2 The classification algorithm

  
2.1 General remarks on classification

Generally speaking, classification is a process of pattern recognition which usually has to deal with noisy data. Mathematically, a classifier is a function, which is mapping a feature vector of a measured object characteristics onto a discriminant vector, that contains the object's likelihoods for belonging to the different available classes. Any classification relies on the feature space being chosen such that different classes cover different volumes and overlap as little as possible to avoid ambiguities.

If a survey is designed without class definitions in mind, it will be difficult to choose a set of measurable features for a tailored classification. Also, only unsupervised classifiers (= working without knowledge input) can be used to work on measured object lists. In this case, a classifier can find distinguishable classes, e.g. by cluster analysis. This process leads to a definition of new class terms which depends strongly on the visible features taken in account.

For any classification problem, it is of great advantage, if class terms are defined a priori and encyclopedic knowledge is available about measurable features and their typical values. Then models of the classes representing this knowledge can be constructed to serve as an essential input to a supervised classifier (= using input knowledge as a guide). When selecting the features, two potential problems should be avoided: One is the use of well-known but hardly discriminating features, which will obviously not improve the classification but just increase the effort. The other is using features which are not well-known and therefore can easily cause mistakes in the classification. Especially, with high measurement accuracy this can lead to apparent unclassifiability when an object looks different than expected.

Two different types of class models can be distinguished depending on the uniqueness of the classification answer:

1.

In one type of models geometric rules are used to delimit sectors in feature space covered by the classes in competition. These models assign just one class to the measurement uniquely and definitely, which is the one containing the feature vector within its geometric limits. Effectively, the discriminant vector does not contain likelihood values in a statistical sense but instead a single entry "1'' for the class decided on and zeros for the other classes (while nearest-neighbor classifications can define rather complicated boundary shapes in feature space, they also belong to this type);

2.

Another type are statistical class models rendered as likelihood functions which are defined across the entire feature space range. Only these provide discriminant vectors with relative likelihoods of class membership for an object, thereby following a "fuzzy logic'' approach.

While classes are discrete entities, a statistical classification can also work on continuous parameters. The discriminant vector then becomes a likelihood function of the parameter value. Based on this distinction classification problems can be considered as decision problems for discrete variables and estimation problems for continuous variables (Melsa & Cohen 1978a). In either case, a definite statistical classification containes two consecutive steps: First, the discriminant vector is determined (see Sect. 2.2) and second, it is mapped either by decision to a final class or to a parameter estimate (see Sect. 2.3).

  
2.2 Step 1: Determining discriminant vectors

We assume an object with m features being measured by any device, thus displaying the feature vector $\vec q = (q_1, \ldots q_m)$. We consider n classes $c_1, \ldots c_n$ as a possible nominal interpretation and denote the likelihood of this object to belong to the class c_i as $p(c_i\vert\vec q)$. A true member of class c_i has an a priori probability of displaying the features $\vec q$ given by $p(\vec q\vert c_i)$.

Initially, we assume a simple case of uniquely defined class models, where all members of a single class c_i have the same intrinsic features $\vec q_{c_i}$, so that any spread in measured $\vec q$ values arises solely from measurement errors. Assuming a Gaussian error distribution for every single feature, it follows (Melsa & Cohen 1978b), that

$$p(\vec q \vert c_i) = C \exp \left(-\frac{1}{2} (\vec q-\vec q_{c_i}) V^{-1}_k (\vec q-\vec q_{c_i})^t \right) ,$$ (1)

where $(\vec q-\vec q_{c_i})$ is the measurement error in case the object does belong to c_i and $(\vec q-\vec q_{c_i})^t$ is its transposed version. Each feature q_kis measured with its own error variance $\sigma^2_k$, which are the diagonal elements in the variance-covariance matrix V. If all the features are statistically independent, the off-diagonal elements vanish. The normalisation factor C is

$$C = \frac{1}{\sqrt{(2 \pi)^m \vert V_k\vert}}\cdot$$ (2)

As contained in the discriminant vector, the likelihood for an object observed with $\vec q$ to belong to class c_i is then

 

$$p(c_i \vert \vec q) = p(\vec q \vert c_i) / \sum^n_{l=1} p(\vec q \vert c_l) .$$ (3)

However, in realistic cases the classes themselves are extended in feature space and their volume might have rather complicated shapes. In the spirit of Parzen's kernel estimator (Parzen 1963) the extended class c_i can be represented by a dense cloud of individual uniquely defined (point shape) members c_ij. Every member accounts for some a-priori probability to display $\vec q$, given as $p(\vec q\vert c_{ij})$, just as if it were a "class'' on its own. The complete class c_i is now rendered as a superposition of its N_i members and adds up to a total probability of

 

$$p(\vec q \vert c_i) = \frac{1}{N_i} \sum_j p(\vec q \vert c_{ij}) .$$ (4)

In an estimation problem the probability functions have the same form, except for changes in the notion: $\theta$ denotes the parameter to be estimated, and ideally the class model c_i had a continuous shape covering the range of expected values. The discriminant vector would then be a function $p(\theta\vert\vec q)$. Again, the class model can be approximated by a discrete set of members sampling the $\theta$ range of interest at sufficient density.

The astronomical application discussed in this paper poses a mixture of decision and estimation problems which can be realized simultaneously with a unified approach: the decision may choose from the three classes c₁ = stars, c₂ = galaxies and c₃ = quasars, and an estimation process takes care of the parameters redshift and different spectral energy distributions (SED). The internal structure of every class c_i is then spanned by its individual parameter set $\vec \theta_i = \{\theta_{i\vec j} \}$, either following a grid design or being unsorted if no parameter structure is needed.

If one chooses to approximate the spatial extension of a class by a dense grid sampling discrete parameter values, two problems are solved at once: on the one hand, an internal structure is present for estimating parameters, and on the other hand, the class is well represented for calculating its total probability $p(\vec q\vert c_i)$. Altogether, the probability function with internal parameters $\theta_{i\vec j}$being resembled by class members $c_{i \vec j}$ is then

$$p(\vec q \vert \theta_{i \vec j}) = C \exp \left(-\frac{1}{2} (\v... ...vec q(\theta_{i \vec j})) V^{-1}_k (\vec q-\vec q(\theta_{i \vec j}))^t \right)$$ (5)

with the total probability for class c_i being

 

$$p(\vec q \vert c_i) = \frac{1}{N_i} \sum_{\vec j} p(\vec q \vert \theta_{i\vec j}) ,$$ (6)

and the equation for the class likelihood function still being

$$p(c_i \vert \vec q) = p(\vec q \vert c_i) / \sum^n_{l=1} p(\vec q \vert c_l) .$$ (7)

Based on these probability functions the classification can perform a decision between object classes and estimations of redshift and other object parameters at once. Two different analyses are integrated into one paradigm and calculated efficiently by evaluating the same probability density function.

  
2.3 Step 2: Decision and estimation

Decision rules are functions mapping a discriminant vector $\vec p(c_i\vert\vec q)$ to a decision value d. The value d_i denotes a decision in favor of class c_i, i.e. the object displaying features $\vec q$ is then assumed to belong to this class. The most simple decision rule is the maximum likelihood (ML) scheme, which decides for the one class with the highest likelihood p. In case of two classes existing this means

$$\hspace{-1mm} {\mbox{if \quad $p(c_1 \vert \vec q) > p(c_2 \vert ... ...{1mm} \mbox{if \quad $p(c_1 \vert \vec q) < p(c_2 \vert \vec q)$ , then}~~d_2.}$$ (8)

A more compact notion for the same rule is

$$p(c_1 \vert \vec q) \begin{array}{c} {\scriptstyle d}_1 \\ ^>_< \\ {\scriptstyle d}_2 \end{array} p(c_2 \vert \vec q) .$$ (9)

Depending on the purpose of the classification tailored improvements can be made to this rule. The probability of error (PoE) method, e.g., attempts to minimize the rate of misclassifications by including the a-priori-probability for observing a member of a given class. Following Bayes theorem these "priors'', denoted P(c₁) and P(c₂), are just the relative abundance of the class in the whole sample. The PoE decision rule is then

$$p(c_1 \vert \vec q) P(c_1) \begin{array}{c} {\scriptstyle d}_1 \\ ^>_< \\ {\scriptstyle d}_2 \end{array}p(c_2 \vert \vec q) P(c_2) ,$$ (10)

which causes somewhat ambiguous objects to be preferentially classified as belonging to the more common class. Rare objects are then less likely to be found at all, but the overall performance of the classifier improves. A general approach uses any type of priors for trimming the classification towards specific goals, so every decision rule compares the likelihood ratio $\Lambda$ with a threshold T and follows the form (with T=1 for ML decision)

$$\Lambda (\vec q) = \frac{p(c_1 \vert \vec q)}{p(c_2 \vert \vec q)... ...gin{array}{c} {\scriptstyle d}_1 \\ ^>_< \\ {\scriptstyle d}_2 \end{array} T.$$ (11)

Estimation rules are functions mapping a discriminant vector $\vec p(\theta\vert\vec q)$to an estimated value $\tilde\theta$. The most simple estimation rule is again the maximum likelihood (ML) rule, which chooses the one parameter value with the highest likelihood p, i.e., the ML estimator is given by

$$p(\tilde\theta_{{\rm ML}} \vert \vec q) \geq p(\theta \vert \vec q) \qquad \forall \theta .$$ (12)

The Bayesian approach can also be applied to continuous variables, whereas one special case is of particular interest: if the error distribution of the feature measurement is Gaussian, and if the goal is to minimize the variance of the true estimation error, then the optimum estimation rule can be derived analytically (Melsa & Cohen 1978b). This minimum error variance (MEV) estimator is given by

$$\tilde\theta_{{\rm MEV}} = \frac{\int \theta p(\theta \vert \vec ... ...eta) \,{\rm d}\theta} {\int p(\theta \vert \vec q) P(\theta) \,{\rm d}\theta} ,$$ (13)

and it is equivalent to interpreting the discriminant vector as a statistical ensemble and determining the mean of the distribution. It is also dubbed mean square estimator or conditional mean estimator. Note that, if $p(\theta\vert\vec q)$ is symmetric in $\theta$ and unimodal, the MEV estimator is identical to the ML estimator.

  
2.4 Application to astronomical multi-color surveys

Deep extragalactic surveys usually contain mostly galaxies, fewer stars and a tiny fraction of quasars, with relative numbers on the order of 100:10:1. A survey at galactic latitudes above with a limiting magnitude of R=23and an area of 1 $\ifmmode\hbox{\rlap{$\sqcap$ }$\sqcup$ }\else{\unskip\nobreak\hfil \penalty50\h... ... }$\sqcup$ } \parfillskip=0pt\finalhyphendemerits=0\endgraf}\fi\hbox{$^\circ$ }$, e.g., should contain roughly 30000 galaxies (Metcalfe et al. 1995), some 3000 to 6000 stars (Bahcall & Soneira 1981; Phleps et al. 2000), and about 400 quasars including Seyfert-1 galaxies (Hartwick & Schade 1990). Any classification would ideally be capable of distinguishing all three classes of objects. Only in surveys, which do not care about the rare quasars, their class could be dropped and the classification needed to separate only stars from galaxies.

In addition to the class itself, plenty of physical parameters could potentially be recovered from an object's photometric spectrum. Most importantly, we would like to determine redshift estimates for galaxies and quasars. In addition, the spectral energy distribution of galaxies contains information about their star formation rate and the age of their stellar populations. A photometric spectrum of sufficiently high spectral resolution can even allow to estimate the intensity of emission-lines. Finally, the spectra of stars tell mostly their effective temperature, but also their metallicity and their surface gravity.

The literature provides abundant knowledge of spectral properties for all three object classes. Synthetic photometry can use published spectra together with efficiency curves from the survey filter set in order to obtain predicted colors of objects. Sometimes, model assumptions are needed to fill in data gaps present in the literature, which could either be gaps on the spectral wavelength axis or gaps on physical parameter ranges, e.g. star-formation rate. Eventually, systematic multi-color class models can be calculated from published libraries covering various physical parameters. These can serve for later comparison with observed data. Therefore, we decided to build a statistical classification based on published spectral libraries and a limited number of model assumptions (see Sect.3).

In a multi-color survey the dominant information gathered are the object fluxes in the different filters. We decided to use the color indices as an input to the classification rather than the fluxes themselves, which eliminates one dimension from the problem by omitting the need for any flux normalisation, that remains as an additional fit parameter in template fitting procedures. It will be shown in Sect.2.5, that the color-based approach is equivalent to the flux-based one under certain constraints.

Morphological information is typically also available to some extent and can be included in the classification based on the assumption that only galaxies are capable of showing spatial extent. But this should be done carefully, since luminous host galaxies can render quasars as extended. Also, if the image quality varies across the observed field, the morphological analysis is of limited use for not clearly extended sources.

We define the color q_g-h as a magnitude difference between the flux measurements in two filters F_g and F_h:

$$q_{g-h} = m_g-m_h = -2.5\log\frac{F_g}{F_h}\cdot$$ (14)

Obviously, the color system depends on the filter set chosen and also on the flux normalisation used. As long as the flux errors are relatively small, the linear approximation of the logarithm can be used to express magnitude errors as $\sigma_{m_i} \approx \sigma_{F_i} / F_i$, so that the error of the color is

$$\sigma_{q_{g-h}} = \sqrt{\sigma^2_{m_g} + \sigma^2_{m_h}} \approx \sqrt{ (\sigma_{F_g} / F_g)^2 + (\sigma_{F_h} / F_h)^2} .$$ (15)

Since the likelihoods determined for the classification depend sensitively on the colors $\vec q$ and their errors $\vec \sigma_q$, both values must be carefully calibrated. If any color offset is present between measurement and model, the classification will go wrong systematically. If errors are underestimated, the likelihood function could focus on a wrong interpretation, rather than including the full range of likely ones. Overestimated errors will obviously diffuse the likelihoods and give away focus which is originally present in the data. The approximation of errors as presented will only work well with flux detections of at least 5$\sigma$ to 10$\sigma$, but at lower levels the classification is likely to fail anyway, so we ignore this concern.

Given $\vec q$ and $\vec \sigma_q$ a measured object is represented by a Gaussian error distribution rather than a single color vector. If colors are measured very accurately and the object is rendered as a narrow distribution, it could possibly fall between two grid steps of a discrete class model and "get lost'' for the classification. In this case low likelihoods would be derived despite the spatial proximity of object and model in terms of metric distance. The likelihood function would appear not much different from that of a truely strange object residing off the class in an otherwise empty region of color space. In technical terms, the classification would violate the sampling theorem (Jähne 1991), and the probability functions would not be invertible any more.

For discrete class models the sampling theorem requires that every measurement falling inside the volume of a model should "see'' at least two model members inside of its Gaussian core. Due to practical limitations of computing time and storage space, it does not make sense to develop discrete models with virtually infinite density accounting for arbitrarily sharp measurements. Also, for measurements with low photon noise the dominant source of error will be the limited accuracy of the color calibration.

The solution to the problem is then to design the discrete model with the achievable measurement accuracy in mind, and to smooth the discrete model into a continuous entity by convolving its grid with a continuous function that is wide enough to prevent residual low-density holes between the grid points. A sensible smoothing width would just fulfill the sampling theorem, i.e. the smoothing function should roughly stretch over a couple of discrete points. As a result, even an extremely sharp measurement will be covered by the model and classified correctly.

Higher resolution would only increase the computational efforts while lower resolution would ignore information which is present in the data and therefore potentially worsen the classification. From a different point of view, one could leave the discrete model unchanged and claim the data to have larger effective errors by including the calibration errors thereby limiting the width of the Gaussian data representation to a lower threshold, which will always ensure the sampling theorem on the discrete grid anyway.

Both approaches are mathematically identical, if one chooses to represent the calibration errors as well as the smoothing function by a Gaussian. Due to the symmetry of the Gaussian function, convolving the discrete grid or convolving the error distribution of the data yields the same result. The choice of the Gaussian is computationally very efficient, because the convolution of the Gaussian measurement with the Gaussian calibration error results in another Gaussian of enlarged width. As mentioned in Sect.5.1 and discussed in Paper II, a survey in the visual bands can be calibrated with a relative accuracy on the order of 3% between the different filters. Therefore, we decide to apply a $0\hbox{$.\!\!^{\rm m}$ }03$-Gaussian as a smoothing function.

In summary, we apply the formalism presented in Sect. 2.2 in the following way: the errors $\sigma_{q_i}$ of the colors q_i are convolved with the smoothing $0\hbox{$.\!\!^{\rm m}$ }03$-Gaussian and as a result the effective errors are

$$\sigma^2_i = \sigma^2_{q_i} + (0\hbox{$.\!\!^{\rm m}$ }03)^2 .$$ (16)

For simplicity, we assume the individual colors to be uncorrelated, which is actually not true for filters sharing spectral regions in their transmission. The variance-covariance matrix then becomes diagonal

$$V_k = \left( \begin{array}{cccc} \sigma^2_1 & & & \\ & \sigma^2_2 & & \\ & & \ddots & \\ & & & \sigma^2_m \end{array} \right) ,$$ (17)

and the probability function turns into

$$p(\vec q \vert c_i) = \frac{C}{N_i} \sum_{j=1}^{N_i} \exp \left(-... ...{2} \sum_{k=1}^m \left( \frac{q_l - q_{c_{ij},k}}{\sigma_k} \right)^2 \right) .$$ (18)

Based on the three object classes discussed the likelihood function is

$$p(c_i \vert \vec q) = \frac{p(\vec q \vert c_i)}{p(\vec q \vert c... ... + p(\vec q \vert c_{{\rm galaxies}}) + p(\vec q \vert c_{{\rm quasars}})}\cdot$$ (19)

Considering three classes implies that extremely faint objects with large errors get average probabilities of 33% assigned for all classes. In general applications, we use a decision rule for an object seen as $\vec q$, which requires that one class is at least three times more probable than the other two classes put together, i.e.:

If there is one class with $p(c_i\vert\vec q) > 0.75$, then we assign this class to the object, but if all classes have likelihoods below 0.75, we call it unclassifiable.

For the detection of unusual objects, we look at the color distance of an object to the nearest member of any class model to derive a statistical consistency with the class. The value of this consistency depends on the different color variances and can be calculated from $\chi^2$-statistics. Lacking an analytic expression we use $\chi^2$-tables (Abramowitz & Stegun 1972) to evaluate the statistical consistency between class and object. In practice, the resulting $\chi^2$-values need to be normalised to a plausible scale, since the raw values obtained are enlarged artificially due to the discrete sampling of the library and cosmic variance. We use the following operative criterion for the selection of unusual objects:

If an object is inconsistent at least at a confidence level of 99.73% (i.e. 3$\sigma$ in case of a normal distribution) with all members of all classes, then we call it strange.

Strange objects can formally be classifiable, if the likelihoods still prefer a certain class membership. They have either intrinsically different spectra without counterparts in the class models, or they are reduction artifacts, e.g. when neighboring objects affect their color determination, and this is not taken into proper account for the error calculation.

Apart from the rather trivial ML estimator, we use the MEV estimator to obtain redshifts and SED parameters of galaxies and quasars. Their class models are designed as regular grids (see Sect.3) with members c_ij residing at redshift z_ij. The MEV estimator for the redshift is then

$$\langle z \rangle _{{\rm MEV}} = \frac{\sum_j z_{ij} p(\vec q \vert c_{ij})} {\sum_j p(\vec q \vert c_{ij})}\cdot$$ (20)

It is applied to the class models for galaxies and quasars independently and for each class interpretation an independent redshift estimate is obtained. There is also an assessment for the likely error of the z estimate given by the variance of the distribution $p(\vec q\vert z)$:

$$\sigma^2_z = \frac{\sum_j (z_{ij}-\langle z \rangle _{{\rm MEV}})^2 p(\vec q \vert c_{ij})} {\sum_j p(\vec q \vert c_{ij})} \cdot$$ (21)

This estimation scheme would be sufficient, if models had a rather simple shape in color space, i.e. if color space and model parameter space could easily be mapped onto each other. In fact, the class model for galaxies and particularly the one for quasars can have very complicated folded shapes in color space, so that the distribution $p(\vec q\vert z)$ can have a correspondingly complicated structure that is not at all well described by mean and variance.

Therefore, we distinguish three cases: unimodal (single peaked), bimodal (double peaked) and broad distributions. In unimodal cases $\langle z \rangle _{{\rm MEV}}$ and $\sigma_z$ are appropriate reductions of $p(\vec q\vert z)$. In bimodal cases we split the redshift axis in two intervals delimited at $\langle z \rangle _{{\rm MEV}}$ and obtain two alternative unimodal solutions with relative probabilities given by the p sums in the two intervals. If the distribution is so broad, that it starts to resemble a uniform distribution, $\langle z \rangle _{{\rm MEV}}$ approaches the mean z value of the model and $\sigma_z$ approaches $\sqrt{1/12} (z_{{\rm max}}-z_{{\rm min}})$. In order to keep our statistics clean from such mean redshift contaminants, we cut off the estimator at some uncertainty:

If an object has $\sigma_z > 1/8 (z_{{\rm max}}-z_{{\rm min}})$, then we ignore the MEV estimate and call its redshift uncertain.

In particular, it is possible, that an object has a bimodal distribution with one peak (result) and one broad (uncertain) component. In the following, we denote this extended scheme of MEV estimates accounting for possible bimodalities as our MEV+ estimate. In Sect.4.4 we will compare the performance of all three estimators, ML versus MEV and MEV+.

An effort was made to implement a classification code optimized for short computing time. The use of precalculated class models eliminates any synthetic photometry from a typical fitting procedure. Furthermore, the use of colors instead of fluxes eliminates the need for finding a flux normalisation. In terms of CPU time, the classification of one object contains mainly the calculation of the probability $p(\vec q\vert c_{ij})$ for every class member, which involves first adding up all $\sigma^2_i$-scaled squared color differences and second evaluating an exponential function of the resulting sum that is already a measure of strangeness. Summing up the $p(\vec q\vert c_{ij})$ to obtain class likelihoods and deriving mean and variance of the internal class parameters should take less time than calculating the probability density function, if more then ten color axes are taken into account. With class models containing about 50000 members and 13 colors, the full classification of one object takes about 0.3sec when running on a 200MHz Ultra Sparc CPU inside a SUN workstation. Since different survey applications might require different sample selection schemes, we decided to calculate and store discriminant vectors for all objects and select subcatalogs for further analysis later.

  
2.5 Equivalence of flux-based and color-based classification

We now show, that the color-based classification yields the same best fit as a flux based template fitting algorithm. Lanzetta et al. (1996), e.g., calculate a likelihood function depending on redshift z, a spectral energy distribution and a flux normalisation parameter A, following the form:

$$L_{{\rm model}} = \exp \left(-\frac{1}{2} \sum_k \left( \frac{F_{k,{\rm obs}} - A \tilde F_{k,{\rm model}}}{\sigma_{F_k}}\right)^2 \right) .$$ (22)

Basically, the likelihood determination relies on the squared photometric distance d between observation and model, resulting from the flux differences $\Delta F_k$in each filter:

 

$$d^2 = \sum_k^n \chi_k^2 \quad \mbox{with} \quad \chi_k = \frac{F_... ...obs}} - F_{k,{\rm model}}}{\sigma_{F_k}} = \frac{\Delta F_k}{\sigma_{F_k}}\cdot$$ (23)

In the color based approach there are n-1 color indices contributing distance components and we assume the single constraint, that there is one particular base filter approximately free of flux errors, e.g. a deeply exposed broad-band filter. The color indices are made by comparing any filter to this base filter ensuring optimum errors for the colors. In this scheme, any errors in the relative calibration are absorbed into the color indices. Therefore, it is very important, that the base filter is not wrongly calibrated with respect to the other wavebands, since the error would spread into the entire vector of color indices.

We then look only at a range of good fits, and do not mind rather crude $\chi$-approximations for relatively bad fits which are anyway ruled out as solutions. Also, we consider only measurements with , which allows the assumption of Gaussian color errors and a linear approximation of the logarithm. The distance components are:

$$\chi_k \frac{(m_k-m_{{\rm base}})_{{\rm obs}}-(m_k-m_{{\rm base}})_{{\rm model}}}{\sigma_{m_k-m_{{\rm base}}}}$$ =

$$2.5 \frac{{\rm log} (F_k/F_{{\rm base}})_{{\rm obs}} - {\rm log} ... ...} {\sqrt{(\sigma_{F_k}/F_k)^2+(\sigma_{F_{{\rm base}}}/F_{{\rm base}})^2}}\cdot$$ = (24)

Using the terms $\Delta F_k$ and $\sigma_{F_{{\rm base}}}/F_{{\rm base}} \approx 0$, we obtain

$${\chi_k \approx 2.5 \frac{F_k}{\sigma_{F_k}} \cdot}$$

$$\left\{ {\rm log} \left(1+\frac{\Delta F_k}{F_{k,{\rm model}}} \r... ...} \left(1+\frac{\Delta F_{{\rm base}}}{F_{{\rm base,model}}} \right) \right\} .$$ (25)

Expanding the logarithm for $\Delta F_k/F_k \ll 1$, we get

$$\chi_k \approx \frac{F_k}{\sigma_{F_k}} \left\{ \frac{\Delta F_k}{F_{k,{\rm model}}} - \frac{\Delta F_{{\rm base}}}{F_{{\rm base,model}}} \right\}$$

$$\approx \frac{\Delta F_k}{\sigma_{F_k}} + \frac{\Delta F_k}{\sigma_{F_k}}... ...frac{\Delta F_{{\rm base}}}{\sigma_{F_k}} \frac{F_k}{F_{{\rm base,model}}}\cdot$$ (26)

The first term is typically on the order of 1, while the second term is on the order of $\sigma_{F_k}/F_k \ll 1$ and the third one of $\sigma_{F_{{\rm Base}}}/\sigma_{F_k} \ll 1$. Therefore, the last two terms can be dropped and the expression for $\chi_k$reduces to

$$\chi_k \approx \frac{\Delta F_k}{\sigma_{F_k}} ,$$ (27)

which is identical to the expression used in the flux template fitting method shown in Eq. (23).

2.6 System of color indices

In the previous section, we had discussed the relevance of a common base filter for the various color indices, which is supposed to have relatively small flux errors in order to keep the color errors as low as possible. Our multiband survey applications usually involve a smaller number of broad bands as well as a larger number of medium-band observations. For these, we decided to form color indices from broad bands neigboring on the wavelength axis, i.e. U-B, B-V, V-R and R-I, which we assume to be the optimum solution for comparably deep bands. Each of the shallower medium bands we combine with the most nearby broad-band in terms of wavelength, which then serves as a base filter for the medium-band color indices, e.g. B-486 or 605-R, where letters denote broad bands and numbers represent the central wavelength of medium-band filters measured in nanometers.

In terms of flux template fitting, this scheme of color indices means, that we use a few deep broad bands to fit the global shape of the SED, and then use a few groups of medium bands around each deep broad-band to fit the smaller-scale shape locally. The $\chi^2$-values of the global fit and the several local fits are then just added up to the total $\chi^2$. This scheme has a particular advantage over a solely global flux fitting: the local fits can well trace spectral structures, even if the global distribution of the object differs from the template (e.g. as it could be caused by extinction). Therefore, it is not too dependent on accurate global template shapes and it can use the ability of the medium bands to discriminate narrow spectral features for a more accurate classification. Of course, this advantage vanishes immediately for a pure broad-band survey, where local structures in the spectrum are not traced, and therefore no local fits are available for the $\chi^2$-sum.