In Vapnik–Chervonenkis theory, the Vapnik–Chervonenkis (VC) dimension is a measure of the size (capacity, complexity, expressive power, richness, or flexibility) of a class of sets. The notion can be extended to classes of binary functions. It is defined as the cardinality of the largest set of points that the function class can shatter—that is, for which all possible binary labelings can be realized by some function in the class. It was originally defined by Vladimir Vapnik and Alexey Chervonenkis. Informally, the capacity of a classification model is related to how complicated it can be. For example, consider the thresholding of a high-degree polynomial: if the polynomial evaluates above zero, that point is classified as positive, otherwise as negative. A high-degree polynomial can be wiggly, so that it can fit a given set of training points well. Such a polynomial has a high capacity. A much simpler alternative is to threshold a linear function. This function may not fit the training set well, because it has a low capacity. This notion of capacity is made rigorous below. == Definitions == === VC dimension of a set-family === Let C = { C } C ∈ C {\displaystyle {\mathcal {C}}=\{C\}_{C\in {\mathcal {C}}}} be a family of sets (also called set family, collection of sets or set of sets) and X {\displaystyle X} a set. Their intersection is defined as the following set family: C ∩ X := { C ∩ X ∣ C ∈ C } . {\displaystyle {\mathcal {C}}\cap X:=\{C\cap X\mid C\in {\mathcal {C}}\}.} Here typically X {\displaystyle X} and each C ∈ C {\displaystyle C\in {\mathcal {C}}} are subsets of a big "universe" of possibilities U {\displaystyle U} where intersection takes place. We say that a set X {\displaystyle X} is shattered by C {\displaystyle {\mathcal {C}}} if P ( X ) = C ∩ X {\displaystyle {\mathcal {P}}(X)={\mathcal {C}}\cap X} i.e. the set of intersections contains (hence is equal to) all the subsets of X {\displaystyle X} . For finite sets X {\displaystyle X} this is equivalent to | C ∩ X | = 2 | X | . {\displaystyle |{\mathcal {C}}\cap X|=2^{|X|}.} The VC dimension D {\displaystyle D} of C {\displaystyle {\mathcal {C}}} is the cardinality of the largest set that is shattered by C {\displaystyle {\mathcal {C}}} . If arbitrarily large sets can be shattered, the VC dimension of C {\displaystyle {\mathcal {C}}} is ∞ {\displaystyle \infty } . === VC dimension of a classification model === A binary classification model f {\displaystyle f} with some parameter vector θ {\displaystyle \theta } is said to shatter a set of generally positioned data points ( x 1 , x 2 , … , x n ) {\displaystyle (x_{1},x_{2},\ldots ,x_{n})} if, for every assignment of labels to those points, there exists a θ {\displaystyle \theta } such that the model f {\displaystyle f} makes no errors when evaluating that set of data points. The VC dimension of a model f {\displaystyle f} is the maximum number of points that can be arranged so that f {\displaystyle f} shatters them. More formally, it is the maximum cardinal D {\displaystyle D} such that there exists a generally positioned data point set of cardinality D {\displaystyle D} that can be shattered by f {\displaystyle f} . == Examples == f {\displaystyle f} is a constant classifier (with no parameters); Its VC dimension is 0 since it cannot shatter even a single point. In general, the VC dimension of a finite classification model, which can return at most 2 d {\displaystyle 2^{d}} different classifiers, is at most d {\displaystyle d} (this is an upper bound on the VC dimension; the Sauer–Shelah lemma gives a lower bound on the dimension). f {\displaystyle f} is a single-parametric threshold classifier on real numbers; i.e., for a certain threshold θ {\displaystyle \theta } , the classifier f θ {\displaystyle f_{\theta }} returns 1 if the input number is larger than θ {\displaystyle \theta } and 0 otherwise. The VC dimension of f {\displaystyle f} is 1 because: (a) It can shatter a single point. For every point x {\displaystyle x} , a classifier f θ {\displaystyle f_{\theta }} labels it as 0 if θ > x {\displaystyle \theta >x} and labels it as 1 if θ < x {\displaystyle \theta
Speech segmentation
Speech segmentation is the process of identifying the boundaries between words, syllables, or phonemes in spoken natural languages. The term applies both to the mental processes used by humans, and to artificial processes of natural language processing. In the field of automatic pronunciation assessment, the process of segmenting an utterance against expected word(s) is called forced alignment. Speech segmentation is a subfield of general speech perception and an important subproblem of the technologically focused field of speech recognition, and cannot be adequately solved in isolation. As in most natural language processing problems, one must take into account context, grammar, and semantics, and even so the result is often a probabilistic division (statistically based on likelihood) rather than a categorical one. Though it seems that coarticulation—a phenomenon which may happen between adjacent words just as easily as within a single word—presents the main challenge in speech segmentation across languages, some other problems and strategies employed in solving those problems can be seen in the following sections. This problem overlaps to some extent with the problem of text segmentation that occurs in some languages which are traditionally written without inter-word spaces, like Chinese and Japanese, compared to writing systems which indicate speech segmentation between words by a word divider, such as the space. However, even for those languages, text segmentation is often much easier than speech segmentation, because the written language usually has little interference between adjacent words, and often contains additional clues not present in speech (such as the use of Chinese characters for word stems in Japanese). == Lexical recognition == In natural languages, the meaning of a complex spoken sentence can be understood by decomposing it into smaller lexical segments (roughly, the words of the language), associating a meaning to each segment, and combining those meanings according to the grammar rules of the language. Though lexical recognition is not thought to be used by infants in their first year, due to their highly limited vocabularies, it is one of the major processes involved in speech segmentation for adults. Three main models of lexical recognition exist in current research: first, whole-word access, which argues that words have a whole-word representation in the lexicon; second, decomposition, which argues that morphologically complex words are broken down into their morphemes (roots, stems, inflections, etc.) and then interpreted and; third, the view that whole-word and decomposition models are both used, but that the whole-word model provides some computational advantages and is therefore dominant in lexical recognition. To give an example, in a whole-word model, the word "cats" might be stored and searched for by letter, first "c", then "ca", "cat", and finally "cats". The same word, in a decompositional model, would likely be stored under the root word "cat" and could be searched for after removing the "s" suffix. "Falling", similarly, would be stored as "fall" and suffixed with the "ing" inflection. Though proponents of the decompositional model recognize that a morpheme-by-morpheme analysis may require significantly more computation, they argue that the unpacking of morphological information is necessary for other processes (such as syntactic structure) which may occur parallel to lexical searches. As a whole, research into systems of human lexical recognition is limited due to little experimental evidence that fully discriminates between the three main models. In any case, lexical recognition likely contributes significantly to speech segmentation through the contextual clues it provides, given that it is a heavily probabilistic system—based on the statistical likelihood of certain words or constituents occurring together. For example, one can imagine a situation where a person might say "I bought my dog at a ____ shop" and the missing word's vowel is pronounced as in "net", "sweat", or "pet". While the probability of "netshop" is extremely low, since "netshop" isn't currently a compound or phrase in English, and "sweatshop" also seems contextually improbable, "pet shop" is a good fit because it is a common phrase and is also related to the word "dog". Moreover, an utterance can have different meanings depending on how it is split into words. A popular example, often quoted in the field, is the phrase "How to wreck a nice beach", which sounds very similar to "How to recognize speech". As this example shows, proper lexical segmentation depends on context and semantics which draws on the whole of human knowledge and experience, and would thus require advanced pattern recognition and artificial intelligence technologies to be implemented on a computer. Lexical recognition is of particular value in the field of computer speech recognition, since the ability to build and search a network of semantically connected ideas would greatly increase the effectiveness of speech-recognition software. Statistical models can be used to segment and align recorded speech to words or phones. Applications include automatic lip-synch timing for cartoon animation, follow-the-bouncing-ball video sub-titling, and linguistic research. Automatic segmentation and alignment software is commercially available. == Phonotactic cues == For most spoken languages, the boundaries between lexical units are difficult to identify; phonotactics are one answer to this issue. One might expect that the inter-word spaces used by many written languages like English or Spanish would correspond to pauses in their spoken version, but that is true only in very slow speech, when the speaker deliberately inserts those pauses. In normal speech, one typically finds many consecutive words being said with no pauses between them, and often the final sounds of one word blend smoothly or fuse with the initial sounds of the next word. The notion that speech is produced like writing, as a sequence of distinct vowels and consonants, may be a relic of alphabetic heritage for some language communities. In fact, the way vowels are produced depends on the surrounding consonants just as consonants are affected by surrounding vowels; this is called coarticulation. For example, in the word "kit", the [k] is farther forward than when we say 'caught'. But also, the vowel in "kick" is phonetically different from the vowel in "kit", though we normally do not hear this. In addition, there are language-specific changes which occur in casual speech which makes it quite different from spelling. For example, in English, the phrase "hit you" could often be more appropriately spelled "hitcha". From a decompositional perspective, in many cases, phonotactics play a part in letting speakers know where to draw word boundaries. In English, the word "strawberry" is perceived by speakers as consisting (phonetically) of two parts: "straw" and "berry". Other interpretations such as "stra" and "wberry" are inhibited by English phonotactics, which does not allow the cluster "wb" word-initially. Other such examples are "day/dream" and "mile/stone" which are unlikely to be interpreted as "da/ydream" or "mil/estone" due to the phonotactic probability or improbability of certain clusters. The sentence "Five women left", which could be phonetically transcribed as [faɪvwɪmɘnlɛft], is marked since neither /vw/ in /faɪvwɪmɘn/ nor /nl/ in /wɪmɘnlɛft/ are allowed as syllable onsets or codas in English phonotactics. These phonotactic cues often allow speakers to easily distinguish the boundaries in words. Vowel harmony in languages like Finnish can also serve to provide phonotactic cues. While the system does not allow front vowels and back vowels to exist together within one morpheme, compounds allow two morphemes to maintain their own vowel harmony while coexisting in a word. Therefore, in compounds such as "selkä/ongelma" ('back problem') where vowel harmony is distinct between two constituents in a compound, the boundary will be wherever the switch in harmony takes place—between the "ä" and the "ö" in this case. Still, there are instances where phonotactics may not aid in segmentation. Words with unclear clusters or uncontrasted vowel harmony as in "opinto/uudistus" ('student reform') do not offer phonotactic clues as to how they are segmented. From the perspective of the whole-word model, however, these words are thought be stored as full words, so the constituent parts would not necessarily be relevant to lexical recognition. == In infants and non-natives == Infants are one major focus of research in speech segmentation. Since infants have not yet acquired a lexicon capable of providing extensive contextual clues or probability-based word searches within their first year, as mentioned above, they must often rely primarily upon phonotactic and rhythmic cues (with prosody being the dominant cue), all
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Keyword (linguistics)
In corpus linguistics a key word is a word which occurs in a text more often than we would expect to occur by chance alone. Key words are calculated by carrying out a statistical test (e.g., loglinear or chi-squared) which compares the word frequencies in a text against their expected frequencies derived in a much larger corpus, which acts as a reference for general language use. Keyness is then the quality a word or phrase has of being "key" in its context. Combinations of nouns with parts of speech that human readers would not likely notice, such as prepositions, time adverbs, and pronouns can be a relevant part of keyness. Even separate pronouns can constitute keywords. Compare this with collocation, the quality linking two words or phrases usually assumed to be within a given span of each other. Keyness is a textual feature, not a language feature (so a word has keyness in a certain textual context but may well not have keyness in other contexts, whereas a node and collocate are often found together in texts of the same genre so collocation is to a considerable extent a language phenomenon). The set of keywords found in a given text share keyness, they are co-key. Words typically found in the same texts as a key word are called associates. == Sociological aspects == In politics, sociology and critical discourse analysis, the key reference for keywords was Raymond Williams (1976), but Williams was resolutely Marxist, and Critical Discourse Analysis has tended to perpetuate this political meaning of the term: keywords are part of ideologies and studying them is part of social criticism. Cultural studies has tended to develop along similar lines. This stands in stark contrast to present day linguistics which is wary of political analysis, and has tended to aspire to non-political objectivity. The development of technology, new techniques and methodology relating to massive corpora have all consolidated this trend. === Translatability === There are, however, numerous political dimensions that come into play when keywords are studied in relation to cultures, societies and their histories. The Lublin Ethnolinguistics School studies Polish and European keywords in this fashion. Anna Wierzbicka (1997), probably the best known cultural linguist writing in English today, studies languages as parts of cultures evolving in society and history. And it becomes impossible to ignore politics when keywords migrate from one culture to another. Underhill and Gianninoto demonstrate the way political terms like, "citizen" and "individual" are integrated into the Chinese worldview over the course of the 19th and 20th century. They argue that this is part of a complex readjustment of conceptual clusters related to "the people". Keywords like "citizen" generate various translations in Chinese, and are part of an ongoing adaptation to global concepts of individual rights and responsibilities. Understanding keywords in this light becomes crucial for understanding how the politics of China evolves as Communism emerges and as the free market and citizens' rights develop. Underhill and Gianninoto argue that this is part of the complex ways ideological worldviews interact with the language as an ongoing means of perceiving and understanding the world. Barbara Cassin studies keywords in a more traditional manner, striving to define the words specific to individual cultures, in order to demonstrate that many of our keywords are partially "untranslatable" into their "equivalents. The Greeks may need four words to cover all the meanings English-speakers have in mind when speaking of "love". Similarly, the French find that "liberté" suffices, while English-speakers attribute different associations to "liberty" and "freedom": "freedom of speech" or "freedom of movement", but "the Statue of Liberty". == Software-assisted identification == Keywords are identified by software that compares a word-list of the text with a word-list based on a larger reference corpus. Software such as e.g. WordSmith, lists keywords and phrases and allows plotting their occurrence as they appear in texts.
Uniform convergence in probability
Uniform convergence in probability is a form of convergence in probability in statistical asymptotic theory and probability theory. It means that, under certain conditions, the empirical frequencies of all events in a certain event-family uniformly converge to their theoretical probabilities. Uniform convergence in probability has applications to statistics as well as machine learning as part of statistical learning theory. Specifically, the Glivenko-Cantelli theorem and the homonymous classes of functions are fundamentally related to uniform convergence. The law of large numbers says that, for each single event A {\displaystyle A} , its empirical frequency in a sequence of independent trials converges (with high probability) to its theoretical probability. In many application however, the need arises to judge simultaneously the probabilities of events of an entire class S {\displaystyle S} from one and the same sample. Moreover, it, is required that the relative frequency of the events converge to the probability uniformly over the entire class of events S {\displaystyle S} . The Uniform Convergence Theorem gives a sufficient condition for this convergence to hold. Roughly, if the event-family is sufficiently simple (its VC dimension is sufficiently small) then uniform convergence holds. == Definitions == For a class of predicates H {\displaystyle H} defined on a set X {\displaystyle X} and a set of samples x = ( x 1 , x 2 , … , x m ) {\displaystyle x=(x_{1},x_{2},\dots ,x_{m})} , where x i ∈ X {\displaystyle x_{i}\in X} , the empirical frequency of h ∈ H {\displaystyle h\in H} on x {\displaystyle x} is Q ^ x ( h ) = 1 m | { i : 1 ≤ i ≤ m , h ( x i ) = 1 } | . {\displaystyle {\widehat {Q}}_{x}(h)={\frac {1}{m}}|\{i:1\leq i\leq m,h(x_{i})=1\}|.} The theoretical probability of h ∈ H {\displaystyle h\in H} is defined as Q P ( h ) = P { y ∈ X : h ( y ) = 1 } . {\displaystyle Q_{P}(h)=P\{y\in X:h(y)=1\}.} The Uniform Convergence Theorem states, roughly, that if H {\displaystyle H} is "simple" and we draw samples independently (with replacement) from X {\displaystyle X} according to any distribution P {\displaystyle P} , then with high probability, the empirical frequency will be close to its expected value, which is the theoretical probability. Here "simple" means that the Vapnik–Chervonenkis dimension of the class H {\displaystyle H} is small relative to the size of the sample. In other words, a sufficiently simple collection of functions behaves roughly the same on a small random sample as it does on the distribution as a whole. The Uniform Convergence Theorem was first proved by Vapnik and Chervonenkis using the concept of growth function. == Uniform Convergence Theorem == The statement of the Uniform Convergence Theorem is as follows: If H {\displaystyle H} is a set of { 0 , 1 } {\displaystyle \{0,1\}} -valued functions defined on a set X {\displaystyle X} and P {\displaystyle P} is a probability distribution on X {\displaystyle X} then for ε > 0 {\displaystyle \varepsilon >0} and m {\displaystyle m} a positive integer, we have: P m { | Q P ( h ) − Q x ^ ( h ) | ≥ ε for some h ∈ H } ≤ 4 Π H ( 2 m ) e − ε 2 m / 8 . {\displaystyle P^{m}\{|Q_{P}(h)-{\widehat {Q_{x}}}(h)|\geq \varepsilon {\text{ for some }}h\in H\}\leq 4\Pi _{H}(2m)e^{-\varepsilon ^{2}m/8}.} In the above, for any x ∈ X m , {\displaystyle x\in X^{m},} Q P ( h ) = P { ( y ∈ X : h ( y ) = 1 } , {\displaystyle Q_{P}(h)=P\{(y\in X:h(y)=1\},} Q ^ x ( h ) = 1 m | { i : 1 ≤ i ≤ m , h ( x i ) = 1 } | {\displaystyle {\widehat {Q}}_{x}(h)={\frac {1}{m}}|\{i:1\leq i\leq m,h(x_{i})=1\}|} and | x | = m . {\displaystyle |x|=m.} P m {\displaystyle P^{m}} indicates that the probability is taken over x {\displaystyle x} consisting of m {\displaystyle m} i.i.d. draws from the distribution P . {\displaystyle P.} Finally, the growth function Π H {\displaystyle \Pi _{H}} is defined in the following way, for any { 0 , 1 } {\displaystyle \{0,1\}} -valued functions H {\displaystyle H} over X {\displaystyle X} and for any natural number m {\displaystyle m} : Π H ( m ) = max | { h ∩ D : D ⊆ X , | D | = m , h ∈ H } | . {\displaystyle \Pi _{H}(m)=\max |\{h\cap D:D\subseteq X,|D|=m,h\in H\}|.} From the point of view of Learning Theory one can consider H {\displaystyle H} to be the Concept/Hypothesis class defined over the instance set X {\displaystyle X} . Crucially, the Sauer–Shelah lemma implies that Π H ( m ) ≤ m d {\displaystyle \Pi _{H}(m)\leq m^{d}} , where d {\displaystyle d} is the VC dimension of H {\displaystyle H} . == Proof of the Uniform Convergence Theorem == and are the sources of the proof below. Before we get into the details of the proof of the Uniform Convergence Theorem we will present a high level overview of the proof. Symmetrization: We transform the problem of analyzing | Q P ( h ) − Q ^ x ( h ) | ≥ ε {\displaystyle |Q_{P}(h)-{\widehat {Q}}_{x}(h)|\geq \varepsilon } into the problem of analyzing | Q ^ r ( h ) − Q ^ s ( h ) | ≥ ε / 2 {\displaystyle |{\widehat {Q}}_{r}(h)-{\widehat {Q}}_{s}(h)|\geq \varepsilon /2} , where r {\displaystyle r} and s {\displaystyle s} are i.i.d samples of size m {\displaystyle m} drawn according to the distribution P {\displaystyle P} . One can view r {\displaystyle r} as the original randomly drawn sample of length m {\displaystyle m} , while s {\displaystyle s} may be thought as the testing sample which is used to estimate Q P ( h ) {\displaystyle Q_{P}(h)} . Permutation: Since r {\displaystyle r} and s {\displaystyle s} are picked identically and independently, so swapping elements between them will not change the probability distribution on r {\displaystyle r} and s {\displaystyle s} . So, we will try to bound the probability of | Q ^ r ( h ) − Q ^ s ( h ) | ≥ ε / 2 {\displaystyle |{\widehat {Q}}_{r}(h)-{\widehat {Q}}_{s}(h)|\geq \varepsilon /2} for some h ∈ H {\displaystyle h\in H} by considering the effect of a specific collection of permutations of the joint sample x = r | | s {\displaystyle x=r||s} . Specifically, we consider permutations σ ( x ) {\displaystyle \sigma (x)} which swap x i {\displaystyle x_{i}} and x m + i {\displaystyle x_{m+i}} in some subset of 1 , 2 , . . . , m {\displaystyle {1,2,...,m}} . The symbol r | | s {\displaystyle r||s} means the concatenation of r {\displaystyle r} and s {\displaystyle s} . Reduction to a finite class: We can now restrict the function class H {\displaystyle H} to a fixed joint sample and hence, if H {\displaystyle H} has finite VC Dimension, it reduces to the problem to one involving a finite function class. We present the technical details of the proof. It should be stressed that this proof glosses over details like the measurability of the events V {\displaystyle V} and R {\displaystyle R} ; measurability is granted in the case of H {\displaystyle H} being finite or countable, but this is not normally the case in standard applications of the theorem (e.g. for statistical learning theory or to prove the Glivenko-Cantelli theorem). To get measurability, one needs to use a notion of separability of the underlying space, possibly related to H {\displaystyle H} . === Symmetrization === Lemma: Let V = { x ∈ X m : | Q P ( h ) − Q ^ x ( h ) | ≥ ε for some h ∈ H } {\displaystyle V=\{x\in X^{m}:|Q_{P}(h)-{\widehat {Q}}_{x}(h)|\geq \varepsilon {\text{ for some }}h\in H\}} and R = { ( r , s ) ∈ X m × X m : | Q r ^ ( h ) − Q ^ s ( h ) | ≥ ε / 2 for some h ∈ H } . {\displaystyle R=\{(r,s)\in X^{m}\times X^{m}:|{\widehat {Q_{r}}}(h)-{\widehat {Q}}_{s}(h)|\geq \varepsilon /2{\text{ for some }}h\in H\}.} Then for m ≥ 2 ε 2 {\displaystyle m\geq {\frac {2}{\varepsilon ^{2}}}} , P m ( V ) ≤ 2 P 2 m ( R ) {\displaystyle P^{m}(V)\leq 2P^{2m}(R)} . Proof: By the triangle inequality, if | Q P ( h ) − Q ^ r ( h ) | ≥ ε {\displaystyle |Q_{P}(h)-{\widehat {Q}}_{r}(h)|\geq \varepsilon } and | Q P ( h ) − Q ^ s ( h ) | ≤ ε / 2 {\displaystyle |Q_{P}(h)-{\widehat {Q}}_{s}(h)|\leq \varepsilon /2} then | Q ^ r ( h ) − Q ^ s ( h ) | ≥ ε / 2 {\displaystyle |{\widehat {Q}}_{r}(h)-{\widehat {Q}}_{s}(h)|\geq \varepsilon /2} . Therefore, P 2 m ( R ) ≥ P 2 m { ∃ h ∈ H , | Q P ( h ) − Q ^ r ( h ) | ≥ ε and | Q P ( h ) − Q ^ s ( h ) | ≤ ε / 2 } = ∫ V P m { s : ∃ h ∈ H , | Q P ( h ) − Q ^ r ( h ) | ≥ ε and | Q P ( h ) − Q ^ s ( h ) | ≤ ε / 2 } d P m ( r ) = A {\displaystyle {\begin{aligned}&P^{2m}(R)\\[5pt]\geq {}&P^{2m}\{\exists h\in H,|Q_{P}(h)-{\widehat {Q}}_{r}(h)|\geq \varepsilon {\text{ and }}|Q_{P}(h)-{\widehat {Q}}_{s}(h)|\leq \varepsilon /2\}\\[5pt]={}&\int _{V}P^{m}\{s:\exists h\in H,|Q_{P}(h)-{\widehat {Q}}_{r}(h)|\geq \varepsilon {\text{ and }}|Q_{P}(h)-{\widehat {Q}}_{s}(h)|\leq \varepsilon /2\}\,dP^{m}(r)\\[5pt]={}&A\end{aligned}}} since r {\displaystyle r} and s {\displaystyle s} are independent. Now for r ∈ V {\displaystyle r\in V} fix an h ∈ H {\displaystyle h\in H} such that | Q P ( h ) − Q ^ r ( h ) | ≥ ε {\displaystyle |Q_{P}(h)-{\widehat {Q}}_{r}(h)|\geq \varepsilon } . For this h {\displaystyle h} , we shall
Interacting particle system
In probability theory, an interacting particle system (IPS) is a stochastic process ( X ( t ) ) t ∈ R + {\displaystyle (X(t))_{t\in \mathbb {R} ^{+}}} on some configuration space Ω = S G {\displaystyle \Omega =S^{G}} given by a site space, a countably-infinite-order graph G {\displaystyle G} and a local state space, a compact metric space S {\displaystyle S} . More precisely IPS are continuous-time Markov jump processes describing the collective behavior of stochastically interacting components. IPS are the continuous-time analogue of stochastic cellular automata. Among the main examples are the voter model, the contact process, the asymmetric simple exclusion process (ASEP), the Glauber dynamics and in particular the stochastic Ising model. IPS are usually defined via their Markov generator giving rise to a unique Markov process using Markov semigroups and the Hille-Yosida theorem. The generator again is given via so-called transition rates c Λ ( η , ξ ) > 0 {\displaystyle c_{\Lambda }(\eta ,\xi )>0} where Λ ⊂ G {\displaystyle \Lambda \subset G} is a finite set of sites and η , ξ ∈ Ω {\displaystyle \eta ,\xi \in \Omega } with η i = ξ i {\displaystyle \eta _{i}=\xi _{i}} for all i ∉ Λ {\displaystyle i\notin \Lambda } . The rates describe exponential waiting times of the process to jump from configuration η {\displaystyle \eta } into configuration ξ {\displaystyle \xi } . More generally the transition rates are given in form of a finite measure c Λ ( η , d ξ ) {\displaystyle c_{\Lambda }(\eta ,d\xi )} on S Λ {\displaystyle S^{\Lambda }} . The generator L {\displaystyle L} of an IPS has the following form. First, the domain of L {\displaystyle L} is a subset of the space of "observables", that is, the set of real valued continuous functions on the configuration space Ω {\displaystyle \Omega } . Then for any observable f {\displaystyle f} in the domain of L {\displaystyle L} , one has L f ( η ) = ∑ Λ ∫ ξ : ξ Λ c = η Λ c c Λ ( η , d ξ ) [ f ( ξ ) − f ( η ) ] {\displaystyle Lf(\eta )=\sum _{\Lambda }\int _{\xi :\xi _{\Lambda ^{c}}=\eta _{\Lambda ^{c}}}c_{\Lambda }(\eta ,d\xi )[f(\xi )-f(\eta )]} . For example, for the stochastic Ising model we have G = Z d {\displaystyle G=\mathbb {Z} ^{d}} , S = { − 1 , + 1 } {\displaystyle S=\{-1,+1\}} , c Λ = 0 {\displaystyle c_{\Lambda }=0} if Λ ≠ { i } {\displaystyle \Lambda \neq \{i\}} for some i ∈ G {\displaystyle i\in G} and c i ( η , η i ) = exp [ − β ∑ j : | j − i | = 1 η i η j ] {\displaystyle c_{i}(\eta ,\eta ^{i})=\exp[-\beta \sum _{j:|j-i|=1}\eta _{i}\eta _{j}]} where η i {\displaystyle \eta ^{i}} is the configuration equal to η {\displaystyle \eta } except it is flipped at site i {\displaystyle i} . β {\displaystyle \beta } is a new parameter modeling the inverse temperature. == The Voter model == The voter model (usually in continuous time, but there are discrete versions as well) is a process similar to the contact process. In this process η ( x ) {\displaystyle \eta (x)} is taken to represent a voter's attitude on a particular topic. Voters reconsider their opinions at times distributed according to independent exponential random variables (this gives a Poisson process locally – note that there are in general infinitely many voters so no global Poisson process can be used). At times of reconsideration, a voter chooses one neighbor uniformly from amongst all neighbors and takes that neighbor's opinion. One can generalize the process by allowing the picking of neighbors to be something other than uniform. === Discrete time process === In the discrete time voter model in one dimension, ξ t ( x ) : Z → { 0 , 1 } {\displaystyle \xi _{t}(x):\mathbb {Z} \to \{0,1\}} represents the state of particle x {\displaystyle x} at time t {\displaystyle t} . Informally each individual is arranged on a line and can "see" other individuals that are within a radius, r {\displaystyle r} . If more than a certain proportion, θ {\displaystyle \theta } of these people disagree then the individual changes her attitude, otherwise she keeps it the same. Durrett and Steif (1993) and Steif (1994) show that for large radii there is a critical value θ c {\displaystyle \theta _{c}} such that if θ > θ c {\displaystyle \theta >\theta _{c}} most individuals never change, and for θ ∈ ( 1 / 2 , θ c ) {\displaystyle \theta \in (1/2,\theta _{c})} in the limit most sites agree. (Both of these results assume the probability of ξ 0 ( x ) = 1 {\displaystyle \xi _{0}(x)=1} is one half.) This process has a natural generalization to more dimensions, some results for this are discussed in Durrett and Steif (1993). === Continuous time process === The continuous time process is similar in that it imagines each individual has a belief at a time and changes it based on the attitudes of its neighbors. The process is described informally by Liggett (1985, 226), "Periodically (i.e., at independent exponential times), an individual reassesses his view in a rather simple way: he chooses a 'friend' at random with certain probabilities and adopts his position." A model was constructed with this interpretation by Holley and Liggett (1975). This process is equivalent to a process first suggested by Clifford and Sudbury (1973) where animals are in conflict over territory and are equally matched. A site is selected to be invaded by a neighbor at a given time.