text: → (X wj) [i, X, j]P(X|wj) =i P˜(X|wj)C1 i i → (X wi1X wj ) [i , X , j ]Φ ∧ [i, X, j]P(X|iwj)1=j1h(θ +1 θ P1 (X1 |w1j1))C1 C2 i 0 1 1 i1 → (X wi1X wi2X wj ) [i , X , j ]Φ [i , X , j ]Φ ∧ [i, X, j]Pi(X|w1 j)j1= 2h(θj2 + θ 1P(X1 |w1j1)1+ θ2P(X2 |w2 j2)2)C1 C2 i 0 1 1 i1 2 2 i2 → (X X X ) [i, X , k]Φ [k, X , j]Φ [i, X, j]P(X|w1j)2= X| 1 P(XX|1|wik)P(XX|w2| kj) 2X| C1 i P( wk)P( wj)+P( wk)P( wj) i k i k E → ( wj)◦ E ◦i [i, , j] → E E ◦ (X X ) [i, , k] [k, X , j]Φ [i, X, j1]P(X|wj) = P(X|w1j) 1 C1 i k → E E ◦ (X X ) [i, X , k]Φ [k, , j] [i, X,1j]P(X|wj1) = P1(X|wk) C1 i i 1 ◦ where h(x) = {− } is a logistic function, represents the absence, and X, X , X 1 + exp x 1 2 X| − X| represent N or P. As speciﬁed in the polarity model, we have P( wj) = 1 P( wj). i i Figure 6 Inference rules with the polarity model and constraints. scorer = φ(wj, r)Tψ, where φ(wj, r) is the feature vector of using the rule r for the span i i wj, and ψ is the weight vector of the ranking model. The k highest score trees satisfying i the constraints are stored in score[, , ] for decoding the longer text spans. After ﬁnishing the CYK parsing, arg maxX∈{N P} score[0, n, X] is regarded as the polarity label of input , sentence. The time complexity is the same as the standard CYK’s. 4. Model Learning We have described the statistical sentiment parsing framework in the above section. We present the model learning process in this section. The learning process consists of two steps. First, the sentiment grammar and the polarity model are learned from data. In other words, the rules and the parameters used to compute polarity values are learned. These basic sentiment building blocks are then used to build the parse trees. Second, we estimate the parameters of the ranking model using the sentence and polarity label pairs. In this stage, we concentrate on learning how to score the parse trees based on the learned sentiment grammar and polarity model. Section shows the features and the parameter estimation algorithm used in the ranking model. Section illustrates how to learn the sentiment grammar and the polarity model. 18 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: Algorithm 3 Dictionary Rule Learning D G Input: : Dataset, C: Combination rules, τ : Frequency threshold G f Output: D: Dictionary rules D G 1: function MINEDICTIONARYRULES( , C) G (cid:48) ← {} 2: D L D · · · L 3: for (s, ) in do (cid:46) s : w w w| |− , : Polarity label of s s ≤ ≤ | | 0 1 s 1 s · · · 4: for all i, j s.t. 0 i < j s do (cid:46) wj : w w w − G i i i+1 j 1 5: if no negation rule in C covers wj then L i 6: #(wj, ) ++ i Gs (cid:48) 7: add wj to D G ← {} i 8: D G (cid:48) 9: for f in D do · ≥ 10: if #(f, ) τ then f N| P| 11: compute P( f) and P( f) using Equation (21) → G X| 12: add dictionary rule (L f) to D (cid:46) L = arg max ∈{ } P( f) f f X N,P G 13: return D quence of a frequent fragment must also be frequent. This is similar to the key insight in the Apriori algorithm (Agrawal and Srikant 1994). When we learn the dictionary rules, we can count the sentiment fragments from short to long, and prune the infrequent fragments in the early stages if any subsequence is not frequent. This pruning method accelerates the dictionary rule learning process and makes the procedure ﬁt in memory. G The combination rules C are generalizations for the dictionary rules. They are used to handle the compositionality and process unseen phrases. The learning of combina- tion rules is based on the learned dictionary rules and their polarity values. The sen- timent fragments are generalized to combination rules by replacing the subsequences of dictionary rules with their polarity labels. For instance, as shown in Figure 8, the fragments “is not (good/as expected/funny/well done)” are all negative. After replacing the sub-spans “good”, “as expected”, “funny”, and “well done” with their polarity label P, we → can learn the negation rule N is not P. We present the combination rule learning approach in Algorithm 4. Speciﬁcally, the ﬁrst step is to generate combination rule candidates. For every sub-sequence wj i of sentiment fragment f, we replace it with the corresponding non-terminal L if wj L | | | i P( wj) is larger than the threshold τ , and we can get wiL w f . Next, we compare wj i p 0 wj j i L L L (cid:54) L →i | | the polarity with . If = , we regard the rule L wiL w f as a negation wj f f wj f 0 wj j rule. Otherwisei, we further compareitheir polarity values. If this rule mi akes the polarity value become larger (or smaller), it will be treated as a strengthen (or weaken) rule. To obtain the contrast rules, we replace two sub-sequences with their polarity labels in a similar way. If the polarities of these two sub-sequences are different, we categorize this rule to the contrast type. Notably, these two non-terminals can not be next to each other. G (cid:48) After the above steps, we get the rule candidate set C and the occurrence number of each rule. We then ﬁlter the rule candidates whose occurrence frequencies are too small, 24 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: 85 80 y c a ur 75 c c A 70 SVM-1 LM-1 SVM-5 LM-5 MNB-1 s.parser MNB-5 65 0 105 2×105 3×105 4×105 Data Size Figure 9 The curves show the test accuracy as the number of training examples increases. Our method s.parser signiﬁcantly outperforms the other methods, which indicates s.parser can leverage data more effectively and beneﬁt more from larger data. of the datasets, but they harm the accuracy of LM on PL05-C. The high-order n-grams can partially solve compositionality in a brute-force way. 5.3 Effect of Training Data Size We further investigate the effect of the size of training data for different sentiment classiﬁcation methods. This is meaningful as the number of the publicly available reviews is increasing dramatically nowadays. The methods that can take advantage of more training data will be even more useful in practice. We report the results of s.parser compared with SVM, MNB, and LM on the dataset RT-C using different training data size. In order to make the ﬁgure clear, we only present the results of SVM/MNB/LM-1/5 here. As shown in Figure 9, we ﬁnd that the size of training data plays an important role for all these sentiment classiﬁcation methods. The basic conclusion is that the performances of all the methods rise as the data size increases. To be speciﬁc, when the size is small, the performances ascend sharply. It meets our intuition that the size of data is the key factor when the size is relative small. When the size of data is larger, the growth of accuracy becomes slower. The performances of the baseline methods start to converge after the data size is larger than 200,000. The comparisons illustrate that s.parser signiﬁcantly outperforms these base- lines. And the performance of s.parser still becomes better when the data size increases. The convergence of s.parser’s performance is slower than the others. It indicates that s.parser leverages data more effectively and beneﬁts more from a larger dataset. With more training data, s.parser learns more dictionary rules and combination rules. These rules enhance the generalization ability of our model. Furthermore, it estimates more reliable parameters for the polarity model and ranking model. In contrast, the bag-of- words based approaches (such as SVM, MNB, and LM) cannot make full use of high- order information in the dataset. The generalization ability of the combination rules of s.parser leads to better performance, and take advantage of larger data. It should be noted that there are similar trends on other datasets. 34 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## 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text: Table 6 Comparing our dictionary rule learning method with naive counting. The dictionary rules which N are assigned different polarities by these two methods are presented. represents negative, and P represents positive. The polarity probabilities of fragments are shown in this table, and they demonstrate our method learns more intuitive results than counting directly. Naive Count s.parser Fragment N P N P Polarity Polarity N P are fun 0.54 0.46 0.11 0.89 N P a very good movie 0.61 0.39 0.19 0.81 N P looks gorgeous 0.56 0.44 0.17 0.83 N P to enjoy the movies 0.53 0.47 0.14 0.86 P N is corny 0.43 0.57 0.83 0.17 P N ’ s ﬂawed 0.32 0.68 0.63 0.37 P N a difﬁcult ﬁlm to 0.43 0.57 0.67 0.33 P N disappoint 0.39 0.61 0.77 0.23 is learned as a negative phrase. This makes sense for the data distribution, while it may lead to the mismatch for the combination rules. In Figure 13, we show the polarity model of some combination rules learned from the dataset RT-C. The ﬁrst two examples are negation rules. We ﬁnd that both switch negation and shift negation exist in data, instead of using only one negation type in previous work (Saurí 2008; Choi and Cardie 2008; Taboada et al. 2011). For the rule → “N i do not P”, we ﬁnd that it is a switch negation rule. This rule reverses the polarity and the corresponding polarity strength. For instance, the “i do not like it very much” is → more negative than the “i do not like it”. As shown in Figure 13b, the “N is not P.” is a shift negation which reduces a ﬁxed polarity strength to reverse the original polarity. Speciﬁcally, the “is not good” is more negative than the “is not great” as described in Section . We have a similar conclusion for the next two weaken rules. As illustrated → in Figure 13c, the “P P actress” describes one aspect of a movie, hence it is more likely to decrease the polarity intensity. We ﬁnd that this rule is a ﬁxed intensiﬁcation → rule which reduces the polarity probability by a ﬁxed value. The “N a bit of N” is a percentage intensiﬁcation rule, which scales polarity intensity by a percentage. It reduces more strength for stronger polarity. The last two rules in Figure 13e and → → Figure 13f are strengthen rules. Both “P lot of P” and “N N terribly” increase the polarity strength of the sub-fragments. These cases indicate that it is necessary to learn how the context performs compositionality from data. In order to capture the compositionality for different rules, we deﬁne the polarity model and learn parameters for each rule. This also agrees with the models of Socher et al. (2012) and Dong et al. (2014), which use multiple composition matrices to make compositions speciﬁc and improves over the recursive neural network which employs one composition matrix. 6. Conclusion and Future Work In this article, we propose a statistical parsing framework for sentence-level senti- ment classiﬁcation, which provides a novel approach to designing sentiment classiﬁers from a new perspective. It directly analyzes the sentiment structure of a sentence other than relying on syntactic parsing results as in existing literature. We show that 38 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Table

text: Comparing

text: our

text: dictionary

text: rule

text: learning

text: method

text: with

text: naive

text: counting.

text: The

text: dictionary

text: rules

text: which

text: N

text: are

text: assigned

text: different

text: polarities

text: by

text: these

text: two

text: methods

text: are

text: presented.

text: represents

text: negative,

text: and

text: P

text: represents

text: positive.

text: The

text: polarity

text: probabilities

text: of

text: fragments

text: are

text: shown

text: in

text: this

text: table,

text: and

text: they

text: demonstrate

text: our

text: method

text: learns

text: more

text: intuitive

text: results

text: than

text: counting

text: directly.

text: Naive

text: Count

text: s.parser

text: Fragment

text: N

text: P

text: N

text: P

text: Polarity

text: Polarity

text: N

text: P

text: are

text: fun

text: N

text: P

text: a

text: very

text: good

text: movie

text: N

text: P

text: looks

text: gorgeous

text: N

text: P

text: to

text: enjoy

text: the

text: movies

text: P

text: N

text: is

text: corny

text: P

text: N

text: s

text: ﬂawed

text: P

text: N

text: a

text: difﬁcult

text: ﬁlm

text: to

text: P

text: N

text: disappoint

text: is

text: learned

text: as

text: a

text: negative

text: phrase.

text: This

text: makes

text: sense

text: for

text: the

text: data

text: distribution,

text: while

text: it

text: may

text: lead

text: to

text: the

text: mismatch

text: for

text: the

text: combination

text: rules.

text: In

text: Figure

text: we

text: show

text: the

text: polarity

text: model

text: of

text: some

text: combination

text: rules

text: learned

text: from

text: the

text: dataset

text: RT-C.

text: The

text: ﬁrst

text: two

text: examples

text: are

text: negation

text: rules.

text: We

text: ﬁnd

text: that

text: both

text: switch

text: negation

text: and

text: shift

text: negation

text: exist

text: in

text: data,

text: instead

text: of

text: using

text: only

text: one

text: negation

text: type

text: in

text: previous

text: work

text: (Saurí

text: Choi

text: and

text: Cardie

text: Taboada

text: et

text: al.

text: For

text: the

text: rule

text: “N

text: i

text: do

text: not

text: P”,

text: we

text: ﬁnd

text: that

text: it

text: is

text: a

text: switch

text: negation

text: rule.

text: This

text: rule

text: reverses

text: the

text: polarity

text: and

text: the

text: corresponding

text: polarity

text: strength.

text: For

text: instance,

text: the

text: “i

text: do

text: not

text: like

text: it

text: very

text: much”

text: is

text: more

text: negative

text: than

text: the

text: “i

text: do

text: not

text: like

text: it”.

text: As

text: shown

text: in

text: Figure

text: the

text: “N

text: is

text: not

text: P.”

text: is

text: a

text: shift

text: negation

text: which

text: reduces

text: a

text: ﬁxed

text: polarity

text: strength

text: to

text: reverse

text: the

text: original

text: polarity.

text: Speciﬁcally,

text: the

text: “is

text: not

text: good”

text: is

text: more

text: negative

text: than

text: the

text: “is

text: not

text: great”

text: as

text: described

text: in

text: Section

text: We

text: have

text: a

text: similar

text: conclusion

text: for

text: the

text: next

text: two

text: weaken

text: rules.

text: As

text: illustrated

text: in

text: Figure

text: the

text: “P

text: P

text: actress”

text: describes

text: one

text: aspect

text: of

text: a

text: movie,

text: hence

text: it

text: is

text: more

text: likely

text: to

text: decrease

text: the

text: polarity

text: intensity.

text: We

text: ﬁnd

text: that

text: this

text: rule

text: is

text: a

text: ﬁxed

text: intensiﬁcation

text: rule

text: which

text: reduces

text: the

text: polarity

text: probability

text: by

text: a

text: ﬁxed

text: value.

text: The

text: “N

text: a

text: bit

text: of

text: N”

text: is

text: a

text: percentage

text: intensiﬁcation

text: rule,

text: which

text: scales

text: polarity

text: intensity

text: by

text: a

text: percentage.

text: It

text: reduces

text: more

text: strength

text: for

text: stronger

text: polarity.

text: The

text: last

text: two

text: rules

text: in

text: Figure

text: and

text: Figure

text: are

text: strengthen

text: rules.

text: Both

text: “P

text: lot

text: of

text: P”

text: and

text: “N

text: N

text: terribly”

text: increase

text: the

text: polarity

text: strength

text: of

text: the

text: sub-fragments.

text: These

text: cases

text: indicate

text: that

text: it

text: is

text: necessary

text: to

text: learn

text: how

text: the

text: context

text: performs

text: compositionality

text: from

text: data.

text: In

text: order

text: to

text: capture

text: the

text: compositionality

text: for

text: different

text: rules,

text: we

text: deﬁne

text: the

text: polarity

text: model

text: and

text: learn

text: parameters

text: for

text: each

text: rule.

text: This

text: also

text: agrees

text: with

text: the

text: models

text: of

text: Socher

text: et

text: al.

text: and

text: Dong

text: et

text: al.

text: which

text: use

text: multiple

text: composition

text: matrices

text: to

text: make

text: compositions

text: speciﬁc

text: and

text: improves

text: over

text: the

text: recursive

text: neural

text: network

text: which

text: employs

text: one

text: composition

text: matrix.

text: Conclusion

text: and

text: Future

text: Work

text: In

text: this

text: article,

text: we

text: propose

text: a

text: statistical

text: parsing

text: framework

text: for

text: sentence-level

text: senti-

text: ment

text: classiﬁcation,

text: which

text: provides

text: a

text: novel

text: approach

text: to

text: designing

text: sentiment

text: classiﬁers

text: from

text: a

text: new

text: perspective.

text: It

text: directly

text: analyzes

text: the

text: sentiment

text: structure

text: of

text: a

text: sentence

text: other

text: than

text: relying

text: on

text: syntactic

text: parsing

text: results

text: as

text: in

text: existing

text: literature.

text: We

text: show

text: that

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: complicated phenomena in sentiment analysis, such as negation, intensiﬁcation, and contrast, can be handled the same as simple and straightforward sentiment expressions in a uniﬁed and probabilistic way. We provide a formal model to represent the sentiment grammar built upon CFGs (Context-Free Grammars). The framework consists of: (1) a parsing model to analyze the sentiment structure of a sentence; (2) a polarity model to calculate sentiment strength and polarity for each text span in the parsing process; and (3) a ranking model to select the best parsing result from a list of candidate sentiment parse trees. We show that the sentiment parser can be trained from the examples of sentences annotated only with sentiment polarity labels but without using any syntactic or sentiment annotations within sentences. We evaluate the proposed framework on standard sentiment classiﬁcation datasets. The experimental results show the statis- tical sentiment parsing notably outperforms the baseline sentiment classiﬁcation ap- proaches. We believe the work on statistical sentiment parsing can be advanced from many different perspectives. First, statistical parsing has been a well-established research ﬁeld, in which many different grammars and parsing algorithms have been proposed in previously published literature. It will be a very interesting direction to apply and adjust more advanced models and algorithms from the syntactic parsing and the semantic parsing to our framework. We leave it as a line of future work. Second, we can incor- porate target and aspect information in the statistical sentiment parsing framework to facilitate the target-dependent and aspect-based sentiment analysis. Intuitively, this can be done by introducing semantic tags of targets and aspects as new non-terminals in the sentiment grammar and revising grammar rules accordingly. However, acquiring training data will be an even more challenging task as we need more ﬁne-grained information. Third, as the statistical sentiment parsing produces more ﬁne-grained information (e.g., the basic sentiment expressions from the dictionary rules as well as the sentiment structure trees), we will have more opportunities to generate better opinion summaries. Moreover, we are interested in jointly learning parameters of the polarity model and the parsing model from data. Last but not the least, we are interested in investigating the domain adaptation which is a very important and challenging prob- lem in sentiment analysis. Generally, we may need to learn domain-speciﬁc dictionary rules for different domains while we believe combination rules are mostly generic across different domains. This is also worth consideration for further study in future works. Acknowledgments Computational Linguistics. This research was partly supported by NSFC Agrawal, Rakesh and Ramakrishnan Srikant. (Grant No. 61421003) and the fund of the 1994. Fast algorithms for mining State Key Lab of Software Development association rules in large databases. In Environment (Grant No. Proceedings of the 20th International SKLSDE-2015ZX-05). We gratefully Conference on Very Large Data Bases, VLDB acknowledge helpful discussions with Dr. ’94, pages 487–499, San Francisco, CA, Nan Yang and the anonymous reviewers. USA. Morgan Kaufmann Publishers Inc. Artzi, Yoav and Luke Zettlemoyer. 2013. Weakly supervised learning of semantic parsers for mapping instructions to References actions. Transactions of the Association for Agarwal, Apoorv, Boyi Xie, Ilia Vovsha, Computational Linguistics, 1(1):49–62. Owen Rambow, and Rebecca Passonneau. Bao, Junwei, Nan Duan, Ming Zhou, and 2011. Sentiment analysis of twitter data. In Tiejun Zhao. 2014. Knowledge-based Proceedings of the Workshop on Languages in question answering as machine Social Media, LSM ’11, pages 30–38, translation. In Proceedings of the 52nd Stroudsburg, PA, USA. Association for 40 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: complicated

text: phenomena

text: in

text: sentiment

text: analysis,

text: such

text: as

text: negation,

text: intensiﬁcation,

text: and

text: contrast,

text: can

text: be

text: handled

text: the

text: same

text: as

text: simple

text: and

text: straightforward

text: sentiment

text: expressions

text: in

text: a

text: uniﬁed

text: and

text: probabilistic

text: way.

text: We

text: provide

text: a

text: formal

text: model

text: to

text: represent

text: the

text: sentiment

text: grammar

text: built

text: upon

text: CFGs

text: (Context-Free

text: Grammars).

text: The

text: framework

text: consists

text: of:

text: a

text: parsing

text: model

text: to

text: analyze

text: the

text: sentiment

text: structure

text: of

text: a

text: sentence;

text: a

text: polarity

text: model

text: to

text: calculate

text: sentiment

text: strength

text: and

text: polarity

text: for

text: each

text: text

text: span

text: in

text: the

text: parsing

text: process;

text: and

text: a

text: ranking

text: model

text: to

text: select

text: the

text: best

text: parsing

text: result

text: from

text: a

text: list

text: of

text: candidate

text: sentiment

text: parse

text: trees.

text: We

text: show

text: that

text: the

text: sentiment

text: parser

text: can

text: be

text: trained

text: from

text: the

text: examples

text: of

text: sentences

text: annotated

text: only

text: with

text: sentiment

text: polarity

text: labels

text: but

text: without

text: using

text: any

text: syntactic

text: or

text: sentiment

text: annotations

text: within

text: sentences.

text: We

text: evaluate

text: the

text: proposed

text: framework

text: on

text: standard

text: sentiment

text: classiﬁcation

text: datasets.

text: The

text: experimental

text: results

text: show

text: the

text: statis-

text: tical

text: sentiment

text: parsing

text: notably

text: outperforms

text: the

text: baseline

text: sentiment

text: classiﬁcation

text: ap-

text: proaches.

text: We

text: believe

text: the

text: work

text: on

text: statistical

text: sentiment

text: parsing

text: can

text: be

text: advanced

text: from

text: many

text: different

text: perspectives.

text: First,

text: statistical

text: parsing

text: has

text: been

text: a

text: well-established

text: research

text: ﬁeld,

text: in

text: which

text: many

text: different

text: grammars

text: and

text: parsing

text: algorithms

text: have

text: been

text: proposed

text: in

text: previously

text: published

text: literature.

text: It

text: will

text: be

text: a

text: very

text: interesting

text: direction

text: to

text: apply

text: and

text: adjust

text: more

text: advanced

text: models

text: and

text: algorithms

text: from

text: the

text: syntactic

text: parsing

text: and

text: the

text: semantic

text: parsing

text: to

text: our

text: framework.

text: We

text: leave

text: it

text: as

text: a

text: line

text: of

text: future

text: work.

text: Second,

text: we

text: can

text: incor-

text: porate

text: target

text: and

text: aspect

text: information

text: in

text: the

text: statistical

text: sentiment

text: parsing

text: framework

text: to

text: facilitate

text: the

text: target-dependent

text: and

text: aspect-based

text: sentiment

text: analysis.

text: Intuitively,

text: this

text: can

text: be

text: done

text: by

text: introducing

text: semantic

text: tags

text: of

text: targets

text: and

text: aspects

text: as

text: new

text: non-terminals

text: in

text: the

text: sentiment

text: grammar

text: and

text: revising

text: grammar

text: rules

text: accordingly.

text: However,

text: acquiring

text: training

text: data

text: will

text: be

text: an

text: even

text: more

text: challenging

text: task

text: as

text: we

text: need

text: more

text: ﬁne-grained

text: information.

text: Third,

text: as

text: the

text: statistical

text: sentiment

text: parsing

text: produces

text: more

text: ﬁne-grained

text: information

text: (e.g.,

text: the

text: basic

text: sentiment

text: expressions

text: from

text: the

text: dictionary

text: rules

text: as

text: well

text: as

text: the

text: sentiment

text: structure

text: trees),

text: we

text: will

text: have

text: more

text: opportunities

text: to

text: generate

text: better

text: opinion

text: summaries.

text: Moreover,

text: we

text: are

text: interested

text: in

text: jointly

text: learning

text: parameters

text: of

text: the

text: polarity

text: model

text: and

text: the

text: parsing

text: model

text: from

text: data.

text: Last

text: but

text: not

text: the

text: least,

text: we

text: are

text: interested

text: in

text: investigating

text: the

text: domain

text: adaptation

text: which

text: is

text: a

text: very

text: important

text: and

text: challenging

text: prob-

text: lem

text: in

text: sentiment

text: analysis.

text: Generally,

text: we

text: may

text: need

text: to

text: learn

text: domain-speciﬁc

text: dictionary

text: rules

text: for

text: different

text: domains

text: while

text: we

text: believe

text: combination

text: rules

text: are

text: mostly

text: generic

text: across

text: different

text: domains.

text: This

text: is

text: also

text: worth

text: consideration

text: for

text: further

text: study

text: in

text: future

text: works.

text: Acknowledgments

text: Computational

text: Linguistics.

text: This

text: research

text: was

text: partly

text: supported

text: by

text: NSFC

text: Agrawal,

text: Rakesh

text: and

text: Ramakrishnan

text: Srikant.

text: (Grant

text: No.

text: and

text: the

text: fund

text: of

text: the

text: Fast

text: algorithms

text: for

text: mining

text: State

text: Key

text: Lab

text: of

text: Software

text: Development

text: association

text: rules

text: in

text: large

text: databases.

text: In

text: Environment

text: (Grant

text: No.

text: Proceedings

text: of

text: the

text: International

text: We

text: gratefully

text: Conference

text: on

text: Very

text: Large

text: Data

text: Bases,

text: VLDB

text: acknowledge

text: helpful

text: discussions

text: with

text: Dr.

text: pages

text: San

text: Francisco,

text: CA,

text: Yang

text: and

text: the

text: anonymous

text: reviewers.

text: USA.

text: Morgan

text: Kaufmann

text: Publishers

text: Inc.

text: Artzi,

text: Yoav

text: and

text: Luke

text: Zettlemoyer.

text: Weakly

text: supervised

text: learning

text: of

text: semantic

text: parsers

text: for

text: mapping

text: instructions

text: to

text: References

text: actions.

text: Transactions

text: of

text: the

text: Association

text: for

text: Agarwal,

text: Apoorv,

text: Boyi

text: Xie,

text: Ilia

text: Vovsha,

text: Computational

text: Linguistics,

text: Owen

text: Rambow,

text: and

text: Rebecca

text: Passonneau.

text: Bao,

text: Junwei,

text: Duan,

text: Ming

text: Zhou,

text: and

text: Sentiment

text: analysis

text: of

text: twitter

text: data.

text: In

text: Tiejun

text: Zhao.

text: Knowledge-based

text: Proceedings

text: of

text: the

text: Workshop

text: on

text: Languages

text: in

text: question

text: answering

text: as

text: machine

text: Social

text: Media,

text: LSM

text: pages

text: translation.

text: In

text: Proceedings

text: of

text: the

text: Stroudsburg,

text: PA,

text: USA.

text: Association

text: for

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: polynomial approximation. In this aim, we compute the mean maximum values of η: 2.4 and 4. Indeed, when this coefﬁcient value of that difference for each η. Then, the comparison of is not better than 0.99, the correlation is usually considered as this value with a polynome of degree 1 in η is sufﬁcient to not good. achieve an excellent approximation. The result turns out to be interesting: the linear expression in η ﬁts the curve with the same coefﬁcients, whatever the value of η. The SINR calculated is corrected by the linear expression, and in this case, we establish the formula: − SINRfitted = SINR (aη + b) (6) Fluid Fluid which yields ≈ CDF CDFfitted (7) Poisson Fluid where a = 3 and b = -6. As observed on some examples (Fig. 5 and 6), for η = 2.8, 3, 3.6, 3.8, the CDF of the SINR established by the ﬁtted Fluid Fig. 4. CDF of the SINR with η = 2.8 (left) and η = 3.6 (right), for a Poisson model and by the Poisson model are very close: the differences model network and a Fluid model network. between them are less than 0.4 dB (let us notice that, for an outage probability of 5%, this difference may reach 0.9 dB for some values of η). F. Impact of the density of BS It is interesting to observe that the ﬁtting does not depend on the density of BS. Fig. 7 and 8 show that when the density is multiplied by a factor 10 or divided by a factor 10, the CDF are identical, and the ﬁtting is the same. As a consequence, these results can be used whatever the density of the network, i.e. whatever the intersites distance between base stations. It is easy to√understand this property when we realize that, since ρ = 3 , the expression of the SINR (5) given by the ﬂuid BS 6R2 model, canc be expressed as: − − √6 η 2 x η γ(x) = − − (8) 3 2π (2 x)2 η Fig. 5. CDF of the ﬁtted SINR with η = 2.8 (left) and η = 3 (right), for a Poisson model network and a Fluid model network. where x = r represents the relative distance of a UE to its serving basRe cstation. This expression does not explicitely depend on the ISD (distance between neighboring BS). This η Correlation coefﬁcient ζ 2.2 0.9755 expression holds whatever the ISD. Moreover, it does not 2.4 0.9961 depend on the density of BS. 2.6 0.99909 G. Correlation coefﬁcient 2.8 0.99966 3 0.99952 We compare the correlation coefﬁcient (9) computed be- 3.2 0.99941 tween the CDF curves of the ﬁtted ﬂuid model and of the 3.4 0.99934 Poisson model. We recall that, given X and Y , two samples 3.6 0.99897 3.8 0.99861 of length n with res(cid:80)pective means X¯ and Y¯ , the correlation 4 0.99824 coefﬁcient ζ is(cid:113)given by: (cid:113) 4.2 0.99683 (cid:80) (cid:80) n (X − X¯)(Y − Y¯ ) TABLE I ζ = i=1 i i (9) CORRELATION COEFFICIENT FOR THE COMPARISON OF THE CDF OF THE n (X − X¯)2 n (Y − Y¯ )2 SINR FROM THE FITTED FLUID NETWORK AND THE POISSON NETWORK i=1 i i=1 i WITH RESPECT TO η This comparison is synthetized in Table I illustrating that these correlation coefﬁcients are excellent. The linear ap- H. Model limitations proximation becomes less accurate for the minimum and the ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: polynomial

text: approximation.

text: In

text: this

text: aim,

text: we

text: compute

text: the

text: mean

text: maximum

text: values

text: of

text: η:

text: and

text: Indeed,

text: when

text: this

text: coefﬁcient

text: value

text: of

text: that

text: difference

text: for

text: each

text: η.

text: Then,

text: the

text: comparison

text: of

text: is

text: not

text: better

text: than

text: the

text: correlation

text: is

text: usually

text: considered

text: as

text: this

text: value

text: with

text: a

text: polynome

text: of

text: degree

text: in

text: η

text: is

text: sufﬁcient

text: to

text: not

text: good.

text: achieve

text: an

text: excellent

text: approximation.

text: The

text: result

text: turns

text: out

text: to

text: be

text: interesting:

text: the

text: linear

text: expression

text: in

text: η

text: ﬁts

text: the

text: curve

text: with

text: the

text: same

text: coefﬁcients,

text: whatever

text: the

text: value

text: of

text: η.

text: The

text: SINR

text: calculated

text: is

text: corrected

text: by

text: the

text: linear

text: expression,

text: and

text: in

text: this

text: case,

text: we

text: establish

text: the

text: formula:

text: SINRfitted

text: SINR

text: (aη

text: b)

text: Fluid

text: Fluid

text: which

text: yields

text: CDF

text: CDFfitted

text: Poisson

text: Fluid

text: where

text: a

text: and

text: b

text: As

text: observed

text: on

text: some

text: examples

text: (Fig.

text: and

text: for

text: η

text: the

text: CDF

text: of

text: the

text: SINR

text: established

text: by

text: the

text: ﬁtted

text: Fluid

text: Fig.

text: CDF

text: of

text: the

text: SINR

text: with

text: η

text: (left)

text: and

text: η

text: (right),

text: for

text: a

text: Poisson

text: model

text: and

text: by

text: the

text: Poisson

text: model

text: are

text: very

text: close:

text: the

text: differences

text: model

text: network

text: and

text: a

text: Fluid

text: model

text: network.

text: between

text: them

text: are

text: less

text: than

text: dB

text: (let

text: us

text: notice

text: that,

text: for

text: an

text: outage

text: probability

text: of

text: this

text: difference

text: may

text: reach

text: dB

text: for

text: some

text: values

text: of

text: η).

text: F.

text: Impact

text: of

text: the

text: density

text: of

text: BS

text: It

text: is

text: interesting

text: to

text: observe

text: that

text: the

text: ﬁtting

text: does

text: not

text: depend

text: on

text: the

text: density

text: of

text: BS.

text: Fig.

text: and

text: show

text: that

text: when

text: the

text: density

text: is

text: multiplied

text: by

text: a

text: factor

text: or

text: divided

text: by

text: a

text: factor

text: the

text: CDF

text: are

text: identical,

text: and

text: the

text: ﬁtting

text: is

text: the

text: same.

text: As

text: a

text: consequence,

text: these

text: results

text: can

text: be

text: used

text: whatever

text: the

text: density

text: of

text: the

text: network,

text: i.e.

text: whatever

text: the

text: intersites

text: distance

text: between

text: base

text: stations.

text: It

text: is

text: easy

text: to√understand

text: this

text: property

text: when

text: we

text: realize

text: that,

text: since

text: ρ

text: the

text: expression

text: of

text: the

text: SINR

text: given

text: by

text: the

text: ﬂuid

text: BS

text: model,

text: canc

text: be

text: expressed

text: as:

text: √6

text: η

text: x

text: η

text: γ(x)

text: 2π

text: η

text: Fig.

text: CDF

text: of

text: the

text: ﬁtted

text: SINR

text: with

text: η

text: (left)

text: and

text: η

text: (right),

text: for

text: a

text: Poisson

text: model

text: network

text: and

text: a

text: Fluid

text: model

text: network.

text: where

text: x

text: r

text: represents

text: the

text: relative

text: distance

text: of

text: a

text: UE

text: to

text: its

text: serving

text: basRe

text: cstation.

text: This

text: expression

text: does

text: not

text: explicitely

text: depend

text: on

text: the

text: ISD

text: (distance

text: between

text: neighboring

text: BS).

text: This

text: η

text: Correlation

text: coefﬁcient

text: ζ

text: expression

text: holds

text: whatever

text: the

text: ISD.

text: Moreover,

text: it

text: does

text: not

text: depend

text: on

text: the

text: density

text: of

text: BS.

text: G.

text: Correlation

text: coefﬁcient

text: We

text: compare

text: the

text: correlation

text: coefﬁcient

text: computed

text: be-

text: tween

text: the

text: CDF

text: curves

text: of

text: the

text: ﬁtted

text: ﬂuid

text: model

text: and

text: of

text: the

text: Poisson

text: model.

text: We

text: recall

text: that,

text: given

text: X

text: and

text: Y

text: two

text: samples

text: of

text: length

text: n

text: with

text: means

text: X¯

text: and

text: Y¯

text: the

text: correlation

text: coefﬁcient

text: ζ

text: by:

text: n

text: (X

text: X¯)(Y

text: Y¯

text: TABLE

text: I

text: ζ

text: i

text: i

text: CORRELATION

text: COEFFICIENT

text: FOR

text: THE

text: COMPARISON

text: OF

text: THE

text: CDF

text: OF

text: THE

text: n

text: (X

text: n

text: (Y

text: Y¯

text: SINR

text: FROM

text: THE

text: FITTED

text: FLUID

text: NETWORK

text: AND

text: THE

text: POISSON

text: NETWORK

text: i

text: i

text: WITH

text: RESPECT

text: TO

text: η

text: This

text: comparison

text: is

text: synthetized

text: in

text: Table

text: I

text: illustrating

text: that

text: these

text: correlation

text: coefﬁcients

text: are

text: excellent.

text: The

text: linear

text: ap-

text: H.

text: Model

text: limitations

text: proximation

text: becomes

text: less

text: accurate

text: for

text: the

text: minimum

text: and

text: the

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

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text: 6 (cid:2) (cid:3) (cid:88) (cid:88) (cid:88) | − | P(x) − E f 1(f (Xn)) ρ x∈X Q(x)1−α = x(cid:18)∈(cid:88)X P(x)Q(x(cid:19))α(cid:18)1(cid:88) (cid:19) (61) =n (cid:88)n Pn((cid:88)xn)|fn−1(fn((cid:6)Xn))|ρ (cid:7) (67) − ∈X (cid:80)≤ P(x)α α1 Q(x)α αα1 . (62) ≤ xn n (cid:88) Pn(xn) 2n(H(Q)−R(cid:88)+δn(cid:48) ) ρ (68) ∈X ∈X x x ∈P X ∈ Q n( ) xn TQ Dividing by P(x)/Q(x)1−α and raising to the power < 1 + 2ρ (cid:88) 2nρ(H(Q)−R+δn(cid:48) ) Pn(xn) (69) − x || ≥ ∈P X ∈ of α/(α 1) shows that ∆α(P Q) 0. Equality holds if, ≤ Q n( ) − − −xn TQ|| − (cid:48) and only if, P = Q. 1 + 2ρ 2 nρ(R H(Q)+ρ 1D(Q P) δn) (70) ∈P X ≤ −Q n−( ) − 1 + 2 nρ(R Hρ˜(X1) δn). (71) V. UNIVERSAL ENCODERS FOR IID SOURCES Here (68) follows from the construction of f ; (69) follows In Section the direct part of Theorem I.2 is proved using n from (26); (70) follows because the probability of the source the upper bound in Theorem I.1. It is pointed out in Section − || emitting a sequence of type Q is at most 2 nD(Q P); and (71) that the construction of the encoder in the proof of this upper follows from the identity (see [10]) bound requires knowledge of the distribution of X. As the next result shows, however, for IID sources we do not need to − − || H (X ) = max H(Q) ρ 1D(Q P), (72) know the distribution of the source to construct good encoders. ρ˜ 1 ∈P X X Q ( ) Theorem V.1. Let be a ﬁnite set, and let ρ > 0. For every X → { } |P X | |X| rate R > 0 there exist enc{ode}rs∞fn : 1, . . . X, 2nR such and the fact that n( ) < (n + 1) . that for every IID source X with alphabet , (cid:2) (cid:3) i i=1 VI. TASKS WITH SIDE-INFORMATION | − | − − − E f 1(f (Xn)) ρ < 1 + 2 nρ(R Hρ˜(X1) δn), (63) In this section we generalize Theorems I.1, I.2, and V.1 to n n account for side-information: A task X and side-information X ×Y Y are drawn according to a joint PMF P on , where where both X and Y are ﬁnite, and where thXe,Yside-information is available to both the task describer (encoder) and the tasks − |X| 1 + (1 + ρ 1) log(n + 1) performer. The encoder is now of the form δ = . (64) n n X × Y → { } f : 1, . . . , M . (73) In particular, (cid:2) (cid:3) If the realization of (X, Y ) is (x, y) and f(x, y) = m, then | − | lim E f 1(f (Xn)) ρ = 1, (65) all the tasks in the set →∞ n n n − (cid:44) { ∈ X } f 1(m, y) x : f(x, y) = m (74) whenever H (X ) < R. ρ˜ 1 X Proof. We ﬁrst partition n into the different type classes are performed. As in Section , we seek to minimize for a |X| (cid:2) (cid:3) T , of which there are less than (n + 1) . We then par- given M the ρ-th moment of the number of performed tasks Q − (cid:48) (cid:88) (cid:88) t(cid:100)|cTia|tTiQrodQ|ni|n≤2ae−lain2cty(hnRHa−Tt(QδQmn(cid:48))),oi(cid:101)nsettaowchh2(cid:6)enxr(eRn δ∈δn(cid:48)nT) =Qsutbhnsue−sts1(cid:7)e|Xonfd|slcoaugrpd(ninina+laity1s)ua.btsSemitnocoseft E |f−1(f(X=, Y∈)X, Y∈)Y|ρPX,Y (x, y)|f−1(f(x, y), y)|ρ. (75) x y The key quantity here is a conditional version of Re´nyi entropy − (cid:48) 2n(H(Q) R+δn) . (66) (proposed by Arimoto [18](cid:18)): (cid:19) (cid:88) (cid:88) noNt oextecethedat 2thneR.toWtale ncuomnbsterrucotf fsub:sXets →in th{e1,p.a.r.t,it2ionnR}dobeys Hρ˜(X|Y ) = ρ1 log ∈Y ∈X PX,Y (x, y)1+1ρ 1+ρ. (76) n y x enumerating the subsets in the partition with the numbers in { } ∈ { } 1, . . . , 2nR and by mapping to m 1, . . . , 2nR the xn’s Theorem I.1 can be generalized as follows. { }∞ that comprise the m-th subset. Suppose now that X is Theorem VI.1. Let (X, Y ) be a pair of chance variables i i=1 X × Y IID P and observe that taking value in the ﬁnite set , and let ρ > 0. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: equation

text: P(x)

text: E

text: f

text: (Xn))

text: ρ

text: x∈X

text: =n

text: ∈X

text: P(x)α

text: α1

text: Q(x)α

text: αα1

text: xn

text: n

text: Pn(xn)

text: ρ

text: ∈X

text: ∈X

text: x

text: x

text: ∈P

text: X

text: Q

text: n(

text: xn

text: TQ

text: Dividing

text: by

text: and

text: raising

text: to

text: the

text: power

text: 2ρ

text: Pn(xn)

text: x

text: ∈P

text: X

text: of

text: α/(α

text: shows

text: that

text: ∆α(P

text: Q)

text: Equality

text: holds

text: if,

text: Q

text: n(

text: −xn

text: TQ||

text: and

text: only

text: if,

text: P

text: Q.

text: 2ρ

text: nρ(R

text: H(Q)+ρ

text: P)

text: δn)

text: ∈P

text: X

text: −Q

text: n−(

text: nρ(R

text: δn).

text: V.

text: UNIVERSAL

text: ENCODERS

text: FOR

text: IID

text: SOURCES

text: Here

text: follows

text: from

text: the

text: construction

text: of

text: f

text: follows

text: In

text: Section

text: the

text: direct

text: part

text: of

text: Theorem

text: is

text: proved

text: using

text: n

text: from

text: follows

text: because

text: the

text: probability

text: of

text: the

text: source

text: the

text: upper

text: bound

text: in

text: Theorem

text: It

text: is

text: pointed

text: out

text: in

text: Section

text: emitting

text: a

text: sequence

text: of

text: type

text: Q

text: is

text: at

text: most

text: nD(Q

text: P);

text: and

text: that

text: the

text: construction

text: of

text: the

text: encoder

text: in

text: the

text: proof

text: of

text: this

text: upper

text: follows

text: from

text: the

text: identity

text: (see

text: bound

text: requires

text: knowledge

text: of

text: the

text: distribution

text: of

text: X.

text: As

text: the

text: next

text: result

text: shows,

text: however,

text: for

text: IID

text: sources

text: we

text: do

text: not

text: need

text: to

text: H

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text: 2 sidered. Perturbations instabilities will be examined and the analysis will focus on models where dynamical sys- with a(t) the scale factor, while the scalar ﬁeld is tem techniques can be sucessfully applied. Sec. and Sec. taken to be spatially homogeneous φ = φ(t). The mat- will then be dedicated to the study of square and square ter energy-momentum tensor will be of the perfect ﬂuid root kinetic corrections to the canonical scalar ﬁeld La- form with ρ(t) and p(t) its energy density and pressure, grangian. For these simple models the full dynamical respectively. A linear EoS p = wρ, with w the EoS pa- features can be obtained and the background cosmolog- rameter ranging from 0 (dust) to 1/3 (radiation), will be ical evolution can be determined for any initial condi- assumed. tion. The analysis of these two sections will show that a With these assumptions, from the gravitational equa- rich phenomenology can be obtained with these extended tions (3) we obtain the F(cid:18)riedmann cons(cid:19)traint scalar ﬁelds. Finally results and conclusions will be dis- cussed in Sec. 1 II. THE CANONICAL SCALAR FIELD 3H2 = κ2 ρ + φ˙2 + V , (6) 2 In this section we review the cosmological dynamics of and the acceleration equat(cid:18)ion (cid:19) a canonical scalar ﬁeld following the analysis ﬁrst per- formed in [19]. This will serve as an introduction to the dynamical system techniques one can apply in or- − 1 − der to completely determine the cosmological evolution 2H˙ + 3H2 = κ2 p + φ˙2 V , (7) 2 of speciﬁc models. Moreover this section will be helpful in deﬁning notation and conventions. where H = a˙/a is the Hubble parameter and an over- The action of a minimally coupled canonical scalar ﬁeld is given by (cid:90) (cid:20) (cid:21) dot denotes diﬀerentiation with respect to the time t. On the other hand the scalar ﬁeld equation (4) gives √ − R L L S = d4x g + + , (1) ∂V 2κ2 φ m φ¨ + 3Hφ˙ + = 0 . (8) ∂φ where g is the determinant of the metric, R is the Ricci L scalar, κ2 = 8πG/c4, is the matter Lagrangian, and The energy density and pressure of the canonical scalar m ﬁeld are given by the scalar ﬁeld Lagrangian is deﬁned as Lφ = −12∂φ2 − V (φ) , (2) ρφ = 12φ˙2 + V , (9) 1 − p = φ˙2 V , (10) with ∂φ2 = ∂ φ∂µφ and V a general potential for φ. φ 2 µ The variation with respect to g produces the following µν gravitational(cid:18)equations (cid:19) and its EoS parameter, deﬁned as the ratio between its pressure and energy density, is − 1 − G = κ2 T + ∂ φ∂ φ g ∂φ2 g V , (3) − µν µν µ ν 2 µν µν p 1φ˙2 V w = φ = 2 . (11) − φ ρφ 1φ˙2 + V where G = R 1/2g R is the Einstein tensor and 2 µν µν µν T the matter energy momentum tensor. The variation (cid:29) µν For V φ˙2 this approaches a cosmological constant with respect to φ gives the Klein-Gordon equation − (cid:28) EoS with w = 1, while for V φ˙2 this describe a stiﬀ φ ﬂuid with w = 1. φ − ∂V At this point, following [19], we introduce new dimen- (cid:50)φ = 0 , (4) ∂φ sionless variables as ∇ ∇ with (cid:50)φ = µφ. µ κ2φ˙2 κ2V κ2ρ In what follows we will analyze the background cos- x2 = , y2 = , σ2 = . (12) mological evolution of this model. The metric tensor will 6H2 3H2 3H2 be assumed to be of the Friedmann-Robertson-Walker (FRW) type with vanishing spatial curvature These variables are largely employed in scalar ﬁeld cos- mology since not only allow to rewrite Eqs. (6)-(8) as an − autonomous system of equations, but can also be gener- g = diag( 1, a(t)2, a(t)2, a(t)2) , (5) µν alized in diﬀerent contexts such as nonminimally coupled ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: sidered.

text: Perturbations

text: instabilities

text: will

text: be

text: examined

text: and

text: the

text: analysis

text: will

text: focus

text: on

text: models

text: where

text: dynamical

text: sys-

text: with

text: a(t)

text: the

text: scale

text: factor,

text: while

text: the

text: scalar

text: ﬁeld

text: is

text: tem

text: techniques

text: can

text: be

text: sucessfully

text: applied.

text: Sec.

text: and

text: Sec.

text: taken

text: to

text: be

text: spatially

text: homogeneous

text: φ

text: φ(t).

text: The

text: mat-

text: will

text: then

text: be

text: dedicated

text: to

text: the

text: study

text: of

text: square

text: and

text: square

text: ter

text: energy-momentum

text: tensor

text: will

text: be

text: of

text: the

text: perfect

text: ﬂuid

text: root

text: kinetic

text: corrections

text: to

text: the

text: canonical

text: scalar

text: ﬁeld

text: La-

text: form

text: with

text: ρ(t)

text: and

text: p(t)

text: its

text: energy

text: density

text: and

text: pressure,

text: grangian.

text: For

text: these

text: simple

text: models

text: the

text: full

text: dynamical

text: respectively.

text: A

text: linear

text: EoS

text: p

text: wρ,

text: with

text: w

text: the

text: EoS

text: pa-

text: features

text: can

text: be

text: obtained

text: and

text: the

text: background

text: cosmolog-

text: rameter

text: ranging

text: from

text: (dust)

text: to

text: (radiation),

text: will

text: be

text: ical

text: evolution

text: can

text: be

text: determined

text: for

text: any

text: initial

text: condi-

text: assumed.

text: tion.

text: The

text: analysis

text: of

text: these

text: two

text: sections

text: will

text: show

text: that

text: a

text: With

text: these

text: assumptions,

text: from

text: the

text: gravitational

text: equa-

text: rich

text: phenomenology

text: can

text: be

text: obtained

text: with

text: these

text: extended

text: tions

text: we

text: obtain

text: the

text: scalar

text: ﬁelds.

text: Finally

text: results

text: and

text: conclusions

text: will

text: be

text: dis-

text: cussed

text: in

text: Sec.

text: II.

text: THE

text: CANONICAL

text: SCALAR

text: FIELD

text: κ2

text: ρ

text: φ˙2

text: V

text: In

text: this

text: section

text: we

text: review

text: the

text: cosmological

text: dynamics

text: of

text: and

text: the

text: acceleration

text: a

text: canonical

text: scalar

text: ﬁeld

text: following

text: the

text: analysis

text: ﬁrst

text: per-

text: formed

text: in

text: This

text: will

text: serve

text: as

text: an

text: introduction

text: to

text: the

text: dynamical

text: system

text: techniques

text: one

text: can

text: apply

text: in

text: or-

text: der

text: to

text: completely

text: determine

text: the

text: cosmological

text: evolution

text: κ2

text: p

text: φ˙2

text: V

text: of

text: speciﬁc

text: models.

text: Moreover

text: this

text: section

text: will

text: be

text: helpful

text: in

text: deﬁning

text: notation

text: and

text: conventions.

text: where

text: H

text: a˙/a

text: is

text: the

text: Hubble

text: parameter

text: and

text: an

text: over-

text: The

text: action

text: of

text: a

text: minimally

text: coupled

text: canonical

text: scalar

text: ﬁeld

text: is

text: given

text: by

text: dot

text: denotes

text: diﬀerentiation

text: with

text: respect

text: to

text: the

text: time

text: t.

text: On

text: the

text: other

text: hand

text: the

text: scalar

text: ﬁeld

text: equation

text: gives

text: R

text: L

text: L

text: S

text: g

text: ∂V

text: φ

text: m

text: φ¨

text: 3Hφ˙

text: ∂φ

text: where

text: g

text: is

text: the

text: determinant

text: of

text: the

text: metric,

text: R

text: is

text: the

text: Ricci

text: L

text: scalar,

text: κ2

text: is

text: the

text: matter

text: Lagrangian,

text: and

text: The

text: energy

text: density

text: and

text: pressure

text: of

text: the

text: canonical

text: scalar

text: m

text: ﬁeld

text: are

text: given

text: by

text: the

text: scalar

text: ﬁeld

text: Lagrangian

text: is

text: deﬁned

text: as

text: Lφ

text: −12∂φ2

text: V

text: (φ)

text: ρφ

text: V

text: p

text: φ˙2

text: V

text: with

text: ∂φ2

text: φ∂µφ

text: and

text: V

text: a

text: general

text: potential

text: for

text: φ.

text: φ

text: µ

text: The

text: variation

text: with

text: respect

text: to

text: g

text: produces

text: the

text: following

text: µν

text: and

text: its

text: EoS

text: parameter,

text: deﬁned

text: as

text: the

text: ratio

text: between

text: its

text: pressure

text: and

text: energy

text: density,

text: is

text: G

text: κ2

text: T

text: φ∂

text: φ

text: g

text: ∂φ2

text: g

text: V

text: µν

text: µν

text: µ

text: ν

text: µν

text: µν

text: p

text: 1φ˙2

text: V

text: w

text: φ

text: φ

text: ρφ

text: 1φ˙2

text: V

text: where

text: G

text: R

text: R

text: is

text: the

text: Einstein

text: tensor

text: and

text: µν

text: µν

text: µν

text: T

text: the

text: matter

text: energy

text: momentum

text: tensor.

text: The

text: variation

text: µν

text: For

text: V

text: φ˙2

text: this

text: approaches

text: a

text: cosmological

text: constant

text: with

text: respect

text: to

text: φ

text: gives

text: the

text: Klein-Gordon

text: equation

text: EoS

text: with

text: w

text: while

text: for

text: V

text: φ˙2

text: this

text: describe

text: a

text: stiﬀ

text: φ

text: ﬂuid

text: with

text: w

text: φ

text: ∂V

text: At

text: this

text: point,

text: following

text: we

text: introduce

text: new

text: dimen-

text: ∂φ

text: sionless

text: variables

text: as

text: with

text: µφ.

text: µ

text: κ2φ˙2

text: κ2ρ

text: In

text: what

text: follows

text: we

text: will

text: analyze

text: the

text: background

text: cos-

text: σ2

text: mological

text: evolution

text: of

text: this

text: model.

text: The

text: metric

text: tensor

text: will

text: be

text: assumed

text: to

text: be

text: of

text: the

text: Friedmann-Robertson-Walker

text: (FRW)

text: type

text: with

text: vanishing

text: spatial

text: curvature

text: These

text: variables

text: are

text: largely

text: employed

text: in

text: scalar

text: ﬁeld

text: cos-

text: mology

text: since

text: not

text: only

text: allow

text: to

text: rewrite

text: Eqs.

text: as

text: an

text: autonomous

text: system

text: of

text: equations,

text: but

text: can

text: also

text: be

text: gener-

text: g

text: diag(

text: µν

text: alized

text: in

text: diﬀerent

text: contexts

text: such

text: as

text: nonminimally

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