A statistical approach to the study of AGN emission versus activity (with the detailed analysis of Mrk421)

0.4 MB·English

by  Stefano Ansoldi

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A STATISTICAL APPROACH TO THE STUDY
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OF AGN EMISSION VERSUS ACTIVITY
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J (with the detailed analysis of Mrk421)

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1 STEFANOANSOLDI

International CenterforRelativistic Astrophysics (ICRA), and
]
M IstitutoNazionale di Fisica Nucleare (INFN), Sezione di Trieste,and
Dipartimento di Matematica e Informatica, Universita`degli Studi di Udine,
I via delle Scienze 206 Udine (UD)I-33100, Italy
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h E-mail: [email protected]
p
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o We discuss the theory and implementation of statistically rigorous fits to synchrotron
r self Compton models for datasets obtained from multi-wavelength observations of ac-
t tive galactic nuclei spectral energy distributions. The methods and techniques that we
s
a presentare,then,exemplifiedreportingonarecentstudyofanearbyandwellobserved
[ extragalactic source,Markarian421.

1 Keywords: Active Galactic Nuclei (AGN); Chi-square minimization; Kolmogorov-
v Smirnovtest;Markarian421.
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2 Pure logical thinkingcannot yield us any knowledge of the empirical world:
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all knowledge of reality starts from experience and ends in it.
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Albert Einstein, 1954.
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: Introduction
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X In the last 20 years the window on our universe has opened to an unprecedented
r level,allowingustobridgealargefractionofthegapthatexistedbetweenobserva-
a
tional fields, like astrophysicsand cosmology,and more experimental ones, like,for
instance, high energy particle physics. A marked difference between observational
andexperimentaldisciplineswhichis oftenemphasizedis thatthe formeronesusu-
ally lack direct control on the object that we wish to study, which is usually not
directly accessible to us. A consequence of this is that, contrary to experimental
disciplines (where we can mostly prepare the system under study to suit our needs
and tackle the data acquisition and analysis under what we judge to be the most
appropriate and fruitful conditions) in cosmology and astrophysics we usually do
not have this convenience. The lack of control on the systems under study and on
their environments can then result in uncertainties which are usually higher than
the ones obtained in experimental situations and make more challenging to single
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out discrepancies between models and data. Although this stereotype is true to
some extent, the situation has now radically changed, as witnessed, for instance,
by precision measurements of the cosmic microwave background radiation, or by
millisecond pulsartiming. If it is true that, in general,we have no directcontrolon
the phenomena thattake placein ouruniverse,ithas neverthelessto be recognized
that they are often probing the laws of physics in such extreme situations, as we
will never be able to realize in a human controlled experiment. So, while on the
theoretical side we are still struggling to get a unified picture of fundamental in-
teractions at the quantum level, it is undoubted that nowadays this understanding
has to also, if not primarily, face the challenge to be consistent with astrophysical
and cosmological data, in addition to those provided by human-scale experiments:
in this sense, a synergic interplay between large and small scale physics is already
a reality.
With this higher perspective in mind, we will more modestly present, in this
contribution,anexampleofhowcurrentobservationshavethepotentialtoconstrain
emissionmodelsforActiveGalacticNuclei.Ouremphasiswillbeontheimportance
of a solid statistical analysis, a precious approach to obtain a quantitative insight
and guide us in challenging and, hopefully, disproving existing models in favor of
more refined and realistic ones.
Ourpresentationwilltrytobereasonablyself-contained,inthesensethatwewill
cover all the required topics in what follows: we will, however,not be able to cover
them withthe requireddepth, someofwhich canbe obtainedstartingfromthe list
ofreferences.We willstartinSec.1,summarizingsomekeyfactsaboutthe sources
that we will consider here, namely active galactic nuclei. Some more details will be
given for blazars and their emission models in Subsec. 1.1. We will then continue
with a concise presentation of algorithms for (least square) minimization in Sec. 2,
to finally introduce a solid standard, which is the Levenberg-Marquardt approach
(Subsec. 2.3). The problem of statistical significance is then briefly addressed in
Sec.3,wheretheKolmogorov-Smirnovtestisanalyzed(Subsec.3.3):othermethods
to test how bad a fit isa do exist, but will not be discussed here. Following these
prerequisites, Sec. 4 presents details of a recent analysis in which multi-wavelength
datasetsoftheactivegalacticnucleusMarkarian421wherefittedtoagivenemission
model: this section contains three subsections, corresponding to each of the topics
that are presented in the three background sectionsb: Subsec. 4.1 describes the
source and the chosen datasets, Subsec. 4.2 describes the fit algorithm and the fit
results, Subsec. 4.3 describes the statistical significance of the results.
aThecommonlyusednameforthesetestsisgoodness offit tests,although,technically,themean-
ingful results are those inwhich these tests fail, showing inthis way that the model needs to be
refined.
bNumbering has be chosen so that each subsection index matches the number of the section in
whichthecorrespondingtopicisdiscussedingeneral.
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Statistical study of emission versus activity of AGNs 3
1. Active Galactic Nuclei
Active galactic Nuclei are galaxies characterized by a core that appears to be
brighter and more energetic than that of other galaxies. It is, indeed, rather com-
monthatthecoreluminositycompetes,orevenexceeds,theoneofthehostgalaxy:
insome casesit seemsthat aluminosity ofthe orderof104 times the one ofastan-
dard galaxy can be associated with a region with a linear size significantly smaller
than 1pc. In addition, non-thermal AGNs emission covers a very broad range of
the spectrum.
It is usually believed that the engine at the core of AGNs is a black-hole with a
totalmassofmillionsorbillionsofsolarmasses:suchanobjectiscalledasupermas-
siveblackhole.Althoughthereisonlyindirectevidencethatthissupermassivecore
(SMC) is a black hole, there is usually agreement about the fact that the energy
necessary to sustain a luminosity as large as the one mentioned above is coming
fromtheinfallofsurroundingmatterontotheSMC:thisprocessisgenericallycalled
accretion.The details of matter accretiononto the SMC are still to be understood.
AnotherstrikingfeatureofAGNsis,insomecases,thepresenceoftwopowerful,
highlycollimatedjets thatshootoutinoppositedirections.Thewayinwhichthese
jets are formed and other fundamental aspects, for instance related to their com-
position and to the details of the processes involved in the accelerationof particles
inside these jets, are also subject of active research.
From this short,qualitative,introduction it appearsthat there is still a lot that
we have to understand about AGNs. Nevertheless,severalimportant steps forward
have been made and we will concentrate, in what follows, on what we think we do
actually understand. From an historical perspective, it is important to recall that
what nowadays we call AGNs comprises a very wide class of extragalactic objects,
which observationally can (and do) appear very different one from the other. For
this reason, we will report below a standard classification scheme.1
There are several characteristics that can be used to classify AGNs, and their
commongroundisthattheyarenotusuallyfoundinstandardgalaxies.Wealready
mentioned one of them, which is the high luminosity. AGNs can reach luminosities
oftheorderof1048erg s−1,whichis10,000timestheaverageluminosityofagalaxy
·
(intended as the characteristic luminosity of the field galaxy distribution). This is,
the apparentluminosity,whichiswhatwesee afterthe radiationemittedhasbeen,
for instance, absorbedby the circum-objectmedium: the intrinsic luminosity could
then be even higher. The energy output is also distinctive, and characterized by a
broadbandcontinuumemission.Standardgalaxieshaveaspectrumwhich,atzero-th
order can be considered the sum of the black body spectra of all the stars compos-
ing the galaxy: since each star spectrum is approximately a black-body spectrum
with characteristic temperature that equals the surface temperature of the star,
and since the surface temperature of stars spans a range of about one decade, then
a typical galaxy power output is emitted within about one decade of frequency.
On the contrary, AGNs can have a spectrum ranging from the mid-infrared to the
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hardest X-rays, and such that a narrower frequency band dominating the emission
is missing. AGN emission can be moreover characterized by prominent emission
lines, another element of contrast with most standard galaxies. The broadness of
AGNsspectrallinescanbeboth,i)verysharporii)presentbroadwings(withvari-
ability spanning two ordersof magnitudes,depending on the source).The emission
alsopresentsamarkedvariability athighfrequencies(whereas,intheopticalband,
variability is about 10% on human life timescale): there is, nevertheless, a small
class of AGNs (which will be mainly interested in what follows) having a much
more marked, i.e. with much shorter timescales, variability. Although the polariza-
tion of AGNs emission is weak, it can be statistically appreciated when compared
with that of a standard galaxy:a frequency dependence of polarizationis also usu-
ally detected. Finally, AGNs can also be characterized by a strong radio emission,
a phenomenology that has been extensively studied since the beginning of radio
astronomical observations.
An AGN is a source that presents some (not necessarily all) of the above prop-
erties. For instance, the vast majority of AGNs have a spectrum characterized by
a broadband continuum emission, with strong emission lines, some variability and
weak polarization.Many ofthem alsopresenta smallangularsize. Finally,in some
cases radio emission has been detected and, occasionally, the variability and the
polarization are both strong.
Severalobjectsfallwithintheabovedefinitionsandwewillseelateronthatitis
possibletodeviseatentativeunificationscheme,thesocalledunifiedmodel.Wewill
come to it after a concise description of the diverse observational phenomenology.
A first kind of objects that shows properties typical of AGNs are the so-called
quasi stellar sources, or QUASARs. QUASARs show two evident relativistic jets,
buttheirmaincharacteristicisaveryluminousandunresolvednucleuswithangular
size smallerthanthe arc-second.Radioemissionmaybe alsopresent,inwhichcase
they are called, radio QUASARs. QUASARs without radio emission are instead
calledradio quiet QUASARsandtheyareabout20timesmorecommonthanradio
QUASARs. Both radio and radio quiet QUASARs are found at high redshift and
without signs of a surrounding galaxy.
A second kind of objects in the AGN class are radio galaxies. Radio galaxies
are characterized by a radio emission which is thousands times the radio emission
of a standard galaxy: the radio emission is also apparent in two lobes, that extend
in opposite directions outside a bright radio core, to which they appear to be con-
nectedbyemissionjets.Instandardgalaxiestheradioemissioncanbetracedtothe
production of relativistic electrons in supernovae explosions. In radio galaxies, in-
stead, the radio emission, which is characteristicallynon-thermal, can be identified
with synchrotron emission from ultra-relativistic particles. The spectrum, which is
a broad continuum, is markedly non-thermal also in the optical range, where it
especially features superimposed emission lines.
WithsomepropertiessimilartothoseofradioquietQUASARs,Seyfert galaxies
arealsoAGNs,buthavingalowerluminosity.AfurtherclassificationwithinSeyfert
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Statistical study of emission versus activity of AGNs 5
galaxiescanbemadebasedontheirspectrum,thatcanfeaturei)broadbandregions
aswellas narrowlines (these objects arecalledSeyfert1 galaxies)or ii)only broad
lines(thesegalaxiesarenamedSeyfert2).Seyfertgalaxiesaresoclosetoradioquiet
QUASARs that most of the times it is the way in which they are discovered that
makes them members of one class or the other: in particular, in presence of a high
redshift and the absence of a surrounding galaxy the objects tends to be classified
as a QUASAR, while broad emission cores in known galaxies are usually classified
as Seyfert galaxies.
Finally wecome tothe lastgroupofobjectsinthe AGNfamily,namely blazars.
Blazarsarethemostenergeticsourcesthatcanbeobservedinthesky.Theyarealso
characterizedbytwopowerfuljetsshootingoutinoppositedirectionsatrelativistic
speed: their peculiarity is that we are able to view one of them at a small angle to
theobjectaxis(whichisalsothejetemissiondirection),whichmeansthatthejetis
practicallypointedatus.Theemissionspectrumisextremelybroad,fromradiotoγ
frequenciesandallowsanadditionaldistinction:i)whenthespectrumiscompletely
missing emission lines we have the, so-called, BL Lacertae (BL Lac) objects, which
are radio loud objects with marked variability and strong optical polarization; ii)
when,instead,the spectrumalsofeaturesstrongbroademissionlines inthe optical
range, we have what are called Optical Violent Variable objects, which in the radio
band look similar to radio quasars.
We will discuss in more detail some features of AGNs emission, with particu-
lar emphasis on blazars, in subsection 1.1. Before moving to this topic, we would
like, however, remark that despite their apparent difference form the observational
point of view, a unified model to interpretall these observationalfeatures has been
proposed; according to this unified model, all the different features of AGNs are
related to the orientation of the source with respect to the observer.2 All AGNs
would then be compact supermassive objects and they would be surrounded by
an accretion disk composed of hot plasma emitting thermal radiation. The broad
emissionlines thataredetectedinsomespectra,mostlyinthe opticalrange,would
be caused by the presence of a toroidal shape region containing dense molecular
clouds (called the broad emission lines region): the broad emission lines regioncan
obscure the view of the central part. At a larger distance from the central object
there is also a narrow emission lines region also filled with molecular clouds but of
densitylowerthanthe onepresentinthe broademissionlinesregion.Inadirection
perpendicular tothe plane ofthe accretiondisk andinopposite directions,two jest
ofultra-relativisticparticlesemergefromthecentralobjectandextendseveralkilo-
parsecs away from the core. As we anticipated this model allows for explanation
of the AGNs phenomenology, once we consider the orientation of the object with
respectto the line ofsightofthe observer.RadiogalaxiesandSeyfertgalaxieshave
the jets nearlyorthogonaltothe lineofsightofthe observer:the torussurrounding
the core of the object obscures it and reprocesses the radiation coming from the
disk and from the broad emission lines region. The lobes define the region where
the jets loose collimation, and both are responsible for a strong radio emission. A
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completely different picture of the same model is obtained if we change the orien-
tationsothatthe jets nowpoint inthe directionofthe observer’sline ofsight:now
the boostedjet emissionpointing directlytothe observerisdominatingoverallthe
other components, and it is the physics of the jet that is mostly detected. At inter-
mediate angles between the directions in which the line of sightis perpendicular or
parallel to the jets direction, both the central objects and the surrounding regions
are seen: these objects are quasars, and their characteristic emission lines are the
result of the light coming from both the broad and narrow emission line regions.
We will now provide more details on the emission model, with particular em-
phasis on blazars.
1.1. Blazars and emission models
Among the realizations of AGNs that we have briefly described above a very im-
portantroleis playedby blazars.Althoughonly in.10%ofthe casesit is possible
to detect a jet structure in AGNs, a lot ofeffort has been put in understanding the
origin of the jets and the physical processes that allow for particle acceleration in
suchhighly collimatedbeams.Afull modellingofAGNsandAGNsfeaturesforma-
tion and evolution, is still one of the fundamental open problems in astrophysics;
however, it is currently accepted that jets consist of low entropy flows associated
to regions of internal/external shocks, within which the jets dissipate part of their
energy. Since a relativistic jet that moves in a direction that forms a small angle
with the line of sight of the observer is greatly amplified by relativistic beaming,
blazars observations are an exceptional tool to investigate the nature and proper-
ties of AGNs relativistic jets: indeed blazars’ emission is dominated by the jet and
makes these objects the major extragalactic source of γ-rays, despite the fact that
they represent only a minority of the AGNs population. Two major subclasses can
be identified within blazars of the BL Lac type: while all of them show a spectral
energydistribution(SED) withtwopronouncedbumps,thesebumps peak i)inthe
infraredthefirstandaroundGeVfrequenciesthesecondorii)intheX-raythefirst
and around TeV frequencies the second. In particular, BL Lac objects of the first
kind are called low frequency peaked BL Lac, while BL Lac objects of the second
kind are called high frequency peaked BL Lac (HBL).
It is generally agreed that the low energy peak is produced by synchrotron
emission. Models can differ, instead, in the description of the very high energy γ-
ray bump, and can be classified in two qualitative different groups: leptonic models
and hadronic models.
Leptonic models. These models explain the very high energy γ-ray radiation by
inverse Compton scattering of photons off relativistic electrons/positrons.
Ifthescatteredphotonsaresynchrotronphotonscreatedbytheelectronsof
the jet, the models are called synchrotron self Compton (SSC) models.3–5
If, instead, the scattered photons are ambient photons or photons com-
ing from the environment, the models are called external inverse Compton
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Statistical study of emission versus activity of AGNs 7
(EIC) models. The absence of emission lines in the blazars spectra seems
to favor SSC models. SSC models can invoke one region in which the rele-
vant interactions occur (one-zone SSC models), or be refined to take into
account more than one region. In the following we are going to test a one
zone SSC model on a set of nine simultaneous multi-wavelength SEDs.
Hadronic models. In this case the models take advantage from the fact that a
proton component in the jet would be subjected to less synchrotron losses
as compared to electrons and could then be accelerated more efficiently.6
The low energy peak is again explained by synchrotron radiation, so in
these models there is an electron components that is accelerated together
with protons. The higher energy bump is instead produced by the interac-
tionoftheacceleratedprotonswithmatterand/orambientphotonsandor
magnetic fields.7–15 Protoninduced cascades, that can in turn induce elec-
tromagnetic cascades, and/or proton synchrotron models have also been
considered.7,14,15
Itis alsonotexcluded thatboth processescouldcoexistinthe jet.16,17 In whatfol-
lowswewillbeinterestedinaleptonicmodel,specificallyaonezoneSSCmodel.18,19
This model has shown a good agreement with HBL broadband emission in both,
the ground and excited states.20 Moreover, in one zone SSC models, since there is
only one population of electrons that generates the doubly peaked emission, there
is naturally a correlation between the X-ray and the very high energy γ-ray vari-
ability. The emission zone is assumed to be a spherical blob of radius R, moving
with a bulk Lorentz factor Γ in a direction forming an angle θ with respect to the
observer viewing direction. Special relativistic effects can then be described by a
singleparameterδ =(Γ(1 βcosθ))−1.The modelassumesthatthe sphericalblob
−
region is uniformly filled by electrons with density n and by a uniform, tangled,
e
magnetic field B. Five more parameters complete the model providing a descrip-
tion of the relativistic electrons’ spectrum. This is characterized by a smoothed,
broken power-law in γ, the Lorentz factor of the electrons, which is bounded by
γ < γ . The transition between the two power-laws takes place at γ . The
min max br
energy slope at low and high energies are, respectively, n and n . Altogether the
1 2
model has nine free parameters: three of them (δ, R and B) describe the emitting
blob; the other six (n , n , γ , γ , γ , n ) describe the energy distribution of
1 2 min max br e
the electrons’ plasma.
It is crucial to realize that only simultaneous, multi-wavelength observations
allow the determination of all the parameters of the model. In particular, if we
did not have the very high energy γ-ray observations that became available with
modern Cherenkov telescopes, we would have only knowledge of the synchrotron
peak. This would give us information about the electrons’ distribution, but not on
theotherparametersofthemodel,andcertainlyitwouldnothelpustoremove,for
example, the residual degeneracy between the intensity of the magnetic field and
the electron density. This simple example already gives an idea of the importance
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8 Stefano Ansoldi
of simultaneous multi-wavelength observations across all the emission spectrum of
the object.
To determine the parameters of the model that we just described, we will use
a rigorous statistical approach, by fitting the SSC model to several simultaneous
multi-wavelengthdatasetscorrespondingtodifferentactivitystatesofthesource.In
this waywe willbe abletoconstrain,withinsomesignificancelevel,the SSC model
in each emission state (this is the primary goal of this contribution). In turn, this
allows also an analysis of the behavior of the parameters of the model in different
emission states, a point which is discussed in detail elsewhere.21
Before continuing with the analysis anticipated above, we will introduce in the
following sections some basic ideas about non-linear fits and their significance.
2. Nonlinear χ2 fitting
In this section we discuss a technique that allows to identify local minima of real
valued functions and that we will use in the context of non-linear least squares
problems: in particular, our final goal will be to use this technique to minimize
the sum of the squares of the deviations between a given set = (x ,y ,σ )i =
i i i
O { |
1,...,N of measured data points (in which σ is the uncertainty in the measured
i
}
quantityy anduncertaintiesinx areconsideredsmallenoughtobeneglected)and
i i
a model function f(x;p) that depends from a set p=(p ) of n parameters.
j j=1,...,n
Under suitable assumptions it can be proved22 that the optimal values for the
parametersaccordingto the observeddata canbe obtainedbyminimizing the Chi-
Square function, i.e.
1,N 2
1 y f(x ;p)
χ2(p)= i− i .
2 (cid:20) σ (cid:21)
Xi i
Whenthe functionf isalinearfunctionofthe parameters,aclosedformulaforthe
minimization ofχ2 canbe obtained.Here we areinstead interestedin the situation
in which f is a non-linear function of the p . The minimization process can then
j
be performed numerically in several iterations, the goal of each iteration being to
find a perturbation δp of the current values of the parameters p that results in
j j
a lower value of χ2. Several methods can be developed to find the values of the
parameters p that minimize χ2. In view of our final goal, we will concentrate on
j
two of these methods, namely the steepest descent method and the Gauß-Newton
or Inverse Hessian method.
2.1. The steepest descent method
The steepest descent method23,24 is based on the evaluation of the gradient of the
objectivefunction(the χ2 inourcase)withrespecttothe parametersp .Themain
j
idea ofthe methodis thatthe mostdirectpathinthe directionofalocalminimum
is to descend in the direction opposite to the gradient of χ2 with respect to the p .
j
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Statistical study of emission versus activity of AGNs 9
The components of the gradient of χ2, i.e. ∂ χ2 turn out to be
pj
1,N
y f(x ;p)
∂ χ2 = i− i ∂ f(x ;p) .
pj − (cid:20) σ2 pj i (cid:21)
Xi i
For later convenience we will set up a more compact matrix notation for the above
relation, which is
(grad χ2(p))T = (y f)T ΣJ,
p − −
where the gradient is nothing but
grad χ2(p)=(∂ χ2) ,
p pj j=1,...,n
y is the N-dimensional constant vector of the dependent variable observed data,
y=(y ) ,
i i=1,...,N
f is the N dimensionalvector ofthe dependent variable values estimated according
to the model described by f(x,p),
f =f(p)=(f(x ;p)) ,
i i=1,...,N
Σ is the N N diagonal matrix of the weights corresponding to the dependent
×
variable uncertainties in the N measurements ,
O
Σ=diag(σ−2)
i i=1,...,N
and, finallyc, J is the Jacobean matrix of f, i.e.,
J=(∂ f(p)) =(∂ f(x ;p)) .
pj j=1,...,n pj i i=1,...,N
j=1,...,n
A perturbation δp = (δp ) that updates the parameters in the direction of
j j=1,...,n
the steepest descent, i.e. in the direction opposite to the gradient of χ2, can then
be obtained as
δp=µJT Σ(y f),
−
where we used the fact that, since Σ is diagonal, ΣT = Σ. µ is a positive real
number that determines the length of the step in the steepest descent direction.
Based on the above framework, the steepest descent method consists of a se-
quence of parameters updates that are always performed in the direction of the
steepestdescentuntila minimumisfoundwith the prescribedaccuracy.Forsimple
cInthisnotationtheχ2 functioncanbewrittenas
χ2(p) = 1(y−f)TΣ(y−f)
2
= 1yTΣy+yTΣf+ 1fTΣf. (1)
2 2
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objective functions the steepest descent method is recognized as a highly conver-
gentapproachtominimizationand,whenthe number ofparametersishighorvery
high,itcanbeconsideredasthemostreliable,ifnottheonlyviable,method.There
are, nevertheless, weak points of this method, especially for complex models: these
weakpointsaresubstantiallyrelatedtothe factthatthemethoddoesnottakeinto
account the curvature of the surface which, in our case, is the graphic of the χ2.
Because of this, it is possible to make too large steps in steep regions or, too small
stepsinshallowregions:thisclearlyaffectstheconvergenceofthealgorithm.Atthe
same time, particular structures in the χ2 surface, as for instance narrow valleys,
mayalsodamageconvergence.Inthecaseofanarrowvalley,forinstance,wewould
need to move a large step in the direction that points along the flat base of the
valley, but only a small one in the direction perpendicular to the valley walls. If it
is true that second order information, i.e. the use of curvature information about
the χ2 surface, would definitely help to improve the method, it is often (and as we
will see later on,in our casein particular)computationallyexpensive to accessthis
secondorderinformation.Wewillseethatagoodcompromisecanbefound:tothis
end, we need to first analyze another approach to minimization, which we will do
in the following subsection.
2.2. The inverse Hessian method
Anotherapproachthatcanbeusedtodetermineaminimum(inparticularoftheχ2
function described above) is the inverse Hessian (or Gauss–Newton) methodd. To
give a sound motivation to this approach24,25 let us firstconsider a particular case,
i.e. the one in which the dependence of the model function from the parameters is
linear. The model function can then be written as
1,n
f(x,p)=f(0)(x)+(L(1)(x))Tpd=ef.f(0)(x)+ L(1)(x)p ,
j j
Xj
where L(1)(x) is the vector L(1)(x)=(L(1)(x)) . For this linear model
j j=1,...,n
f = f(0)(x )+(L(1)(x ))Tp
i i
(cid:16) (cid:17)i=1,...,N
and
J=((L(1)(x ))T)
i i=1,...,N
so that
f =f(0)+Jp, f(0) d=ef.(f(0)(x )) .
i i=1,...,N
The χ2 is then a quadratic function of the parameters,
1 1
χ2(p)= (y f(0))TΣ(y f(0)) (y f(0))TΣJp+ pTJTΣJp
2 − − − − 2
dWewillusethefirstdenominationhere,althoughthesecondisalsoquitewidespread.