.695~2.697之间,而集群数目对EUC-Hc 算法的影响相对比较明显㊂图2为CCA-Hc 在Oryza sativa L.数据集的聚类树状图,可以自行在所需的层级对树状图进行 剪枝 操作以获得合适的聚类效果[21]㊂图2㊀CCA-Hc 在Oryza sativa L.数据集的聚类树状图Fjg.2㊀Clustering dendrogram of CCA-Hc in Oryzasativa L.dataset3㊀结㊀论本文为了充分有效地挖掘基因表达数据所蕴含的生物机制,提出一种基于CCA -层次聚类的基因聚类算法(CCA-Hc)㊂把典型相关分析方法引入到凝聚层次聚类中来进行多特征基因的聚类分析,成为本文的创新之处㊂该算法利用典型相关分析方法度量基因之间的相关性程度,能够充分考虑基因的多个特征信息㊂同时采用凝聚层次聚类可自主选择聚类集群数目,直观显示聚类结果㊂基于Oryza sativa L.(水稻)的基因表达数据集,本文对比了CCA-Hc 和EUC-Hc 的聚类效果,使用BHI 和FOM 两个评价指标进行衡量,结果表明CCA-Hc 的鲁棒性和聚类准确性均更好,更有利于去探索基因表达数据潜在的生物机制㊂参考文献:[1]㊀欧阳玉梅.基因表达数据聚类分析技术及其软件工具[J].生物信息学,2010,8(2):104.OUYANG 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---------------------------------------------------------------最新资料推荐------------------------------------------------------偏最小二乘法(pls)简介（Partial least squares(PLS) Introduction）偏最小二乘法(pls)简介（Partial least squares (PLS) Introduction）偏最小二乘法(pls)简介（Partial least squares (PLS) Introduction） Partial least squares (PLS). The news about | | | | | Software Research Report training | knowledge sharing | customer list | Forum Partial least squares (PLS). Author: CIC reading: 14122 times the release date: 2004-12-30 Brief introduction The partial least squares method is a new type of multivariate statistical data analysis method, it was founded in 1983 by Wood (S.Wold) and Abano (C.Albano), is proposed for the first time. In recent decades, it has been developed rapidly in theory, method and application. The partial least squares method For a long time, the boundaries between the methods of model type and understanding of the share is very clear. The partial least squares rule them organically, in an algorithm, can simultaneously achieve regression (multivariate linear regression), simplify the structure of the data (principal component analysis) and the correlation between the two sets of variables analysis (canonical correlation analysis). This is a leap in the multivariate1/ 9statistical analysis of data. The importance of partial least squares method in statistical applications are reflected in the following aspects: Partial least squares regression modeling method is a multi variable for multiple variables. The partial least squares method can solve many previous ordinary regression can not solve the problem. The partial least squares method is called the second generation regression method, because it can realize the comprehensive application of various data analysis methods. The main purpose of principal component regression is to extract relevant information hidden in the matrix X, and then used to predict the value of the variable Y. This approach can ensure that we use only those independent variables, the noise will be eliminated, so as to improve the quality of the prediction model. However, the principal component regression still has some defects, when the correlation between some useful variables is very small, we are in the selection of the main components is easy to put them out, make the reliability prediction model for final decline, if we choose for each component, it is too difficult. Partial least squares regression can solve this problem. It adopts the way of decomposition of the variables X and Y, while extracting components from the variables X and---------------------------------------------------------------最新资料推荐------------------------------------------------------Y (usually called factor), then the factor according to the correlation between them in order from large to small. Now, we want to build a model, as long as we choose several factors involved in modeling it The basic concept Partial least squares regression is an expansion of the multivariate linear regression model, in its simplest form, only a linear model to describe the relationship between the independent variables and Y variables X: Y = B0 + b1X1 + b2X2 + bpXp +... In the equation, B0 is intercept, the value of Bi is 1 to the data points of regression coefficient P. For example, we can think of a person’s weight is a function of his height, gender, and from their respective sample points to estimate the regression coefficient, we can generally predict the weight of someone from the measured height and gender. Many of the methods of data analysis, data describing the biggest problem accurately and make a reasonable forecast of new observational data. The multiple linear regression model to deal with more complex data analysis problems, extended some other algorithms, like discriminant analysis, principal component regression, correlation analysis and so on, are multivariate statistical method with multiple linear regression model based. These3/ 9multivariate statistical methods there are two important characteristics, namely the data binding: The factor variable X and variable Y must be extracted from the X’X and Y’Y matrix, these factors cannot be said at the same time correlation between the variables X and Y. The number of prediction equation can never be more than the variable Y with variable X. Partial least squares regression from multiple linear regression and its expansion did not need these data constraints. In the partial least squares regression, prediction equation by the factor extracted from the matrix Y’XX’Y to describe; in order to be more representative, the maximum number of the number of prediction equations extracted may be greater than the variable X and Y. In short, partial least squares regression may be all multivariate calibration methods for the least variable constraint method, this flexibility makes it suitable for many occasions in the traditional multivariate calibration method is not applicable, for example, some observational data less than when the number of predictor variables. Moreover, partial least squares regression can be used as an exploratory analysis tool, before using the traditional linear regression model, first on the number of variables required for appropriate prediction and---------------------------------------------------------------最新资料推荐------------------------------------------------------ removal of noise. Therefore, the partial least squares regression is widely used in many fields of modeling, such as chemical, medicine, economics, psychology and pharmaceutical science and so on, especially it can according to arbitrarily set the variable this advantage is more prominent. In chemometrics, partial least squares regression has been used as a standard multivariate modeling tool. The calculation process The basic model As a method of multiple linear regression, partial least squares regression main purpose is to establish a linear model: Y=XB+E, where Y is the response matrix with m variables, n points, X is the prediction matrix with P variables, n sample, B regression coefficient matrix, E noise the calibration model, which has the same dimension with Y. In general, the variables X and Y was then used to calculate the standard, i.e. minus their average value and standard deviation divided by the. Partial least squares regression and principal component regression, using factor score as the original prediction variable as a linear combination of the basis, so used to establish prediction model between factor scores must be linearly independent. For example, if we have a set of response variables (Y matrix) and now a large number5/ 9of predictor variables X (matrix), some serious variable linear correlation, we use factor extracted from this set of data extraction method is used to calculate the factor, factor score matrix T=XW, then calculate the weight matrix W right then, a linear regression model: Y=TQ+E, where Q is the regression coefficient matrix of T matrix, E matrix error. Once the Q is calculated after the previous equation is equivalent to the Y=XB+E, the B=WQ, it can be directly used as predictive regression model. The difference of partial least squares regression and principal component regression in different extraction methods of factor score in short weight matrix W principal component regression generated reflects the prediction covariance between variables X, partial least squares W regression weight matrix generated reflects the covariance between the predictor variables and the response variable Y X. In the model, partial least squares regression produces weight matrix W PXC, column vector matrix for W score matrix T column vector calculation variable X nxc. The calculation of these weights the response covariance between the corresponding factor score reaches maximum constant. Ordinary least squares regression matrix Q regression on T when calculating Y, the matrix Y load factor (or weight), are used---------------------------------------------------------------最新资料推荐------------------------------------------------------to establish the regression equation: Y=TQ+E. Once calculated Q, we can get the equation: Y=XB+E, B=WQ, the final prediction model is built up. The nonlinear iterative partial least squares method For a standard algorithm for computing partial least squares regression is a nonlinear iterative partial least squares (NIPALS), there are many variables in this algorithm, some have been standardized, some are not. The following algorithm is considered to be one of the most effective method in the nonlinear iterative partial least squares. On C, A0=X’Y and h=1..., M0=X’X, C0=I, C of known variables. Calculation of QH, the principal eigenvectors of Ah’Ah. Wh=GhAhqh, wh=wh/||wh||, and wh as the column vector of W. Ph=Mhwh, ch=wh’Mhwh, ph=ph/ch, and pH as the column vect or of P. Qh=Ah’wh/ch, and QH as the column vector of Q. Ah+1=Ah - chphqh’, Bh+1=Mh - chphph’ Ch+1=Ch - whph’ Factor score matrix T can be calculated: T=XW, partial least squares regression coefficient B can be calculated by the formula B=WQ. SIMPLS algorithm There is a method to estimate the regression component of partial least squares, called SIMPLS algorithm. On C, A0=X’Y and h=1..., M0=X’X, C0=I, C of known variables. Calculation of QH, the principal eigenvectors of Ah’Ah.7/ 9Wh=Ahqh, ch=wh’Mhwh, wh=wh/sqrt (CH), and wh as the column vector of W. Ph=Mhwh, and pH as the column vector of P. Qh=Ah’wh, and QH as the column vector of Q. Vh=Chph, vh=vh/||vh|| Ch+1=Ch - vhvh’, Mh+1=Mh - phph’ Ah+1=ChAh The same as NIPALS, SIMPLS T T=XW calculated by the formula, B formula by B=WQ’calculation. Related literature Xu Lu, chemometrics methods, Science Press, Beijing, 1995. Wang Huiwen, the partial least squares regression method and application, National Science and Technology Press, Beijing, 1996. Chin, W. W., and Newsted, P. R. Structural Equation (1999). Modeling analysis with Small Samples Using Partial Least Squares. In Rick Hoyle (Ed.), Statistical Strategies for Small Sample Research, Sage Publications. Chin, W. W. (1998) The partial least squares approach for. Structural equation modelling. In George A. Marcoulides (Ed.), Modern Methods for Business Research, Lawrence Erlbaum Associates. Barclay, D., C. Higgins and R. Thompson The Partial (1995). Least Squares (PLS) Approach to Causal Modeling: Personal Computer Adoption and Use as an Illustration. Technology Studies, Volume 2, issue 2, 285-309. Chin, W. W. (1995). Partial Least Squares Is To LISREL As 主成分分析是常见的因素分析。

数学词汇中英文对照abbreviation 简写符号；简写abscissa 横坐标absolute complement 绝对补集absolute error 绝对误差absolute inequality 绝不等式absolute maximum 绝对极大值absolute minimum 绝对极小值absolute monotonic 绝对单调absolute value 绝对值accelerate 加速acceleration 加速度acceleration due to gravity 重力加速度; 地心加速度accumulation 累积accumulative 累积的accuracy 准确度act on 施于action 作用; 作用力acute angle 锐角acute-angled triangle 锐角三角形add 加addition 加法addition formula 加法公式addition law 加法定律addition law(of probability) （概率）加法定律additive inverse 加法逆元; 加法反元additive property 可加性adjacent angle 邻角adjacent side 邻边adjoint matrix 伴随矩阵algebra 代数algebraic 代数的algebraic equation 代数方程algebraic expression 代数式algebraic fraction 代数分式；代数分数式algebraic inequality 代数不等式algebraic number 代数数algebraic operation 代数运算algebraically closed 代数封闭algorithm 算法系统; 规则系统alternate angle （交）错角alternate segment 内错弓形alternating series 交错级数alternative hypothesis 择一假设; 备择假设; 另一假设altitude 高；高度；顶垂线；高线ambiguous case 两义情况；二义情况amount 本利和；总数analysis 分析；解析analytic geometry 解析几何angle 角angle at the centre 圆心角angle at the circumference 圆周角angle between a line and a plane 直 与平面的交角angle between two planes 两平面的交角angle bisection 角平分angle bisector 角平分线 ；分角线angle in the alternate segment 交错弓形的圆周角angle in the same segment 同弓形内的圆周角angle of depression 俯角angle of elevation 仰角angle of friction 静摩擦角; 极限角angle of greatest slope 最大斜率的角angle of inclination 倾斜角angle of intersection 相交角；交角angle of projection 投射角angle of rotation 旋转角angle of the sector 扇形角angle sum of a triangle 三角形内角和angles at a point 同顶角angular displacement 角移位angular momentum 角动量angular motion 角运动angular velocity 角速度annum(X% per annum) 年（年利率X%）anti-clockwise direction 逆时针方向；返时针方向anti-clockwise moment 逆时针力矩anti-derivative 反导数; 反微商anti-logarithm 逆对数；反对数anti-symmetric 反对称apex 顶点approach 接近；趋近approximate value 近似值approximation 近似；略计；逼近Arabic system 阿刺伯数字系统arbitrary 任意arbitrary constant 任意常数arc 弧arc length 弧长arc-cosine function 反余弦函数arc-sin function 反正弦函数arc-tangent function 反正切函数area 面积Argand diagram 阿根图, 阿氏图argument (1)论证; (2)辐角argument of a complex number 复数的辐角argument of a function 函数的自变量arithmetic 算术arithmetic mean 算术平均；等差中顶；算术中顶arithmetic progression 算术级数；等差级数arithmetic sequence 等差序列arithmetic series 等差级数arm 边array 数组; 数组arrow 前号ascending order 递升序ascending powers of X X 的升幂assertion 断语; 断定associative law 结合律assumed mean 假定平均数assumption 假定；假设asymmetrical 非对称asymptote 渐近asymptotic error constant 渐近误差常数at rest 静止augmented matrix 增广矩阵auxiliary angle 辅助角auxiliary circle 辅助圆auxiliary equation 辅助方程average 平均；平均数；平均值average speed 平均速率axiom 公理axiom of existence 存在公理axiom of extension 延伸公理axiom of inclusion 包含公理axiom of pairing 配对公理axiom of power 幂集公理axiom of specification 分类公理axiomatic theory of probability 概率公理论axis 轴axis of parabola 拋物线的轴axis of revolution 旋转轴axis of rotation 旋转轴axis of symmetry 对称轴back substitution 回代bar chart 棒形图；条线图；条形图；线条图base （1）底；（2）基；基数base angle 底角base area 底面base line 底线base number 底数；基数base of logarithm 对数的底basis 基Bayes´theorem 贝叶斯定理bearing 方位（角）；角方向（角）bell-shaped curve 钟形图belong to 属于Bernoulli distribution 伯努利分布Bernoulli trials 伯努利试验bias 偏差；偏倚biconditional 双修件式; 双修件句bijection 对射; 双射; 单满射bijective function 对射函数; 只射函数billion 十亿bimodal distribution 双峰分布binary number 二进数binary operation 二元运算binary scale 二进法binary system 二进制binomial 二项式binomial distribution 二项分布binomial expression 二项式binomial series 二项级数binomial theorem 二项式定理bisect 平分；等分bisection method 分半法；分半方法bisector 等分线 ；平分线Boolean algebra 布尔代数boundary condition 边界条件boundary line 界（线）；边界bounded 有界的bounded above 有上界的；上有界的bounded below 有下界的；下有界的bounded function 有界函数bounded sequence 有界序列brace 大括号bracket 括号breadth 阔度broken line graph 折线图calculation 计算calculator 计算器；计算器calculus (1) 微积分学; (2) 演算cancel 消法；相消canellation law 消去律canonical 典型; 标准capacity 容量cardioid 心脏Cartesian coordinates 笛卡儿坐标Cartesian equation 笛卡儿方程Cartesian plane 笛卡儿平面Cartesian product 笛卡儿积category 类型；范畴catenary 悬链Cauchy sequence 柯西序列Cauchy´s principal value 柯西主值Cauchy-Schwarz inequality 柯西- 许瓦尔兹不等式central limit theorem 中心极限定理central line 中线central tendency 集中趋centre 中心；心centre of a circle 圆心centre of gravity 重心centre of mass 质量中心centrifugal force 离心力centripedal acceleration 向心加速度centripedal force force 向心力centroid 形心；距心certain event 必然事件chain rule 链式法则chance 机会change of axes 坐标轴的变换change of base 基的变换change of coordinates 坐标轴的变换change of subject 主项变换change of variable 换元；变量的换characteristic equation 特征(征)方程characteristic function 特征(征)函数characteristic of logarithm 对数的首数; 对数的定位部characteristic root 特征(征)根chart 图；图表check digit 检验数位checking 验算chord 弦chord of contact 切点弦circle 圆circular 圆形；圆的circular function 圆函数；三角函数circular measure 弧度法circular motion 圆周运动circular permutation 环形排列; 圆形排列; 循环排列circumcentre 外心；外接圆心circumcircle 外接圆circumference 圆周circumradius 外接圆半径circumscribed circle 外接圆cissoid 蔓叶class 区；组；类class boundary 组界class interval 组区间；组距class limit 组限；区限class mark 组中点；区中点classical theory of probability 古典概率论classification 分类clnometer 测斜仪clockwise direction 顺时针方向clockwise moment 顺时针力矩closed convex region 闭凸区域closed interval 闭区间coaxial 共轴coaxial circles 共轴圆coaxial system 共轴系coded data 编码数据coding method 编码法co-domain 上域coefficient 系数coefficient of friction 摩擦系数coefficient of restitution 碰撞系数; 恢复系数coefficient of variation 变差系数cofactor 余因子; 余因式cofactor matrix 列矩阵coincide 迭合；重合collection of terms 并项collinear 共线collinear planes 共线面collision 碰撞column (1)列；纵行；(2) 柱column matrix 列矩阵column vector 列向量combination 组合common chord 公弦common denominator 同分母；公分母common difference 公差common divisor 公约数；公约common factor 公因子；公因子common logarithm 常用对数common multiple 公位数；公倍common ratio 公比common tangent 公切 commutative law 交换律comparable 可比较的compass 罗盘compass bearing 罗盘方位角compasses 圆规compasses construction 圆规作图compatible 可相容的complement 余；补余complement law 补余律complementary angle 余角complementary equation 补充方程complementary event 互补事件complementary function 余函数complementary probability 互补概率complete oscillation 全振动completing the square 配方complex conjugate 复共轭complex number 复数complex unmber plane 复数平面complex root 复数根component 分量component of force 分力composite function 复合函数; 合成函数composite number 复合数；合成数composition of mappings 映射构合composition of relations 复合关系compound angle 复角compound angle formula 复角公式compound bar chart 综合棒形图compound discount 复折扣compound interest 复利；复利息compound probability 合成概率compound statement 复合命题; 复合叙述computation 计算computer 计算机；电子计算器concave 凹concave downward 凹向下的concave polygon 凹多边形concave upward 凹向上的concentric circles 同心圆concept 概念conclusion 结论concurrent 共点concyclic 共圆concyclic points 共圆点condition 条件conditional 条件句；条件式conditional identity 条件恒等式conditional inequality 条件不等式conditional probability 条件概率cone 锥；圆锥（体）confidence coefficient 置信系数confidence interval 置信区间confidence level 置信水平confidence limit 置信极限confocal section 共焦圆锥曲congruence (1)全等；(2)同余congruence class 同余类congruent 全等congruent figures 全等图形congruent triangles 全等三角形conic 二次曲 ; 圆锥曲conic section 二次曲 ; 圆锥曲conical pendulum 圆锥摆conjecture 猜想conjugate 共轭conjugate axis 共轭conjugate diameters 共轭轴conjugate hyperbola 共轭(直)径conjugate imaginary / complex number 共轭双曲 conjugate radical 共轭虚/复数conjugate surd 共轭根式; 共轭不尽根conjunction 合取connective 连词connector box 捙接框consecutive integers 连续整数consecutive numbers 连续数；相邻数consequence 结论；推论consequent 条件；后项conservation of energy 能量守恒conservation of momentum 动量守恒conserved 守恒consistency condition 相容条件consistent 一贯的；相容的consistent estimator 相容估计量constant 常数constant acceleration 恒加速度constant force 恒力constant of integration 积分常数constant speed 恒速率constant term 常项constant velocity 怛速度constraint 约束；约束条件construct 作construction 作图construction of equation 方程的设立continued proportion 连比例continued ratio 连比continuity 连续性continuity correction 连续校正continuous 连续的continuous data 连续数据continuous function 连续函数continuous proportion 连续比例continuous random variable 连续随机变量contradiction 矛盾converge 收敛convergence 收敛性convergent 收敛的convergent iteration 收敛的迭代convergent sequence 收敛序列convergent series 收敛级数converse 逆(定理)converse of a relation 逆关系converse theorem 逆定理conversion 转换convex 凸convex polygon 凸多边形convexity 凸性coordinate 坐标coordinate geometry 解析几何；坐标几何coordinate system 坐标系系定理；系；推论coplanar 共面coplanar forces 共面力coplanar lines 共面co-prime 互质; 互素corollary 系定理; 系; 推论correct to 准确至；取值至correlation 相关correlation coefficient 相关系数correspondence 对应corresponding angles (1)同位角；(2)对应角corresponding element 对应边corresponding sides 对应边cosecant 余割cosine 余弦cosine formula 余弦公式cost price 成本cotangent 余切countable 可数countable set 可数集countably infinite 可数无限counter clockwise direction 逆时针方向；返时针方向counter example 反例counting 数数；计数couple 力偶Carmer´s rule 克莱玛法则criterion 准则critical point 临界点critical region 临界域cirtical value 临界值cross-multiplication 交叉相乘cross-section 横切面；横截面；截痕cube 正方体；立方；立方体cube root 立方根cubic 三次方；立方；三次(的)cubic equation 三次方程cubic roots of unity 单位的立方根cuboid 长方体；矩体cumulative 累积的cumulative distribution function 累积分布函数cumulative frequecy 累积频数；累积频率cumulative frequency curve 累积频数曲cumulative frequcncy distribution 累积频数分布cumulative frequency polygon 累积频数多边形；累积频率直方图curvature of a curve 曲线的曲率curve 曲线curve sketching 曲线描绘(法)curve tracing 曲线描迹(法)curved line 曲线curved surface 曲面curved surface area 曲面面积cyclic expression 输换式cyclic permutation 圆形排列cyclic quadrilateral 圆内接四边形cycloid 旋输线; 摆线cylinder 柱；圆柱体cylindrical 圆柱形的damped oscillation 阻尼振动data 数据De Moivre´s theorem 棣美弗定理De Morgan´s law 德摩根律decagon 十边形decay 衰变decay factor 衰变因子decelerate 减速decelaration 减速度decile 十分位数decimal 小数decimal place 小数位decimal point 小数点decimal system 十进制decision box 判定框declarative sentence 说明语句declarative statement 说明命题decoding 译码decrease 递减decreasing function 递减函数；下降函数decreasing sequence 递减序列；下降序列decreasing series 递减级数；下降级数decrement 减量deduce 演绎deduction 推论deductive reasoning 演绎推理definite 确定的；定的definite integral 定积分definition 定义degenerated conic section 降级锥曲线degree (1) 度; (2) 次degree of a polynomial 多项式的次数degree of accuracy 准确度degree of confidence 置信度degree of freedom 自由度degree of ODE 常微分方程次数degree of precision 精确度delete 删除; 删去denary number 十进数denominator 分母dependence (1)相关; (2)应变dependent event(s) 相关事件; 相依事件; 从属事件dependent variable 应变量; 应变数depreciation 折旧derivable 可导derivative 导数derived curve 导函数曲线derived function 导函数derived statistics 推算统计资料; 派生统计资料descending order 递降序descending powers of x x的降序descriptive statistics 描述统计学detached coefficients 分离系数(法)determinant 行列式deviation 偏差; 变差deviation from the mean 离均差diagonal 对角线diagonal matrix 对角矩阵diagram 图; 图表diameter 直径diameter of a conic 二次曲线的直径difference 差difference equation 差分方程difference of sets 差集differentiable 可微differential 微分differential coefficient 微商; 微分系数differential equation 微分方程differential mean value theorem 微分中值定理differentiate 求...的导数differentiate from first principle 从基本原理求导数differentiation 微分法digit 数字dimension 量; 量网; 维(数)direct impact 直接碰撞direct image 直接像direct proportion 正比例direct tax, direct taxation 直接税direct variation 正变(分)directed angle 有向角directed line 有向直线directed line segment 有向线段directed number 有向数direction 方向; 方位direction angle 方向角direction cosine 方向余弦direction number 方向数direction ratio 方向比directrix 准线Dirichlet function 狄利克来函数discontinuity 不连续性discontinuous 间断(的);连续(的); 不连续(的) discontinuous point 不连续点discount 折扣discrete 分立; 离散discrete data 离散数据; 间断数据discrete random variable 间断随机变数discrete uniform distribution 离散均匀分布discriminant 判别式disjoint 不相交的disjoint sets 不相交的集disjunction 析取dispersion 离差displacement 位移disprove 反证distance 距离distance formula 距离公式distinct roots 相异根distincr solution 相异解distribution 公布distributive law 分配律diverge 发散divergence 发散(性)divergent 发散的divergent iteration 发散性迭代divergent sequence 发散序列divergent series 发散级数divide 除dividend (1)被除数；(2)股息divisible 可整除division 除法division algorithm 除法算式divisor 除数；除式；因子divisor of zero 零因子dodecagon 十二边形domain 定义域dot 点dot product 点积double angle 二倍角double angle formula 二倍角公式double root 二重根dual 对偶duality (1)对偶性; (2) 双重性due east/ south/ west /north 向东/ 南/ 西/ 北dynamics 动力学eccentric angle 离心角eccentric circles 离心圆eccentricity 离心率echelon form 梯阵式echelon matrix 梯矩阵edge 棱；边efficient estimator 有效估计量effort 施力eigenvalue 本征值eigenvector 本征向量elastic body 弹性体elastic collision 弹性碰撞elastic constant 弹性常数elastic force 弹力elasticity 弹性element 元素elementary event 基本事件elementary function 初等函数elementary row operation 基本行运算elimination 消法elimination method 消去法；消元法ellipse 椭圆ellipsiod 椭球体elliptic function 椭圆函数elongation 伸张；展empirical data 实验数据empirical formula 实验公式empirical probability 实验概率；经验概率empty set 空集encoding 编码enclosure 界限end point 端点energy 能; 能量entire surd 整方根epicycloid 外摆线equal 相等equal ratios theorem 等比定理equal roots 等根equal sets 等集equality 等(式)equality sign 等号equation 方程equation in one unknown 一元方程equation in two unknowns (variables) 二元方程equation of a straight line 直线方程equation of locus 轨迹方程equiangular 等角(的)equidistant 等距(的)equilateral 等边(的)equilateral polygon 等边多边形equilateral triangle 等边三角形equilibrium 平衡equiprobable 等概率的equiprobable space 等概率空间equivalence 等价equivalence class 等价类equivalence relation 等价关系equivalent 等价(的)error 误差error allowance 误差宽容度error estimate 误差估计error term 误差项error tolerance 误差宽容度escribed circle 旁切圆estimate 估计；估计量estimator 估计量Euclidean algorithm 欧几里德算法Euclidean geometry 欧几里德几何Euler´s formula 尤拉公式；欧拉公式evaluate 计值even function 偶函数even number 偶数evenly distributed 均匀分布的event 事件exact 真确exact differential form 恰当微分形式exact solution 准确解；精确解；真确解exact value 法确解；精确解；真确解example 例excentre 外心exception 例外excess 起exclusive 不包含exclusive disjunction 不包含性析取exclusive events 互斥事件exercise 练习exhaustive event(s) 彻底事件existential quantifier 存在量词expand 展开expand form 展开式expansion 展式expectation 期望expectation value, expected value 期望值；预期值experiment 实验；试验experimental 试验的experimental probability 实验概率explicit function 显函数exponent 指数exponential function 指数函数exponential order 指数阶; 指数级express…in terms of…以………表达expression 式；数式extension 外延；延长；扩张；扩充extension of a function 函数的扩张exterior angle 外角external angle bisector 外分角external point of division 外分点extreme point 极值点extreme value 极值extremum 极值face 面factor 因子；因式；商factor method 因式分解法factor theorem 因子定理；因式定理factorial 阶乘factorization 因子分解；因式分解factorization of polynomial 多项式因式分解fallacy 谬误FALSE 假(的)falsehood 假值family 族family of circles 圆族family of concentric circles 同心圆族family of straight lines 直线族feasible solution 可行解；容许解Fermat´s last theorem 费尔马最后定理Fibonacci number 斐波那契数；黄金分割数Fibonacci sequence 斐波那契序列fictitious mean 假定平均数figure (1)图(形)；(2)数字final velocity 末速度finite 有限finite dimensional vector space 有限维向量空间finite population 有限总体finite probability space 有限概率空间finite sequence 有限序列finite series 有限级数finite set 有限集first approximation 首近似值first derivative 一阶导数first order differential equation 一阶微分方程first projection 第一投影; 第一射影first quartile 第一四分位数first term 首项fixed deposit 定期存款fixed point 定点fixed point iteration method 定点迭代法fixed pulley 定滑轮flow chart 流程图focal axis 焦轴focal chord 焦弦focal length 焦距focus(foci) 焦点folium of Descartes 笛卡儿叶形线foot of perpendicular 垂足for all X 对所有Xfor each /every X 对每一Xforce 力forced oscillation 受迫振动form 形式；型formal proof 形式化的证明format 格式；规格formula(formulae) 公式four leaved rose curve 四瓣玫瑰线four rules 四则four-figure table 四位数表fourth root 四次方根fraction 分数；分式fraction in lowest term 最简分数fractional equation 分式方程fractional index 分数指数fractional inequality 分式不等式free fall 自由下坠free vector 自由向量; 自由矢量frequency 频数；频率frequency distribution 频数分布；频率分布frequency distribution table 频数分布表frequency polygon 频数多边形；频率多边形friction 摩擦; 摩擦力frictionless motion 无摩擦运动frustum 平截头体fulcrum 支点function 函数function of function 复合函数；迭函数functional notation 函数记号fundamental theorem of algebra 代数基本定理fundamental theorem of calculus 微积分基本定理gain 增益；赚；盈利gain perent 赚率；增益率；盈利百分率game （1）对策；（2）博奕Gaussian distribution 高斯分布Gaussian elimination 高斯消去法general form 一般式；通式general solution 通解；一般解general term 通项generating function 母函数; 生成函数generator (1)母线; (2)生成元geoborad 几何板geometric distribution 几何分布geometric mean 几何平均数；等比中项geometric progression 几何级数；等比级数geometric sequence 等比序列geometric series 等比级数geometry 几何；几何学given 给定；已知global 全局; 整体global maximum 全局极大值; 整体极大值global minimum 全局极小值; 整体极小值golden section 黄金分割grade 等级gradient (1)斜率；倾斜率；(2)梯度grand total 总计graph 图像；图形；图表graph paper 图表纸graphical method 图解法graphical representation 图示；以图样表达graphical solution 图解gravitational acceleration 重力加速度gravity 重力greatest term 最大项greatest value 最大值grid lines 网网格线group 组；grouped data 分组数据；分类数据grouping terms 并项；集项growth 增长growth factor 增长因子half angle 半角half angle formula 半角公式half closed interval 半闭区间half open interval 半开区间harmonic mean (1) 调和平均数; (2) 调和中项harmonic progression 调和级数head 正面（钱币）height 高（度）helix 螺旋线hemisphere 半球体；半球heptagon 七边形Heron´s formula 希罗公式heterogeneous (1)参差的; (2)不纯一的hexagon 六边形higher order derivative 高阶导数highest common factor(H.C.F) 最大公因子；最高公因式；最高公因子Hindu-Arabic numeral 阿刺伯数字histogram 组织图；直方图；矩形图Holder´s Inequality 赫耳德不等式homogeneous 齐次的homogeneous equation 齐次方程Hooke´s law 虎克定律horizontal 水平的；水平horizontal asymptote 水平渐近线horizontal component 水平分量horizontal line 横线 ；水平线 horizontal range 水平射程hyperbola 双曲线hyperbolic function 双曲函数hypergeometric distribution 超几何分布hypocycloid 内摆线hypotenuse 斜边hypothesis 假设hypothesis testing 假设检验hypothetical syllogism 假设三段论hypotrochoid 次内摆线idempotent 全幂等的identical 全等；恒等identity 等（式）identity element 单位元identity law 同一律identity mapping 恒等映射identity matrix 恒等矩阵identity relation 恒等关系式if and only if/iff 当且仅当；若且仅若if…, then 若….则；如果…..则illustration 例证；说明image 像点；像image axis 虚轴imaginary circle 虚圆imaginary number 虚数imaginary part 虚部imaginary root 虚根imaginary unit 虚数单位impact 碰撞implication 蕴涵式；蕴含式implicit definition 隐定义implicit function 隐函数imply 蕴涵；蕴含impossible event 不可能事件improper fraction 假分数improper integral 广义积分; 非正常积分impulse 冲量impulsive force 冲力incentre 内力incircle 内切圆inclination 倾角；斜角inclined plane 斜面included angle 夹角included side 夹边inclusion mapping 包含映射inclusive 包含的；可兼的inclusive disjunction 包含性析取；可兼析取inconsistent 不相的(的)；不一致(的) increase 递增；增加increasing function 递增函数increasing sequence 递增序列increasing series 递增级数increment 增量indefinite integral 不定积分idenfinite integration 不定积分法independence 独立；自变independent equations 独立方程independent event 独立事件independent variable 自变量；独立变量indeterminate (1)不定的；(2)不定元；未定元indeterminate coefficient 不定系数；未定系数indeterminate form 待定型；不定型index,indices 指数；指index notation 指数记数法induced operation 诱导运算induction hypothesis 归纳法假设inelastic collision 非弹性碰撞inequality 不等式；不等inequality sign 不等号inertia 惯性；惯量infer 推断inference 推论infinite 无限；无穷infinite dimensional 无限维infinite population 无限总体infinite sequence 无限序列；无穷序列infinite series 无限级数；无穷级数infinitely many 无穷多infinitesimal 无限小；无穷小infinity 无限(大)；无穷(大)inflection (inflexion) point 拐点；转折点inherent error 固有误差initial approximation 初始近似值initial condition 原始条件；初值条件initial point 始点；起点initial side 始边initial value 初值；始值initial velocity 初速度initial-value problem 初值问题injection 内射injective function 内射函数inner product 内积input 输入input box 输入inscribed circle 内切圆insertion 插入insertion of brackets 加括号instantaneous 瞬时的instantaneous acceleration 瞬时加速度instantaneous speed 瞬时速率instantaneous velocity 瞬时速度integer 整数integrable 可积integrable function 可积函数integral 积分integral index 整数指数integral mean value theorem 积数指数integral part 整数部份integral solution 整数解integral value 整数值integrand 被积函数integrate 积；积分；......的积分integrating factor 积分因子integration 积分法integration by parts 分部积分法integration by substitution 代换积分法；换元积分法integration constant 积分常数interaction 相互作用intercept 截距；截段intercept form 截距式intercept theorem 截线定理interchange 互换interest 利息interest rate 利率interest tax 利息税interior angle 内角interior angles on the same side of the transversal 同旁内角interior opposite angle 内对角intermediate value theorem 介值定理internal bisector 内分角internal division 内分割internal energy 内能internal force 内力internal point of division 内分点interpolating polynomial 插值多项式interpolation 插值inter-quartile range 四分位数间距intersect 相交intersection (1)交集；(2)相交；(3)交点interval 区间interval estimation 区间估计；区域估计intuition 直观invalid 失效；无效invariance 不变性invariant (1)不变的；(2)不变量；不变式inverse 反的；逆的inverse circular function 反三角函数inverse cosine function 反余弦函数inverse function 反函数；逆函数inverse cosine function 反三角函数inverse function 反函数；逆映射inverse mapping 反向映射；逆映射inverse matrix 逆矩阵inverse problem 逆算问题inverse proportion 反比例；逆比例inverse relation 逆关系inverse sine function 反正弦函数inverse tangent function 反正切函数inverse variation 反变(分)；逆变(分)invertible 可逆的invertible matrix 可逆矩阵irrational equation 无理方程irrational number 无理数irreducibility 不可约性irregular 不规则isomorphism 同构isosceles triangle 等腰三角形iterate (1)迭代值; (2)迭代iteration 迭代iteration form 迭代形iterative function 迭代函数iterative method 迭代法jet propulsion 喷气推进joint variation 联变(分)；连变(分)kinetic energy 动能kinetic friction 动摩擦known 己知L.H.S. 末项L´Hospital´s rule 洛必达法则Lagrange interpolating polynomial 拉格朗日插值多项代Lagrange theorem 拉格朗日定理Lami´s law 拉密定律Laplace expansion 拉普拉斯展式last term 末项latent root 本征根; 首通径lattice point 格点latus rectum 正焦弦; 首通径law 律；定律law of conservation of momentum 动量守恒定律law of indices 指数律；指数定律law of inference 推论律law of trichotomy 三分律leading coefficient 首项系数leading diagonal 主对角线least common multiple, lowest common multiple (L.C.M) 最小公倍数；最低公倍式least value 最小值left hand limit 左方极限lemma 引理lemniscate 双纽线length 长(度)letter 文字；字母like surd 同类根式like terms 同类项limacon 蜗牛线limit 极限limit of sequence 序列的极限limiting case 极限情况limiting friction 最大静摩擦limiting position 极限位置line 线；行line of action 作用力线line of best-fit 最佳拟合line of greatest slope 最大斜率的直 ；最大斜率line of intersection 交线line segment 线段linear 线性；一次linear convergence 线性收敛性linear differeantial equation 线性微分方程linear equation 线性方程；一次方程linear equation in two unknowns 二元一次方程；二元线性方程linear inequality 一次不等式；线性不等式linear momentum 线动量linear programming 线性规划linearly dependent 线性相关的linearly independent 线性无关的literal coefficient 文字系数literal equation 文字方程load 负荷loaded coin 不公正钱币loaded die 不公正骰子local maximum 局部极大(值)local minimum 局部极小(值)locus, loci 轨迹logarithm 对数logarithmic equation 对数方程logarithmic function 对数函数logic 逻辑logical deduction 逻辑推论；逻辑推理logical step 逻辑步骤long division method 长除法loop 回路loss 赔本；亏蚀loss per cent 赔率；亏蚀百分率lower bound 下界lower limit 下限lower quartile 下四分位数lower sum 下和lower triangular matrix 下三角形矩阵lowest common multiple(L.C.M) 最小公倍数machine 机械Maclaurin expansion 麦克劳林展开式Maclaurin series 麦克劳林级数magnitude 量；数量；长度；大小major arc 优弧；大弧major axis 长轴major sector 优扇形；大扇形major segment 优弓形；大弓形mantissa 尾数mantissa of logarithm 对数的尾数；对数的定值部many to one 多个对一个many-sided figure 多边形many-valued 多值的map into 映入map onto 映上mapping 映射marked price 标价Markov chain 马可夫链mass 质量mathematical analysis 数学分析mathematical induction 数学归纳法mathematical sentence 数句mathematics 数学matrix 阵; 矩阵matrix addition 矩阵加法matrix equation 矩阵方程matrix multiplication 矩阵乘法matrix operation 矩阵运算maximize 极大maximum absolute error 最大绝对误差maximum point 极大点maximum value 极大值mean 平均(值)；平均数；中数mean deviation 中均差；平均偏差mean value theorem 中值定理measure of dispersion 离差的量度measurement 量度mechanical energy 机械能median (1)中位数；(2)中线meet 相交；相遇mensuration 计量；求积法method 方法method of completing square 配方法method of interpolation 插值法; 内插法method of least squares 最小二乘法; 最小平方法method of substitution 代换法；换元法method of successive substitution 逐次代换法; 逐次调替法method of superposition 迭合法metric unit 十进制单位mid-point 中点mid-point formula 中点公式mid-point theorem 中点定理million 百万minimize 极小minimum point 极小点minimum value 极小值Minkowski Inequality 闵可夫斯基不等式minor (1)子行列式；(2)劣；较小的minor arc 劣弧；小弧minor axis 短轴minor of a determinant 子行列式minor sector 劣扇形；小扇形minor segment 劣弓形；小弓形minus 减minute 分mixed number(fraction) 带分数modal class 众数组mode 众数model 模型modulo (1)模; 模数; (2)同余modulo arithmetic 同余算术modulus 模; 模数modulus of a complex number 复数的模modulus of elasticity 弹性模(数)moment arm (1)矩臂; (2)力臂moment of a force 力矩moment of inertia 贯性矩momentum 动量monomial 单项式monotone 单调monotonic convergence 单调收敛性monotonic decreasing 单调递减monotonic decreasing function 单调递减函数monotonic function 单调函数monotonic increasing 单调递增monotonic increasing function 单调递增函数motion 运动movable pulley 动滑轮multinomial 多项式multiple 倍数multiple angle 倍角multiple-angle formula 倍角公式multiple root 多重根multiplicand 被乘数multiplication 乘法multiplication law (of probability) (概率)乘法定律multiplicative inverse 乘法逆元multiplicative property 可乘性multiplicity 重数multiplier 乘数；乘式multiply 乘multi-value 多值的mulually disjoint 互不相交mutually exclusive events 互斥事件mutually independent 独立; 互相独立mutually perpendicular lines 互相垂直n factorial n阶乘n th derivative n阶导数n th root n次根；n次方根n the root of unity 单位的n次根Napierian logarithm 纳皮尔对数; 自然对数natural logarithm 自然对数natural number 自然数natural surjection 自然满射necessary and sufficient condition 充要条件necessary condition 必要条件negation 否定式negative 负negative angle 负角negative binomial distribution 负二项式分布negative index 负指数negative integer 负整数negative number 负数negative vector 负向量; 负矢量neighborhood 邻域net 净(值)net force 净力Newton-Cote´s rule 牛顿- 高斯法则Newton-Raphson´s method 牛顿- 纳逊方法Newton´s formula 牛顿公式Newton´s law of motion 牛顿运动定律Newton´s method 牛顿方法n-gon n边形nonagon 九边形non-collinear 不共线non-commutative 非交换的non-linear 非线性。

对算法时代的看法英语作文Title: Perspectives on the Algorithmic Era。

In the contemporary landscape of technological advancements, the algorithmic era has emerged as a defining epoch, reshaping various aspects of human existence. From the way we communicate and consume information to how we conduct business and make decisions, algorithms wield significant influence. This essay aims to delve into the multifaceted dimensions of the algorithmic era, examining its impacts on society, economy, and individual lives.At the heart of the algorithmic era lies the omnipresence of algorithms, intricate sets of instructions designed to perform specific tasks or solve problems. These algorithms permeate our daily lives, powering search engines, social media platforms, recommendation systems, and even financial markets. They analyze vast amounts of data, extract patterns, and generate insights with unprecedented speed and accuracy. Consequently, algorithmshave become indispensable tools for navigating the complexities of the digital age.One of the most notable domains transformed by algorithms is the realm of communication and information dissemination. Social media algorithms, for instance, curate personalized feeds based on users' preferences, behavior, and demographics. While this enhances user experience by presenting relevant content, it also engenders concerns regarding filter bubbles and echo chambers, where individuals are exposed only to information that aligns with their existing beliefs, potentially exacerbating polarization and misinformation.Moreover, algorithms wield profound influence in the realm of commerce and industry. E-commerce platformsutilize recommendation algorithms to suggest products tailored to users' preferences, thereby enhancing sales and customer satisfaction. Similarly, in the financial sector, algorithmic trading algorithms execute high-speed transactions based on complex mathematical models, reshaping market dynamics and posing challenges fortraditional traders.However, the algorithmic era is not without its challenges and ethical dilemmas. One pressing concern is the issue of algorithmic bias, where algorithms inadvertently reflect and perpetuate existing societal inequalities. For example, facial recognition algorithms have been found to exhibit higher error rates for certain demographic groups, leading to discriminatory outcomes in law enforcement and surveillance practices. Addressing these biases requires concerted efforts to enhance algorithmic fairness and transparency, incorporating diverse perspectives in the development and deployment of algorithms.Furthermore, the algorithmic era raises profound questions about privacy and data security. With algorithms processing vast amounts of personal data, concerns regarding data breaches, surveillance, and algorithmic manipulation have escalated. Safeguarding privacy rights and ensuring data protection necessitate robust regulatory frameworks and ethical guidelines to govern the collection,storage, and usage of data in algorithmic systems.Despite these challenges, the algorithmic era presents immense opportunities for innovation and progress. Machine learning algorithms, for instance, hold promise in revolutionizing healthcare by facilitating early disease detection, personalized treatment plans, and drug discovery. Likewise, in transportation and urban planning, algorithms can optimize traffic flow, reduce congestion, and enhance sustainability through smart city initiatives.In conclusion, the algorithmic era signifies a paradigm shift in how we perceive and interact with technology.While algorithms offer unprecedented capabilities toanalyze data and automate tasks, they also pose complex challenges regarding fairness, transparency, and privacy. Navigating the algorithmic era requires a balanced approach that harnesses the benefits of algorithms while mitigating their potential harms. By fostering collaboration between technologists, policymakers, and society at large, we can shape a future where algorithms serve as tools forempowerment and advancement, enriching the human experience in the digital age.。

C++C++11中头⽂件numeric的使⽤<numeric>是C++标准程序库中的⼀个头⽂件，定义了C++ STL标准中的基础性的数值算法(均为函数模板)：(1)、accumulate: 以init为初值，对迭代器给出的值序列做累加，返回累加结果值，值类型必须⽀持”+”算符。

它还有⼀个重载形式。

(2)、adjacent_difference:对输⼊序列，计算相邻两项的差值，写⼊到输出序列中。

它还有⼀个重载形式。

(3)、inner_product:计算两个输⼊序列的内积。

它还有⼀个重载形式。

(4)、partial_sum:计算输⼊序列从开始位置到当前位置的累加值，写⼊输出序列中。

它还有⼀个重载形式。

(5)、iota: 向序列中写⼊以val为初值的连续值序列。

The algorithms are similar to the Standard Template Library (STL) algorithms and are fully compatible with the STL but are part of the C++ Standard Library rather than the STL. Like the STL algorithms, they are generic because they can operate on a variety of data structures. The data structures that they can operate on include STL container classes, such as vector and list, and program-defined data structures and arrays of elements that satisfy the requirements of a particular algorithm. The algorithms achieve this level of generality by accessing and traversing the elements of a container indirectly through iterators. The algorithms process iterator ranges that are typically specified by their beginning or ending positions. The ranges referred to must be valid in the sense that all pointers in the ranges must be dereferenceable and within the sequences of each range, the last position must be reachable from the first by means of incrementation.下⾯是从其它⽂章中copy的<numeric>测试代码，详细内容介绍可以参考对应的reference：#include "numeric.hpp"#include <iostream>#include <functional>#include <numeric>// reference: /reference/numeric/namespace numeric_ {/*std::accumulate: The behavior of this function template is equivalent to :template <class InputIterator, class T>T accumulate (InputIterator first, InputIterator last, T init){while (first!=last) {init = init + *first; // or: init=binary_op(init,*first) for the binary_op version++first;}return init;}*/static int myfunction(int x, int y) { return x + 2 * y; }struct myclass {int operator()(int x, int y) { return x + 3 * y; }} myobject;int test_numeric_accumulate(){int init = 100;int numbers[] = { 10, 20, 30 };std::cout << "using default accumulate: ";std::cout << std::accumulate(numbers, numbers + 3, init); // 160std::cout << '\n';std::cout << '\n';std::cout << "using functional's minus: ";std::cout << std::accumulate(numbers, numbers + 3, init, std::minus<int>()); // 40 std::minus => x - y std::cout << '\n';std::cout << "using custom function: ";std::cout << std::accumulate(numbers, numbers + 3, init, myfunction); // 220std::cout << '\n';std::cout << "using custom object: ";std::cout << std::accumulate(numbers, numbers + 3, init, myobject); // 280std::cout << '\n';return 0;}////*std::adjacent_difference: The behavior of this function template is equivalent totemplate <class InputIterator, class OutputIterator>OutputIterator adjacent_difference (InputIterator first, InputIterator last, OutputIterator result){if (first!=last) {typename iterator_traits<InputIterator>::value_type val,prev;*result = prev = *first;while (++first!=last) {val = *first;*++result = val - prev; // or: *++result = binary_op(val,prev)prev = val;}++result;}return result;}*/static int myop(int x, int y) { return x + y; }int test_numeric_adjacent_difference(){int val[] = { 1, 2, 3, 5, 9, 11, 12 };int result[7];std::adjacent_difference(val, val + 7, result);std::cout << "using default adjacent_difference: ";for (int i = 0; i<7; i++) std::cout << result[i] << ' '; // 1 1 1 2 4 2 1std::cout << '\n';std::adjacent_difference(val, val + 7, result, std::multiplies<int>()); // x * ystd::cout << "using functional operation multiplies: ";for (int i = 0; i<7; i++) std::cout << result[i] << ' '; // 1 2 6 15 45 99 132std::cout << '\n';std::adjacent_difference(val, val + 7, result, myop); // 1 3 5 8 14 20 23std::cout << "using custom function: ";for (int i = 0; i<7; i++) std::cout << result[i] << ' ';std::cout << '\n';return 0;}/*std::inner_product: The behavior of this function template is equivalent to:template <class InputIterator1, class InputIterator2, class T>template <class InputIterator1, class InputIterator2, class T>T inner_product (InputIterator1 first1, InputIterator1 last1, InputIterator2 first2, T init) {while (first1!=last1) {init = init + (*first1)*(*first2);// or: init = binary_op1 (init, binary_op2(*first1,*first2));++first1; ++first2;}return init;}*/static int myaccumulator(int x, int y) { return x - y; }static int myproduct(int x, int y) { return x + y; }int test_numeric_inner_product(){int init = 100;int series1[] = { 10, 20, 30 };int series2[] = { 1, 2, 3 };std::cout << "using default inner_product: ";std::cout << std::inner_product(series1, series1 + 3, series2, init); // 240std::cout << '\n';std::cout << "using functional operations: ";std::cout << std::inner_product(series1, series1 + 3, series2, init,std::minus<int>(), std::divides<int>()); // 70std::cout << '\n';std::cout << "using custom functions: ";std::cout << std::inner_product(series1, series1 + 3, series2, init,myaccumulator, myproduct); // 34std::cout << '\n';return 0;}///*std::partial_sum: The behavior of this function template is equivalent to:template <class InputIterator, class OutputIterator>OutputIterator partial_sum (InputIterator first, InputIterator last, OutputIterator result) {if (first!=last) {typename iterator_traits<InputIterator>::value_type val = *first;*result = val;while (++first!=last) {val = val + *first; // or: val = binary_op(val,*first)*++result = val;}++result;}return result;}*/static int myop_(int x, int y) { return x + y + 1; }int test_numeric_partial_sum(){int val[] = { 1, 2, 3, 4, 5 };int result[5];std::partial_sum(val, val + 5, result);std::cout << "using default partial_sum: ";std::cout << "using default partial_sum: ";for (int i = 0; i<5; i++) std::cout << result[i] << ' '; // 1 3 6 10 15std::cout << '\n';std::partial_sum(val, val + 5, result, std::multiplies<int>());std::cout << "using functional operation multiplies: ";for (int i = 0; i<5; i++) std::cout << result[i] << ' '; // 1 2 6 24 120std::cout << '\n';std::partial_sum(val, val + 5, result, myop_);std::cout << "using custom function: ";for (int i = 0; i<5; i++) std::cout << result[i] << ' '; // 1 4 8 13 19std::cout << '\n';return 0;}////*std::iota: The behavior of this function template is equivalent to:template <class ForwardIterator, class T>void iota (ForwardIterator first, ForwardIterator last, T val){while (first!=last) {*first = val;++first;++val;}}*/int test_numeric_iota(){int numbers[10];std::iota(numbers, numbers + 10, 100);std::cout << "numbers:";for (int& i : numbers) std::cout << ' ' << i; // 100 101 102 103 104 105 106 107 108 109 std::cout << '\n';return 0;}} // namespace numeric_。

KCF：High-SpeedTrackingwithKernelizedCorrelati。

High-Speed Tracking with Kernelized Correlation Filters 的翻译与分析基于核相关滤波器的⾼速⽬标跟踪⽅法,简称KCF 写在前⾯，之所以对这篇⽂章进⾏精细的阅读，是因为这篇⽂章极其重要，在⽬标跟踪领域⽯破天惊的⼀篇论⽂，后来在此论⽂基础上⼜相继出现了很多基于KCF的⽂章，因此⽂章好⽐作⼤厦的基⽯，深度学习，长短记忆等框架⽹络也可以在KCF上进⾏增添模块，并能够达到较好的效果，因此我将深⼊学习这篇⽂章，并在此与⼤家分享，由于学识有限，难免有些谬误，还请斧正。

High-Speed Tracking with Kernelized Correlation Filters:Abstract---The core component of most modern trackers is discriminative classifier, tasked with distinguishing between the target and the surrounding environment. To cope with natural image changes, this classifier is typically trained with translated and scaled sample patches. Such sets of samples are riddled with redundancies --any overlapping pixels are constrained to be the same. Based on this simple observation, we can diagonalize it with the Discrete Fourier Transform, reducing both storage and computation by several orders of magnitude. Interestingly, for linear regression our formulation is equivalent to a correlation Filter (KCF), that unlike other kernel algorithms has the exact same complexity as its linear counterpart. Building on it ,we also propose a fast multi-channel extension of linear correlation filters ,via a linear kernel ,which we call Dual Correlation Filter(DCF).Both KCF and DCF outperform top-ranking trackers such as Struck of TLD on a 50videos benchmark, despite running at hundreds of frames-per -second , and being implemented in a few lines ofcode(Algorithm1). To encourage further developments, our tracking framework was made open-source.Index Terms——Visual tracking, circulant matrices, discrete Fourier transform, kernel methods, ridge regression, correlation filters.摘要：当前最前沿的跟踪器的核⼼组成是⼀个具有分辨能⼒的分类器，负责将⽬标和其背景环境区分开来。