Improving CMD Areal Density Analysis: Algorithms and Strategies

Journal of Astronomy and Space Sciences.
2014.
Jun,
31(2):
121-130

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

- Received : May 20, 2014
- Accepted : May 30, 2014
- Published : June 15, 2014

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Essential ideas, successes, and difficulties of
Areal Density Analysis
(
ADA
) for color-magnitude diagrams (
CMD
's) of resolved stellar populations are examined, with explanation of various algorithms and strategies for optimal performance. A
CMD
-generation program computes theoretical datasets with simulated observational error and a solution program inverts the problem by the method of
Differential Corrections
(
DC
) so as to compute parameter values from observed magnitudes and colors, with standard error estimates and correlation coefficients.
ADA
promises not only impersonal results, but also significant saving of labor, especially where a given dataset is analyzed with several evolution models. Observational errors and multiple star systems, along with various single star characteristics and phenomena, are modeled directly via the
Functional Statistics Algorithm
(
FSA
). Unlike
Monte Carlo
,
FSA
is not dependent on a random number generator. Discussions include difficulties and overall requirements, such as need for fast evolutionary computation and realization of goals within machine memory limits. Degradation of results due to influence of pixelization on derivatives,
Initial Mass Function
(
IMF
) quantization,
IMF
steepness, low Areal Densities (𝒜), and large variation in 𝒜 are reduced or eliminated through a variety of schemes that are explained sufficiently for general application. The Levenberg-Marquardt and MMS algorithms for improvement of solution convergence are contained within the DC program. An example of convergence, which typically is very good, is shown in tabular form. A number of theoretical and practical solution issues are discussed, as are prospects for further development.
Areal Density Analysis
(
ADA
) (
Wilson 2001
,
Wilson & Hurley 2003
, hereafter, WH03) evaluates age, metallicity, binary fraction, and other parameters from color-magnitude diagrams (
CMD’
s)
1
of star clusters and other populations
2
. Present software consists of a direct program that accepts parameters and computes a
CMD
and an inverse program that evaluates parameters of an observed
CMD
according to the Least Squares criterion via the
Differential Corrections
(
DC
) algorithm. Although
DC
is well known, with many applications in the sciences,
ADA’
s application of
DC
turns out to be rather intricate with many sub-schemes needed for viable operation. Day to day progress and setbacks are reminiscent of internal combustion engine development in earlier times, with adventures and frustrations as one tries to come up with a new kind of engine and coordinate its many parts so that the engine runs well. The incentive is what can be done when the engine does run well - quantify basic information on resolved stellar populations impersonally, and with reduced human effort. The
ADA
engine now runs and gives good results for synthetic data but real star populations must be approached with caution.
In its present form, ADA
The last point is important, as results are sensitive to photometric conversion accuracy.
Not crucial but convenient is window specification, which allows a given data file to serve for experiments on a variety of
CMD
sub-regions. Data windows now are defined by the corner coordinates of rectangles, although only minor revision will be needed to permit more general shapes.
ADA is not
So
ADA
is for follow-up on numerical experimentation, fine tuning, impersonal results, standard error estimates, and correlation coefficients. Future astrophysically informative experiments include analysis of several regions within a globular cluster. Usually those will be similar enough for one set of starting parameter values to serve for all regions, so preliminary experimentation need be done only once. In any case, the direct program makes experimentation easy.
ADA
currently has the following two problems:
CMD
distribution, as conceptualized in the familiar wire-like isochrone, but also in the number density of stars within the form. This point is well recognized (
e.g
.
Tolstoy & Saha 1996
,
Dolphin 1997
,
Vandenberg 2000
), but the kind of analysis described here differs in major ways from all others, for example by direct inclusion of multiple star systems and by generation of standard errors and parameter correlation coefficients. It is specifically geared to impersonal extraction of numbers such as age, metallicity, and binary fraction.
Furthermore, not only does number density vary along the form, but the form is not one dimensional - its sequences have intrinsic widths for several reasons and are not symmetrical in width because of multiple star systems. The isochrone notion could be generalized to encompass the whole distribution, but that is not its ordinary meaning, so
ADA
essentially abandons the isochrone idea. The Least Squares criterion allows a variety of choices for the solution scheme (
e.g. DC, Simplex, Steepest Descent,
many others). Consideration of these points indicates
CMD Areal Density, 𝒜,
as the logical observable quantity for analysis, by standard
Least Squares
.
In comparison with other schemes for reaching a solution,
DC
has attractive characteristics of fast convergence, straightforward generation of correlation coeffcients, and standard error estimates that come from the covariance matrix in the usual way and do not require auxiliary numerical experiments. Its iterations can lead into a local minimum of variance that may not be the deepest minimum, but other solution algorithms have the same problem. Practical counter-measures are willingness to re-start from several points in parameter space and good intuition for starting points.
ADA
’s version of
DC
solves a main parameter set and any of its subsets for the same computational cost as for the main set alone, thereby allowing easy changes in the fitted parameters as a solution progresses.
DC
is well tested, with astronomical applications at least as far back as the 1930’s, and its operation within
ADA
is explained in
Wilson (2001)
and WH03. Other minimization algorithms would be satisfactory for
ADA
but do not produce standard errors and correlation coefficients. Most alternatives also are slower overall than
DC
, due to slow convergence.
DC
so that part of the
ADA
engine is easy to change. Of course the evolution routine must not only be accurate and reliable, but also very
fast
, as enormous numbers of stars are to be evolved. Why are so many stars evolved?
ADA DC
program can adjust any combination of the following 20 parameters, although experimentation has mainly been limited to subsets of six or fewer. The parameters are:
The six parameters for which there is a significant basis in experience are
Z, A_{V}, A_{wide}, V − M_{V}, T,
and
N_{stars}
. The
Z, A_{V},
and binary mass ratio distributions are not handled by Monte Carlo techniques but by the far more efficient
Functional Statistics Algorithm
(
FSA
), which is explained in
Wilson (2001)
and WH03. FSA achieves the statistical aims of Monte Carlo
without random numbers
, and its efficiency keeps typical
ADA
iterations within the range of 15 to 30 minutes rather than the days to weeks that would be needed under Monte Carlo.
DC
’s numerical derivatives not be unduly affected by pixel quantization. The
CMD
dots that represent star systems can be visualized to be in motion through the pixel structure as parameters (
p
) are varied, and
DC
’s derivatives ∂𝒜/∂
p
are supposed to quantify that motion. However the contribution of a star system to the various ∂𝒜/∂
p
’s will be zero under simple binning unless the dot crosses a pixel boundary, in which case there will be a discontinuous jump. Of course
DC
’s residuals, δ𝒜 = 𝒜
_{o}
− 𝒜
_{c}
, also will be adversely affected. The numerous zero contributions with occasional big jumps may tend to average out for pixels with thousands of stars, but certainly they introduce enormous noise. However that noise is avoidable through pixel sharing, which leads to proper contributions to ∂𝒜/∂
p
for within-pixel motion and completely eliminates boundary-crossing discontinuities. In its simplest version, pixel sharing replaces the dot of ordinary binning with a plotting element that has the size and shape of a pixel. Theoretical 𝒜's thereby become smoother and their derivatives become
entirely free of noise due to boundary crossings
, thus greatly improving solution convergence and stability.
i.e.
low, medium, and high mass), with each regime characterized by a specific exponent, denoted by
p
.
ADA
’s generation of zero age primaries is by uniform spacing of
n_{bin}
stars within a given mass bin in each of the three regimes according to
where
n_{regime}
is the number of stars in the mass regime,
m
_{1}
and
m
_{2}
are the regime boundaries, and
m_{bottom}
and
m_{top}
are the bin boundaries. A serious quantization problem can arise because, although the formula predicts
n_{bin}
as a floating point number, the actual number of stars to be evolved must be rounded to an integer. Consequences will be that the stars thereby evolved do not fairly represent the IMF, and that jumps in 𝒜 can occur as parameters
p
and
N_{stars}
vary. The situation is somewhat complicated, as each of the many combinations of primaries and companions can evolve to its own
CMD
pixel. The fix-up is to record the ratio of rounded to unrounded
n_{bin}
’s and correct the appropriate 𝒜’s separately for each system and its (up to 4) pixels. These corrections can be very important, especially where mass bin counts are low, but the corrections effectively compensate for
IMF
quantization.
N_{stars}
, as defined above. Although nothing can be done to improve the observational situation, theoretical counts can be increased by spacing stars more finely on the IMF and thereby increasing their number by a re-scaling factor 𝒜 that can be made large enough for theoretical 𝒜’s to be precise. Division of directly produced 𝒜’s by ℛ then produces correct 𝒜’s. The practical limit on ℛ will be set by computation time, so the idea is to have the product ℛ
N_{stars}
large enough for departures from smoothness in pixel 𝒜’s to be unimportant. The opposite difficulty can be encountered for globular clusters, where
N_{stars}
may be large and lead to excessive computation time. The simple expedient is to set ℛ smaller than unity for rich clusters, and perhaps even much smaller. Given that the observations are definite and need only minor computations, overall computational needs are set by the requirement of having smooth theoretical 𝒜’s, which ℛ can control.
IMF
. Suppose the number of theoretical stars to be evolved is very large, perhaps because the re-scaling factor ℛ is something like 100, or perhaps just because
N_{stars}
is large. Without further action, the number of low mass stars will be extremely large. An ultra-simple remedy could be to have the same number of stars in all
IMF
bins and then convert the resulting 𝒜 contributions according to a realistic
IMF
, but there is a better way - the situation is made more flexible by operating with a fictitious
IMF
, to generate theoretical stars, in addition to the real physical
IMF
. A correction for each
IMF
bin is easily computed, tagged onto each star system, and subsequently used to correct its 𝒜 contributions to the appropriate
CMD
pixels. For example, WH03 adopted exponents
p
_{1,2,3}
= −0.30 for all three
IMF
regions in generating theoretical stars and then corrected each star’s 𝒜
’s
to what they would be with
KTG
’s recommended exponents of
p
_{1}
= −1.3,
p
_{2}
= −2.2, and
p
_{3}
= −2.7. The same exponents were used for this paper’s experiment.
ADA
pixels.
Typical distribution of 𝒜 residuals. Each dot represents a pixel.
Note that the distribution is significantly asymmetrical, as is most easily seen by comparing its two tails. The reason is that areal density is logically positive, so observed
ADA
is capped at a lower value of zero but is not capped at any definite upper value. Theoretical
ADA
’s have been made as smooth as we can manage via the schemes discussed above (pixel sharing, re-scaling, rounding within mass bins) so they have neither large positive nor large negative excursions, therefore having small positive values in lightly populated pixels. Statistical fluctuations of the
observed
areal densities thereby make a long tail on the positive residual side that is not matched on the negative side. Suppose one were apply pixel sharing to the observed points. In ordinary pixel sharing where the plotting element has fixed size, the
δ
𝒜 distribution can be made more nearly symmetrical by use of larger pixels, thereby smoothing the 𝒜
_{o}
’s, but at the cost of diminished resolution in high density areas where asymmetry of
δ
𝒜’s is not a significant problem. The general remedy is to have the plotting element size vary with
CMD
location, being large in low density areas and small in high density areas. An optional refinement on
DC
’s pixel sharing has been to allow the plotting element for observed points to be larger or smaller than a pixel, a feature that can make residual distributions more nearly Gaussian while preserving resolution. The acronym is
VDF
, for
Variable Density Factor
, with the local
VDF
based on a pixel’s observational counts. Where the plotting element turns out to be small (high density pixels), it is nearly like plotting a dot for an observed point (see
Fig. 2
), so loss of resolution is negligible in high density regions.
Illustration of plotting elements for ordinary pixel sharing (left) and Variable Density pixel sharing (right). The main purpose of pixel sharing is to allow accurate numerical differentiation of areal density with respect to solution parameters. Variable Density pixel sharing operates with large plotting elements in low density areas for good averaging, and small elements in high density areas for good resolution. Although the larger illustrated element is only modestly larger than a pixel, actual situations may call for very large elements.
DC
iterations can diverge, although that is true of many iterative Least Squares solution algorithms. The problem is not ordinarily caused by inaccurate derivatives,
∂f/∂p
(where
f
is an observable function and
p
is a parameter), but by absence of second and higher derivatives in usual formulations, coupled with parameter correlations. In pictorial terms, standard
DC
tries to traverse a curved surface in multi-dimensional parameter space by linear algebra. That will work in short steps, but correlations can lure
DC
into taking long steps. Introduction of higher derivatives usually is not feasible, especially if one considers the need for cross-derivatives,
∂
^{2}
f
/
∂p
_{1}
∂p
_{2}
. Maintainance of good accuracy in numerical second derivatives also is difficult.
A very effective way to combat the problem is the
Levenberg-Marquardt
(
L-M
) scheme (
Levenberg 1944
,
Marquardt 1963
), which effectively merges
DC
with the
Steepest Descent
solution method so as to exploit the best convergence characteristics of both. With
w
written for weight and
r
for (observed - computed) residual, the Method of
Steepest Descent
examines the local variance, Σ
wr
^{2}
, and follows its negative gradient.
Steepest Descent
can be very slow but is resistant to convergence troubles, whereas
DC
’s convergence is fast but trouble is a possibility.
Marquardt (1963)
introduced a (typically small) parameter
λ
that is added to each diagonal element of the normal equation matrix after the equations have been rescaled so as to have diagonal elements of unity (so the diagonal elements become 1 +
λ
). Marquardt also gave an intuitive view of the merged scheme, whose iterations are fast and greatly improved in reliability. Some authors continually revise
λ
as a solution progresses, with the aim of reducing it to zero at the end, but experiment shows that a very wide range of
λ
(say 10
^{−4}
to 10
^{−10}
) gives essentially the same parameter corrections, regardless of whether the operating point is close to or far from the Least Squares minimum. Accordingly,
λ
was 10
^{−5}
throughout the iterations of this paper.
Another very good way to strengthen convergence is the
Method of Multiple Subsets
, or
MMS
(
Wilson & Biermann 1976
), whose idea is to simplify the correlation matrix by breaking it into smaller pieces. Correlations are not necessarily smaller under
MMS
, but
DC
needs to deal with fewer correlations. As before, the Least Squares problem must cope with non-linearities, correlations, and many dimensions. The
MMS
strategy is to reduce the number of dimensions and thus reduce overall complexity. All that need be done is to operate with some number of subsets of the full parameter set. For an example with six parameters,
DC
might alternate between two subsets of three parameters each. There seems to be a misconception that the
MMS
may be needed
away from the Least Squares
minimum but not at the minimum (
De Landsheer 1981
). However that is not true - if
MMS
is needed, it is needed all the way, including exactly at the minimum (
Wilson 1983
).
Sometimes neither
L-M
nor
MMS
is needed and sometimes one or the other is enough, while especially difficult situations require both. The two schemes are in
ADA
’s
DC
program, which includes
L-M
in the way mentioned above, and
MMS
via the option of multiple subset solutions as part of each program submission.
CMD
sequences and branches (
e.g.
main sequence, subgiant branch, etc.), and of course the
CMD
as a whole, will differ statistically between observation and theory if noise is omitted from theory. Theoretical patterns will be too narrow and there will be no chance of a meaningful match if this issue is ignored. Photometric errors enter the problem differently than do statistical star count fluctuations. Star count statistics appear as
δ
𝒜 fluctuations, whereas photometric errors show up as wrong
CMD
locations, perhaps in the wrong pixel, so the pattern broadenings discussed here are caused by photometric errors. Photometric noise has several essentially Gaussian sources such as sky transparency variations and photon counting statistics (both simulated in the present
ADA
model), with the latter dominant in magnitudes and color indices of faint stars. Errors due to unresolved optical multiples also change
CMD
locations and have distinctly non-Gaussian statistics. The analogous pattern mis-match situation in ordinary
x,y
graphs is uncommon, but that is because many
x,y
problems that involve counts (and therefore bins) have time as the
“x”
variable, and time can be measured very accurately. If not time, then frequency or wavelength (also measured precisely) may be the
“x”
variable, so pattern mis-matches tend to be unusual in x,y fitting and may seem characteristic of higher dimensions. In reality the mis-matches are un-related to dimensionality. Accordingly,
ADA
needs to model photometric errors and apply them to
CMD
locations. A model parameter (or more than one) is needed for noise in each photometric band - for example two parameters for
V vs. V-I
. A convenient choice is the set [
N
_{V=0}
;
N_{I}
_{=0}
], the numbers of photon counts for a star of zero magnitude in each observed band.
Comparison of an observed test case (left) and the ADA fit (right) in V vs. V-I. The CMD’s are shown side by side rather than in one panel to avoid need for a plot in color. Agreement is not very good due to deficiencies in the adopted theoretical evolution tracks, although there was strong convergence to a definite solution. Use of tabulated evolution tracks will fix this problem. The boxed region is the fitting window.
How can photometric noise be made part of the model? As we are dealing with distributions (in this case, error distributions), an efficient way is via
FSA
, as explained in Section 3 of
Wilson (2001)
and Section 6.3 of WH03, where 9 × 9 distributions were applied (9 magnitude errors, 9 color index errors). Error distributions require little computation time because no evolution is involved and because the Gauss error function (
ERF
) that computes the abscissas needs to be entered only a few times, as that computation is the same for every star system. Only very modest memory allocation (9 × 9 × 36 = 2916 in WH03’s example with binaries and triples) is required for theoretical magnitudes and colors, with each
IMF
primary represented by 36 star systems of various mass ratios - a single star, 10 binaries, and 25 triples. For this paper’s example with only single stars and binaries, there are only 6 star systems for each
IMF
primary.
CMD
location to another. Note that red giant to white dwarf transitions can be a problem from the side of theoretical modeling even with no red giants in the window when a red giant has a companion. Whereas a single red giant that is well outside the
CMD
window typically jumps to another place outside the window upon becoming a white dwarf, one with an unevolved or mildly evolved companion can easily jump into the window, thus suddenly appearing “from nowhere” and causing discontinuous
Areal Density
behavior. That is - the giant effectively disappears but the companion is still there. Fortunately such events have been rare in experience to date, but one should recognize their existence and be alert for corresponding convergence degrradtion.
Three iterations from the middle of a solution are tabulated below so that readers can have a partial experience of examining the overall process. The preceding
DC
iterations began from a fit that was found subjectively (trial and error) and subsequent iterations went on until all corrections were smaller than the standard errors, including a few iterations at the very end to be sure that the operating point was not creeping slowly along (
i.e.
that the corrections were not consistently of the same sign). Convergence in iterations 3, 4, and 5 is almost at the final stage where corrections (column 3) are small compared to the standard errors (column 5). Departures from perfect arithmetic in the iterations are due to rounding in making the tables.
DC
can find a Least Squares solution very accurately, as shown by
Fig. 2
to 6 of WH03, where there is no discernable departure of derived parameters from the Least Squares minimum. Each illustrated curve was computed with 5 of the 6 parameters at the solution value, and the weighted variance was generated as the other parameter varied.
DC
helps one to explore parameter uncertainties intuitively.
Figure 4
shows observed minus computed residuals, 𝒜
_{o}
− 𝒜
_{c}
, graphed vs. cluster age (
T
) derivatives, ∂𝒜
c
/∂
T
. Each dot is for a pixel, and both the residuals and the derivatives were weighted according to the inverse variance of the photometry. The figure isolates one of the 6 parameter dimensions and is for a converged solution, so it corresponds to the end of an iterative process in which the machine has successfully made the residual distribution flat. That is,
DC
has searched for a solution in which the residuals are uncorrelated with the derivatives. The widely distributed points mainly represent the subgiant branch. Readers will notice a dense concentration of points near the origin, which are main sequence and turnoff region pixels. There the ∂𝒜
_{c}
/∂
T
’s are small because evolutionary motion is very slow and the residuals are small because the large numbers of stars lead to good statistical averaging.
Fig. 5
magnifies the region around the origin and shows that flatness has been attained there also, so the fit for the main sequence turnoff is also a solution for the entire
CMD
window. Such diagrams for the other five parameters also are flat for this solution so
DC
has achieved its goal according to eye inspection. These two figures include what might be called 1
σ
lines. The horizontal axis represents the solution and the 1
σ
lines correspond to the solution being off by ±1
σ_{T}
, thus giving a visual impression of the uncertainty of fitting. In this solution and others, one can imagine trying to fit a horizontal line by eye, judge how well the machine has done its job, and look for irregularities and inconsistencies.
One dimension (cluster age) of the 6 dimensional Least Squares solution in graphical form. DC seeks to make the residuals uncorrelated with the derivatives (flat distribution). The slightly diverging straight lines are 1σ age limits. See text for further explanation.
A detail of the inner core of points in the preceding graph. These are main sequence and turnoff region pixels. The 1σ age limits are shown again, but their divergence is too small to see over such a small range of the abscissa.
ADA
programs. Those parameters characterize mixing length in superadiabatic convection zones, convective overshooting, helium abundance, specific abundances such as “
α
” elements
O
,
Ne, Mg, Si, S,
and
Ar,
and other physical conditions. Future
ADA
applications will allow selection among numerous fast evolution subroutines developed by various persons. Armed with such an
ADA
program and having arrived close to a solution, whether by a future automatic algorithm or by trial and error, one will carry out
impersonal
solutions according to several or all of the evolution routines. This is where the labor saving characteristic of
ADA
will show to best advantage,
as one has to get close only once
, whereupon the machine can take over.
So the main problem areas are evolutionary accuracy and need for further adjustable parameters. Otherwise
ADA
works well. Simulations show that solutions for cluster age, binary fraction, distance modulus, metallicity, and interstellar extinction recover known results from synthetic clusters when the same evolution model is used for simulation and solution. Convergence of the iterative solutions is usually fast and gives essentially stable final results, accurately at a Least Squares minimum in multidimensional parameter space. Correlation matrices show that overall correlation problems tend not to be unduly severe. Typically most correlations are within ±0.2 or so, perhaps with a few around ±0.6 to ±0.7, and with none close to unity, although experience to date is limited to 6 or 7 parameters. The routines that convert observable flux and temperature to magnitude and color index have been thoroughly tested and are reliable, although now limited to 25 standard photometric bands.
1 For simplicity, the terms color-magnitude diagram and Hertzsprung-Russell diagram (HRD ) are considered synonyms here, although historically HRD’ s graph magnitude vs. spectral type rather than vs. color index.
2 This contribution was originally to be printed in the proceedings of the meeting Resolved Stellar Populations that took place in Cozumel, Mexico in 2005, April 18-23. However the proceedings have never appeared and will not appear, according to the organizer.

Color-magnitude diagrams
;
stellar evolution
;
stellar statistics
;
star clusters
;
distance estimation

1. INTRODUCTION

- 1. is essentially objective;
- 2. has direct inclusion of binaries (also triples);
- 3. allows straightforward utilization of standard Least Squares, and consequently generates standard error estimates and correlation matrices;
- 4. saves investigator time and work (just let it run);
- 5. has many effective algorithms to deal with pixel structure, low areal density, and error modeling;
- 6. converges quickly to known answers for self-generated synthetic data (i.e.is internally consistent);
- 7. includes analysis of luminosity functions as a special case, as discussed in WH03;
- 8. has accurate and fast conversion from [log T, log g,chemical composition] to 25 photometric bands (Van Hamme & Wilson, 2003).

- 1. a substitute for good astrophysical modeling (it requires an embedded evolution model - any such model may need improvement);
- 2. for getting close, where experience is still needed. Automated “getting close” schemes certainly can be developed, but lie in the future.

- I. it requires ultra-fast evolutionary computation (options are limited at present);
- II. its present evolution subroutine produces outlier CMD’s, even around the main sequence turnoff. Interpolation in tabulated evolution tracks offers a remedy.

- 1.1 The Essential Idea of ADA

Information on a cluster’s evolutionary state and history resides not only in the form of its
- 1.2 Speed and Storage

The call to an evolution subroutine is only one program line within
- 1. The number of stars to be evolved is large even for sparse clusters, as it is set by the requirement to have accuratetheoreticalareal density, not by the number of observed stars.
- 2. Multiple star systems consist not merely of some number of stars, but have mass distributions that must be represented by a reasonable number of points. Five points on each companion mass ratio distribution have been used in experiments to date. The program distinguishes close from wide binaries, although that may be an unnecessary refinement in practice because of the relative rarity of close binaries, so altogether 11 stars may need to be evolved (1+5+5) for each primary on the Initial Mass Function (IMF). If only single stars and wide binaries are modeled, then only 6 evolutions need be carried out for each primary.
- 3.DCrequires evaluation of numerical derivatives (∂𝒜/∂p, wherepis a parameter), so we need not just one evolutionary contribution to 𝒜 but (np+ 1) contributions per star system, as a given system contributes once for each residual, 𝒜o- 𝒜c, andnptimes to formnppartial derivatives.
- 4. A parameter that affects evolution has a distribution in a real population and may need one in a solution. For example, metallicityZmight be assignednZvalues on each side of a Gaussian distribution plus a center value, thereby multiplying the number of stars to be evolved and the corresponding computation time by 2nZ+1. A parameter that is unrelated to evolution, such as interstellar extinction, may also be assigned a distribution, but with little effect on computation time.
- 5. Parameter solutions are iterative, with typically of order 10 iterations per solution. Readers may wonder how areal densities and their derivatives can be stored for millions of theoretical systems (with binary and triple star distributions, metallicity distributions and error distributions) and thousands of pixels. Memory is economized ifADAonly accumulates the contributions to 𝒜 for each system and each pixel. The contributions associated with eachIMFprimary are added to the 𝒜'sfor all pixels (most pixel contributions for a given system being zero, of course), and the memory arrays for system-specific quantities are then overwritten for the next IMF primary.

- 1.3 Parameters

The present version of the
- Z(fractional abundance by mass of elements 3 and higher);
- Gaussian standard deviation ofZ;
- AV, interstellar extinction inV(or other vertical coordinate magnitude);
- Gaussian standard deviation ofAV;
- Ratio ofAVto color excess inB-V(or another color index);
- InterceptAclosein mass ratio proba bility lawPclose=Aclose+Bcloseqclosefor close binaries (q is mass ratio);
- Slope,Bwide, in above distribution;
- InterceptAwidein mass ratio probability lawPwide=Awide+Bwideqwidefor wide binaries;
- Slope,Bwide, in above distribution;
- Reimers mass loss parameterη(winds);
- Binary enhanced mass loss parameter (Tout & Eggleton 1988);
- Lowest mass for which theIMFis defined;
- Mass at the first of two breaks in theIMF;
- Mass at the second break in theIMF;
- Exponent (of mass) in theKroupa, et al. (1993; KTG)IMFfor low mass stars;
- Exponent (of mass) in theKTG IMFfor intermediate mass stars;
- Exponent (of mass) in theKTG IMFfor high mass stars;
- ClusterVband distance modulus,V - MV;
- Cluster age,T;
- Number of stars above the low massIMFcutoff (in the sampled part of cluster),Nstars.

2. DEALING WITH QUANTIZATION

- 2.1. Pixel Sharing for Theoretical Areal Density

Regular pixel structure provides a simple means of binning data in two dimensions, although quantization problems can defeat the overall goals if binning is too simplistic, so the imprint of pixel structure should be reduced as much as possible. Theoretical areal densities should vary smoothly from pixel to pixel, but much more important for solution stability is that
- 2.2. Mass Bin Rounding Corrections

KTG recommend a mass function in three regimes (
PPT Slide

Lager Image

- 2.3. Areal Density Re-scaling

An obvious difficulty is that open clusters typically have fewer stars than one would like for good statistics, with the relevant parameter being
- 2.4. IMF Steepness and Computation Time

Computional efficiency can be improved by recognizing that statistics are far better for low than high mass stars, due to the steepness of the
- 2.5. Low Density (Observational) Areas and VDF

Justification for the Least Squares fitting criterion is based on distributions of observables being Gaussian and the method works best when that is realistically the case.
Fig. 1
shows a distribution of residuals for
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- 2.6. Levenberg-Marquardt and MMS Algorithms

3. PATTERN MIS-MATCHES DUE TO OBSERVATIONAL NOISE

The various
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4. CONVERGENCE WITH AN EXAMPLE

The convergence situation is almost entirely good. From a reasonable intuitive start, one or two iterations often are enough to reach the essential answers, after which typically 5 or 6 refinement iterations reach a stable Least Squares minimum. Sometimes minor jiggles continue, but ordinarily they are well below levels of astrophysical significance. Causes of small jiggles may be essentially discontinuous evolutionary events, such as red giants becoming white dwarfs and jumping from one
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- 4.1. The Minimum Variance in Parameter Space Illustrated

With these refinements,
5. STANDARD ERRORS

Graphical examination of actual Least Squares problems solved by
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6. PROSPECTS

Much work with clear goals remains on evolutionary accuracy. The idea is to develop fast versions of evolution programs by a mix of approximation functions and tabular interpolation, and to allow variation of important parameters that have fixed values in the one fast evolution program now embedded in the
Acknowledgements

This work was done under U.S. National Science Foundation grant 0307561.

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Citing 'Improving CMD Areal Density Analysis: Algorithms and Strategies
'

@article{ OJOOBS_2014_v31n2_121}
,title={Improving CMD Areal Density Analysis: Algorithms and Strategies}
,volume={2}
, url={http://dx.doi.org/10.5140/JASS.2014.31.2.121}, DOI={10.5140/JASS.2014.31.2.121}
, number= {2}
, journal={Journal of Astronomy and Space Sciences}
, publisher={The Korean Space Science Society}
, author={Wilson, R. E.}
, year={2014}
, month={Jun}