#  Determination of the superoutburst cycle lengths of 206 SU UMa type dwarf
novae

N. Vogt,1 E. C. Puebla,1 and A. Contreras-Quijada1

1Instituto de Física y Astronomía, Universidad de Valparaíso, Avda. Gran
Bretaña 1111, Valparaíso, Chile
E-mail<EMAIL_ADDRESS>(NV)

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

SU UMa stars are characterized by “superoutbursts" which are brighter at
maximum light and which last much longer than the more frequent “ordinary"
outbursts of these dwarf novae. Although there are now more than 1180 SU UMa
type dwarf novae catalogued, our knowledge on their superoutburst cycle length
Cso was hitherto limited to about 6$\%$ of the entire sample of known SU UMa
stars. Using public data bases we have determined new Cso values for a total
of 206 additional SU UMa stars in the range 17 d < Cso < 4590 d (including
some ER UMa and WZ Sge type representants) within total time intervals between
2 and 57 years, and with an estimated uncertainty of $\pm$11$\%$. This way, we
are increasing our present knowledge of Cso values by a factor $\sim$3.8. Its
distribution is characterized by a broad maximum around Cso $\approx$ 270
days, and slowly decreasing numbers till Cso $\approx$ 800 d. The domain Cso >
450 d was unexplored until now; we add here 106 cases ($\sim$51$\%$ of our
total sample) in this range of long cycles, implying a better statistical
basis for future studies of their distribution. Our sample contains 16 known
WZ Sge stars, and we propose WZ Sge membership for 5 others hitherto
classified as ordinary SU UMa stars. Individual superoutburst timings deviate
in average about $\pm$7$\%$ of the cycle length from their overall linear
ephemeris, confirming a pronounced quasi-periodic repeatability of
superoutbursts. All relevant parameters are listed with their errors, and a
table with individual superoutburst epochs of our targets is given, enabling
future researchers to combine our results with other (past or future)
observations.

###### keywords:

stars: novae, dwarf novae, cataclysmic variables.

††pubyear: 2020††pagerange:  Determination of the superoutburst cycle lengths
of 206 SU UMa type dwarf novae– Determination of the superoutburst cycle
lengths of 206 SU UMa type dwarf novae

##  1 Introduction

Dwarf novae (DNe) are cataclysmic variables (CVs) , i.e. interacting binaries,
whose low-mass main sequence secondary is filling its Roche lobe, causing a
permanent mass flow towards the white dwarf primary. Conservation of angular
momentum of the transferred mass leads to the formation of an accretion disk.
Instabilities within this disk are the principal cause of DN outbursts
(Warner, 1995). Among the DNe the so-called SU UMa sub-class is characterized
by two very distinct outbursts types: short ones (lasting a few days) and
superoutbursts which can last two weeks or longer in their rather bright
“plateau" phase. Typical superoutburst cycle lengths of these “ordinary" SU
UMa stars range from 100 to 500 days. Two additional sub-groups can be
distinguished: WZ Sge stars have longer superoutburst cycle lengths (several
thousand days) and lack the normal, short outbursts, while ER UMa type dwarf
novae are characterized by very short superoutburst cycles (20-40 days), and
even more frequent normal outbursts in between.

More than four decades ago, periodic “superhumps" have been detected in the SU
UMa star VW Hyi (Vogt, 1974; Warner, 1975) during a plateau phase; later they
turned out to be a characteristic feature of all SU UMa stars, but they appear
occasionally also in other CV types, as classical nova remnants and nova-like
stars (Patterson, 1999). Since then, and till now most observational efforts
on SU UMa stars have been concentrated to detect superhumps and to determine
their properties (Kato et al., 2017, and previous papers of a series), widely
neglecting a systematic coverage of the general outburst behavior of these and
other dwarf novae.

The semi-regular outburst behavior of DNe has been an appealing item of
studies by amateur astronomers for a long time, organized by their societies,
as the American Association of Variable Star Observers (AAVSO), the British
Astronomical Association (BAA) and the Royal Astronomical Society of New
Zealand, Variable Star Section (RASNZ,VSS), and very valuable long-term light
curve studies have already been published decades ago, containing catalogues
of individual outbursts with their main properties as maximum brightness,
amplitude, width and classification according to light curve shape, as well as
some statistical analysis of these properties. As typical examples, we can
mention here extensive studies on SS Cyg (Mattei et al., 1985), U Gem (Mayall,
1957; Mattei et al., 1987) and VW Hyi (Bateson, 1977). To our knowledge, no
long-term studies like those of the 1960ies and 1970ies have been published
more recently.

Simultaneously, there was and still is a boom of detections of new CVs,
especially dwarf novae. The VSX AAVSO catalogue (Watson, 2006) lists more than
9000 DNe, but does not give any cycle length information. The latest on-line
version of the CV catalogue of Ritter & Kolb (2003, version 7.24, February
2017) list a total of 852 DNe with known orbital period, 631 of them (74$\%$)
are classified as SU UMa, ER UMa or WZ Sge stars. Only for 121 (19$\%$) some
cycle length data are given. Discounting the doubtful cases, Ritter & Kolb
(2003) give only for a total of 73 cases (12$\%$) reliable information on the
lengths of the superoutburst cycle (hereafter “super cycle length"). However,
no references are given for the cycle lengths, and it is often very difficult
and time consuming to check these data. Therefore, we here determine super
cycle length values of other SU UMa stars, for which this knowledge is
lacking, or for which only uncertain published information on their
superoutburst cadence is available.

##  2  Methods and data sources

The superoutbursts of most SU UMa stars show a pronounced quasi-periodicity,
occurring rather regularly one linear ephemeris with a standard deviation of
only 5-10$\%$ of the corresponding period (Vogt, 1980). This property allows
determining a rough value of the cycle length even if only few superoutbursts
are recorded. In the present study we limited to those dwarf novae for which
no reliable cycle length values are published in the latest on-line version of
the catalogue of Ritter & Kolb (2003). The main sources were light curves
determined by AAVSO members and available in the LCG data base of this
organization, as well in the photometric catalogue All Sky Automated Survey
ASAS (Pojmanski, 2002) and in the Catalina Sky Survey CSS (Drake et al.,
2009). For a few cases this information was complemented from ASAS-SN
(Kochanek et al., 2017) and Gaia alerts (Gaia Collaboration et al., 2018).

The AAVSO archive contains many time-resolved light curves whose details will
not be considered in our context. Therefore, we limited our analysis to the V
band pass, calculating an average V magnitude value from all observations
within the same Julian day. Since ASAS and CSS contain only observations
coinciding accidentally with outburst timings, sometimes it is difficult to
decide whether a particular brightening refers to a short eruption or a
superoutburst. Criteria are the brightness near maximum, compared to other
superoutbursts of the same star, the presence of a continuous bright state
during several days, and the expected time differences to previous or
following superoutbursts. We include only the epoch (Julian day) of the
outburst maximum in our analysis, no other outburst properties as their
maximum magnitude, duration, width or decline rate have been determined. Often
an observation that was recorded in one of the three above mentioned sources
could be confirmed by another one. In each case, a sequence of the cycle count
number E was established, beginning with E = 0 for the first recorded
superoutburst timing, and assigning the remaining E values based on the
minimal time difference between two subsequent records as the basic unit. A
linear least square fit Heliocentric Julian Date (Maximum brightness) vs. E.

$T_{\mathrm{max}}=T_{\mathrm{0}}+C_{\mathrm{so}}E$ (1)

leads to the super cycle Cso, its error $\sigma$(Cso) and the standard
deviation which we call here ${\rm\overline{O-C}}$ because it is a measure for
the mean difference between the observed superoutburst epoch from the linear
ephemeris.

##  3 The new catalogue of super cycle length values

Among the SU UMa type dwarf novae considered here there is only one case
(J1749+1917) that belongs to the ER UMa subclass. A total of 16 stars are of
WZ Sge type, most of our remaining targets are ordinary SU UMa stars; five of
them could belong to the WZ Sge subclass, as discussed in section 4.3.

At least three superoutburst timings N have to be available for any star which
is included in our analysis. The average number of superoutbursts per target
observed is $\overline{N}$ = 6.3, the range is 3 $\leq$ N $\leq$ 18\. The
results are given in Table 1, which lists the 206 dwarf novae with new super
cycle values arranged according to right ascension. We use the coordinate
identification of Ritter & Kolb (2003) in cases without official names given
in the General Catalogue of Variable Stars (GCVS). Table 1 gives all relevant
parameters determined according eq. 1 and their errors, as well as the
standard deviations $\mathrm{\overline{O-C}}$ and some remarks, mainly
referring to WZ Sge classification. Table 5 lists some properties of five
stars which also could belong to the WZ Sge class.

Figure 1:  Superoutburst cycle length Cso vs. orbital period P. Filled
circles: new Cso values determined here. Open circles: previously known cycle
length values.

We present other two tables as on-line data: Table 6 gives additional
information on available data used by us for the statistics of Section 4.1 and
Fig. 5 while Table 7 contains all individual superoutburst maximum epochs
$T_{\mathrm{max}}$ and the corresponding cycle numbers E used in our analysis
for each of our 206 targets. This table will enable future researchers to
combine our results with other(past or future) observations, in order to
improve or correct our(presumably in some of our cases)rather preliminary
cycle Cso values.

##  4 Results and discussion

###  4.1 $C_{\rm so}$ vs. orbital period and vs. short outburst cycle length

As mentioned above, only for 73 cases there are reliable superoutburst cycle
length values available from the literature. Our study is adding 206 new
cases, augmenting the previous knowledge by a factor 2.8. In addition, apart
from Cso and its error, we give here data on the detailed ephemeris of recent
superoutbursts. Similar information is available from the analysis of 26 well
observed SU UMa stars analyzed by A. Contreras-Quijada in his Master thesis,
using a much longer time basis compared to our targets. All these 26 cases
form part of the 73 stars listed by Ritter & Kolb (2003), implying that
finally only 47 Cso values are taken from the published literature, and the
remaining 232 Cso values were determined by our group.

Figure 2: Ratio $C_{\mathrm{so}}$/$C_{\mathrm{n}}$ vs. orbital period P (top)
and vs. Cso (bottom).

On the other hand, no attempt was made by us to determine the cycle length Cn
of the “normal" short outbursts of SU UMa stars. In order to include them in
our statistics we use the Cn values given by Ritter & Kolb (2003),
complementing them in few cases by cadence data of Coppejans et al. (2016).

Figure 3:  Distribution of the superoutburst numbers N used in the
determination of the new cycle length parameters listed in Table 1.

Fig. 1 shows the super cycle length values vs. orbital period. There is no
clear correlation, but the WZ Sge stars can be distinguished in this diagram
in the period range 0.05-0-06 d, due to their large Cso values. However, there
are also ordinary SU UMa stars with shorter super cycle values in this period
range. In addition, some of our targets show ultra-short periods (0.035 < Porb
< 0.045 d) well separated from the remaining SU UMa stars with Porb > 0.05 d.
They belong to the AM CVn class with orbital periods less than $\sim$ 1 hour,
i.e. CVs containing an evolved donor star, typically a Roche lobe filling low
mass white dwarf or a semi-degenerate He-rich star (Ramsay et al., 2018).
Despite of their rather different underlying physics (their accretion disks
are mainly of helium) their average Cso values coincide with those of the
remaining SU UMa stars with hydrogen rich disks.

In Fig. 2 we show the ratio Cso/Cn vs. orbital period and vs. Cso. For Cso <
400 d seems to be a slight tendency to diminish the frequency of short
outbursts for cases with larger superoutburst cycle length, but there is a
large scatter in both diagrams.

Figure 4:  Top: Distribution of the total time intervals $\Delta$T covered by
data used for the determination of the new cycle length parameters listed in
Table 1 (binning 500d). Bottom: Distribution of the time intervals of constant
Cso values, found in a study of the long-term cycle length variability of 22
well observed SU UMa stars. Both distributions agree within the expected
statistical uncertainties.

###  4.2 Statistical analysis of other parameters

In this section we analyze mainly our own data, with emphasis on those
parameters that are not available from other published sources. Fig. 3 shows
the distribution of the numbers N of superoutbursts used for our targets,
revealing a broad peak around 5 observed superoutburst epochs per star. This
could be considered as a rather low number, but, based on the pronounced
periodic repeatability of SU UMa type superoutbursts (see below), it permits a
valid first estimation of their cycle length. Fig. 4 (upper panel) gives a
histogram of the time intervals $\Delta$T between the first and the last
observed and catalogued superoutbursts for each star, revealing a maximum
around 3000 – 4000 days and a rather symmetric distribution between 1500 and
7000 days, followed by few $\Delta$T values up to more than 20000 days (total
range 2 < $\Delta$T < 57 years). In addition, Fig. 4 compares this
distribution with results of a recent Master thesis (Contreras-Quijada $\&$
Vogt, in preparation), referring to a study of long-term supercylce length
variations of 23 well observed SU UMa stars with mean Cso values between 88
and 531 days, determined within total time intervals between 19 and 88 years.
In 22 cases of this study (96$\%$) significant changes in Cso in time scales
of several years were found, establishing a total of 63 partial time intervals
$\Delta$T with constant Cso, whose distribution is shown in the lower panel of
Fig. 4. It becomes evident that the distributions in both panels of Fig. 4 are
similar, confirming, that our procedure of linear fits in eq.(1) is valid for
a preliminary estimation of Cso, based on our time intervals $\Delta$T.

This same study permits to estimate the expected accuracy of our Cso
determination. For this purpose, we calculated the individual supercycle
lengths values within the time intervals of constant Cso, for each of the
above mentioned 22 SU UMa stars with known long-term behavior, and determined
the standard deviation $\delta$(Cso) from the general mean supercycle length
${\rm\overline{C_{So}}}$ in each case. The ratio
$\delta$(Cso)/${\rm\overline{C_{So}}}$ is a measure for the reliability of our
method, because the time intervals covered by us here correspond to those of
typical linear portions in the over-all ${\rm\overline{O-C}}$ vs. E behavior
of Contreras-Quijada $\&$ Vogt´s targets, as shown in Fig. 4. The resulting
mean value of the ratio $\delta$(Cso)/${\rm\overline{C_{So}}}$ is 0.109 $\pm$
0.009, with a standard deviation of 0.046, and extremes 0.044 and 0.217.
Accordingly, our Cso values are expected to deviate from those based on long-
term studies by $\pm$11$\%$ in average, and should never exceed 22$\%$,
implying that our results are valid first approximations.

The upper panel of Fig. 5 shows the Cso distribution of our targets (with Cso
< 1300 d); in the lower panel of this figure we add to this distribution the
hitherto known data, obtaining a statistically significant overall Cso
distribution for the first time. It reveals a remarkable maximum around Cso
$\approx$ 270 d which was not present in the earlier data. Generally, there
are rapidly increasing numbers for Cso $\approx$ 100 - 250 d and subsequently
slowly declining numbers up to Cso $\approx$ 800 d. For larger Cso values the
statistics are still very poor. The narrow maxima around 365 and 730 d could
be caused by seasonal selection effects (observational bias of 1 and 2 years).
The domain Cso > 450 d was unexplored until now; we add a total of 106 cases
(51$\%$ of our total sample) to this range of large Cso values, implying a
better statistical basis for future studies of their distribution.

Figure 5:  Top: Distribution of the new Cso values listed in Table 1, limited
for Cso < 1300 d (binnig 20 d). Bottom: this distribution (in red color) is
compared and added to that of all previously known Cso values (in gris color).
It becomes evident that the additional data presented here allow statistical
relevant information on the real Cso distribution for the first time.

Figure 6:  Distribution of the $\mathrm{\overline{O-C}}$ values (top: binning
5 d ) and of the ratio $\mathrm{\overline{O-C}}$/Cso of all targets listed in
Table 1 (bottom, binning 0.005d).

Figure 6 gives histograms of the standard deviations $\mathrm{\overline{O-C}}$
and the ratio $\mathrm{\overline{O-C}}$/Cso. The $\mathrm{\overline{O-C}}$
distribution is characterized by a broad maximum between 20 and 60 days, with
a rapid decline for larger values. The ratio $\mathrm{\overline{O-C}}$/Cso has
a relatively rapid increase up to a maximum around 0.07 and a slower decline
ending at 0.17, with sparse isolated values up to 0.3. This ratio reflects the
precision of the “clock" of quasi-periodic repeatability of superoutbursts,
and confirms, that individual superoutburst timings deviate from the overall
linear ephemeris in average about $\pm$7$\%$ of the cycle length.

Figure 7 gives information about the errors $\sigma$(Cso) of the super
outburst cycle values. The upper panel shows that these errors are generally
very small, mostly < 5 d, with a strong decline until $\sim$ 15 d, and a few
outliers with larger values. The lower panel shows the ratio
$\sigma$(Cso)/Cso, i.e. the distribution of the relative errors of our super
cycle length determination and confirm that most ratios $\sigma$(Cso)/Cso are
rather small with a peak around 0.007 and declining until 0.03, with few
larger outlier values $\leq$ 0.06.

Finally, we investigate in Figure 8 how the mean standard deviation
$\mathrm{\overline{O-C}}$ is correlated to the super cycle length Cso and its
error $\sigma$(Cso). Four groups of numbers N of observations are indicated by
dots of different colors. The upper panel of this figure shows how
$\mathrm{\overline{O-C}}$ is correlated to Cso; small numbers N mainly refer
to the longest Cso values, with weak or absent correlation to Cso. In
contrast, larger numbers N include also shorter Cso values, developing
systematic trends in the sense that larger cycle lengths are correlated with
larger $\mathrm{\overline{O-C}}$ values. According to the lower panel of
Figure 8 a general tendency of significant correlations between $\sigma$(Cso)
and $\mathrm{\overline{O-C}}$ seems to be valid for all groups of N, now with
similar slopes; however these groups are well separated in their zero points:
for any given $\mathrm{\overline{O-C}}$ value, group members with N = 3 are
characterized by about 10 times larger errors $\sigma$(Cso) than those with 9
$\leq$ N $\leq$ 18, and the groups of intermediate N values are located
between these two extrema. We believe that these features are consequences of
the unavoidable inhomogeneity of our data: more observed superoutburst epochs
reveal more precise results, underlining the importance to publish always the
corresponding error together with any derived parameter.

Figure 7:  Distribution of $\sigma$(Cso) limited to values < 110 d (top:
binning 2 d ) and of the ratio $\sigma$(Cso)/Cso of all targets listed in
Table 1 (bottom, binning 0.001 d.).

###  4.3 WZ Sge type stars

This subclass is characterized by long cycle lengths Cso (between 500 d and up
to several decades), large amplitudes in V up to 8m and very rare occurrence
or absence of short outbursts, revealing only superoutbursts of rather long
duration. Kato (2015) gives some additional properties of WZ Sge stars, as
early superhumps, appearing before the ordinary ones, and long or multiple
rebrightenings after the main eruption. He discusses many properties of WZ Sge
stars in detail, but with little attention to the outburst cycle lengths Cso;
he mentions in his Table 6 only outburst years of 58 WZ Sge stars.

Our targets contain a total of 16 WZ Sge stars, known according to Kato (2015)
and the VSX, (marked “WZ” in Table 1) with Cso values between 2.7 and 12.6
years. On the other hand, we detected five possible WZ Sge candidates among
our SU UMa targets, whose most important data are listed in Table 5
(particularly large amplitudes and cycle lengths). Most of the established WZ
Sge stars have short orbital periods between 0.04 and 0.06 days, and none of
them > 0.077 days, in contrast to two stars in Table 5 (EF Peg and V444 Peg)
that reveal longer orbital periods. Especially V444 Peg could be an
interesting case, since it would be the first WZ Sge star within the period
gap between 2 and 3 hours. However, all 5 candidates in Table 5 should be
confirmed before drawing further conclusions.

##  5 Conclusions

We have determined individual cycle length values of a total of 206 SU UMa
stars whose superoutburst cadence was unknown previously. Using data bases
with public access, we were able to increase the number of SU UMa stars with
reliable information concerning this crucial parameter by a factor $\sim$ 3.8.
To our knowledge, similar studies have not been published for a long time. The
dwarf nova cadence statistics by Coppejans et al. (2016) cannot be compared to
our results because it refers to all types of dwarf novae, and it does not
distinguish between short eruptions and superoutbursts in case of SU UMa
stars.

Figure 8: The new super cycle length values Cso (top) and their errors
$\sigma$(Cso) (bottom) vs. $\mathrm{\overline{O-C}}$. Different numbers N are
represented by different colors: N=3 (black), N=4 (blue), N=5 (green), N=6-8
(red) and N=9-18 (yellow)

Superoutburst cycle length values are now known for a total of 279 SU UMa type
dwarf novae. On the other hand, a total of more than 1180 SU UMa stars
(including the ER UMa and WZ Sge subtypes) are listed in the AAVSO VSX
catalogue, implying still an actual deficit of 76$\%$ when referring to our
knowledge of the parameter Cso. On the other hand, we here can present, for
the first time, a realistic superoutburst cycle length distribution (Fig. 5),
with a peak around Cso $\approx$ 270 days. Among our targets there are 16 WZ
Sge stars for which we give new Cso values, and other 5 DNe hitherto
classified of SU UMa type, whose long cycles and large amplitudes suggest
belonging also to the WZ Sge subgroup. This, however, requires confirmation,
especially because two of them seem to exceed the range of orbital periods,
typical for WZ Sge stars.

There are striking differences between our study and most similar ones,
published recently. In fact, a previous reliable Cso catalogue does not exist.
Sparse information is given by Ritter & Kolb (2003), but without errors and
references, and without any indication of possible Cso variability over years
and decades. A first attempt to investigate these items was made by Vogt
(1980) for few SU UMa stars observed during sufficiently long time spans at
that epoch. Recently, this was redone by Contreras-Quijada $\&$ Vogt (in
preparation) comprising more SU UMa stars and longer time spans now available;
this study confirms most of the preliminary results of Vogt (1980). Based on
them, we here compare in Fig. 4 our $\Delta$T distribution with that of
typical time intervals with constant Cso cycle lengths determined in well
observed SU UMa stars in the course of decades, and find that most SU UMa
stars maintain constant Cso values within similar time intervals as our rather
short $\Delta$T values. According to our experience in the above-mentioned
study, these “short-lived” Cso values deviate in average $\pm$11$\%$ from
their long-term average in each case. We expect that our 206 new Cso values
comprise uncertainties of this order of magnitude, whenever they will be
compared to the (not yet known) long-term behavior of our targets.

Our results are presented in a consciously reproducible way, aimed at enabling
any present or future researcher to verify, complement, or modify them, given
that they are necessarily limited by the information available today. We
present here “early” data, for use in further studies on superoutburst cycles.
We regard it as important that there is transparency in the presentation of
the data, to avoid the black-box approach of some other studies that renders
them difficult to combine with later results.

There is some hope for improvements during coming years. It will be possible
to determine many more superoutburst cycle length values whenever future
patrol projects will cover sufficiently large time intervals. In particular,
the Vera C. Rubin Observatory ( former LSST, Large Synoptic Survey Telescope,
Kessler et al., 2019) will soon be operational, a crucial step forward in our
knowledge of the outburst cadence values of thousands of dwarf novae,
including SU UMa stars. Some patience will be necessary, but after several
years of operation of the LSST and other patrol projects, the results
presented here could be combined with future data, extending this way the
total time intervals covered.

## Acknowledgements

We acknowledge financial support by the Centro de Astrofísica de Valparaíso
(CAV). Part of this work is based on the light curve data base of the American
Association of Variable Star Observers (AAVSO) whose rich archival material,
mainly based on continuous amateur astronomy efforts, is of great value also
for the research by professional astronomers. We thank the referee for
constructive comments and advices enabling us to improve the presentation of
the paper.

## Data Availability Statements

The full data underlying this article are available as online supplementary
material.

Table 1: New cycle length values of 206 SU UMa type outbursts. N is the number of superoutbursts covered within the time interval $\Delta$T. The remaining parameters refer to results of the linear fit eq.(1): T0 is the epoch (in JD-2450000) of the first superoutburst, $\sigma$(T0) its error, Cso the supercycle length, $\sigma$(Cso) its error and $\mathrm{\overline{O-C}}$ the mean standard deviation of the actual superoutburst epochs from the linear ephemeris eq.(1). “WZ” refers WZ Sge type stars, “WZ?” to WZ Sge type candidates, also listed in Table 6. The table is available online.  Name | RA (J2000) | DEC | N |  $\Delta$T | T0 |  $\sigma$(T0) | Cso |  $\sigma$(Cso) | $\mathrm{\overline{O-C}}$ | Remarks
---|---|---|---|---|---|---|---|---|---|---
|  |  |  | (d) | (d) | (d) | (d) | (d) | (d) |
J0000+3325 | 00 00 24.6 | +33 25 43 | 6 | 3245 | 4436 | 21 | 247.4 | 2.4 | 26.4 |
BE Oct | 00 00 48.8 | -77 18 57 | 6 | 3816 | 4091 | 44 | 476.3 | 7.9 | 49.9 |
J0009-1210 | 00 09 38.2 | -12 10 16 | 3 | 1822 | 3642 | 47 | 603.0 | 22.5 | 48.6 |
J0015+2636 | 00 15 38.2 | +26 36 57 | 5 | 3596 | 3689 | 29 | 724.7 | 8.9 | 34.1 |
J0021-5719 | 00 21 30.9 | -57 19 22 | 5 | 2611 | 2115 | 39 | 529.7 | 12.1 | 50.0 |
J0024-6635 | 00 24 30.7 | -66 35 53 | 6 | 6303 | 2178 | 48 | 563.7 | 6.5 | 67.1 |
J0036+2151 | 00 36 35.8 | +21 51 26 | 4 | 3959 | 3329 | 12 | 246.8 | 1.4 | 17.7 |
PT And | 00 40 24.4 | +41 04 03 | 4 | 10513 | -2461 | 157 | 3528.1 | 83.9 | 187.6 | WZ
LL And | 00 41 51.5 | +26 37 21 | 3 | 7274 | -609 | 134 | 3637.0 | 103.7 | 146.6 | WZ
J0042-5609 | 00 42 14.2 | -56 09 20 | 6 | 4254 | 2757 | 20 | 328.9 | 3.2 | 32.3 |
J0045+5032 | 00 45 27.5 | +50 32 16 | 3 | 762 | 6546 | 32 | 251.0 | 15.3 | 33.1 |
GV Psc | 01 13 06.7 | +21 52 51 | 4 | 3702 | 3667 | 46 | 529.9 | 11.1 | 57.6 |
GZ Cet | 01 37 01.1 | -09 12 35 | 5 | 5685 | 2050 | 35 | 952.1 | 9.9 | 47.4 |
FZ Cet | 01 42 25.3 | -22 15 58 | 5 | 5099 | 2261 | 58 | 733.8 | 11.6 | 62.7 |
J0150+3326 | 01 50 51.5 | +33 26 22 | 3 | 1837 | 3994 | 34 | 609.2 | 16.3 | 35.1 |
FI Cet | 01 51 18.6 | -02 23 01 | 3 | 4645 | 2141 | 85 | 1557.5 | 46.8 | 101.1 | WZ
BG Ari | 01 51 51.9 | +14 00 47 | 4 | 2123 | 4121 | 15 | 235.7 | 2.6 | 17.2 |
J0153+3408 | 01 53 21.5 | +34 08 56 | 5 | 2936 | 3650 | 29 | 741.7 | 11.9 | 37.7 |
J0211+1716 | 02 11 10.2 | +17 16 24 | 8 | 5490 | 429 | 16 | 273.2 | 1.5 | 26.2 |
AX For | 02 19 28.0 | -30 45 46 | 11 | 5225 | 2225 | 24 | 319.5 | 2.5 | 47.0 |
WY Tri | 02 25 00.5 | +32 59 56 | 6 | 5751 | 1805 | 48 | 522.5 | 6.6 | 62.7 |
PU Per | 02 42 16.1 | +35 40 46 | 7 | 6207 | 1044 | 58 | 779.3 | 12.1 | 89.7 |
PV Per | 02 42 53.5 | +38 04 04 | 6 | 4104 | 12 | 21 | 240.5 | 1.8 | 28.0 |
J0243-1603 | 02 43 54.1 | -16 03 14 | 3 | 2786 | 3784 | 29 | 700.0 | 12.3 | 36.3 |
BB Ari | 02 44 57.7 | +27 31 09 | 4 | 5199 | 3312 | 9 | 399.5 | 1.0 | 10.0 |
UW Tri | 02 45 17.3 | +33 31 26 | 3 | 9178 | -4543 | 297 | 4589.1 | 229.9 | 325.1 | WZ
RU Hor | 02 46 07.5 | -63 35 47 | 6 | 6269 | 86 | 24 | 566.0 | 3.5 | 33.8 |
V368 Per | 02 47 32.6 | +34 58 28 | 3 | 3188 | 3752 | 8 | 638.5 | 2.6 | 9.6 |
KY Eri | 03 10 51.7 | -07 55 00 | 4 | 1776 | 5127 | 36 | 299.1 | 8.6 | 39.7 |
QY Per | 03 15 36.8 | +42 28 14 | 5 | 7693 | -326 | 71 | 966.8 | 15.2 | 92.2 |
FT Cam | 03 21 14.4 | +61 05 27 | 6 | 4480 | 1077 | 29 | 247.3 | 2.6 | 44.0 |
J0334-0710 | 03 34 49.9 | -07 10 48 | 6 | 5118 | 1968 | 37 | 564.1 | 7.1 | 55.4 |
V701 Tau | 03 44 01.9 | +21 57 08 | 8 | 6993 | 43 | 29 | 409.7 | 2.9 | 43.4 |
V1212 Tau | 03 51 57.0 | +25 25 28 | 7 | 3633 | 4133 | 27 | 275.9 | 3.3 | 41.0 |
V1024 Per | 04 02 39.0 | +42 50 45 | 4 | 1669 | 5625 | 30 | 424.8 | 11.6 | 36.8 |
V1389 Tau | 04 06 59.8 | +00 52 44 | 5 | 2564 | 3342 | 39 | 637.6 | 15.8 | 49.9 |
XZ Eri | 04 11 25.8 | -15 23 24 | 6 | 6506 | 812 | 55 | 927.7 | 11.9 | 69.4 |
J0411-0907 | 04 11 33.6 | -09 07 29 | 6 | 2894 | 4109 | 27 | 358.7 | 5.6 | 37.7 |
J0416+2928 | 04 16 36.9 | +29 28 06 | 7 | 4079 | 3247 | 25 | 291.7 | 2.7 | 34.4 |
V342 Cam | 04 23 32.9 | +74 52 50 | 6 | 3997 | 3492 | 31 | 500.3 | 6.1 | 41.1 |
J0426+3541 | 04 26 09.3 | +35 41 45 | 5 | 1544 | 6209 | 5 | 119.0 | 0.8 | 8.1 |
J0442-0023 | 04 42 16.0 | -00 23 34 | 4 | 1920 | 3657 | 22 | 383.0 | 7.2 | 26.0 |
HV Aur | 04 53 16.8 | +38 16 29 | 10 | 4365 | 3031 | 18 | 274.2 | 1.8 | 30.4 |
V1208 Tau | 04 59 44.0 | +19 26 23 | 7 | 4285 | 2790 | 43 | 327.7 | 5.6 | 72.5 |
J0507+1253 | 05 07 16.2 | +12 53 15 | 13 | 4017 | 3685 | 7 | 133.4 | 0.4 | 13.8 |
GR Ori | 05 21 35.0 | +01 10 10 | 3 | 3638 | 2651 | 101 | 1819.2 | 78.5 | 111.0 | WZ
J0529+1848 | 05 29 58.8 | +18 48 10 | 13 | 4914 | 2932 | 7 | 95.2 | 0.2 | 20.3 |
V391 Cam | 05 32 33.9 | +62 47 52 | 7 | 4381 | 3408 | 26 | 555.4 | 6.4 | 42.0 |
AR Pic | 05 49 45.4 | -49 21 56 | 10 | 5466 | 1949 | 23 | 220.5 | 1.5 | 43.2 |
J0557+6832 | 05 57 18.5 | +68 32 27 | 4 | 3390 | 4074 | 17 | 488.2 | 3.8 | 22.9 |
AD Men | 06 04 30.9 | -71 25 23 | 12 | 5152 | 2789 | 23 | 160.6 | 1.3 | 40.0 |
CI Gem | 06 30 05.9 | +22 18 51 | 6 | 5392 | 1214 | 54 | 597.5 | 11.3 | 77.3 |
UV Gem | 06 38 44.1 | +18 16 11 | 14 | 6817 | 882 | 23 | 147.9 | 0.8 | 43.0 |
PU CMa | 06 40 47.7 | -24 23 14 | 11 | 5774 | 1731 | 15 | 337.8 | 2.0 | 31.5 |
Table 2: continued Name | RA (J2000) | DEC | N |  $\Delta$T | T0 |  $\sigma$(T0) | Cso |  $\sigma$(Cso) | $\mathrm{\overline{O-C}}$ | Remarks
---|---|---|---|---|---|---|---|---|---|---
|  |  |  | (d) | (d) | (d) | (d) | (d) | (d) |
J0647+4915 | 06 47 25.7 | +49 15 43 | 6 | 3478 | 4384 | 15 | 217.6 | 1.3 | 16.6 |
CG CMa | 07 04 05.2 | -23 45 34 | 3 | 6988 | 1187 | 102 | 3494.0 | 78.8 | 111.4 | WZ
AQ CMi | 07 14 34.8 | +08 48 06 | 7 | 7323 | 113 | 21 | 433.1 | 1.8 | 30.2 |
FQ Mon | 07 16 41.2 | -06 56 49 | 5 | 3611 | 3053 | 23 | 721.9 | 7.0 | 26.9 |
V496 Aur | 07 27 52.3 | +40 46 54 | 5 | 3327 | 3691 | 21 | 303.0 | 3.1 | 25.2 |
J0732+4130 | 07 32 08.1 | +41 30 09 | 6 | 3236 | 3809 | 24 | 248.9 | 3.1 | 34.1 |
J0741-0945 | 07 41 12.7 | -09 45 56 | 5 | 3982 | 3135 | 19 | 283.1 | 2.2 | 23.2 |
J0746+1734 | 07 46 40.6 | +17 34 13 | 4 | 3201 | 3835 | 17 | 318.9 | 2.6 | 22.2 |
J0800+1924 | 08 00 33.9 | +19 24 16 | 6 | 3977 | 3460 | 58 | 506.1 | 10.6 | 73.2 |
J0803+2516 | 08 03 03.9 | +25 16 27 | 7 | 2897 | 3463 | 29 | 260.9 | 4.6 | 44.1 |
J0803+2848 | 08 03 07.0 | +28 48 56 | 7 | 3707 | 3708 | 11 | 154.6 | 0.8 | 17.0 |
KK Cnc | 08 07 14.3 | +11 38 13 | 3 | 2995 | 4483 | 107 | 987.0 | 58.8 | 127.1 | WZ
LX Cnc | 08 12 07.6 | +13 18 24 | 5 | 2391 | 3652 | 28 | 486.9 | 8.9 | 36.8 |
J0813-0103 | 08 13 18.5 | -01 03 28 | 6 | 5056 | 2677 | 24 | 268.5 | 2.2 | 34.1 |
J0814+0907 | 08 14 08.4 | +09 07 59 | 5 | 3323 | 4050 | 47 | 547.1 | 11.9 | 61.5 |
J0814-0050 | 08 14 18.9 | -00 50 22 | 6 | 4685 | 2713 | 13 | 187.4 | 1.0 | 20.8 |
J0819+1915 | 08 19 36.1 | +19 15 40 | 3 | 2214 | 4883 | 24 | 740.1 | 11.5 | 24.7 |
FL Lyn | 08 24 09.7 | +49 31 24 | 7 | 3167 | 4113 | 41 | 321.2 | 6.2 | 59.8 |
J0838+4910 | 08 38 45.2 | +49 10 56 | 4 | 1545 | 4893 | 9 | 385.2 | 3.3 | 10.6 |
RX Vol | 08 39 32.3 | -66 17 39 | 8 | 4002 | 3089 | 18 | 232.4 | 1.8 | 30.0 |
J0841+2100 | 08 41 27.4 | +21 00 54 | 8 | 3130 | 4352 | 13 | 210.4 | 1.5 | 21.4 |
EG Cnc | 08 43 04.0 | +27 51 50 | 4 | 12617 | -4395 | 211 | 2529.8 | 68.6 | 247.3 | WZ
V498 Hya | 08 45 55.1 | +03 39 30 | 3 | 2947 | 4521 | 109 | 992.3 | 52.2 | 112.8 |
CT Hya | 08 51 07.4 | +03 08 34 | 7 | 4977 | 4039 | 31 | 419.2 | 4.0 | 46.8 |
J0851+3444 | 08 51 13.4 | +34 44 49 | 4 | 2912 | 4783 | 13 | 731.6 | 5.1 | 16.1 |
BZ UMa | 08 53 44.1 | +57 48 41 | 9 | 15198 | -8904 | 33 | 303.4 | 1.0 | 47.4 |
PU UMa | 09 01 03.9 | +48 09 11 | 4 | 3037 | 4426 | 41 | 501.3 | 12.1 | 55.4 |
MM Hya | 09 14 14.0 | -06 47 45 | 9 | 5498 | 1310 | 24 | 362.3 | 2.7 | 44.7 |
GZ Cnc | 09 15 51.7 | +09 00 50 | 9 | 5064 | 2120 | 31 | 281.9 | 2.6 | 46.9 |
J0943-2720 | 09 43 27.3 | -27 20 30 | 7 | 5170 | 1928 | 36 | 470.6 | 5.7 | 59.3 |
J0945-1944 | 09 45 51.0 | -19 44 01 | 5 | 3972 | 1987 | 32 | 567.4 | 8.1 | 41.8 |
J0947+0610 | 09 47 59.8 | +06 10 44 | 3 | 5367 | 3577 | 186 | 2682.9 | 144.4 | 204.2 | WZ
J0959-1601 | 09 59 26.5 | -16 01 47 | 7 | 4414 | 1877 | 16 | 232.9 | 1.4 | 21.3 |
J1005+1911 | 10 05 15.4 | +19 11 08 | 5 | 4482 | 2994 | 21 | 897.1 | 7.5 | 28.9 |
CP Dra | 10 15 39.8 | +73 26 05 | 8 | 4655 | 1947 | 16 | 245.9 | 1.7 | 26.7 |
NSV 4838 | 10 23 20.2 | +44 05 09 | 5 | 3718 | 3774 | 10 | 372.8 | 1.9 | 15.2 |
J1026+1920 | 10 26 16.1 | +19 20 45 | 5 | 2973 | 3817 | 19 | 427.3 | 4.4 | 25.1 |
J1027-4343 | 10 27 05.8 | -43 43 41 | 5 | 1756 | 4184 | 11 | 291.0 | 2.8 | 14.3 |
J1028-0819 | 10 28 42.9 | -08 19 27 | 8 | 3143 | 2796 | 9 | 211.1 | 1.0 | 13.7 |
RX Cha | 10 36 26.3 | -80 02 48 | 5 | 3575 | 1296 | 29 | 717.9 | 9.2 | 38.2 |
J1100+1315 | 11 00 14.7 | +13 15 52 | 5 | 3795 | 3372 | 15 | 472.0 | 2.8 | 20.2 |
J1100-1156 | 11 00 38.1 | -11 56 47 | 4 | 4402 | 2735 | 9 | 339.0 | 1.3 | 13.3 |
TU Crt | 11 03 36.5 | -21 37 46 | 9 | 4697 | 2058 | 26 | 465.1 | 4.7 | 45.4 |
J1122-1110 | 11 22 53.3 | -11 10 38 | 3 | 685 | 4666 | 1 | 228.2 | 0.7 | 1.6 |
RZ Leo | 11 37 22.3 | +01 48 58 | 5 | 11353 | -3796 | 104 | 1888.1 | 29.6 | 141.3 | WZ
V359 Cen | 11 58 15.3 | -41 46 08 | 7 | 5030 | 1707 | 13 | 716.0 | 3.3 | 19.2 |
J1200-1526 | 12 00 52.9 | -15 26 21 | 5 | 4413 | 1957 | 14 | 366.2 | 1.6 | 16.4 |
J1202+4503 | 12 02 31.0 | +45 03 49 | 5 | 5203 | 1880 | 16 | 274.0 | 1.6 | 25.0 |
GP CVn | 12 27 40.8 | +51 39 25 | 5 | 4227 | 3754 | 24 | 467.4 | 4.3 | 34.4 |
AL Com | 12 32 25.9 | +14 20 43 | 4 | 13542 | -7419 | 134 | 2358.9 | 34.0 | 148.7 | WZ
V591 Cen | 12 42 18.0 | -33 34 07 | 5 | 4338 | 1417 | 36 | 431.8 | 5.5 | 49.9 |
GO Com | 12 56 37.1 | +26 36 44 | 7 | 3608 | 2842 | 46 | 607.9 | 12.9 | 68.0 |
V485 Cen | 12 57 23.3 | -33 12 07 | 18 | 14976 | -7724 | 21 | 282.4 | 0.7 | 46.4 |
HV Vir | 13 21 03.2 | +01 53 29 | 4 | 6193 | -1015 | 321 | 2840.5 | 171.6 | 383.6 | WZ
LY Hya | 13 31 53.8 | -29 40 59 | 5 | 5964 | 870 | 56 | 1010.5 | 14.3 | 74.0 | WZ?
Table 3: continued Name | RA (J2000) | DEC | N |  $\Delta$T | T0 |  $\sigma$(T0) | Cso |  $\sigma$(Cso) | $\mathrm{\overline{O-C}}$ | Remarks
---|---|---|---|---|---|---|---|---|---|---
|  |  |  | (d) | (d) | (d) | (d) | (d) | (d) |
J1343-4426 | 13 43 37.2 | -44 26 42 | 10 | 5082 | 2044 | 15 | 130.5 | 0.7 | 28.8 |
HS Vir | 13 43 38.4 | -08 14 03 | 14 | 6545 | 973 | 22 | 361.0 | 2.0 | 45.7 |
J1425+1515 | 14 25 48.1 | +15 15 01 | 4 | 3806 | 3795 | 39 | 635.6 | 10.1 | 43.7 |
OU Vir | 14 35 00.1 | -00 46 07 | 6 | 5910 | 1613 | 57 | 735.2 | 10.6 | 70.1 |
V362 Lib | 14 43 41.9 | -17 55 49 | 4 | 2489 | 2410 | 66 | 619.5 | 24.4 | 72.2 |
UZ Boo | 14 44 01.3 | +22 00 56 | 5 | 12725 | -6225 | 40 | 1157.8 | 5.0 | 44.3 | WZ
J1459+3548 | 14 59 21.8 | +35 48 05 | 5 | 1844 | 4560 | 28 | 305.0 | 7.1 | 36.5 |
NZ Boo | 15 02 41.0 | +33 34 24 | 5 | 3093 | 4203 | 31 | 763.7 | 12.6 | 39.8 |
V418 Ser | 15 14 53.6 | +02 09 35 | 5 | 2734 | 4500 | 21 | 456.4 | 5.3 | 27.2 |
DM Dra | 15 34 12.1 | +59 48 32 | 7 | 4352 | 1817 | 22 | 438.7 | 3.6 | 35.0 |
AB Nor | 15 49 15.5 | -43 04 49 | 4 | 1772 | 752 | 4 | 221.2 | 0.9 | 5.8 |
V493 Ser | 15 56 44.3 | -00 09 51 | 6 | 3758 | 3815 | 34 | 752.8 | 11.2 | 46.9 |
QZ Ser | 15 56 54.5 | +21 07 20 | 5 | 5385 | 858 | 90 | 784.9 | 20.4 | 117.4 |
VW CrB | 16 00 03.8 | +33 11 15 | 7 | 5085 | 2073 | 24 | 723.6 | 6.3 | 37.0 |
V893 Sco | 16 15 15.1 | -28 37 31 | 5 | 2243 | 4313 | 22 | 557.8 | 8.8 | 27.8 |
V589 Her | 16 22 07.2 | +19 22 36 | 8 | 3479 | 1430 | 36 | 314.3 | 5.1 | 54.2 |
V699 Oph | 16 25 14.8 | -04 40 26 | 8 | 5610 | 1219 | 59 | 337.9 | 5.4 | 77.9 |
J1625+1203 | 16 25 20.3 | +12 03 09 | 4 | 1728 | 5398 | 20 | 247.0 | 4.8 | 26.5 |
V1032 Oph | 16 26 09.7 | -03 53 25 | 7 | 2691 | 4921 | 2.7 | 72.6 | 0.1 | 4.4 |
J1628+0653 | 16 28 06.2 | +06 53 15 | 5 | 2947 | 3617 | 13 | 296.9 | 2.4 | 20.4 |
J1628+2402 | 16 28 30.9 | +24 02 59 | 7 | 3064 | 3454 | 15 | 382.3 | 3.4 | 23.7 |
FL TrA | 16 30 36.6 | -61 50 21 | 8 | 3563 | 3571 | 31 | 269.6 | 3.7 | 47.3 |
J1631+1031 | 16 31 20.9 | +10 31 34 | 4 | 2555 | 4570 | 45 | 506.6 | 12.8 | 48.0 |
J1639+1224 | 16 39 42.7 | +12 24 14 | 6 | 2870 | 3565 | 6 | 205.7 | 0.7 | 8.7 |
V611 Her | 16 44 49.4 | +19 59 40 | 7 | 3436 | 3678 | 7 | 114.4 | 0.3 | 9.5 |
J1649+0358 | 16 49 50.4 | +03 58 35 | 5 | 3944 | 3235 | 40 | 648.6 | 11.0 | 56.9 |
DV Sco | 16 50 27.9 | -28 07 59 | 5 | 2191 | 2514 | 45 | 555.3 | 18.4 | 58.1 |
V1227 Her | 16 53 59.1 | +20 10 10 | 16 | 4098 | 3593 | 14 | 117.1 | 0.6 | 27.0 |
V1239 Her | 17 02 13.3 | +32 29 54 | 6 | 3595 | 3655 | 24 | 720.0 | 8.0 | 33.3 |
V877 Ara | 17 16 53.9 | -65 32 52 | 5 | 3432 | 2491 | 38 | 567.5 | 10.8 | 51.6 |
V2527 Oph | 17 22 04.4 | -19 49 09 | 3 | 1502 | 3202 | 13 | 751.1 | 10.4 | 14.7 |
J1730+6247 | 17 30 08.3 | +62 47 55 | 11 | 5341 | 2143 | 23 | 358.5 | 2.6 | 39.0 |
MM Sco | 17 30 45.3 | -42 11 41 | 15 | 20625 | -13188 | 42 | 711.6 | 2.6 | 85.3 |
J1740+4147 | 17 40 33.5 | +41 47 56 | 3 | 3900 | 4308 | 163 | 1949.9 | 126.2 | 178.4 | WZ
V660 Her | 17 42 09.2 | +23 48 31 | 11 | 5209 | 1333 | 28 | 372.3 | 3.6 | 54.4 |
J1743+2311 | 17 43 05.7 | +23 11 08 | 4 | 4342 | 4232 | 5 | 334.0 | 0.7 | 7.4 |
J1748+5050 | 17 48 27.9 | +50 50 40 | 4 | 2417 | 4747 | 11 | 603.9 | 4.5 | 14.2 |
J1749+1913 | 17 49 02.1 | +19 13 30 | 13 | 3649 | 2836 | 1 | 17.8 | 0.01 | 2.2 |
NY Her | 17 52 52.6 | +29 22 19 | 13 | 4402 | 3559 | 2 | 63.7 | 0.04 | 2.9 |
J1821+1709 | 18 21 13.9 | +17 09 16 | 7 | 1975 | 2759 | 23 | 180.2 | 3.3 | 32.2 |
J1830+1233 | 18 30 01.8 | +12 33 48 | 10 | 4995 | 2814 | 16 | 172.3 | 1.0 | 30.3 |
DT Oct | 18 40 52.6 | -83 43 10 | 10 | 4369 | 2642 | 7 | 311.6 | 0.8 | 12.5 |
J1900-4930 | 19 00 05.3 | -49 30 34 | 6 | 5722 | 2885 | 35 | 358.8 | 2.9 | 41.2 |
V419 Lyr | 19 10 13.9 | +29 06 14 | 14 | 3684 | 2890 | 7 | 115.2 | 0.4 | 13.8 |
V585 Lyr | 19 13 58.4 | +40 44 09 | 4 | 4966 | 2250 | 47 | 622.3 | 9.3 | 62.1 |
V344 Pav | 19 16 09.7 | -62 35 53 | 10 | 6058 | 1468 | 15 | 287.7 | 1.3 | 29.2 |
J1927-4847 | 19 27 10.1 | -48 47 52 | 10 | 2974 | 2437 | 14 | 176.6 | 1.5 | 23.0 |
V2176 Cyg | 19 27 11.6 | +54 17 52 | 3 | 5252 | 672 | 27 | 1753.4 | 14.7 | 31.7 | WZ
KX Aql | 19 33 53.7 | +14 17 47 | 6 | 14826 | -7859 | 85 | 819.2 | 6.9 | 114.9 |
V1006 Cyg | 19 48 47.2 | +57 09 22 | 8 | 4008 | 3958 | 20 | 361.6 | 2.7 | 30.5 |
V1047 Aql | 19 51 31.1 | +10 57 22 | 12 | 4815 | 3092 | 13 | 108.2 | 0.5 | 22.3 |
DO Vul | 19 52 10.7 | +19 34 43 | 4 | 4673 | 2804 | 55 | 944.2 | 20.1 | 75.3 |
V405 Vul | 19 53 05.0 | +21 14 49 | 8 | 3130 | 2773 | 15 | 388.0 | 3.4 | 23.9 |
V1454 Cyg | 19 53 38.5 | +35 21 46 | 5 | 6647 | 333 | 47 | 949.5 | 9.5 | 51.1 |
V725 Aql | 19 56 45.1 | +10 49 32 | 7 | 5904 | -174 | 37 | 537.8 | 5.7 | 62.6 |
Table 4: continued Name | RA (J2000) | DEC | N |  $\Delta$T | T0 |  $\sigma$(T0) | Cso |  $\sigma$(Cso) | $\mathrm{\overline{O-C}}$ | Remarks
---|---|---|---|---|---|---|---|---|---|---
|  |  |  | (d) | (d) | (d) | (d) | (d) | (d) |
AW Sge | 19 58 37.1 | +16 41 29 | 9 | 9927 | -3372 | 52 | 471.1 | 3.7 | 79.2 |
V337 Cyg | 19 59 52.9 | +39 14 00 | 8 | 7460 | 299 | 41 | 389.7 | 3.3 | 61.1 |
V1028 Cyg | 20 00 56.6 | +56 56 37 | 12 | 10145 | -4165 | 22 | 375.4 | 1.3 | 31.8 |
V550 Cyg | 20 05 05.2 | +32 21 21 | 5 | 5652 | -327 | 34 | 707.9 | 7.2 | 45.9 |
AX Cap | 20 08 57.0 | -17 16 38 | 5 | 3365 | 3221 | 77 | 654.7 | 24 | 100.3 |
J2009-6326 | 20 09 24.1 | -63 26 23 | 5 | 5081 | 2987 | 26 | 504.8 | 3.6 | 30.1 | WZ?
V1316 Cyg | 20 12 13.6 | +42 45 52 | 12 | 5568 | 2118 | 12 | 136.2 | 0.6 | 24.1 |
J2027-2240 | 20 27 31.2 | -22 40 01 | 4 | 2634 | 2020 | 35 | 527.7 | 11.7 | 44.9 |
SY Cap | 20 29 47.5 | -15 54 38 | 13 | 4879 | 2037 | 17 | 163.2 | 0.8 | 27.3 |
HO Del | 20 36 55.5 | +14 0310 | 8 | 7411 | -391 | 48 | 823.2 | 8.9 | 74.7 |
IL Vul | 20 38 32.7 | +22 42 17 | 6 | 3360 | 3554 | 21 | 484.4 | 4.9 | 30.0 |
J2054-1940 | 20 54 08.2 | -19 40 27 | 5 | 2584 | 3993 | 37 | 517.3 | 11.4 | 43.7 |
QU Aqr | 21 00 14.1 | +00 44 46 | 16 | 4023 | 3589 | 5 | 111.0 | 0.3 | 12.1 |
J2102+0258 | 21 02 05.7 | +02 58 34 | 7 | 3955 | 3693 | 18 | 180.4 | 1.4 | 24.0 |
AO Oct | 21 05 08.2 | -75 21 02 | 13 | 8733 | -1416 | 26 | 292.5 | 1.5 | 44.8 |
V364 Peg | 21 12 29.7 | +12 32 04 | 7 | 6512 | 798 | 16 | 361.0 | 1.4 | 24.2 |
EF Peg | 21 15 04.1 | +14 03 49 | 5 | 6640 | -1385 | 110 | 1110.7 | 34.9 | 160.5 | WZ?
J2120+1941 | 21 20 25.2 | +19 41 57 | 3 | 2838 | 4048 | 79 | 937.7 | 43.1 | 93.0 |
J2123-1539 | 21 23 25.6 | -15 39 54 | 4 | 6067 | 2252 | 98 | 1214.9 | 30.3 | 124.8 | WZ?
J2125-1026 | 21 25 21.8 | -10 26 28 | 4 | 3213 | 3714 | 13 | 200.4 | 1.4 | 16.7 |
V630 Cyg | 21 34 59.2 | +40 40 19 | 15 | 8021 | -3613 | 20 | 215.3 | 1.0 | 41.6 |
V632 Cyg | 21 36 04.2 | +40 26 19 | 11 | 15345 | -9491 | 53 | 491.4 | 2.5 | 70.2 |
V444 Peg | 21 37 01.8 | +07 14 46 | 3 | 2923 | 4760 | 33 | 1461.4 | 25.7 | 26.4 | WZ?
V627 Peg | 21 38 06.7 | +26 19 57 | 3 | 3321 | 3643 | 25 | 1660.6 | 19.5 | 27.6 | WZ
J2144+2220 | 21 44 26.4 | +22 20 25 | 3 | 2667 | 3532 | 26 | 891.4 | 12.7 | 27.4 |
J2147+2445 | 21 47 38.4 | +24 45 54 | 6 | 3850 | 3530 | 28 | 382.1 | 3.8 | 30.5 |
V476 Peg | 21 54 33.7 | +35 50 17 | 8 | 3958 | 3683 | 21 | 358.8 | 3.3 | 24.5 |
J2158+0947 | 21 58 15.3 | +09 47 10 | 6 | 4865 | 2909 | 42 | 341.1 | 4.6 | 59.7 |
J2206+3014 | 22 06 41.1 | +30 14 36 | 4 | 3376 | 3653 | 17 | 421.1 | 3.0 | 18.6 |
J2212+1601 | 22 12 32.0 | +16 01 40 | 7 | 4623 | 2899 | 33 | 255.7 | 3.0 | 45.6 |
V521 Peg | 22 21 44.8 | +18 40 08 | 9 | 10591 | 2881 | 17 | 164.1 | 1.0 | 30.6 |
J2234-0355 | 22 34 18.4 | -03 55 30 | 4 | 2355 | 3525 | 7 | 391.8 | 1.7 | 7.7 |
V650 Peg | 22 43 48.5 | +08 09 27 | 4 | 3609 | 3664 | 16 | 719.0 | 4.6 | 17.8 |
TY PsA | 22 49 39.9 | -27 06 54 | 8 | 3172 | 1883 | 18 | 224.4 | 2.2 | 31.6 |
J2258-0949 | 22 58 31.2 | -09 49 32 | 6 | 3730 | 2874 | 35 | 246.0 | 4.2 | 56.7 |
V368 Peg | 22 58 43.5 | +11 09 12 | 8 | 6269 | 1401 | 39 | 452.5 | 4.7 | 65.2 |
HY Psc | 23 03 51.7 | +01 06 52 | 5 | 4128 | 2480 | 19 | 457.5 | 3.8 | 28.3 |
CC Scl | 23 15 31.9 | -30 48 46 | 8 | 5960 | 1733 | 32 | 423.4 | 4.0 | 54.1 |
V776 And | 23 19 36.1 | +36 47 00 | 4 | 3234 | 3692 | 30 | 538.6 | 7.7 | 33.2 |
EG Aqr | 23 25 19.2 | -08 18 19 | 5 | 3210 | 4044 | 55 | 818.5 | 22.5 | 71.3 |
EI Psc | 23 29 54.3 | +06 28 11 | 7 | 5421 | 2257 | 31 | 671.7 | 6.2 | 47.3 |
J2333-1557 | 23 33 13.0 | -15 57 44 | 3 | 2152 | 4392 | 27 | 719.8 | 12.7 | 27.5 |

Table 5: New WZ Sge type candidate stars. Po and Ps refer to orbital and
superhump periods resp., A to the outburst amplitude in V. N and Cso as in
Table 1. References:(1) Still et al. (1994), (2) Howell et al. (1993),
(3) Kato et al. (2014).

Name | Other name | Po | Ps | A | N | Cso | Ref
---|---|---|---|---|---|---|---
|  | (d) | (d) | (mag) |  | (d) |
LY Hya | 1329-294 | 0.0748 | 0.077 | 5.3 | 5 | 1011 | (1)
J2009-6326 | ASASSN-14lk | 0.0600 | 0.0614 | 7.5 | 5 | 505 | -
EF Peg | AN 143.1935 | 0.0837 | 0.087 | 7.8 | 5 | 1111 | (2)
J2123-1539 | ASASSN-14dh | 0.072 | 0.0736 | 7.3 | 4 | 1215 | -
V444 Peg | Var Peg 2008 | 0.0947 | 0.0995 | 6.4 | 3 | 1461 | (3)

Table 6: Data used for our statistical comparison with published information. Cn refer to cycle length values of normal (short) outbursts of SU UMa stars, Cso to cycle lengths of their superoutbursts. The references in the last column are: (1) this work, only for Cso; (2) Contreras-Quijada et al, in preparation, only for Cso; (3) Coppejans et al. (2016) only for Cn; (4) Ritter & Kolb (2003, version 7.24, February 2017) for Cn and/or Cso in the remaining cases. The full table is available online.  Name | RA (J2000) | DEC | Cn | Cso | Ref
---|---|---|---|---|---
|  |  | (d) | (d) |
J0000+3325 | 00 00 24.6 | +33 25 43 | - | 247 | 1
BE Oct | 00 00 48.8 | -77 18 57 | - | 476 | 1
J0009-1210 | 00 09 38.2 | -12 10 16 | - | 603 | 1
V402 And | 00 11 07.3 | +30 32 36 | - | 400 | 4
J0015+2636 | 00 15 38.2 | +26 36 57 | - | 725 | 1
J0021-5719 | 00 21 30.9 | -57 19 22 | - | 530 | 1
Table 7: Cycle count numbers E and epochs Tmax (JD-2450000) of individual superoutbursts used for the calculation of the parameters in Table 1. The full table is available online.  Name | E | Tmax |
---|---|---|---
|  | (d) |
J0000+3325 | 0 | 4417.5 |
| 4 | 5449.4 |
| 6 | 5944.6 |
| 9 | 6628.4 |
| 12 | 7399.6 |
| 13 | 7662.56 |
BE Oct | 0 | 4123 |
| 4 | 5919 |
| 5 | 6482 |
| 6 | 6975.1 |
| 7 | 7397.1 |
| 8 | 7938.8 |
J0009-1210 | 0 | 3654.5 |
| 2 | 4808.5 |
| 3 | 5476.5 |
J0015+2636 | 0 | 3709.5 |
| 2 | 5092.4 |
| 3 | 5866.5 |
| 4 | 6617.6 |
| 5 | 7305.4 |
J0021-5719 | 0 | 2168.7 |
| 1 | 2578.7 |
| 3 | 3703.7 |
| 4 | 4230.9 |
| 5 | 4779.6 |

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claimClaim remarkRemark hypothesisHypothesis

    Supervised Learning of Probabilistic GMs for Density EstimationY. Liu, M. Yang, Z. Zhang, F. Bao, Y. Cao, G. Zhang

    # Diffusion-Model-Assisted Supervised Learning of Generative Models for Density Estimation††thanks: Notice: This manuscript has been authored by UT-Battelle, LLC, under contract DE-AC05-00OR22725 with the US Department of Energy (DOE). The US government retains and the publisher, by accepting the article for publication, acknowledges that the US government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for US government purposes. DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan.

    Yanfang Liu
    Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA.

    Minglei Yang
    Fusion Energy Science Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA.

    Zezhong Zhang22footnotemark: 2

    Feng Bao
    Department of Mathematics, Florida State University, Tallahassee, FL 32306, USA.

    Yanzhao Cao
    Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, USA.

    Guannan Zhang22footnotemark: 2

    ###### Abstract

    We present a supervised learning framework of training generative models for density estimation. Generative models, including generative adversarial networks, normalizing flows, variational auto-encoders, are usually considered as unsupervised learning models, because labeled data are usually unavailable for training. Despite the success of the generative models, there are several issues with the unsupervised training, e.g., requirement of reversible architectures, vanishing gradients, and training instability.
    To enable supervised learning in generative models, we utilize the score-based diffusion model to generate labeled data. Unlike existing diffusion models that train neural networks to learn the score function, we develop a training-free score estimation method. This approach uses mini-batch-based Monte Carlo estimators to directly approximate the score function at any spatial-temporal location in solving an ordinary differential equation (ODE), corresponding to the reverse-time stochastic differential equation (SDE). This approach can offer both high accuracy and substantial time savings in neural network training. Once the labeled data are generated, we can train a simple fully connected neural network to learn the generative model in the supervised manner.
    Compared with existing normalizing flow models, our method does not require to use reversible neural networks and avoids the computation of the Jacobian matrix. Compared with existing diffusion models, our method does not need to solve the reverse-time SDE to generate new samples. As a result, the sampling efficiency is significantly improved. We demonstrate the performance of our method by applying it to a set of 2D datasets as well as real data from the UCI repository.

    ###### keywords:

    Score-based diffusion models, density estimation, curse of dimensionality, generative models, supervised learning

    ##
    1 Introduction

    Density estimation involves the approximation of the probability density function of a given set of observation data points. The overarching goal is to characterize the underlying structure of the observation data. Generative models belong to a class of machine learning (ML) models designed to model the underlying probability distribution of a given dataset, enabling the generation of new samples that are statistically similar to the original data. Many methods for generative models
    have been proposed over the past decade, including variational auto-encoders (VAEs) [13], generative adversarial networks (GANs) [6], normalizing flows [14], and diffusion models [32].
    Generative models have been successfully used in a variety of applications, including image synthesis [28, 3, 10, 20, 6], image denoising [18, 9, 27, 15], anomaly detection [25, 21] and natural language processing [1, 11, 16, 24, 33, 19]. The key idea behind recent generative models is to exploit the superior expressive power and flexibility of deep neural networks to detect and characterize complicated geometries embedded in the possibly high-dimensional observational data.

    Generative models for density estimation are usually considered as unsupervised learning, primarily due to the absence of labeled data. Various unsupervised loss functions have been defined to train the underlying neural network models in generative models, including adversarial loss for GANs [6], the maximum likelihood loss for normalizing flows [14], and the score matching losses for diffusion models
    [12, 31, 29]. Despite the success of the generative models, there are several issues resulted from the unsupervised training approach. For example, the training of GANs may suffer from mode collapse, vanishing gradients, and training instability [23]. The maximum likelihood loss used in normalizing flows requires efficient computation of the determinant of the Jacobian matrix, which places severe restrictions on networks’ architectures. If labeled data can be created based on the observational data without complicated training, the generator in the generative models, e.g., the decoder in VAEs or normalizing flows, can be trained in a supervised manner, which can circumvent the issues in unsupervised training.

    In this work, we propose a supervised learning framework of training generative models for density estimation. The key idea is to use the score-based diffusion model to generate labeled data and then use the simple mean squared loss (MSE) to train the generative model.
    A diffusion model can transport the standard Gaussian distribution to a complex target data distribution through a reverse-time diffusion process in the form of a stochastic differential equation (SDE), and the score function in the drift term guides the reverse-time SDE towards the data distribution. Since the standard Gaussian distribution is independent of the target data distribution, the information of the data distribution is fully stored in the score function. Thus, we use the reverse-time SDE to generate the labeled data.
    Unlike existing diffusion models that train neural networks to learn the score function [30, 2, 26],
    we develop a training-free score estimation that uses mini-batch-based Monte Carlo estimators to directly approximate the score function at any spatial-temporal location in solving the reverse-time SDE.
    Numerical examples in Section 4 demonstrate that the training-free score estimation approach offers sufficient accuracy and save significant computing cost on training neural networks in the meantime. Once the labeled data are generated, we can train a simple fully connected neural network to learn the generator in the supervised manner.
    Compared with existing normalizing flow models, our method does not require to use reversible neural networks and avoids the computation of the Jacobian matrix. Compared with existing diffusion models, our method does not need to solve the reverse-time SDE to generate new samples. This way, the sampling efficiency is significantly improved.

    The rest of this paper is organized as follows. In Section 2, we briefly introduce the density estimation problem under consideration. In Section 3, we provide a comprehensive discussion of the proposed method. Finally in Section 4, we demonstrate the performance of our method by applying it to a set of 2D datasets as well as real data from the UCI repository.

    ##
    2 Problem setting

    We are interested in learning how to generate unrestricted number of samples of a target $d$-dimensional random variable, denoted by

    (1)

    $X\in\mathbb{R}^{d}\text{ and }X\sim p(x),$

    where $p$ is the probability density function of $X$.
    We aim to achieve this from a given dataset, denoted by $\mathcal{X}=\{x_{1},x_{2},\ldots,x_{J}\}\subset\mathbb{R}^{d}$, and from its probability density function (PDF) $p$. To this end, we aim at building a parameterized generative model, denoted by

    (2)

    $X=F(Y;\theta)\;\text{ with }\;Y\in\mathbb{R}^{d},$

    which is a transport model that maps a reference variable $Y$ (usually following a standard Gaussian distribution) to the target random variable $X$. Once the optimal value for the parameter $\theta$ is obtained by training the model, we can generate samples of $X$ by drawing samples of $Y$ and pushing them through the trained generative model.

    This problem has been extensively studied in the machine learning community using normalizing flow models [14, 22, 4, 21, 5, 7, 8], where $F(y;\theta)$ is defined as a bijective function. One major challenge for training the generative model $F(y;\theta)$ is the lack of labeled training data, i.e., there is no corresponding sample of $Y$ for each sample $x_{j}\in\mathcal{X}$. As a result, the model $F(y;\theta)$ cannot be trained via the simplest supervised learning using the mean square error (MSE) loss. Instead, an indirect loss defined by the change of variables formula, i.e., $p(x)=p(F^{-1}(x))|\det\mathbf{D}(F^{-1}(x))|$, is extensively used to train normalizing flow models in an unsupervised manner. The drawback of this training approach is that specially designed reversible architectures for $F(y;\theta)$ need to be used to efficiently perform backpropagation through the computation of $|\det\mathbf{D}(F^{-1}(x))|$. In this paper, we address this challenge by using score-based diffusion models to generate labeled data such that the generative model $F(y;\theta)$ can be trained in a supervised manner and $F^{-1}$ is no longer needed in the training.

    ##
    3 Methodology

    In this section, we present in details the proposed method. We briefly review the score-based diffusion model in Section 3.1. In Section 3.2, we introduce the training-free score estimation approach that is needed for generating the labeled data. The main algorithm for the supervised learning of the generative model $F(y;\theta)$ is presented in Section 3.3.

    ###
    3.1 The score-based diffusion model

    In this subsection, we will provide a brief overview of score-based diffusion models (see [30, 29] for more details). The model under consideration consists of a forward SDE and a reverse-time SDE defined in a standard temporal domain $[0,1]$. The forward SDE, defined by

    (3)

    $dZ_{t}=b(t)Z_{t}dt+\sigma(t)dW_{t}\;\text{ with }\;Z_{0}=X\text{ and }Z_{1}=Y,$

    is used to map the target random variable $X$ (i.e., the initial state $Z_{0}$) to the standard Gaussian random variable $Y\sim\mathcal{N}(0,\mathbf{I}_{d})$ (i.e., the terminal state $Z_{1}$). There is a number of choices for the drift and diffusion coefficients in Eq. (3) to ensure that the terminal state $Z_{1}$ follows $\mathcal{N}(0,\mathbf{I}_{d})$ (see [30, 31, 9, 17] for details). In this work, we choose $b(t)$ and $\sigma(t)$ in Eq. (3) as

    (4)

    $\displaystyle b(t)=\frac{{\rm d}\log\alpha_{t}}{{\rm d}t}\;\;\;\text{ and }\;\;\;\sigma^{2}(t)=\frac{{\rm d}\beta_{t}^{2}}{{\rm d}t}-2\frac{{\rm d}\log\alpha_{t}}{{\rm d}t}\beta_{t}^{2},$

    where the two processes $\alpha_{t}$ and $\beta_{t}$ are defined by

    (5)

    $\alpha_{t}=1-t,\;\;\beta_{t}^{2}=t\;\;\text{ for }\;\;t\in[0,1].$

    Because the forward SDE in (3)
    is linear and driven by an additive noise, the conditional probability density function $Q(Z_{t}|Z_{0})$ for any fixed $Z_{0}$ is a Gaussian. In fact,

    (6)

    $Q(Z_{t}|Z_{0})=\mathcal{N}(\alpha_{t}Z_{0},\beta_{t}^{2}\mathbf{I}_{d}),$

    which leads to $Q(Z_{1}|Z_{0})=\mathcal{N}(0,\mathbf{I}_{d})$. The corresponding reverse-time SDE is defined by

    (7)

    $d{Z}_{t}=\left[b(t){Z}_{t}-\sigma^{2}(t)S(Z_{t},t)\right]dt+\sigma(t)d\mathchoice{\mkern 3.0mu\reflectbox{$\displaystyle\vec{\reflectbox{$\mkern-3.0mu\displaystyle W\mkern 3.0mu$}}$}\mkern-3.0mu}{\mkern 3.0mu\reflectbox{$\textstyle\vec{\reflectbox{$\mkern-3.0mu\textstyle W\mkern 3.0mu$}}$}\mkern-3.0mu}{\mkern 2.0mu\reflectbox{$\scriptstyle\vec{\reflectbox{$\mkern-2.0mu\scriptstyle W\mkern 2.0mu$}}$}\mkern-2.0mu}{\mkern 2.0mu\reflectbox{$\scriptscriptstyle\vec{\reflectbox{$\mkern-2.0mu\scriptscriptstyle W\mkern 2.0mu$}}$}\mkern-2.0mu}_{t}\;\text{ with }\;Z_{0}=X\text{ and }Z_{1}=Y,$

    where $\mathchoice{\mkern 3.0mu\reflectbox{$\displaystyle\vec{\reflectbox{$\mkern-3.0mu\displaystyle W\mkern 3.0mu$}}$}\mkern-3.0mu}{\mkern 3.0mu\reflectbox{$\textstyle\vec{\reflectbox{$\mkern-3.0mu\textstyle W\mkern 3.0mu$}}$}\mkern-3.0mu}{\mkern 2.0mu\reflectbox{$\scriptstyle\vec{\reflectbox{$\mkern-2.0mu\scriptstyle W\mkern 2.0mu$}}$}\mkern-2.0mu}{\mkern 2.0mu\reflectbox{$\scriptscriptstyle\vec{\reflectbox{$\mkern-2.0mu\scriptscriptstyle W\mkern 2.0mu$}}$}\mkern-2.0mu}_{t}$ is the backward Brownian motion and $S(Z_{t},t)$ is the score function. If the score function is defined by

    (8)

    $S(Z_{t},t):=\nabla_{z}\log Q({Z}_{t}),$

    where $Q(Z_{t})$ is the probability density function of $Z_{t}$ in Eq. (3), then the reverse-time SDE maps the standard Gaussian random variable $Y$ to the target random variable $X$. Therefore, if we can evaluate the score function for any $(Z_{t},t)$, then we can directly generate samples of $X$ by pushing samples of $Y$ through the reverse-time SDE. Thus, the central task in training diffusion models is appropriately approximating the score function. There are several established approaches, including score matching [12],
    denoising score matching [31], and sliced score matching [29], etc.
    Recent advances in diffusion models focus on explorations of using neural networks to approximate the score function.
    Compared to the normalizing flow models, a notable drawback of learning the score function is its inefficiency in sampling since generating one sample of $X$ requires solving the reverse-time SDE. To alleviate this challenge, we use the score-based diffusion model as a data labeling approach, which will enable supervised learning of the generative model of interest.

    ###
    3.2 Training-free score estimation

    We now derive the analytical formula of the score function and its approximation scheme. Substituting $Q(Z_{t})=\int_{\mathbb{R}^{d}}Q(Z_{t},Z_{0})dZ_{0}=\int_{\mathbb{R}^{d}}Q({Z}_{t}|Z_{0})Q(Z_{0})dZ_{0}$ into Eq. (8) and exploiting the fact in Eq. (6), we can rewrite the score function as

    (9)

    $\displaystyle S(Z_{t},t)$
    $\displaystyle=\nabla_{z}\log\left(\int_{\mathbb{R}^{d}}Q({Z}_{t}|Z_{0})Q(Z_{0})dZ_{0}\right)$

    $\displaystyle=\frac{1}{\int_{\mathbb{R}^{d}}Q({Z}_{t}|Z^{\prime}_{0})Q(Z^{\prime}_{0})dZ^{\prime}_{0}}\int_{\mathbb{R}^{d}}-\frac{Z_{t}-\alpha_{t}Z_{0}}{\beta^{2}_{t}}Q({Z}_{t}|Z_{0})Q(Z_{0})dZ_{0}$

    $\displaystyle=\int_{\mathbb{R}^{d}}-\frac{Z_{t}-\alpha_{t}Z_{0}}{\beta^{2}_{t}}w_{t}({Z}_{t},Z_{0})Q(Z_{0})dZ_{0},$

    where the weight function $w_{t}({Z}_{t},Z_{0})$ is defined by

    (10)

    $w_{t}({Z}_{t},Z_{0}):=\frac{Q({Z}_{t}|Z_{0})}{\int_{\mathbb{R}^{d}}Q({Z}_{t}|Z^{\prime}_{0})Q(Z^{\prime}_{0})dZ^{\prime}_{0}},$

    satisfying that $\int_{\mathbb{R}^{d}}w_{t}({Z}_{t},Z_{0})Q(Z_{0})dZ_{0}=1$.

    The integrals/expectations in Eq. (9) can be approximated by Monte Carlo estimators using the available samples in $\mathcal{X}=\{x_{1},x_{2},\ldots,x_{J}\}\subset\mathbb{R}^{d}$. According to the definition of the reverse-time SDE in Eq. (7), the samples in $\mathcal{X}$ are also samples from $Q(Z_{0})$. Thus, the integral in Eq. (9) can be estimated by

    (11)

    $S(Z_{t},t)\approx\bar{S}(Z_{t},t):=\sum_{n=1}^{N}-\frac{Z_{t}-\alpha_{t}x_{j_{n}}}{\beta^{2}_{t}}\bar{w}_{t}({Z_{t}},x_{j_{n}}),$

    using a mini-batch of the dataset $\mathcal{X}$ with batch size $N\leq J$, denoted by $\{x_{j_{n}}\}_{n=1}^{N}$, where the weight $w_{t}({Z_{t}},x_{j_{n}})$ is calculated by

    (12)

    $w_{t}({Z_{t}},x_{j_{n}})\approx\bar{w}_{t}({Z_{t}},x_{j_{n}}):=\frac{Q(Z_{t}|x_{j_{n}})}{\sum_{m=1}^{N}Q(Z_{t}|x_{j_{m}})},$

    and $Q(Z_{t}|x_{j_{n}})$ is the Gaussian distribution given in Eq. (6).
    This means $w_{t}({Z_{t}},Z_{0})$ can be estimated by the normalized probability density values
    $\{Q(Z_{t}|x_{j_{n}})\}_{n=1}^{N}$. In practice, the size of the mini-batch $\{x_{j_{n}}\}_{n=1}^{N}$ can be flexibly adjusted to balance the efficiency and accuracy.

    ###
    3.3 Supervised learning of the generative model

    In this subsection, we describe how to use the score approximation scheme given in Section 3.2 to generate labeled data and use such data to train the generative model of interest. Due to the stochastic nature of the reverse-time SDE in Eq. (7), the relationship between the initial state $Z_{0}$ and the terminal state $Z_{1}$ is not deterministic or smooth, as shown in Figure 1. Thus, we cannot directly use Eq. (7) to generate labeled data. Instead, we use the corresponding ordinary differential equation (ODE), defined by

    (13)

    $d{Z}_{t}=\left[b(t){Z}_{t}-\frac{1}{2}\sigma^{2}(t)S(Z_{t},t)\right]dt\;\text{ with }\;Z_{0}=X\text{ and }Z_{1}=Y,$

    whose trajectories share the same marginal probability density functions as the reverse-time SDE in Eq. (7). An illustration of the trajectories of the SDE and the ODE is given in Figure 1. We observe that ODE defines a much smoother function relationship between the initial state and the terminal state than that defined by the SDE. Thus, we use the ODE in Eq. (13) to generate labeled data.

    Figure 1: Illustration of the trajectories of the ODE model in Eq. (13) and the SDE in Eq. (7) using a simple one-dimensional example. We observe that the ODE model creates a much smoother function relationship between the initial state and the terminal state, which indicates that the ODE model is more suitable for generating the labeled data for supervised learning of the generative model of interest.

    Specifically, we first draw $M$ random samples of $Y$, denoted by $\mathcal{Y}=\{y_{1},\ldots,y_{M}\}$ from the standard Gaussian distribution. For $m=1,\ldots,M$, we solve the ODE in Eq. (13) from $t=1$ to $t=0$ and collect the state $Z_{0}|y_{m}$, where the score function is computed using Eq. (11), Eq. (12), and the dataset $\mathcal{X}=\{x_{1},\ldots,x_{J}\}$. The labeled training dataset is denoted by

    (14)

    $\mathcal{D}_{\rm train}:=\{(y_{m},x_{m}):x_{m}=Z_{0}|y_{m},\;\text{ for }\;m=1,\ldots,M\},$

    where $x_{m}$ is obtained by solving the ODE in Eq. (13). We note that the $x_{m}$’s in $\mathcal{D}_{\rm train}$
    may not belong to $\mathcal{X}$ and $M$ could be arbitrarily large. After obtaining $\mathcal{D}_{\rm train}$ we can use it to train the generative model $F(y;\theta)$ in Eq. (2) using supervised learning with the MSE loss.

    Algorithm 1: supervised learning of generative models

    1:  Input: observation data $\mathcal{X}$, diffusion coefficient $\sigma(t)$, and drift coefficient $b(t)$;

    2:  Output: the trained generative model $F(y;\theta)$;

    3: Draw $M$ samples $\mathcal{Y}=\{y_{1},\ldots,y_{M}\}$ from the standard Gaussian distribution;

    4:  for $m=1,\ldots,M$

    5:   Solve the ODE in Eq. (13) with the score function estimated by
    Eq. (11) and Eq. (12);

    6:   Define a pair of labeled data $(y_{m},x_{m})$ where $y_{m}=Z_{1}$ and $x_{m}=Z_{0}$ in Eq. (13);

    7: end

    8: Train the generative model $x=F(y;\theta)$ with the MSE loss.

    Our method is summarized in Algorithm 1. Compared to the existing normalizing flow models and diffusion models, our method has two significant advantages in performing density estimation tasks. First, it does not require to know $F^{-1}(x;\theta)$, hence it does not require the computation of $|\det\mathbf{D}(F^{-1}(x))|$ in the training process. This enables us to use simpler neural
    network architectures to define $F(y;\theta)$, resulting in a more straightforward training procedure compared to the training of a normalizing flow model.
    Second, after $F(y;\theta)$ is trained, our method does not require solving the reverse-time SDE or ODE to generate samples of $X$. As a result, it significantly enhances the sampling efficiency in comparison with the diffusion model.

    ##
    4 Numerical Experiments

    We demonstrate the performance of the proposed method on several benchmark problems for density estimation.
    To solve the reverse-time ODE for generating the labeled data, we use theexplicit Euler scheme.
    The generative model $F(y;\theta)$ is defined by a fully-connected feed-forward neural network.
    Our method is implemented in Pytorch with GPU acceleration enabled. The source code is publicly available at https://github.com/mlmathphy/supervised_generative_model. The numerical results in this section can be precisely reproduced using the code on Github.

    ###
    4.1 Density Estimation on Toy 2D Data

    We use four two-dimensional datasets [7] to demonstrate and visualize the performance of the proposed method. Each dataset has 1000 data, referred to as the ground truth in Figure 2.
    We use a fully-connected neural network with four hidden layers to define $F(y;\theta)$, each of which has 100 neurons. 100 times steps are used to discretize the reverse-time ODE in Eq. (13) to generate 1000 labeled data. The neural network is then trained with the MSE loss for 5000 epochs using the Adam optimizer with the learning rate chosen as 0.005.

    The results are shown in Figure 2. We observe that the labeled samples are not the same as the ground truth, but they accurately approximate the distribution of the ground truth. In fact, the reverse-time ODE in Eq. (13) can be viewed as a training-free version of the neural-ODE-based normalizing flow [7], which can capture multi-modal and discontinuous distributions. The samples generated by $F(y;\theta)$ also provide an accurate approximation to the ground truth, even though the accuracy is lower than the distribution of the labeled training data. There are scattered samples generated by $F(y;\theta)$ because the used neural networks cannot perfectly approximate the discontinuity among different modes of the target distribution.

    Figure 2: Results on four 2D toy datasets. The left column shows the ground true distribution, i.e., the dataset $\mathcal{X}$; the middle column shows the generated labeled data by solving the ODE in Eq. (13); and the right column shows the samples generated by the trained generative model $F(y;\theta)$.

    ###
    4.2 Density Estimation on Real Data

    We demonstrate the performance of our method using density estimation on four UCI datasets, including POWER (6-dimensional, available at http://archive.ics.uci.edu/ml/datasets/Individual+household+electric+power+consumption), GAS (8-dimensional, available at http://archive.ics.uci.edu/ml/datasets/Gas+sensor+array+under+dynamic+gas+mixtures), HEPMASS (11-dimensional, available at http://archive.ics.uci.edu/ml/datasets/HEPMASS), and MINIBOONE (43-dimensional, available at http://archive.ics.uci.edu/ml/datasets/MiniBooNE+particle+identification). These datasets are taken from the UCI machine learning repository. We follow the approach in [21] to pre-proprocess each dataset. We normalize each dimension of the data by subtracting its mean and dividing its standard deviation, such that all the datasets used to test our method have zero mean and unit standard deviation. Discrete-valued dimensions and every attribute with a Pearson correlation coefficient greater than 0.98 were eliminated.

    The reverse-time ODE in Eq. (13) is solved using the explicit Euler scheme with 500 time steps. All training data is split into mini-batches of size $N=5,000$. We use a fully-connected neural network with one hidden layer to define $F(y;\theta)$. Specifically, the network for POWER has 256 hidden neurons, the network for GAS has 512 hidden neurons, the network for HEPMASS has 1024 neurons, and the network for MINIBOONE has 1400 hidden neurons, respectively. The neural networks are trained by the Adam optimizer with 20,000 epochs.

    Figure 3 to Figure 6 show the comparison among the ground truth data, the labeled data (from the ODE in Eq. (13)) and the generated samples from $F(y;\theta)$ using the following metrics:

      *
    •

    The 1D marginal PDF of a randomly selected dimension;

      *
    •

    The K-L divergences of all the 1D marginal distributions;

      *
    •

    The mean values of all the 1D marginal distributions;

      *
    •

    The standard deviations of all the 1D marginal distributions.

    As expected, the labeled data and the generated samples can accurately approximate the distribution of the ground truth. Table 1 shows the computing costs of generating the labeled data, training the neural network $F(y;\theta)$ and using $F(y;\theta)$ to generate samples. The computational time is obtained by running our code on a workstation with Nvidia RTX A5000 GPU.

    Dataset
    | Data labeling
    |
    Training $F$

    | Synthesizing 100K samples

    ---|---|---|---

    POWER
    | 64.83 seconds
    | 182.51 seconds
    | 0.10 seconds

    GAS
    | 85.78 seconds
    | 426.61 seconds
    | 0.23 seconds

    HEPMASS
    | 109.12 seconds
    | 1940.87 seconds
    | 0.51 seconds

    MINIBOONE
    | 408.36 seconds
    | 2220.79 seconds
    | 0.66 seconds

    Table 1: Wall-clock time of different stages of our method. Data labeling refers to the stage of solving the reverse-time ODE in Eq. (13); Training $F$ refers to the stage of using the labeled data to train the generative model $F(y;\theta)$; Synthesizing 100K samples refers to using the trained model $F(y;\theta)$ to generate 100K new samples of $X$. We observe that our method features a promising efficiency in generating a large number of samples.

    Figure 3: Results on the POWER dataset. (a) The 1D marginal PDF of a randomly selected dimension (b) The K-L divergences of all the 1D marginal distributions; (c) The mean values of all the 1D marginal distributions; (d) The standard deviations of all the 1D marginal distributions.

    Figure 4: Results on the GAS dataset. (a) The 1D marginal PDF of a randomly selected dimension (b) The K-L divergences of all the 1D marginal distributions; (c) The mean values of all the 1D marginal distributions; (d) The standard deviations of all the 1D marginal distributions.

    Figure 5: Results on the HEPMASS dataset. (a) The 1D marginal PDF of a randomly selected dimension (b) The K-L divergences of all the 1D marginal distributions; (c) The mean values of all the 1D marginal distributions; (d) The standard deviations of all the 1D marginal distributions.

    Figure 6: Results on the MINIBOONE dataset. (a) The 1D marginal PDF of a randomly selected dimension (b) The K-L divergences of all the 1D marginal distributions; (c) The mean values of all the 1D marginal distributions; (d) The standard deviations of all the 1D marginal distributions.

    ##
    5 Conclusion

    We introduced a supervised learning framework for training generative models for density estimation. Within this framework, we utilize the score-based diffusion model to generate labeled data and employ simple, fully-connected, neural networks to learn the generative model of interest. The key ingredient is the training-free score estimation that enables data labeling without training the score function. It is important to note that the current algorithm has only been successfully tested using a tabular dataset, and its performance in image/signal synthesis remains to be explored. On the other hand, this algorithm can be applied to a variety of Bayesian sampling problems in scientific and engineering applications, including parameter estimation of physical models, state estimation of dynamical systems (e.g., chemical reactions), surrogate models for particle simulation in physics, all of which will be our future work on this topic.

    ## Acknowledgement

    This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under the contract ERKJ387 at the Oak Ridge National Laboratory, which is operated by UT-Battelle, LLC, for the U.S. Department of Energy under Contract DE-AC05-00OR22725. The author Feng Bao would also like to acknowledge the support from the U.S. National Science Foundation through project DMS-2142672 and the support from the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under Grant DE-SC0022297, and the author Yanzhao Cao would also like to acknowledge the support from the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under Grant DE-SC0022253.

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