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74-511: If RankBrain sees a word or phrase it isn’t familiar with, the program can make a guess as to what words or phrases might have a similar meaning and filter the result accordingly, making it more effective at handling never-before-seen search queries or keywords. Search queries are sorted into word vectors , also known as “distributed representations,” which are close to each other in terms of linguistic similarity. RankBrain attempts to map this query into words (entities) or clusters of words that have

148-526: A web search engine to satisfy their information needs . Web search queries are distinctive in that they are often plain text and boolean search directives are rarely used. They vary greatly from standard query languages , which are governed by strict syntax rules as command languages with keyword or positional parameters . There are three broad categories that cover most web search queries: informational, navigational, and transactional. These are also called "do, know, go." Although this model of searching

222-406: A common alternative to the asymptotic power-law distribution because it naturally captures finite-size effects. The Tweedie distributions are a family of statistical models characterized by closure under additive and reproductive convolution as well as under scale transformation. Consequently, these models all express a power-law relationship between the variance and the mean. These models have

296-498: A constant factor c {\displaystyle c} causes only a proportionate scaling of the function itself. That is, where ∝ {\displaystyle \propto } denotes direct proportionality . That is, scaling by a constant c {\displaystyle c} simply multiplies the original power-law relation by the constant c − k {\displaystyle c^{-k}} . Thus, it follows that all power laws with

370-505: A full characterization of the tail behavior of many well-known probability distributions, including power-law distributions, distributions with other types of heavy tails, and even non-heavy-tailed distributions. Bundle plots do not have the disadvantages of Pareto Q–Q plots, mean residual life plots and log–log plots mentioned above (they are robust to outliers, allow visually identifying power laws with small values of α {\displaystyle \alpha } , and do not demand

444-423: A fundamental role as foci of mathematical convergence similar to the role that the normal distribution has as a focus in the central limit theorem . This convergence effect explains why the variance-to-mean power law manifests so widely in natural processes, as with Taylor's law in ecology and with fluctuation scaling in physics. It can also be shown that this variance-to-mean power law, when demonstrated by

518-442: A hundred power-law distributions have been identified in physics (e.g. sandpile avalanches), biology (e.g. species extinction and body mass), and the social sciences (e.g. city sizes and income). Among them are: A broken power law is a piecewise function , consisting of two or more power laws, combined with a threshold. For example, with two power laws: The pieces of a broken power law can be smoothly spliced together to construct

592-490: A major importance to fully understanding the meaning or intent behind a person’s search query. It’s also able to parse patterns between searches that are seemingly unconnected, to understand how those searches are similar to each other. Once RankBrain's results are verified by Google 's team, the system is updated and goes live again. Google has stated that it uses tensor processing unit (TPU) ASICs for processing RankBrain requests. RankBrain has allowed Google to speed up

666-419: A more convergent estimate than the maximum likelihood method. It has been applied to study probability distributions of fracture apertures. In some contexts the probability distribution is described, not by the cumulative distribution function , by the cumulative frequency of a property X , defined as the number of elements per meter (or area unit, second etc.) for which X  >  x applies, where x

740-404: A particular scaling exponent are equivalent up to constant factors, since each is simply a scaled version of the others. This behavior is what produces the linear relationship when logarithms are taken of both f ( x ) {\displaystyle f(x)} and x {\displaystyle x} , and the straight-line on the log–log plot is often called the signature of

814-471: A power law for a limited range of values, because a pure power law would allow for arbitrarily large or small values. Acoustic attenuation follows frequency power-laws within wide frequency bands for many complex media. Allometric scaling laws for relationships between biological variables are among the best known power-law functions in nature. The power-law model does not obey the treasured paradigm of statistical completeness. Especially probability bounds,

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888-513: A power law over a wide range of magnitudes: these include the sizes of craters on the moon and of solar flares , cloud sizes, the foraging pattern of various species, the sizes of activity patterns of neuronal populations, the frequencies of words in most languages, frequencies of family names , the species richness in clades of organisms, the sizes of power outages , volcanic eruptions, human judgments of stimulus intensity and many other quantities. Empirical distributions can only fit

962-402: A power law relationship, as many non power-law distributions will appear as straight lines on a log–log plot. This method consists of plotting the logarithm of an estimator of the probability that a particular number of the distribution occurs versus the logarithm of that particular number. Usually, this estimator is the proportion of times that the number occurs in the data set. If the points in

1036-505: A power law. With real data, such straightness is a necessary, but not sufficient, condition for the data following a power-law relation. In fact, there are many ways to generate finite amounts of data that mimic this signature behavior, but, in their asymptotic limit, are not true power laws. Thus, accurately fitting and validating power-law models is an active area of research in statistics; see below. A power-law x − k {\displaystyle x^{-k}} has

1110-426: A power of another. The change is independent of the initial size of those quantities. For instance, the area of a square has a power law relationship with the length of its side, since if the length is doubled, the area is multiplied by 2 , while if the length is tripled, the area is multiplied by 3 , and so on. The distributions of a wide variety of physical, biological, and human-made phenomena approximately follow

1184-450: A power-law distribution of the form to the data x ≥ x min {\displaystyle x\geq x_{\min }} , where the coefficient α − 1 x min {\displaystyle {\frac {\alpha -1}{x_{\min }}}} is included to ensure that the distribution is normalized . Given a choice for x min {\displaystyle x_{\min }} ,

1258-469: A power-law known as the Pareto distribution (for example, the net worth of Americans is distributed according to a power law with an exponent of 2). On the one hand, this makes it incorrect to apply traditional statistics that are based on variance and standard deviation (such as regression analysis ). On the other hand, this also allows for cost-efficient interventions. For example, given that car exhaust

1332-438: A significant bias in α ^ {\displaystyle {\hat {\alpha }}} , while choosing it too large increases the uncertainty in α ^ {\displaystyle {\hat {\alpha }}} , and reduces the statistical power of our model. In general, the best choice of x min {\displaystyle x_{\min }} depends strongly on

1406-486: A small finite sample-size bias of order O ( n − 1 ) {\displaystyle O(n^{-1})} , which is small when n  > 100. Further, the standard error of the estimate is σ = α ^ − 1 n + O ( n − 1 ) {\displaystyle \sigma ={\frac {{\hat {\alpha }}-1}{\sqrt {n}}}+O(n^{-1})} . This estimator

1480-450: A small set of universality classes. Similar observations have been made, though not as comprehensively, for various self-organized critical systems, where the critical point of the system is an attractor . Formally, this sharing of dynamics is referred to as universality , and systems with precisely the same critical exponents are said to belong to the same universality class . Scientific interest in power-law relations stems partly from

1554-977: A smoothly broken power law. There are different possible ways to splice together power laws. One example is the following: ln ⁡ ( y y 0 + a ) = c 0 ln ⁡ ( x x 0 ) + ∑ i = 1 n c i − c i − 1 f i ln ⁡ ( 1 + ( x x i ) f i ) {\displaystyle \ln \left({\frac {y}{y_{0}}}+a\right)=c_{0}\ln \left({\frac {x}{x_{0}}}\right)+\sum _{i=1}^{n}{\frac {c_{i}-c_{i-1}}{f_{i}}}\ln \left(1+\left({\frac {x}{x_{i}}}\right)^{f_{i}}\right)} where 0 < x 0 < x 1 < ⋯ < x n {\displaystyle 0<x_{0}<x_{1}<\cdots <x_{n}} . When

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1628-434: A technique traditionally used by librarians can be applied. A user who is looking for documents that cover several topics or facets may want to describe each of them by a disjunction of characteristic words, such as vehicles OR cars OR automobiles . A faceted query is a conjunction of such facets; e.g. a query such as (electronic OR computerized OR DRE) AND (voting OR elections OR election OR balloting OR electoral)

1702-535: A web query is navigational, informational or transactional. A 2011 study found that the average length of queries had grown steadily over time and the average length of non-English language queries had increased more than English ones. Google implemented the hummingbird update in August 2013 to handle longer search queries since more searches are conversational (e.g. "where is the nearest coffee shop?"). With search engines that support Boolean operators and parentheses,

1776-401: A well-defined mean over x ∈ [ 1 , ∞ ) {\displaystyle x\in [1,\infty )} only if k > 2 {\displaystyle k>2} , and it has a finite variance only if k > 3 {\displaystyle k>3} ; most identified power laws in nature have exponents such that the mean is well-defined but

1850-429: A wide variety of quantities seem to follow the power-law form, at least in their upper tail (large events). The behavior of these large events connects these quantities to the study of theory of large deviations (also called extreme value theory ), which considers the frequency of extremely rare events like stock market crashes and large natural disasters . It is primarily in the study of statistical distributions that

1924-436: Is a slowly varying function , which is any function that satisfies lim x → ∞ L ( r x ) / L ( x ) = 1 {\displaystyle \lim _{x\rightarrow \infty }L(r\,x)/L(x)=1} for any positive factor r {\displaystyle r} . This property of L ( x ) {\displaystyle L(x)} follows directly from

1998-542: Is a variable real number. As an example, the cumulative distribution of the fracture aperture, X , for a sample of N elements is defined as 'the number of fractures per meter having aperture greater than x . Use of cumulative frequency has some advantages, e.g. it allows one to put on the same diagram data gathered from sample lines of different lengths at different scales (e.g. from outcrop and from microscope). Although power-law relations are attractive for many theoretical reasons, demonstrating that data does indeed follow

2072-486: Is accumulated, they continue to grow. These power-law probability distributions are also called Pareto-type distributions, distributions with Pareto tails, or distributions with regularly varying tails. A modification, which does not satisfy the general form above, with an exponential cutoff, is In this distribution, the exponential decay term e − λ x {\displaystyle \mathrm {e} ^{-\lambda x}} eventually overwhelms

2146-588: Is also a power-law function, but with a smaller scaling exponent. For data, an equivalent form of the cdf is the rank-frequency approach, in which we first sort the n {\displaystyle n} observed values in ascending order, and plot them against the vector [ 1 , n − 1 n , n − 2 n , … , 1 n ] {\displaystyle \left[1,{\frac {n-1}{n}},{\frac {n-2}{n}},\dots ,{\frac {1}{n}}\right]} . Although it can be convenient to log-bin

2220-431: Is assumed here that a random sample is obtained from a probability distribution, and that we want to know if the tail of the distribution follows a power law (in other words, we want to know if the distribution has a "Pareto tail"). Here, the random sample is called "the data". Pareto Q–Q plots compare the quantiles of the log-transformed data to the corresponding quantiles of an exponential distribution with mean 1 (or to

2294-467: Is close to 0, because Pareto Q–Q plots are not designed to identify distributions with slowly varying tails. On the other hand, in its version for identifying power-law probability distributions, the mean residual life plot consists of first log-transforming the data, and then plotting the average of those log-transformed data that are higher than the i -th order statistic versus the i -th order statistic, for i  = 1, ...,  n , where n

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2368-469: Is convenient to assume a lower bound x m i n {\displaystyle x_{\mathrm {min} }} from which the law holds. Combining these two cases, and where x {\displaystyle x} is a continuous variable, the power law has the form of the Pareto distribution where the pre-factor to α − 1 x min {\displaystyle {\frac {\alpha -1}{x_{\min }}}}

2442-409: Is distributed according to a power-law among cars (very few cars contribute to most contamination) it would be sufficient to eliminate those very few cars from the road to reduce total exhaust substantially. The median does exist, however: for a power law x , with exponent ⁠ k > 1 {\displaystyle k>1} ⁠ , it takes the value 2 x min , where x min

2516-406: Is equivalent to the popular Hill estimator from quantitative finance and extreme value theory . For a set of n integer-valued data points { x i } {\displaystyle \{x_{i}\}} , again where each x i ≥ x min {\displaystyle x_{i}\geq x_{\min }} , the maximum likelihood exponent is the solution to

2590-410: Is likely to find documents about electronic voting even if they omit one of the words "electronic" or "voting", or even both. Power law In statistics , a power law is a functional relationship between two quantities, where a relative change in one quantity results in a relative change in the other quantity proportional to the change raised to a constant exponent : one quantity varies as

2664-488: Is the normalizing constant . We can now consider several properties of this distribution. For instance, its moments are given by which is only well defined for m < α − 1 {\displaystyle m<\alpha -1} . That is, all moments m ≥ α − 1 {\displaystyle m\geq \alpha -1} diverge: when α ≤ 2 {\displaystyle \alpha \leq 2} ,

2738-426: Is the minimum value for which the power law holds. The equivalence of power laws with a particular scaling exponent can have a deeper origin in the dynamical processes that generate the power-law relation. In physics, for example, phase transitions in thermodynamic systems are associated with the emergence of power-law distributions of certain quantities, whose exponents are referred to as the critical exponents of

2812-474: Is the size of the random sample. If the resultant scatterplot suggests that the plotted points tend to stabilize about a horizontal straight line, then a power-law distribution should be suspected. Since the mean residual life plot is very sensitive to outliers (it is not robust), it usually produces plots that are difficult to interpret; for this reason, such plots are usually called Hill horror plots. Log–log plots are an alternative way of graphically examining

2886-626: The method of expanding bins , implies the presence of 1/ f noise and that 1/ f noise can arise as a consequence of this Tweedie convergence effect. Although more sophisticated and robust methods have been proposed, the most frequently used graphical methods of identifying power-law probability distributions using random samples are Pareto quantile-quantile plots (or Pareto Q–Q plots ), mean residual life plots and log–log plots . Another, more robust graphical method uses bundles of residual quantile functions. (Please keep in mind that power-law distributions are also called Pareto-type distributions.) It

2960-522: The power law , or long tail distribution curves. That is, a small portion of the terms observed in a large query log (e.g. > 100 million queries) are used most often, while the remaining terms are used less often individually. This example of the Pareto principle (or 80–20 rule ) allows search engines to employ optimization techniques such as index or database partitioning , caching and pre-fetching. In addition, studies have been conducted into linguistically-oriented attributes that can recognize if

3034-586: The algorithmic testing it does for keyword categories to attempt to choose the best content for any particular keyword search. This means that old methods of gaming the rankings with false signals are becoming less and less effective, and the highest quality content from a human perspective is being ranked higher in Google. RankBrain has helped Google Hummingbird (the 2013 version of the ranking algorithm) provide more accurate results because it can learn words and phrases it may not know. It also learns them specifically for

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3108-437: The average and all higher-order moments are infinite; when 2 < α < 3 {\displaystyle 2<\alpha <3} , the mean exists, but the variance and higher-order moments are infinite, etc. For finite-size samples drawn from such distribution, this behavior implies that the central moment estimators (like the mean and the variance) for diverging moments will never converge – as more data

3182-507: The best chance of matching it. Therefore, RankBrain attempts to guess what people mean and records the results, which adapts the results to provide better user satisfaction. RankBrain is trained offline with batches of past searches. Studies showed how RankBrain better interpreted the relationships between words. This can include the use of stop words in a search query ("the," "and," "without," etc.) – words that were historically ignored previously by Google, but are sometimes of

3256-431: The cdfs of the data and the power law with exponent α {\displaystyle \alpha } , respectively. As this method does not assume iid data, it provides an alternative way to determine the power-law exponent for data sets in which the temporal correlation can not be ignored. This criterion can be applied for the estimation of power-law exponent in the case of scale-free distributions and provides

3330-503: The collection of much data). In addition, other types of tail behavior can be identified using bundle plots. In general, power-law distributions are plotted on doubly logarithmic axes , which emphasizes the upper tail region. The most convenient way to do this is via the (complementary) cumulative distribution (ccdf) that is, the survival function , P ( x ) = P r ( X > x ) {\displaystyle P(x)=\mathrm {Pr} (X>x)} , The cdf

3404-429: The constant C is a scaling factor to ensure that the total area is 1, as required by a probability distribution. More often one uses an asymptotic power law – one that is only true in the limit; see power-law probability distributions below for details. Typically the exponent falls in the range 2 < α < 3 {\displaystyle 2<\alpha <3} , though not always. More than

3478-410: The continuous version should not be applied to discrete data, nor vice versa. Further, both of these estimators require the choice of x min {\displaystyle x_{\min }} . For functions with a non-trivial L ( x ) {\displaystyle L(x)} function, choosing x min {\displaystyle x_{\min }} too small produces

3552-502: The country, as well as language, in which a query is made. So, if one looks up a query with the word boot in it within the United States, one will get information on footwear. However, if the query comes through the UK, then the information could also be in regard to storage spaces in cars. 角鋼 Web search query A web query or web search query is a query that a user enters into

3626-416: The data, or otherwise smooth the probability density (mass) function directly, these methods introduce an implicit bias in the representation of the data, and thus should be avoided. The survival function, on the other hand, is more robust to (but not without) such biases in the data and preserves the linear signature on doubly logarithmic axes. Though a survival function representation is favored over that of

3700-753: The ease with which certain general classes of mechanisms generate them. The demonstration of a power-law relation in some data can point to specific kinds of mechanisms that might underlie the natural phenomenon in question, and can indicate a deep connection with other, seemingly unrelated systems; see also universality above. The ubiquity of power-law relations in physics is partly due to dimensional constraints , while in complex systems , power laws are often thought to be signatures of hierarchy or of specific stochastic processes . A few notable examples of power laws are Pareto's law of income distribution, structural self-similarity of fractals , scaling laws in biological systems , and scaling laws in cities . Research on

3774-591: The estimation of the power-law exponent, which does not assume independent and identically distributed (iid) data, uses the minimization of the Kolmogorov–Smirnov statistic , D {\displaystyle D} , between the cumulative distribution functions of the data and the power law: with where P e m p ( x ) {\displaystyle P_{\mathrm {emp} }(x)} and P α ( x ) {\displaystyle P_{\alpha }(x)} denote

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3848-422: The exponent α {\displaystyle \alpha } (Greek letter alpha , not to be confused with scaling factor a {\displaystyle a} used above) is greater than 1 (otherwise the tail has infinite area), the minimum value x min {\displaystyle x_{\text{min}}} is needed otherwise the distribution has infinite area as x approaches 0, and

3922-800: The function is plotted as a log-log plot with horizontal axis being ln ⁡ x {\displaystyle \ln x} and vertical axis being ln ⁡ ( y / y 0 + a ) {\displaystyle \ln(y/y_{0}+a)} , the plot is composed of n + 1 {\displaystyle n+1} linear segments with slopes c 0 , c 1 , . . . , c n {\displaystyle c_{0},c_{1},...,c_{n}} , separated at x = x 1 , . . . , x n {\displaystyle x=x_{1},...,x_{n}} , smoothly spliced together. The size of f i {\displaystyle f_{i}} determines

3996-439: The identification of power-law probability distributions using random samples has been proposed. This methodology consists of plotting a bundle for the log-transformed sample . Originally proposed as a tool to explore the existence of moments and the moment generation function using random samples, the bundle methodology is based on residual quantile functions (RQFs), also called residual percentile functions, which provide

4070-562: The log likelihood function becomes: The maximum of this likelihood is found by differentiating with respect to parameter α {\displaystyle \alpha } , setting the result equal to zero. Upon rearrangement, this yields the estimator equation: where { x i } {\displaystyle \{x_{i}\}} are the n {\displaystyle n} data points x i ≥ x min {\displaystyle x_{i}\geq x_{\min }} . This estimator exhibits

4144-440: The most reliable techniques are often based on the method of maximum likelihood . Alternative methods are often based on making a linear regression on either the log–log probability, the log–log cumulative distribution function, or on log-binned data, but these approaches should be avoided as they can all lead to highly biased estimates of the scaling exponent. For real-valued, independent and identically distributed data, we fit

4218-418: The name "power law" is used. In empirical contexts, an approximation to a power-law o ( x k ) {\displaystyle o(x^{k})} often includes a deviation term ε {\displaystyle \varepsilon } , which can represent uncertainty in the observed values (perhaps measurement or sampling errors) or provide a simple way for observations to deviate from

4292-416: The origins of power-law relations, and efforts to observe and validate them in the real world, is an active topic of research in many fields of science, including physics , computer science , linguistics , geophysics , neuroscience , systematics , sociology , economics and more. However, much of the recent interest in power laws comes from the study of probability distributions : The distributions of

4366-400: The particular form of the lower tail, represented by L ( x ) {\displaystyle L(x)} above. More about these methods, and the conditions under which they can be used, can be found in . Further, this comprehensive review article provides usable code (Matlab, Python, R and C++) for estimation and testing routines for power-law distributions. Another method for

4440-399: The pdf while fitting a power law to the data with the linear least square method, it is not devoid of mathematical inaccuracy. Thus, while estimating exponents of a power law distribution, maximum likelihood estimator is recommended. There are many ways of estimating the value of the scaling exponent for a power-law tail, however not all of them yield unbiased and consistent answers . Some of

4514-447: The plot tend to converge to a straight line for large numbers in the x axis, then the researcher concludes that the distribution has a power-law tail. Examples of the application of these types of plot have been published. A disadvantage of these plots is that, in order for them to provide reliable results, they require huge amounts of data. In addition, they are appropriate only for discrete (or grouped) data. Another graphical method for

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4588-399: The power-law behavior at very large values of x {\displaystyle x} . This distribution does not scale and is thus not asymptotically as a power law; however, it does approximately scale over a finite region before the cutoff. The pure form above is a subset of this family, with λ = 0 {\displaystyle \lambda =0} . This distribution is

4662-429: The power-law function (perhaps for stochastic reasons): Mathematically, a strict power law cannot be a probability distribution, but a distribution that is a truncated power function is possible: p ( x ) = C x − α {\displaystyle p(x)=Cx^{-\alpha }} for x > x min {\displaystyle x>x_{\text{min}}} where

4736-417: The quantiles of a standard Pareto distribution) by plotting the former versus the latter. If the resultant scatterplot suggests that the plotted points asymptotically converge to a straight line, then a power-law distribution should be suspected. A limitation of Pareto Q–Q plots is that they behave poorly when the tail index α {\displaystyle \alpha } (also called Pareto index)

4810-516: The queries from the Excite search engine, showed some interesting characteristics of web searches: A study of the same Excite query logs revealed that 19% of the queries contained a geographic term (e.g., place names, zip codes, geographic features, etc.). Studies also show that, in addition to short queries (queries with few terms), there are predictable patterns of how users change their queries. A 2005 study of Yahoo's query logs revealed that 33% of

4884-414: The queries from the same users were repeat queries and that in 87% of cases the user would click on the same result. This suggests that many users use repeat queries to revisit or re-find information. This analysis is confirmed by a Bing search engine blog post which stated that about 30% of queries are navigational queries. In addition, research has shown that query term frequency distributions conform to

4958-488: The requirement that p ( x ) {\displaystyle p(x)} be asymptotically scale invariant; thus, the form of L ( x ) {\displaystyle L(x)} only controls the shape and finite extent of the lower tail. For instance, if L ( x ) {\displaystyle L(x)} is the constant function, then we have a power law that holds for all values of x {\displaystyle x} . In many cases, it

5032-594: The sharpness of splicing between segments i − 1 , i {\displaystyle i-1,i} . A power law with an exponential cutoff is simply a power law multiplied by an exponential function: In a looser sense, a power-law probability distribution is a distribution whose density function (or mass function in the discrete case) has the form, for large values of x {\displaystyle x} , where α > 1 {\displaystyle \alpha >1} , and L ( x ) {\displaystyle L(x)}

5106-427: The suspected cause of typical bending and/or flattening phenomena in the high- and low-frequency graphical segments, are parametrically absent in the standard model. One attribute of power laws is their scale invariance . Given a relation f ( x ) = a x − k {\displaystyle f(x)=ax^{-k}} , scaling the argument x {\displaystyle x} by

5180-454: The system. Diverse systems with the same critical exponents—that is, which display identical scaling behaviour as they approach criticality —can be shown, via renormalization group theory, to share the same fundamental dynamics. For instance, the behavior of water and CO 2 at their boiling points fall in the same universality class because they have identical critical exponents. In fact, almost all material phase transitions are described by

5254-413: The tail of a distribution using a random sample. Taking the logarithm of a power law of the form f ( x ) = a x k {\displaystyle f(x)=ax^{k}} results in: which forms a straight line with slope k {\displaystyle k} on a log-log scale. Caution has to be exercised however as a log–log plot is necessary but insufficient evidence for

5328-456: The transcendental equation where ζ ( α , x m i n ) {\displaystyle \zeta (\alpha ,x_{\mathrm {min} })} is the incomplete zeta function . The uncertainty in this estimate follows the same formula as for the continuous equation. However, the two equations for α ^ {\displaystyle {\hat {\alpha }}} are not equivalent, and

5402-405: The variance is not, implying they are capable of black swan behavior. This can be seen in the following thought experiment: imagine a room with your friends and estimate the average monthly income in the room. Now imagine the world's richest person entering the room, with a monthly income of about 1 billion US$ . What happens to the average income in the room? Income is distributed according to

5476-500: Was not theoretically derived, the classification has been empirically validated with actual search engine queries. Search engines often support a fourth type of query that is used far less frequently: Most commercial web search engines do not disclose their search logs, so information about what users are searching for on the Web is difficult to come by. Nevertheless, research studies started to appear in 1998. A 2001 study, which analyzed

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