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Heuristic Continuous Base Flow Separation

Jozsef Szilagyi 1

Abstract:A digital ?ltering algorithm for continuous base ?ow separation is compared to physically based simulations of base ?ow.It is shown that the digital ?lter gives comparable results to model simulations in terms of the multiyear base ?ow index when a ?lter coef?cient is used that replicates the watershed-speci?c time delay of model simulations.This way,the application of the heuristic digital ?lter for practical continuous base ?ow separation can be justi?ed when auxiliary hydrometeorological data ?such as precipitation and air temperature ?typically required for physically based base ?ow separation techniques are not available or not representative of the watershed.The ?lter coef?cient can then be optimized upon an empirical estimate of the watershed-speci?c time delay,requiring only the drainage area of the watershed.

DOI:10.1061/?ASCE ?1084-0699?2004?9:4?311?

CE Database subject headings:Base ?ow;Digital ?lters;Runoff;Stream?ow;Algorithms.

Introduction

Detailed knowledge of groundwater contribution to streams,i.e.,base ?ow,is important in many water management areas:water supply,wastewater dilution,navigation,hydropower generation ?Dingman 1994?and aquifer characterization ?Brutsaert and Nie-ber 1977;Troch et al.1993;Szilagyi et al.1998;Brutsaert and Lopez 1998?.Also,base ?ow can directly be related to aquifer recharge ?Birtles 1978;Wittenberg and Sivapalan 1999;Szilagyi et al.2003?,which is crucial in ascertaining safe yields of water development schemes,such as irrigation planning in the Great Plains ?Sophocleous 2000?.

The importance of having knowledge of base ?ow is re?ected in the number of published works,as reviewed by Tallaksen ?1995?.With the widespread use of PCs,traditional,event-based methods that contain varying degrees of subjectivity,such as graphical base ?ow separation ?Barnes 1939;Hewlett and Hibbert 1963;Szilagyi and Parlange 1998?,have been replaced by auto-mated techniques that can result in continuous base ?ow model-ing.Present-day automated techniques consist mainly of two types:digital ?ltering methods ?Nathan and McMahon 1990;Ar-nold et al.1995;Arnold and Allen 1999?and conceptual hydro-logic models ?e.g.,Jakeman et al.1990;Szilagyi and Parlange 1999?.The former have ‘‘no true physical basis’’?Arnold and Allen 1999?but have the distinct advantage of requiring only stream?ow measurements.The latter are physically based but re-quire precipitation data as a minimum in addition to measured stream?ow.Often,available precipitation data are insuf?cient be-cause the precipitation station is either not located within the watershed,or it is within the watershed but not at a representative location.In larger catchments,more than one station is typically

needed to obtain a good estimate on the amount of water available to runoff.Many times the precipitation record has discontinuities that can easily thwart efforts to perform continuous base ?ow separation using physically based techniques.Clearly,there is a practical need for a technique that uses the most basic information available:stream?ow and the corresponding drainage area.The digital ?ltering technique of Nathan and McMahon ?1990?is such a ‘‘minimalist’’approach.Because their method is not based on any physical law,a question arises whether the ensuring base ?ow hydrograph is realistic at all,or,in other words,can the results be backed by a more complex,physically based approach?Unfortu-nately,there is no trivial way of validating the results of the ?lter algorithm by measurements.Isotope or chemical tracer tech-niques may one day prove useful in validation efforts in spite of the currently existing discrepancy in base ?ow interpretation be-tween physical and tracer techniques ?Rice and Hornberger 1998?.Base?ow recession can generally be described by the follow-ing equation ?Brutsaert and Nieber 1977?:

??aQ ?(1)

where a ?L 3(1??)T ??2?and ?(?)?constants;Q b ?L 3T ?1??the groundwater discharge to the stream.Under simplifying as-sumptions ?Brutsaert and Lopez 1998?,the theoretical value of ?during recession may change from three to unity.When ?reaches unity,the aquifer behaves as a linear reservoir,and a then equals k ?1,the inverse of the storage coef?cient ?T ?in the linear storage equation S ?kQ b ,where S ?L 3?is water volume in storage.Naturally,not all aquifers behave as linear reservoirs,even after a suf?cient period of stream?ow recession ?Brutsaert and Nieber 1977;Szilagyi and Parlange 1998;Troch et al.1993;Wittenberg and Sivapalan 1999?,but many do,as reported by V ogel and Kroll ?1992?;Jakeman and Hornberger ?1993?,and Brutsaert and Lopez ?1998?.The analysis that follows is strictly valid for watersheds that exhibit this latter type of base ?ow recession property,al-though the results and conclusions can straightforwardly be gen-eralized to a fully nonlinear aquifer case as well,where ?is always larger than unity.

Jakeman and Hornberger ?1993?pointed out that the informa-tion content of a rainfall-runoff model allows for only a handful of model parameters to be optimized.Perrin et al.?2001?,in a

Conservation and Survey Division,Institute of Agriculture and Natu-ral Resources,Univ.of Nebraska-Lincoln,114Nebraska Hall,Lincoln,NE 68588-0517.E-mail:jszilagyi1@https://www.wendangku.net/doc/3f2138683.html,

Note.Discussion open until December 1,2004.Separate discussions must be submitted for individual papers.To extend the closing date by one month,a written request must be ?led with the ASCE Managing Editor.The manuscript for this paper was submitted for review and pos-sible publication on December 16,2002;approved on October 28,2003.This paper is part of the Journal of Hydrologic Engineering ,V ol.9,No.4,July 1,2004.?ASCE,ISSN 1084-0699/2004/4-311–318/$18.00.

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study involving429catchments worldwide,demonstrated that ‘‘very simple models can achieve a level of performance almost as high as models with more parameters.’’In fact,‘‘inadequate complexity typically results in model over-parameterization and parameter uncertainty’’?Perrin et al.2001?.In the light of these ?ndings,the simplest possible physically based model for base ?ow simulation was sought.The model of Jakeman et al.?1990?and Jakeman and Hornberger?1993?,from now on referred to as the Jakeman model,meets this criterion.

Methodology

Following Jakeman et al.?1990?and Jakeman and Hornberger ?1993?,any nonlinearity in the rainfall-runoff relationship can be dealt with by the transformation of the observed precipitation series into‘‘excess’’or‘‘effective’’rainfall u?L T?1?via an an-tecedent precipitation index s(?)

s i?c?r i??1???1?r i?1??1???1?2r i?2?....?(2) where r?L T?1??observed rainfall;?(?)?the rate at which the catchment wetness declines in the absence of precipitation;i ?time index?incremented on a daily basis?;and c?T L?1??a normalizing parameter that ensures that the excess rainfall vol-ume equals the volume of total runoff over the calibration period. Excess rainfall is obtained by

u i?r i s i(3) Seasonal changes in evapotranspiration are described by

?i??0e f?30?t i?(4) where f?t?1??a temperature modulation factor;t?temperature ?°C?;and?0?the rate at which the catchment wetness declines at 30°C.

Effective rainfall is routed through two parallel linear reser-voirs representing quick and slow?i.e.base?ow?storm re-sponses.The unit impulse response?h(?)?of a linear reservoir in discrete time i is?O’Connor1976?

h i?

1?k?k1?k

?i i?0,1,2, (5)

from which the impulse response of the two parallel discrete lin-ear reservoirs follows as

h i?h qi?h bi?

v q

1?k q?kq1?k q

?i?v b1?k b?k b1?k b?i i?0,1,2,...

(6) where the subscripts q and b represent quick and base?ow storm responses,respectively.Note that in discrete time the storage co-ef?cients k b and k q become unitless.The volumetric throughput coef?cients v q and v b(?),add up to unity.The model response (Q m?L T?1?)to effective rainfall is obtained via the convolution summation

Q m??i?0

h i u m?i,m?0,1,2,...

(7)

Fig.1.Schematic representation of the Jakeman

model

Fig.2.Model response to?ctive precipitation with arbitrary parameters:k q?1(day),k b?30(day),f?1(°C?1),?0?1,v q?0.5.Solid line is modeled base?ow;intermittent line is modeled total runoff.

;

Altogether,the model has seven parameters ?f ,?0,k q ,k b ,v q ,

v b ,and c ?.A schematic of the system is shown in Fig.1.As a demonstration,the model response to ?ctive precipitation is shown with arbitrarily assigned parameter values in Fig.2.The base ?ow peaks occur almost simultaneously with the total runoff peaks.This is because effective rainfall is split into two parts and routed directly through the two linear reservoirs representing quick and base ?ow responses,without any time delay in the latter case.In reality,there generally is a time lag between the two peaks ?Pilgrim and Cordery 1993;Szilagyi and Parlange 1998?,depending on how long it takes the in?ltrated water to reach the saturation zone.

The present modi?cation of the original Jakeman model can account for this possible time lag by incorporating a third linear reservoir ?with a storage coef?cient k s )representing soil storage ?Besbes and de Marsily 1984;Wu et al.1997;Wittenberg and Sivapalan 1999?.A schematic of the model arrangement can be seen in Fig.3.The unit impulse response of two serial discrete linear reservoirs is obtained via the Z-transform of the difference equation ?Singh 1988?

?1?k s ???1?k b ??Q i ?u i

(8)

where the difference operator ?is for time shifting,i.e.,?g i ?g i ?g i ?1,where g is an arbitrary discrete function.Upon in-verting the resulting transfer function H (z )

H ?z ??z 2

?1?k s ?k b ?k s k b ?z 2??k s ?k b ?2k s k b ?z ?k s k b (9)

the discrete unit impulse response results as

h i ?

?k b

?k b 1?k b ?

i k s ?k s ?k s 1?k s ?i ?k s ?k s 1?k s ?i k b ?k b

?k b

1?k b

k s ?k s 2?k b k s 2?k b ?k b 2?k b 2

k s

(10)

By adding a soil-storage component to the Jakeman model,the number of parameters has increased through k s ,by one,from seven to eight.The soil-storage component delays the base ?ow peak as well as ?attens it,thus making it look more realistic,as is seen in Fig.4,where a k s ?2?day ?was added to the previously prescribed model parameter set.

In the last modi?cation of the model,the changing effect of the exponent in Eq.?1?is being investigated.Right after the start of the base ?ow recession,the exponent may reach a value of three,provided the aquifer became close to full saturation.Fig.5?from Szilagyi 1999?demonstrates this case,with the lower envelopes ?that are thought to represent ‘‘pure ’’groundwater discharge ?of the data points expressing a slope of three and unity.Numerical and analytical solutions of the Boussinesq equation that describe groundwater drainage also con?rm ?Brutsaert and Nieber 1977;Szilagyi 1999?this change of the exponent in Eq.?1?.A time-varying exponent in Eq.?1?can only be modeled via a general nonlinear reservoir,S ?k Q b n ,if n changes with time as well.Alternatively,rather than changing n through time,k may be changed with time in the linear reservoir representation,as was done by Aksoy et al.?2001?.The critical base ?ow

discharge

Fig.3.Schematic representation of the modi?ed Jakeman

model

Fig.4.Response of the modi?ed Jakeman model to ?ctive precipitation with arbitrary parameters:k q ?1(day),k s ?2(day),k b ?30(day),f ?1(°C ?1),?0?1,v q ?0.5.Solid line is modeled base ?ow;intermittent line is modeled total runoff.

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(Q b 0),when this change starts ?Fig.5?,is obtained by solving Eq.?1?simultaneously for the two lower envelope lines as Q b 0?(a 1/a 3)0.5,where a 1and a 3are with ??1and ??3,respec-tively.For convenience,it is assumed here that k changes linearly

from a maximum value of k b (?a 1?1

),when Q b рQ b 0,to a minimum value of 0.5k b when the aquifer becomes close to saturation.Under simplifying assumptions ?Brutsaert and Lopez

1998?,drainable water storage at full saturation,S max ,can be estimated as S max ?1.97?A (a 1a 3)1/2??1,where A ?drainage area of the watershed.Since k is changing with time now,a simple convolution cannot be maintained;instead,base ?ow is simulated ?Fig.6?by numerically solving the linear storage equation with a time-varying storage coef?cient.This means that through the cal-culation of S at each time step,the corresponding k (S )value

Fig.5.Measured daily discharge versus change in discharge between consecutive days,6days after

rain.

Fig.6.Response of the modi?ed Jakeman model with time-varying storage coef?cient to ?ctive precipitation with arbitrary parameters:k q ?1(day),k s ?2(day),k b ?30(day),f ?1(°C ?1),?0?1,v q ?0.5.Solid line is modeled base ?ow;intermittent line is modeled total runoff.Q 0and S max are assumed to be 0.05(mm ?d ?1)and 10?mm ?,respectively.

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obtained with the help of the maximum and minimum values of k ,as k ?c 1S ?c 2,where c 1?k b /2(S 0?S max ),and c 2?k b ?S 0c 1.S 0is the drainable water storage at Q b ?Q b 0.

This last modi?cation of the Jakeman model ?MJ ?will be used for the validation of the digital ?lter algorithm ?Nathan and Mc-Mahon 1990?,which estimates base ?ow (Q b )as

Q bi ?pQ b ?i ?1??1?p

?Q i ?Q i ?1?

(11)

from measured or modeled stream?ow ?Q ?,where p ???is the

?lter parameter.The resulting base ?ow values are constrained by the concurrent stream?ow values,so that whenever Q bi ?Q i ,the Q bi value is replaced by Q i .The validation is done by running

the MJ model with Monte Carlo-simulated daily precipitation val-ues in combination with deterministic daily temperature values,following Milly ?1994?and Szilagyi ?2001?.The daily values of precipitation (P d ?L ?)are assumed to follow an exponential dis-tribution ???

??P d ???e ??P d

(12)

where ??1??P a /(365.25*SF )?,with P a ?L ?denoting the mean annual precipitation,and SF ?T ?1?the mean storm frequency.SF is calculated as 2?P d 2?/var(P d ),where the angular brackets de-note temporal averaging,and var denotes the variance.The num-ber of interstorm days (i d )is assumed to follow a Poisson distri-

bution

Fig.7.?a ?First year of the simulated stream?ow ?intermittent line ?and base ?ow values;?b ?base ?ow hydrographs of the same period.Intermittent line is the ?lter result.Here k q ?1(day),k s ?1(day),k b ?60(day),v q ?

0.2.

Fig.8.?a ?First year of the simulated stream?ow ?intermittent line ?and base ?ow values;?b ?base ?ow hydrographs of the same period.Intermittent line is the ?lter result.Here k q ?1(day),k s ?2(day),k b ?30(day),v q ?0.8.

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P ?i d ?N ??

?N !

e ??

(13)

where ??SF ?1.

Results and Discussion

Rather than ?tting the MJ model to measured stream?ow and comparing the ?lter results to the MJ-model-obtained base ?ow,a Monte Carlo-type simulation with the MJ model was preferred due to the much greater ?exibility the latter approach offers.By making sure that the model-prescribed parameters are physically meaningful and driving the model with realistic precipitation and temperature inputs,realistic model simulations of base ?ow can be expected and compared to ?lter results.The MJ model,even in its original,simplest form,performed quite effectively in simulat-ing daily stream?ow of small catchments in the U.S.,Europe,Asia,and in Australia ?Jakeman et al.1990;Jakeman and Horn-berger 1993?.

The modi?ed MJ model was run in a Monte Carlo simulation mode with daily precipitation and daily mean temperature inputs,characteristic of a mild continental climate of central Europe,with a mean annual precipitation of 600mm evenly distributed ?i.e.,no seasonal cycle ?throughout the year,a mean annual tem-perature of 11°C,and a mean storm frequency of 0.2365/day.A choice of ?0?1and f ?1°C ?1,in combination with a 5th order

polynomial in Eq.?2?,resulted in a 7%runoff ratio,which is typical of the lowland regions in central Europe.The daily mean temperatures ?°C ?followed the mean monthly temperatures in the model starting with January:?1.1,1,5.8,11.8,16.8,20.2,22.2,21.4,17.4,11.3,5.8,and 1.5.Each model simulation represented 10years.The quick storm response parameter k q and the soil storage coef?cient k s were each assigned two values:1and 2d .The base ?ow storage coef?cient k b was allowed to have values of 30and 60d .The volumetric throughput parameter v q was assigned the following values:0.2,0.4,0.6,and 0.8.Note that v b ?1?v q ,correspondingly.The values of the above parameters are representative of the catchments reported by Jakeman and Hornberger ?1993?.With decreasing v q values,groundwater con-tribution to the stream?ow increases,requiring increased subsur-face storage capability in the watershed.This is accommodated for in the model by increasing the value of S max and Q b 0in the model accordingly,such as ?5,0.03?,?10,0.06?,?15,0.09?,and ?20,0.12?,where the ?rst value in each parenthesis is S max ?mm ?,the second one is Q b 0?mm/day ?,and the ?rst parenthesis corre-sponds to v q ?0.8.The S max and Q b 0values are representative of the small catchments of the Washita Experimental Watershed complex in Oklahoma ?Brutsaert and Lopez 1998?.

The three storage coef?cients and the v q values amount to 32different and unique combinations.With each combination of the model parameters,the MJ model was run for 10years in daily time increments.From the resulting base ?ow hydrograph,

the

Fig.9.?a ?First year of the simulated stream?ow ?intermittent line ?and base ?ow values;?b ?base ?ow hydrographs of the same period.Intermittent line is the ?lter result.Here k q ?2(day),k s ?2(day),k b ?30(day),v q ?0.8.

Table 1.Model Simulation and Optimized Filtering Results (N d ,p ,BFI ?lt /BFI).k q (d )1

k s (d )1

k b (d )3060306030603060V q ?0.2 3.34.953.94 3.15.960.99 3.25.955.94 3.15.9591.00 4.75.998.48 4.31.989.87 4.71.998.49 4.29.989.88?0.4 3.39.972.99 3.26.9761.04 3.37.9731.00 3.25.9761.06 4.86.993.78 4.49.993.92 4.83.996.69 4.44.997.80?0.6 3.67.9861.03 3.51.9841.13 3.64.9861.06 3.50.9851.14 5.27.996.82 4.84.996.97 5.22.995.88 4.84.996.99?0.8

4.20.997.95

4.02.9941.21

4.16.9951.10

4.00

.994

1.22

6.21

.998

.93

5.79.999

.95

6.13

.998

.95

5.79

.999

.96

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mean watershed-speci?c time delay ?Linsley et al.1958?N d (d )could be calculated.N d is the mean elapsed time between the peak of stream?ow and the ?rst instant when stream?ow becomes dominated by base ?ow.The critical point when this latter hap-pens was calculated as Q bi /Q i у1?0.25v q ,which results in a Q bi /Q i ratio of 80%when v q is 0.8and 95%when v q is 0.2.A more stringent critical value is necessary when base ?ow domi-nates stream?ow.Note that when v q ?0.2,80%of the stream?ow is made up by base ?ow on a long-term basis,which means that the base ?ow index,BFI (??Q bi ?/?Q i ?,where the angle brackets denote temporal averaging ?is 0.8or 80%,as well.Note also that the use of such a critical value is not necessary with the ?lter algorithm because of the constraint applied there,which makes stream?ow become base ?ow fully ‘‘overnight.’’In the MJ model,this can never happen due to the exponential decay in the quick ?ow component.

With the known N d value from the MJ model,the ?lter param-eter p was systematically changed until the ?lter model gave the closest possible matching value of N d with the MJ model,which was generally within 1%.Figs.7,8,and 9display hydrographs for small (?3.15d ),medium (?4.16d ),and large N d (?6.13d )cases,respectively.The resulting p ,N d ,and BFI ?lt /BFI values are listed in Table 1.As it can be seen,the MJ-simulated watershed-speci?c time delays ranged between 3.15and 6.21d ,the ?lter parameter value p ranged from 0.953to 0.999,and the BFI ?lt /BFI ratios changed between 0.48and 1.22.Fig.10displays the distribution of the values.

Fig.10shows that the long-term base ?ow index,given by the ?lter algorithm is within 20%of the modeled BFI value in 80%of the cases considered,with a mean value of only 6%less than the modeled mean BFI value.This suggests that the ?lter algorithm of Nathan and McMahon ?1990?is of practical value,provided one can estimate the watershed-speci?c time delay N d for real watersheds.Fortunately,this is possible by the application of Lin-sley’s empirical equation ?Linsley et al.1958?N d ?A 0.2,where N d is in days and A ,the drainage area of the watershed,is in square miles.When applying the ?lter algorithm,the ?lter param-eter must be adjusted until the resulting N d value becomes suf?-ciently close to Linsley’s value.This has been done by Szilagyi

et al.?2003?for 100-plus gauging stations in Nebraska where the spatial distribution of the long-term BFI index was of interest.In conclusion,it can be stated that the ?lter algorithm,in spite of its lack of any physical basis,can have its place in practical applications when more complex and/or physically based base ?ow separation methods are hindered by data availability.The ?lter algorithm,with its suggested optimization,based on the watershed-speci?c time delay,requires only the most basic data:stream?ow and the corresponding drainage area.Of course,at best,the practical value of the ?lter algorithm is only as good as the empirical equation of Linsley et al.?1958?,which has been frequently used in a wide variety of applications in the past 4decades.As illustrated previously with the help of model simula-tions,it gave comparable results to a more complex,physically based base ?ow separation technique under a variety of soil and aquifer properties characteristic of small watersheds in Oklahoma and North Carolina.

Acknowledgment

The writer is grateful to Charles Flowerday for his editorial help.The views,conclusions,and opinions expressed in this paper are solely those of the writer and not the University of Nebraska,state of Nebraska or any political subdivision thereof.

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E U N I V E R S I T Y L I B R A R I E o n 10/16/12. C o p y r i g h t A S C E .

F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r v e d .

比较PageRank算法和HITS算法的优缺点

题目：请比较PageRank算法和HITS算法的优缺点，除此之外，请再介绍2种用于搜索引擎检索结果的排序算法，并举例说明。答： 1998年，Sergey Brin和Lawrence Page[1]提出了PageRank算法。该算法基于“从许多优质的网页链接过来的网页，必定还是优质网页”的回归关系，来判定网页的重要性。该算法认为从网页A导向网页B的链接可以看作是页面A对页面B的支持投票，根据这个投票数来判断页面的重要性。当然，不仅仅只看投票数，还要对投票的页面进行重要性分析，越是重要的页面所投票的评价也就越高。根据这样的分析，得到了高评价的重要页面会被给予较高的PageRank值，在检索结果内的名次也会提高。PageRank是基于对“使用复杂的算法而得到的链接构造”的分析，从而得出的各网页本身的特性。 HITS 算法是由康奈尔大学( Cornell University ) 的JonKleinberg 博士于1998 年首先提出。Kleinberg认为既然搜索是开始于用户的检索提问，那么每个页面的重要性也就依赖于用户的检索提问。他将用户检索提问分为如下三种：特指主题检索提问（specific queries，也称窄主题检索提问）、泛指主题检索提问（Broad-topic queries，也称宽主题检索提问）和相似网页检索提问（Similar-page queries）。HITS 算法专注于改善泛指主题检索的结果。 Kleinberg将网页（或网站）分为两类，即hubs和authorities，而且每个页面也有两个级别，即hubs（中心级别）和authorities（权威级别）。Authorities 是具有较高价值的网页，依赖于指向它的页面；hubs为指向较多authorities的网页，依赖于它指向的页面。HITS算法的目标就是通过迭代计算得到针对某个检索提问的排名最高的authority的网页。通常HITS算法是作用在一定范围的，例如一个以程序开发为主题的网页，指向另一个以程序开发为主题的网页，则另一个网页的重要性就可能比较高，但是指向另一个购物类的网页则不一定。在限定范围之后根据网页的出度和入度建立一个矩阵，通过矩阵的迭代运算和定义收敛的阈值不断对两个向量authority 和hub值进行更新直至收敛。从上面的分析可见，PageRank算法和HITS算法都是基于链接分析的搜索引擎排序算法，并且在算法中两者都利用了特征向量作为理论基础和收敛性依据。

pagerank算法实验报告

PageRank算法实验报告一、算法介绍 PageRank是Google专有的算法，用于衡量特定网页相对于搜索引擎索引中的其他网页而言的重要程度。它由Larry Page 和Sergey Brin在20世纪90年代后期发明。PageRank实现了将链接价值概念作为排名因素。 PageRank的核心思想有2点： 1.如果一个网页被很多其他网页链接到的话说明这个网页比较重要，也就是pagerank值会相对较高； 2.如果一个pagerank值很高的网页链接到一个其他的网页，那么被链接到的网页的pagerank值会相应地因此而提高。若页面表示有向图的顶点，有向边表示链接，w(i,j)=1表示页面i存在指向页面j的超链接，否则w(i,j)=0。如果页面A存在指向其他页面的超链接，就将A 的PageRank的份额平均地分给其所指向的所有页面，一次类推。虽然PageRank 会一直传递，但总的来说PageRank的计算是收敛的。实际应用中可以采用幂法来计算PageRank，假如总共有m个页面，计算如公式所示： r=A*x 其中A=d*P+（1-d)*(e*e'/m) r表示当前迭代后的PageRank，它是一个m行的列向量,x是所有页面的PageRank初始值。 P由有向图的邻接矩阵变化而来，P'为邻接矩阵的每个元素除以每行元素之和得到。 e是m行的元素都为1的列向量。二、算法代码实现

三、心得体会在完成算法的过程中，我有以下几点体会： 1、在动手实现的过程中，先将算法的思想和思路理解清楚，对于后续动手实现有很大帮助。 2、在实现之前，对于每步要做什么要有概念，然后对于不会实现的部分代码先查找相应的用法，在进行整体编写。 3、在实现算法后，在寻找数据验证算法的过程中比较困难。作为初学者，对于数据量大的数据的处理存在难度，但数据量的数据很难寻找，所以难以进行实例分析。

PageRank算法的核心思想

如何理解网页和网页之间的关系，特别是怎么从这些关系中提取网页中除文字以外的其他特性。这部分的一些核心算法曾是提高搜索引擎质量的重要推进力量。另外，我们这周要分享的算法也适用于其他能够把信息用结点与结点关系来表达的信息网络。今天，我们先看一看用图来表达网页与网页之间的关系，并且计算网页重要性的经典算法：PageRank。 PageRank 的简要历史时至今日，谢尔盖·布林（Sergey Brin）和拉里·佩奇（Larry Page）作为Google 这一雄厚科技帝国的创始人，已经耳熟能详。但在1995 年，他们两人还都是在斯坦福大学计算机系苦读的博士生。那个年代，互联网方兴未艾。雅虎作为信息时代的第一代巨人诞生了，布林和佩奇都希望能够创立属于自己的搜索引擎。1998 年夏天，两个人都暂时离开斯坦福大学的博士生项目，转而全职投入到Google 的研发工作中。他们把整个项目的一个总结发表在了1998 年的万维网国际会议上（WWW7，the seventh international conference on World Wide Web）（见参考文献[1]）。这是PageRank 算法的第一次完整表述。 PageRank 一经提出就在学术界引起了很大反响，各类变形以及对PageRank 的各种解释和分析层出不穷。在这之后很长的一段时间里，PageRank 几乎成了网页链接分析的代名词。给你推荐一篇参考文献[2]，作为进一步深入了解的阅读资料。

PageRank 的基本原理我在这里先介绍一下PageRank 的最基本形式，这也是布林和佩奇最早发表PageRank 时的思路。首先，我们来看一下每一个网页的周边结构。每一个网页都有一个“输出链接”（Outlink）的集合。这里，输出链接指的是从当前网页出发所指向的其他页面。比如，从页面A 有一个链接到页面B。那么B 就是A 的输出链接。根据这个定义，可以同样定义“输入链接”（Inlink），指的就是指向当前页面的其他页面。比如，页面C 指向页面A，那么C 就是A 的输入链接。有了输入链接和输出链接的概念后，下面我们来定义一个页面的PageRank。我们假定每一个页面都有一个值，叫作PageRank，来衡量这个页面的重要程度。这个值是这么定义的，当前页面I 的PageRank 值，是I 的所有输入链接PageRank 值的加权和。那么，权重是多少呢？对于I 的某一个输入链接J，假设其有N 个输出链接，那么这个权重就是N 分之一。也就是说，J 把自己的PageRank 的N 分之一分给I。从这个意义上来看，I 的PageRank，就是其所有输入链接把他们自身的PageRank 按照他们各自输出链接的比例分配给I。谁的输出链接多，谁分配的就少一些；反之，谁的输出链接少，谁分配的就多一些。这是一个非常形象直观的定义。