大规模数据挖掘：MapReduce与相似性搜索

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"大数据挖掘-Mining of Massive Datasets, 斯坦福教材" 这本书《大数据挖掘-Mining of Massive Datasets》由Anand Rajaraman和Jeﬀrey D. Ullman共同撰写，基于他们在斯坦福大学开设的课程CS345A“Web Mining”的教学材料发展而来。这门课程最初设计为高级研究生课程，但其内容也吸引了众多高级本科生的兴趣。书中的核心内容聚焦在大规模数据挖掘上，特别是那些无法一次性加载到主内存中的海量数据。本书采取了算法视角来探讨数据挖掘，强调运用算法处理数据，而非依赖数据训练机器学习引擎。书中主要涵盖了以下几个关键主题： 1. 分布式文件系统和MapReduce：MapReduce是一种用于创建并行算法的工具，特别适用于处理非常大的数据集。它通过将大任务分解成可独立处理的小任务（Map阶段）并在多台计算机上并行执行，然后将结果合并（Reduce阶段）来实现对大规模数据的高效处理。 2. 相似性搜索：在大数据环境中，寻找相似数据是重要的任务。书中介绍了关键的技术，如余弦相似度、Jaccard相似度等，以及如何在大规模数据中有效地进行近似最近邻搜索，这些技术广泛应用于推荐系统、搜索引擎和图像识别等领域。 3. 数据聚类：包括K-means算法、DBSCAN（密度基空间分割）等聚类方法，它们在发现数据的内在结构和模式时发挥着重要作用。 4. 图挖掘：书中可能涉及网络分析，如社交网络分析，探讨如何在图数据中发现社区结构、路径查找和最短路径算法等。 5. 文本挖掘：介绍如何处理和分析大规模文本数据，包括词频统计、TF-IDF（词频-逆文档频率）计算、主题模型（如LDA，潜在狄利克雷分配）等，这些都是理解和提取文本信息的关键技术。 6. 异常检测：在大量数据中识别异常或离群值，对于欺诈检测、系统故障预测等应用场景至关重要。 7. 机器学习基础：尽管不是主要关注点，但可能会简要介绍监督和无监督学习的基本概念，以及如何在大数据背景下应用这些概念。 8. 实际案例研究：书中可能会包含一些实际的大数据项目案例，如Google的PageRank算法，它展示了如何利用大数据解决实际问题。《大数据挖掘-Mining of Massive Datasets》是一本深入浅出地探讨大规模数据处理和分析的教材，适合对大数据和数据挖掘感兴趣的学者和从业者阅读，提供了一套处理和理解海量数据的实用方法和技术。

4 CHAPTER 1. DATA MINING

The cases clustered around some o f the intersections of roads. These inter-

sections were the locations of wells that had become contaminated; people who

lived nearest these wells got sick, while people who lived nearer to wells that

had not been contaminated did not get sick. Without the ability to cluster the

data, the cause of Cholera would not have been discovered. 2

1.1.5 Feature Extraction

The typical feature-based model looks for the most extreme examples of a phe-

nomenon and represents the data by these examples. If you are familiar with

Bayes nets, a branch of machine learning and a topic we do not cover in this

book, you know how a complex relationship between objects is represented by

ﬁnding the strongest statistical dependencies among these objects and using

only those in representing all s tatistical connections. Some of the important

kinds of feature extraction from large-scale data that we shall study are:

1. Frequent Itemsets. This model makes sense for da ta that consists of “bas-

kets” of small sets of items, as in the market-basket problem that we shall

discuss in Chapter 6. We look for small sets of items that appear together

in many baskets, and these “frequent itemsets” are the characterization of

the data that we seek. The orignal application of this sor t of mining was

true market baskets: the sets of items, such as hamburger and ketchup,

that people tend to buy together when checking out at the cash register

of a store or super market.

2. Similar Items. Often, your data looks like a collection of sets, and the

objective is to ﬁnd pair s of sets that have a relatively large fraction of

their elements in common. An exa mple is treating custo mers a t an on-

line store like Amazon as the se t of items they have bought. In order

for Amazon to recommend something else they might like, Amazo n can

look for “similar” customers and recommend something many of these

customers have bought. This process is called “collaborative ﬁltering.”

If customers were single-minded, tha t is, they bought only o ne kind of

thing, then cluster ing customers might work. However, since customers

tend to have interests in many diﬀerent things, it is more useful to ﬁnd,

for each customer, a small number of other customers who are similar

in their tastes, and represent the data by these connections. We discuss

similarity in Chapter 3.

1.2 Statistical Limits on Data Mining

A common sort of data-mining problem involves discovering unusual events

hidden within massive amounts of data. This section is a discussion of the

problem, including “Bonferroni’s Principle,” a warning against overzealous use

of data mining.

1.2. STATISTICAL LIMITS ON DATA MINING 5

1.2.1 Total Information Awareness

In 2002, the Bush administration put forward a plan to mine all the data it could

ﬁnd, including credit-card receipts, hotel records, travel da ta, and many other

kinds of information in order to track terrorist activity. This idea naturally

caused great concern among privacy advocates, and the project, called TIA,

or Total Information Awareness, was eventually killed by Congress, although

it is unclear whether the project in fact exists under another name. It is not

the purpose of this book to discus s the diﬃcult issue of the privacy-security

tradeoﬀ. However, the prospect of TIA or a system like it does raise technical

questions about its feasibility and the realism of its assumptions.

The concern raised by many is that if you lo ok at so much data, and you try

to ﬁnd within it activities that look like terroris t behavior, are you not going to

ﬁnd many innocent activities – or even illicit activities that are not terrorism –

that will result in visits from the police and maybe worse than just a v isit? The

answer is that it all depends on how narrowly you deﬁne the activities that you

look for. Statisticians have see n this problem in many guises and have a theory,

which we introduce in the next section.

1.2.2 Bonferroni’s Principle

Suppose you have a certain amount of da ta, and you look for events of a cer-

tain typ e within that data. Yo u can expect events of this type to occur, even if

the data is co mpletely random, and the number of occurrences of these events

will grow as the size o f the data grows. These occurrences are “bogus,” in the

sense that they have no cause other than that random data will always have

some number of unusual features that look signiﬁcant but aren’t. A theore m

of statistics, known as the Bonferroni correction gives a statistically sound way

to avoid most of these bogus positive responses to a search through the data.

Without going into the statistical details, we oﬀer an informal version, Bon-

ferroni’s principle, that helps us avoid treating ra ndom occurrences as if they

were real. Calculate the expected number of occurrences of the e vents you are

looking for, on the assumption that data is random. If this number is signiﬁ-

cantly larger than the number of real instances you hop e to ﬁnd, then you must

exp ect almost anything you ﬁnd to be bogus, i.e., a statistical artifact rather

than evidence of what you are looking for. This obse rvation is the informal

statement of Bonferroni’s principle.

In a situation like searching for terrorists, where we expect that there are

few terr orists operating at any one time, Bonferroni’s principle says that we

may o nly detect terrorists by looking for events that are so rare that they are

unlikely to occur in random data. We shall give an extended example in the

next section.

6 CHAPTER 1. DATA MINING

1.2.3 An Example of B onferroni’s Principle

Suppose there are believed to be some “ e vil-doers” out there, and we want

to detect them. Suppose further that we have reason to believe tha t periodi-

cally, evil-doers g ather at a hotel to plot their evil. Le t us ma ke the following

assumptions about the size of the problem:

1. There are one billion people who might be evil-doers.

2. Everyone goes to a hotel one day in 100.

3. A hotel holds 100 people. Hence, there are 100,000 hotels – enough to

hold the 1% of a billion people who visit a hotel on any given day.

4. We sha ll examine hotel records for 1000 days.

To ﬁnd evil-do e rs in this data, we shall look for people who, on two diﬀerent

days, were both at the same hotel. Suppose, however, that there really are no

evil-doers. That is, everyone behaves at random, deciding with probability 0.01

to visit a hotel on any given day, and if so, choosing one of the 10

hotels at

random. Would we ﬁnd any pairs of people who appear to be evil-doers?

We c an do a simple approximate calculation as follows. The probability of

any two pe ople both deciding to visit a hotel on any given day is .0001. The

chance that they will visit the same hotel is this probability divided by 10

the number of hotels. Thus, the chance that they will visit the same hotel on

one given day is 10

−9

. The chance that they will visit the same hotel on two

diﬀerent given days is the square of this number, 10

−18

. Note that the hotels

can be diﬀerent on the two days.

Now, we must consider how many events will indicate evil-doing. An “event”

in this s e nse is a pair of people and a pair of days, such that the two people

were at the same hotel on each of the two days. To simplify the arithmetic, note

that for large n,





is about n

/2. We shall use this approximation in what

follows. Thus, the number of pairs of people is





= 5 × 10

. The number

of pairs of days is



1000



= 5 × 10

. The expected number of events that look

like evil-doing is the product of the number of pairs of people, the number of

pairs of days, and the probability that any one pair of people and pair of days

is an instance of the behavior we are looking for. That number is

5 × 10

× 5 × 10

× 10

−18

= 250, 000

That is, there will be a quarter of a million pairs of people w ho look like evil-

doers, even though they are not.

Now, suppose there r e ally are 10 pairs of evil-doers o ut there. T he police

will need to investigate a quarter of a million other pairs in order to ﬁnd the real

evil-doers. In addition to the intrusion on the lives of half a million innocent

people, the work involved is suﬃciently great that this approach to ﬁnding

evil-doers is probably not feasible.

1.3. THINGS USEFUL TO KNOW 7

1.2.4 Exercises for Section 1.2

Exercise 1.2.1 : Using the information from Section 1.2.3, what would be the

number of suspected pairs if the following changes were made to the data (and

all other numbers remained as they were in that section)?

(a) The number of days of obser vation was rais e d to 2000 .

(b) The number of people observed was raised to 2 billion (a nd there were

therefore 200,000 hotels).

same time on three diﬀerent days.

! Exercise 1.2.2 : Suppose we have information about the supermarket pur-

chases of 100 million people. Each person goes to the supermarket 100 times

in a year and buys 10 of the 1000 items that the s upermarket se lls. We believe

that a pair of terrorists will buy exactly the same set of 10 items (pe rhaps the

ingredients for a bomb?) at some time during the year. If we search for pairs of

people who have bought the s ame set of items, would we expect that any such

people found were truly terrorists?

1.3 Things Useful to Know

In this section, we oﬀer brief introductions to subjects that you may or may

not have seen in your study of other courses. Each will be useful in the study

of data mining. They include:

1. The TF.IDF measure of word importance.

2. Hash functions and their use.

3. Secondary storage (disk) and its eﬀect on running time of algorithms.

4. The base e of natural logarithms and identities involving that constant.

5. Power laws.

1.3.1 Importance of Words in Documents

In several applications of data mining, we shall be faced with the problem of

categorizing documents (sequences of words) by their topic. Typically, topics

are identiﬁed by ﬁnding the s pecial words that characterize documents about

that topic. For instance, articles about baseball would tend to have many

occurrences of words like “ball,” “bat,” “pitch,”, “run,” and so on. Once we

That is, assume our hypothesis that terrorists will surely buy a set of 10 items in common

at some time during the year. We don’t want to address the matter of whether or not terrorists

would necessarily do so.

8 CHAPTER 1. DATA MINING

have classiﬁed documents to determine they are about baseball, it is not hard

to notice that words such as these appear unusually frequently. However, until

we have made the classiﬁcation, it is not possible to identify these words as

characteristic.

Thus, classiﬁcation often star ts by looking at documents, and ﬁnding the

signiﬁcant words in those documents. Our ﬁrst guess might be that the words

appearing most frequently in a document ar e the most signiﬁcant. However,

that intuition is exactly opposite of the truth. The most fr equent words will

most surely b e the common words such as “the” o r “a nd,” which help build

ideas but do not carry any signiﬁcance themselves. In fact, the several hundred

most common words in English (called stop words) are often removed from

documents before any attempt to classify them.

In fac t, the indicators of the topic are r e lative ly rare words. However, not

all rare words are equally us e ful as indicators. There are certain words, for

example “notwithstanding” or “albeit,” that appear rarely in a collection of

documents, yet do not tell us a nything useful. On the other hand, a word like

“chukker” is probably equally rare, but tips us oﬀ that the document is about

the sport of polo. The diﬀerence between rar e words that tell us something and

those that do not has to do with the concentration of the useful words in just a

few documents. Tha t is, the presence of a word like “albeit” in a document does

not make it terribly more likely that it will appear multiple times. However,

if an article mentions “chukker” once , it is likely to tell us what happened in

the “ﬁrst chukker,” then the “second chukker,” and so o n. That is, the word is

likely to be r epeated if it appears at all.

The formal measure of how concentrated into relatively few documents are

the occurrences of a given word is called TF.IDF (Term Frequency times In-

verse Document Frequency). It is normally computed as follows. Suppose we

have a collection of N documents. Deﬁne f

to be the frequency (number of

occurrences) of ter m (word) i in document j. Then, deﬁne the term frequency

to be:

max

That is, the term frequency of term i in document j is f

normalized by dividing

it by the maximum number of occurrences of any term (perhaps excluding s top

words) in the same document. Thus, the most frequent term in document j

gets a TF of 1, and other terms get fractions as their term frequency for this

document.

The IDF for a term is deﬁned as follows. Suppose term i appears in n

of the N documents in the collection. Then IDF

= log

(N/n

). The TF.IDF

score for term i in document j is then deﬁned to be TF

× IDF

. The terms

with the highest TF.IDF sco re are often the terms that best characterize the

topic of the document.

Example 1.3 : Suppose our repository consists of 2

= 1,0 48,576 documents.

Suppose word w appears in 2

= 1024 of these documents. Then IDF

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