11institutetext: Department of Physics and Astronomy, FI-20014 University of
Turku, Finland; 11email<EMAIL_ADDRESS>22institutetext: Armagh Observatory &
Planetarium, College Hill, Armagh BT61 9DG, UK; 33institutetext: Department of
Physics & Astronomy, University of Western Ontario, London, Ontario N6A 3K7,
Canada; 44institutetext: Leibniz-Institut für Sonnenphysik (KIS), Schöneckstr
6, Freibirg, Germany; 55institutetext: IRSOL Istituto Ricerche Solari “Aldo e
Cele Daccò”, Faculty of Informatics, Università della Svizzera italiana, Via
Patocci 57, Locarno, Switzerland; 66institutetext: Euler Institute, Faculty of
Informatics, Università della Svizzera italiana, Via la Santa 1, 6962 Lugano,
Switzerland

# Discovery of magnetic fields in five DC white dwarfs

Andrei V. Berdyugin 11 Vilppu Piirola 11 Stefano Bagnulo 22 John D. Landstreet
2233 Svetlana V. Berdyugina 445566

(Received October 7, 2022; accepted October 27, 2022)

About half of white dwarfs (WDs) evolve to the DC state as they cool; the
others become DQ or (temporarily?) DZ WDs. The recent magnetic survey of the
local 20 pc volume has established a high frequency of magnetic fields among
WDs older than 2–3 Gyr, demonstrating that in low- and average-mass WDs, the
effects of magnetism become more common as they age, and the fields on average
become stronger. However, the available statistics of WDs older than about 5
Gyr do not clearly establish how fields evolve beyond this age. We are
carrying out a survey to clarify the occurrence of magnetism in DC-type WDs in
order to better understand this late evolution. We use broadband filter
polarimetry, arguably the most efficient way to detect magnetic fields in
featureless WDs via continuum circular polarization. Here we report the
discovery of a magnetic field in five DC WDs (of 23 observed), almost doubling
the total sample of known magnetic WDs belonging to the DC spectral class.

###### Key Words.:

White dwarfs – Stars: magnetic fields – polarization

##  1 Introduction

Single stars of $M\la 8M_{\odot}$ evolve to become white dwarfs (WDs). The
descendants of these single stars of intermediate mass provide most of the
population of WDs, concentrated around the mean mass of $0.6M_{\odot}$. A
smaller fraction of current WDs were also formed from close binary systems.
Some of these systems eventually merged to form a single collapsed remnant,
frequently of a significantly larger mass; others ended their nuclear
lifetimes as double WD binaries.

Once formed, the evolution of a WD is normally to cool slowly over several
gigayears. Cooling is a fairly complex process even for single-star evolution,
both to observe and to understand. Observationally, young hot WDs usually show
strong spectra of H (DA WDs), He (DB WDs), or sometimes C (DQs). As they cool,
spectral lines of the dominant elements H or He become weaker: He lines vanish
at about 11000 K, H lines around 5000 K. In parallel with this general
evolution, WDs may (temporarily) show lines of metals such as Mg, Si, Ca,
and/or Fe (DZ, DAZ, and DZA stars), and some have spectra dominated by C.
Below about 5000 K, about one-quarter of WDs have very weak H$\alpha$, another
quarter show spectral lines of metals (especially Ca ii) or of C2, and the
remaining half show essentially featureless spectra (DC WDs; Bagnulo &
Landstreet, 2021, Table 1). It appears that the dominant element(s) in the
atmosphere can change as cooling occurs, for example due to gravitational
diffusion, development of convection, and accretion of circumstellar planetary
debris.

One of the physical effects adding complexity to our efforts to understand WD
evolution is that a significant fraction, about 20-25%, of WDs in the local
volume near the Sun (Bagnulo & Landstreet, 2021) possess detectable surface
magnetic fields (this high frequency was already suggested on the basis of
literature reports of fields in nine WDs in the 13 pc volume by Kawka et al.,
2007). The fields observed at the surface range in strength, measured by the
mean surface field $\langle|B|\rangle$, from tens of kG to hundreds of MG.
Such fields can significantly affect WD evolution by altering or suppressing
surface convection and internal shear, and by transferring angular momentum
between internal layers or during accretion or mass loss (see for example
Tremblay et al., 2015). The fields may also introduce additional forces into
envelope and atmosphere layers, altering their hydrostatic structure from that
expected when magnetic effects are absent (Landstreet, 1987).

For WDs formed by single-star evolution, which generates most of the large
populations of WDs with masses around $0.6M_{\odot}$, it has become clear that
recently formed WDs (with cooling ages of less than, say, 1 Gyr) are very
rarely detectably magnetic, and when they are magnetic, the fields are usually
very weak (Bagnulo & Landstreet, 2022). As WDs cool, fields begin to appear
more frequently and usually become stronger. In WDs older than 3 or 4 Gyr,
megagauss-scale fields are not uncommon (Bagnulo & Landstreet, 2021).

The observed evolution in magnetic field frequency and strength of normal-mass
WDs for the first few gigayears of cooling may be understood as a slow
emergence – as a result of field relaxation to the stellar surface – of the
internal fields present in the degenerate cores of the WD precursors. An
additional contribution to observed surface fields may be due to magnetic
fields generated during cooling by a dynamo that acts during the period when
the core of the WD is crystallising (Isern et al., 2017; Gentile Fusillo et
al., 2018). Beyond the end of crystallisation, the only identified evolution
mechanisms are continued field relaxation and Ohmic decay.

Observationally, however, after about 5 Gyr of WD cooling, we have very
limited information with which to guide and confront theory. Only small survey
samples constrain observed field evolution on cool WDs, such as DQ WDs, where
C2 bands show no polarization in strong fields (Berdyugina et al., 2007).
Particularly little is known about the magnetic fields of DC WDs, in which no
spectral features are seen at all, leading to the questions of whether field
strength begins to decay Ohmically and whether the frequency of surface fields
continues to increase. Data that could help us answer these questions are very
limited. For WDs within 20 pc of the Sun (the 20 pc volume sample), Bagnulo &
Landstreet (2021) showed that of 31 DC WDs, only 4 are magnetic white dwarfs
(MWDs), and that only 4 of 24 WDs of any spectral class older than $\sim 6$
Gyr are magnetic. These data are obviously too limited to clearly describe the
evolution of fields in these old WDs.

Previous surveys have provided almost no information about magnetic fields in
DC WDs. Fields in such stars cannot be detected through the magnetic splitting
of spectral lines. They can only be detected via the observation of continuum
circular polarisation (CCP; Kemp, 1970), a method of observation hardly
employed since the 1970s (Angel et al., 1981). Remarkably, most CCP
observations have led to the discovery of magnetic fields in stars that are
not featureless but in which the magnetic field is strong enough to shift and
broaden spectral lines in a such a way as to make their intensity spectra
unrecognisable. Only seven featureless DC WDs are presently known to be
magnetic. Five of them were discovered only in the last couple of years
(Bagnulo & Landstreet, 2020; Berdyugin et al., 2022). Before these results,
the only known magnetic DC stars were G195-19 and G111-49, discovered
respectively by Angel & Landstreet (1971) and Putney (1995).

To improve our knowledge of the magnetic fields in the latest stages of
stellar evolution, we have started a volume-limited survey of DC stars in the
local 33 pc volume, which is about 4.5 times larger than the previously
explored 20 pc volume and should have a correspondingly larger sample of DCs
and DC MWDs. With this sample we expect to find enough DC MWDs to delineate
the evolution of their magnetic fields, both in the WDs with He-rich
atmospheres that become DCs as soon as their effective temperatures reach
about 11 000 K (‘young’ DCs), and in DC WDs with $T_{\rm eff}$ below about
5000 K, with cooling ages of around 4 Gyr or more (‘old’ DCs).

##  2 Observations

Almost all known MWDs have been discovered via the magnetic (Zeeman) splitting
of spectral lines, observed in stellar flux spectra, or via the Zeeman
polarisation of spectral features (Ferrario et al., 2015). Using these
methods, fields of a few kG up to 1 GG can be reliably detected. However,
these techniques cannot be used to measure fields in WDs that lack spectral
lines. For such stars, it is necessary to rely on continuum polarisation,
which Kemp (1970) showed should occur in radiation from a magnetized emitter.
The value of this effect was confirmed by the discovery of a very strong field
in the bright WD Grw+70 8247 = WD 1900+705 through the detection of broadband
circular polarisation (BBCP) by Kemp et al. (1970).

Broadband circular polarisation is a relatively weak effect. Bagnulo &
Landstreet (2020) have estimated that a field of $\langle B_{z}\rangle\sim 15$
MG is required to produce BBCP of order 1 % in optical radiation from a cool
WD. However, with a sensitive polarimeter, especially one with a very stable
and well-established zero point, it is possible in principle to detect
polarisation of $10^{-4}$ or less, corresponding to $\sim 100$ kG fields in
‘sufficiently bright’ WDs.

Table 1:  Observing log of bright non-polarised standard stars and the highly polarised MWD WD 1900+705. Polarisation values are given assuming as instrumental polarisation the values of 0.0121$\pm$0.0004 %, 0.0109$\pm$0.0005 %, and 0.0084$\pm$0.0004 % in the $B^{\prime}$, $V^{\prime}$, and $R^{\prime}$ filters, respectively. For comparison, we report the polarisation values of WD 1900+705 measured in our 2021 and 2022 runs. STAR | $G$ | DATE | UT | JD – | Exp. |  $V/I$ (%)
---|---|---|---|---|---|---
|  | yyyy-mm-dd | hh:mm | 2400000 | (s) | $B^{\prime}$ | $V^{\prime}$ | $R^{\prime}$
HD 107146 | 6.9 | 2022-06-27 | 21:23 | 59758.391 | 1680 |  $0.0005$$\pm$ | 0.0007 |  $-0.0006$$\pm$ | 0.0007 |  $-0.0001$$\pm$ | 0.0006
HD 115043 | 6.7 | 2022-06-28 | 21:21 | 59759.390 | 1520 |  $0.0005$$\pm$ | 0.0008 |  $-0.0001$$\pm$ | 0.0012 |  $-0.0004$$\pm$ | 0.0004
HD 122652 | 7.0 | 2022-06-29 | 21:18 | 59760.387 | 1520 |  $-0.0012$$\pm$ | 0.0009 |  $-0.0015$$\pm$ | 0.0015 |  $-0.0019$$\pm$ | 0.0014
HD 122676 | 7.1 | 2022-06-30 | 21:17 | 59761.387 | 1520 |  $0.0000$$\pm$ | 0.0011 |  $0.0003$$\pm$ | 0.0010 |  $0.0018$$\pm$ | 0.0008
HD 124694 | 7.0 | 2022-07-01 | 21:16 | 59762.386 | 1520 |  $-0.0012$$\pm$ | 0.0008 |  $0.0014$$\pm$ | 0.0011 |  $0.0002$$\pm$ | 0.0007
HD 135891 | 6.9 | 2022-07-02 | 21:18 | 59763.387 | 1520 |  $0.0014$$\pm$ | 0.0008 |  $0.0006$$\pm$ | 0.0010 |  $-0.0004$$\pm$ | 0.0007
HD 117860 | 7.2 | 2022-07-03 | 21:16 | 59764.386 | 1520 |  $-0.0004$$\pm$ | 0.0010 |  $-0.0002$$\pm$ | 0.0012 |  $0.0005$$\pm$ | 0.0006
WD 1900+705 | 13.2 | 2021-07-02 | 22:22 | 59398.432 | 640 |  $3.756$$\pm$ | 0.016 |  $3.604$$\pm$ | 0.016 |  $3.827$$\pm$ | 0.019
|  | 2022-07-02 | 00:25 | 59762.518 |  |  $3.789$$\pm$ | 0.016 |  $3.602$$\pm$ | 0.019 |  $3.838$$\pm$ | 0.018

To detect and measure broadband continuum polarisation, one uses either
spectropolarimetry or filter polarimetry with broad, photometry-like filters.
It is very difficult to establish the zero point with sufficient accuracy
below polarisation levels of the order of $10^{-3}$ in spectropolarimetric
measurements of the continuum (Fossati et al., 2007; Siebenmorgen et al.,
2014); therefore, in DC WDs, only megagauss-scale fields can be detected in
this way (Bagnulo & Landstreet, 2020). In contrast, broadband filter
polarimeters can be very stable, and instrumental polarisation can be
calibrated at the $10^{-5}$ level, so detections with such instruments of
fields of hundreds of kilogauss are in practice limited by the telescope
aperture and WD brightness (Berdyugin et al., 2022).

The search for magnetic fields of a fraction of 1 MG or stronger in DC WDs
that is reported here was carried out with the DIPol-UF broadband filter
polarimeter (Piirola et al., 2020) mounted on the 2.5 m Nordic Optical
Telescope (NOT) at the Observatorio del Roque de los Muchachos on the island
of La Palma, in the Canaries. This instrument obtains simultaneous circular
polarisation (normalized Stokes V/I) measurements in three filter bands
isolated by dichroic mirrors. The passbands are centred at about 4450 Å (the
$B^{\prime}$ band), 5400 Å (the $V^{\prime}$ band), and 6400 Å (the
$R^{\prime}$ band) with full widths at half maximum (FWHMs) of 1140, 750, and
960 Å , respectively. With this instrument on the NOT, we can detect a
polarisation degree at the $3\sigma$ level of $\sim 10^{-4}$ for the Gaia
G-band magnitude $G\sim 12$, down to a degree of $\sim 10^{-3}$ at $G\sim 17$.
This instrument and the filter system are discussed in more detail in our
previous paper, which reports the results of the first part of our search for
magnetic fields in DC WDs (Berdyugin et al., 2022).

Here we report observations of 23 DC WDs and discovery of 5 new DC MWDs. We
note that our survey includes nine young DCs of He-rich atmospheres with
$11000\la T_{\rm eff}\la 5000$ K and 14 old WDs with $T_{\rm eff}\la 5000$ K
and ages $\tau\ga 4$ Gyr. The stars are selected from available
classifications and with help from Gentile Fusillo et al. (2021). Our new
observations were obtained between June 27 and July 5, 2022.

Figure 1:  Wavelength dependence of circular polarisation detected for five
targets ($>3\sigma$ confidence level). A wide variety of polarisation
behaviour is observed. Horizontal bars in the bottom panel show the FWHM of
the $B^{\prime}V^{\prime}R^{\prime}$ filter passbands.

###  2.1 Instrumental polarisation and alignment of the polarimetric optics

Table 2:  Programme stars and their main physical features. Star names in
boldface identify WDs in which fields were discovered during the observations
reported in this paper (see Table 3).

STAR | $G$ | $d$ | $T_{\rm eff}$ | log $g$ | $M$ | Age | Atmosphere and ref.
---|---|---|---|---|---|---|---
|  |  | (pc) | (K) | c.g.s. | ($M_{\odot}$) | (Gyr) |
WD 0005+395 | LP 240-30 | 16.6 | 34.4 | 4680 | 6.77 | 0.08 | 1.40 | DC, H Prob${}_{\rm WD}=0.70$ (1,3)
WD 0010+543 | LSR J0013+5437 | 18.0 | 32.3 | 4123 | 7.77 | 0.46 | 7.08 | DC, (2, H assumed)
WD 0028+035 | PB 6002 | 16.1 | 27.8 | 6548 | 8.14 | 0.68 | 2.40 | DC, (2, H assumed)
WD 1251+366 | LP 267-311 | 17.2 | 28.5 | 4445 | 7.62 | 0.37 | 3.78 | DC, He (1)
WD 1315+222 | LP 378-956 | 16.7 | 31.8 | 6235 | 8.21 | 0.71 | 3.61 | DCH, He (1)
WD 1346+121 | LP 498-66 | 17.8 | 28.3 | 4150 | 7.88 | 0.50 | 6.58 | DCH, He (1)
WD 1425+495 | CSO 649 | 16.7 | 33.9 | 6895 | 8.41 | 0.85 | 3.77 | DC, (2, H assumed)
WD 1427$-$238 | LP 857-45 | 17.4 | 32.6 | 4866 | 7.90 | 0.52 | 5.40 | DC, (2, H assumed)
WD 1434+437 | LP 221-217 | 17.2 | 27.2 | 4685 | 7.93 | 0.54 | 6.30 | DC, H-He (1)
WD 1533+469 | LP 176-60 | 17.8 | 30.8 | 4310 | 7.83 | 0.48 | 6.45 | DC?, H (1)
WD 1601$-$073 | LP 684-16 | 17.9 | 26.9 | 4920 | 8.55 | 0.94 | 9.83 | DCH, (2, H assumed)
WD 1612+092 | LSPM J1614+0906 | 17.2 | 27.9 | 4775 | 7.90 | 0.52 | 5.57 | DC, H (1)
WD 1702$-$016 | LP 626-29 | 17.3 | 28.3 | 4700 | 7.94 | 0.54 | 6.50 | DC, (2, H assumed)
WD 1737+798 | LP 24-66 | 16.9 | 26.8 | 5535 | 8.28 | 0.75 | 5.72 | DC, He (1)
WD 1746+450 | GD 366 | 15.5 | 29.9 | 9331 | 8.47 | 0.90 | 1.72 | DC, (2, H assumed)
WD 1800+508 | LP 139-38 | 17.4 | 31.0 | 4635 | 7.85 | 0.48 | 5.12 | DC, He-H (1)
WD 1853+775 | LP 25-7 | 17.0 | 30.5 | 4850 | 7.74 | 0.43 | 3.63 | DCH, He (1)
WD 2058+550 | LSR J2059+5517 | 17.1 | 22.7 | 4415 | 7.93 | 0.53 | 7.15 | DC, H-He (1)
WD 2109$-$295 | EC 21096-2934 | 15.1 | 32.8 | 9260 | 7.98 | 0.57 | 0.78 | DC, He-H (3)
WD 2152$-$280 | LP930-61 | 16.3 | 23.5 | 5220 | 7.85 | 0.48 | 3.68 | DC, He (1)
WD 2211+372 | LP 287-35 | 16.8 | 29.2 | 6345 | 8.47 | 0.88 | 4.56 | DC?H, He (1)
WD 2215+368 | LP 287-39 | 16.8 | 20.3 | 4485 | 7.92 | 0.53 | 6.80 | DC, H (1)
WD 2311$-$068 | G 157-34 | 15.3 | 25.9 | 7360 | 7.97 | 0.56 | 1.31 | DC, He (1)

Key to references: 1: Blouin et al. (2019); 2: Gentile Fusillo et al. (2021);
3: Bergeron et al. (2021). Where not found in these references, ages have been
interpolated using the tables from Bédard et al. (2020).

During our observing run we obtained seven observations of seven different
bright nearby stars, which are believed to have zero circular polarisation, to
check for instrumental polarisation. These observations are reported in Table
1.

As in our previous run in July 2021, the high S/N measurements of non-
polarised stars yield the instrumental polarisation to a precision better than
$10^{-5}$. In the $B^{\prime}V^{\prime}R^{\prime}$ bands, the values of Stokes
V/I are 0.0121 $\pm$ 0.0004 %, 0.0109 $\pm$ 0.0005 %, and 0.0084 $\pm$ 0.0004
%, respectively. These are very close to the values obtained in 2021. The
instrumental polarisation was subtracted from the observed polarisation of all
targets, including the measurements of the standard stars reported in Table 1.

In addition, we obtained one measurement of the well-known MWD WD 1900+705,
which appears to show a signal of circular polarisation that is nearly
constant with time (see e.g. Bagnulo & Landstreet, 2019). Our new measurement
is compared in Table 1 to one of the same star that we made during the July
2021 run. The agreement is very satisfactory and demonstrates that we can
obtain measurements that are precise at the 0.02 % level for a $G$ = $13.2$
star with about 10 minutes of exposure time. This shows that the alignment of
our polarimetric optics is stable over a few years.

###  2.2 Results

Table 3:  Observing log of WDs. Detections are marked in boldface.  STAR | DATE | UT | JD – | Exp. |  $V/I$ (%)
---|---|---|---|---|---
| yyyy-mm-dd | hh:mm | 2400000 | (s) | $B^{\prime}$ | $V^{\prime}$ | $R^{\prime}$
WD 0005+395 | 2022-07-06 | 04:38 | 59766.693 | 3900 |  ${\rm-0.017}$$\pm$ | 0.063 |  ${\rm-0.082}$$\pm$ | 0.056 |  ${\rm 0.028}$$\pm$ | 0.044
WD 0010+543 | 2022-07-05 | 04:12 | 59765.675 | 7100 |  ${\rm-0.028}$$\pm$ | 0.140 |  ${\rm-0.011}$$\pm$ | 0.129 |  ${\rm 0.094}$$\pm$ | 0.067
WD 0028+035 | 2002-07-06 | 03:39 | 59766.652 | 3300 |  ${\rm-0.084}$$\pm$ | 0.045 |  ${\rm-0.051}$$\pm$ | 0.050 |  ${\rm-0.010}$$\pm$ | 0.058
WD 1251+366 | 2022-06-27 | 22:41 | 59758.445 | 5200 |  ${\rm-0.155}$$\pm$ | 0.065 |  ${\rm 0.006}$$\pm$ | 0.063 |  ${\rm 0.039}$$\pm$ | 0.038
WD 1315+222 | 2022-06-28 | 22:25 | 59759.435 | 4200 |  ${\rm 0.104}$$\pm$ | 0.045 |  ${\bf 0.182}$$\pm$ | 0.052 |  ${\bf 0.215}$$\pm$ | 0.064
| 2022-07-01 | 22:24 | 59762.434 | 4200 |  ${\rm 0.084}$$\pm$ | 0.040 |  ${\bf 0.253}$$\pm$ | 0.061 |  ${\bf 0.199}$$\pm$ | 0.044
WD 1346+121 | 2022-07-02 | 22:43 | 59763.446 | 6500 |  ${\bf-0.508}$$\pm$ | 0.093 |  ${\rm 0.044}$$\pm$ | 0.091 |  ${\bf-1.074}$$\pm$ | 0.054
| 2022-07-04 | 22:34 | 59765.44 | 6500 |  ${\bf-0.691}$$\pm$ | 0.109 |  ${\rm 0.222}$$\pm$ | 0.098 |  ${\bf-1.256}$$\pm$ | 0.062
WD 1425+495 | 2022-06-29 | 22:22 | 59760.432 | 4200 |  ${\rm 0.012}$$\pm$ | 0.045 |  ${\rm-0.002}$$\pm$ | 0.058 |  ${\rm 0.018}$$\pm$ | 0.038
WD 1427-238 | 2022-06-30 | 23:11 | 59761.466 | 5600 |  ${\rm-0.038}$$\pm$ | 0.114 |  ${\rm 0.102}$$\pm$ | 0.078 |  ${\rm-0.044}$$\pm$ | 0.061
WD 1434+437 | 2022-06-30 | 00:04 | 59760.503 | 5200 |  ${\bf 0.231}$$\pm$ | 0.077 |  ${\rm-0.019}$$\pm$ | 0.067 |  ${\rm-0.019}$$\pm$ | 0.059
WD 1533+469 | 2022-07-01 | 01:02 | 59761.543 | 6600 |  ${\rm-0.065}$$\pm$ | 0.127 |  ${\rm-0.273}$$\pm$ | 0.110 |  ${\bf-0.195}$$\pm$ | 0.052
| 2022-07-03 | 00:42 | 59763.529 | 6600 |  ${\rm 0.139}$$\pm$ | 0.098 |  ${\bf-0.300}$$\pm$ | 0.090 |  ${\rm-0.019}$$\pm$ | 0.042
WD 1601-073 | 2022-07-05 | 22:52 | 59766.453 | 6800 |  ${\bf-0.484}$$\pm$ | 0.111 |  ${\bf-0.386}$$\pm$ | 0.106 |  ${\bf 1.597}$$\pm$ | 0.054
WD 1612+092 | 2022-06-28 | 00:59 | 59758.541 | 5100 |  ${\rm 0.087}$$\pm$ | 0.069 |  ${\rm 0.033}$$\pm$ | 0.052 |  ${\rm 0.097}$$\pm$ | 0.046
WD 1702-016 | 2022-06-30 | 01:59 | 59760.582 | 5300 |  ${\rm 0.215}$$\pm$ | 0.091 |  ${\rm 0.087}$$\pm$ | 0.086 |  ${\rm-0.104}$$\pm$ | 0.053
WD 1737+798 | 2022-06-29 | 01:34 | 59759.566 | 4500 |  ${\rm-0.014}$$\pm$ | 0.082 |  ${\rm-0.085}$$\pm$ | 0.093 |  ${\rm-0.071}$$\pm$ | 0.056
WD 1746+450 | 2022-06-28 | 02:12 | 59758.592 | 2600 |  ${\rm 0.012}$$\pm$ | 0.035 |  ${\rm-0.049}$$\pm$ | 0.032 |  ${\rm-0.074}$$\pm$ | 0.035
WD 1800+508 | 2022-07-05 | 00:59 | 59765.541 | 5600 |  ${\rm-0.157}$$\pm$ | 0.073 |  ${\rm-0.047}$$\pm$ | 0.061 |  ${\rm 0.051}$$\pm$ | 0.053
WD 1853+775 | 2002-07-06 | 01:15 | 59766.552 | 4800 |  ${\rm-0.098}$$\pm$ | 0.066 |  ${\bf-0.680}$$\pm$ | 0.068 |  ${\bf-0.492}$$\pm$ | 0.039
WD 2058+550 | 2022-07-02 | 04:15 | 59762.677 | 5000 |  ${\rm 0.068}$$\pm$ | 0.075 |  ${\rm-0.052}$$\pm$ | 0.070 |  ${\rm 0.064}$$\pm$ | 0.045
WD 2109-295 | 2022-07-01 | 03:51 | 59761.660 | 2200 |  ${\rm 0.011}$$\pm$ | 0.020 |  ${\rm 0.036}$$\pm$ | 0.034 |  ${\rm-0.039}$$\pm$ | 0.037
WD 2152-280 | 2022-06-28 | 03:59 | 59758.666 | 3600 |  ${\rm 0.007}$$\pm$ | 0.040 |  ${\rm-0.055}$$\pm$ | 0.041 |  ${\rm-0.043}$$\pm$ | 0.022
WD 2211+372 | 2022-07-02 | 02:10 | 59762.590 | 4400 |  ${\bf 1.254}$$\pm$ | 0.041 |  ${\bf 0.703}$$\pm$ | 0.054 |  ${\bf 0.446}$$\pm$ | 0.044
| 2022-07-03 | 04:24 | 59763.683 | 4400 |  ${\bf 1.333}$$\pm$ | 0.038 |  ${\bf 0.623}$$\pm$ | 0.044 |  ${\bf 0.285}$$\pm$ | 0.040
WD 2215+368 | 2022-06-30 | 04:12 | 59760.675 | 4400 |  ${\rm 0.187}$$\pm$ | 0.083 |  ${\rm 0.133}$$\pm$ | 0.068 |  ${\rm 0.101}$$\pm$ | 0.044
WD 2311-068 | 2022-06-29 | 04:38 | 59759.693 | 2400 |  ${\rm-0.011}$$\pm$ | 0.028 |  ${\rm 0.011}$$\pm$ | 0.039 |  ${\rm 0.032}$$\pm$ | 0.032

The WDs observed during our 2022 June-July run are listed in Table 2, with
their $G$ magnitudes, distances, physical parameters, cooling ages, and some
comments. Physical parameters were obtained from various studies, cited in the
table’s notes; cooling ages are interpolated from the online cooling data
provided by the Montreal group (Bédard et al., 2020).

The observations are described in the log in Table 3, which gives dates,
integration times, and the polarisation data in the three filter bands for
each WD observation. We list measured BBCP values in boldface if non-zero
polarisation is detected at above the $3\sigma$ level. We consider that real
polarisation has been detected if a consistent picture of detection is found
across the bands, and we highlight star names of WDs in which polarisation is
convincingly detected in boldface in Tables 2 and 3. Of the 23 stars observed,
BBCP has been definitely detected in 5 WDs. The data for these stars are
plotted in Fig. 1.

We observed three of the five WDs in which fields were detected in order to
fully confirm the weak field detections and to check for possible variability.
No variability is detected with confidence. There are in addition two further
WDs, WD 1434+437 and WD 1533+469, in which marginal polarisation detections
have been obtained; these WDs await further observation to confirm (or not)
the fields that may have been detected. However, a single pair of measurements
does not probe all the possible timescales of variation; in particular, our
measurements require integration of the order of one hour and so cannot probe
all the rotation periods that might result from the formation of a MWD from a
close binary.

With these new discoveries, we almost double the number of DC WDs in which
magnetic fields have been detected. One of the new MWDs discovered, LP 684-16
= WD 1601–073, is quite massive compared to most of the rest of the DC WDs
observed. Therefore, because of its relatively small radius, it has cooled
quite slowly, reaching only $T_{\rm eff}$ = 4920 K, but has a computed cooling
time of 9.8 Gyr. It is probably the oldest magnetic WD of any spectral type
discovered so far. For comparison, according to the parameters listed by
Bagnulo & Landstreet (2021), the oldest MWD in the 20 pc volume, in which such
old MWDs are most likely to be discovered, is WD 1008+290 = LHS 2229. It is a
DQpec star with an age of about 7.9 Gyr, almost 2 Gyr younger than LP 684-16.

We note that no really large polarisation signals, such as that exhibited by
WD 1900+705 (see Table 1), are found. However, the observed level of
polarisation in three of the five definite detections reaches the range 1.2 to
1.6%, so some of the fields detected are probably quite strong.

##  3 Discussion and conclusions

We continue to detect MWDs in roughly one-fifth of the DC sample observed.
Considering that only relatively strong fields can be detected in featureless
stars, our results suggest that the frequency of the occurrence of magnetic
fields in older WDs may be as high as 25 or even 30 %, consistent with the
frequency suggested by Bagnulo & Landstreet (2021).

Polarisation levels in the seven DC MWDs discovered by Berdyugin et al. (2022)
and in this paper range from about 0.1 to 1.6 %. Using the order-of-magnitude
estimator of Bagnulo & Landstreet (2020) of a longitudinal field of 15 MG,
which leads to BBCP of the order of 1%, inferred fields $\langle B_{z}\rangle$
are thus estimated to lie between perhaps 1 and 30 MG. From this result, the
fields $\langle|B|\rangle$ that we detect likely lie in the range of roughly 3
to 200 MG.

Some of the fields produce a polarisation with the same sign in the three
filter bands, while in other stars we detect a polarisation that reverses sign
between one filter band and another (Fig. 1). Similar behaviour was found in
our earlier survey data (Berdyugin et al., 2022) as well as in other strongly
magnetic old WDs (Angel & Landstreet, 1971; Angel et al., 1974, 1975; Putney,
1995).

We carried out a second observation for three of our five new discoveries and
for one suspected candidate. The confirming observations were obtained between
one and three days after the discovery observations. For each of these four
stars, the repeated observation confirms the presence of the magnetic field
first detected, except for one $R^{\prime}$ observation of WD 1533+469. In no
case do we detect clearly significant variability; we note, however, that our
repeated measurements probe only a very limited range of timescales.

We find that, in practice, a magnetic field can be reliably detected from BBCP
at the polarization levels of approximately 0.2% with our broadband filter
polarimeter on a 2.5 m telescope in a DC WD of $G\sim 17$. We would gain about
a factor of 3 in precision, or a 2.5 magnitude increase in limiting magnitude,
by going to an 8 m telescope.

To summarise the current statistical situation, we combine the results from
the 20 pc volume magnetic field survey (Bagnulo & Landstreet, 2021), our
exploratory observing run (Berdyugin et al., 2022), and this work. We have
collected literature data on and surveyed 30 young DC stars, of which 7 have
been found to host magnetic fields, and 43 old DCs, of which 6 are magnetic.

As more detections of DC MWDs are made, especially in the context of volume-
limited surveys such as ours, comparisons with magnetic data for other types
of WDs will at first be hampered by our current inability to assign precise
field strength values to magnetic DC WDs. However, a first comparison can be
made between the overall level of polarisation observed in young and old DC
MWDs, which may be taken as an indicator of the evolution of the overall field
strength with cooling age between these two populations. In the currently
small sample, it appears that larger polarisation levels (say, above Stokes
V/I $\ga 1$ %) appear to be at least as common in the old DC MWD population as
in the young group. There is no clear signal of Ohmic field strength decay.
However, the available sample is still very small, and as our sample increases
we can hope to obtain a statistically more significant constraint and carry
out modelling to provide more accurate field strength estimates corresponding
to observed polarisation. Such surveys need to be carried on until a clearer
statistical view of the magnetism in DC WDs is obtained.

###### Acknowledgements.

Based on observations made with the Nordic Optical Telescope, owned in
collaboration by the University of Turku and Aarhus University, and operated
jointly by Aarhus University, the University of Turku and the University of
Oslo, representing Denmark, Finland and Norway, the University of Iceland and
Stockholm University at the Observatorio del Roque de los Muchachos, La Palma,
Spain, of the Instituto de Astrofisica de Canarias. DIPol-UF is a joint effort
between University of Turku (Finland) and Leibniz Institute for Solar Physics
(Germany). We acknowledge support from the Magnus Ehrnrooth foundation and the
ERC Advanced Grant HotMol ERC-2011-AdG-291659. JDL acknowledges the financial
support of the Natural Sciences and Engineering Research Council of Canada
(NSERC), funding reference number 6377-2016.

##  4 Data Availability

All raw data and calibrations are available on request from the authors.

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# Optimal Cut-Point Estimation for functional digital biomarkers: Application
to Continuous Glucose Monitoring

Oscar Lado-Baleato  Marcos Matabuena  Carla Díaz-Louzao  Francisco Gude

###### Abstract

Establish optimal cut points plays a crucial role in epidemiology and
biomarker discovery, enabling the development of effective and practical
clinical decision criteria. While there is extensive literature to define
optimal cut off over scalar biomarkers, there is a notable lack of general
methodologies for analyzing statistical objects in more complex spaces of
functions and graphs, which are increasingly relevant in digital health
applications. This paper proposes a new general methodology to define optimal
cut points for random objects in separable Hilbert spaces. The paper is
motivated by the need for creating new clinical rules for diabetes mellitus
disease, exploiting the functional information of a continuous diabetes
monitor (CGM) as a digital biomarker. More specifically, we provide the
functional cut off to identify diabetes cases with CGM information based on
glucose distributional functional representations.

Keywords: digital health, optimal cut-point, functional data, compositional
data, continuous glucose monitoring, diabetes.

##  1 Introduction

The advances in digital age technology have led to the development of small
devices that track various aspects of our lives at real time. This
technological progress has been particularly beneficial in the field of
medical sciences, where it is now possible to obtain valuable clinical
information continuously from subjects in both hospital and home settings
[17]. Such devices, named as wearable, are becoming very popular in developed
societies as they are provided by several manufactures in the form of
smartphones, smartwatches or monitoring chest straps. These add-up to medical
specific technologies such as Holter [13] or continuous glucose monitoring
trackers [24]. The versatility of biosensor technology allows to get
mechanical, physiological or biochemical information, thereby allowing the
measurement of physical activity [8], heart rate and electrocardiogram
measurements [29], pulse oxymetry [42] or blood glucose levels [50]. Digital
biomarkers are a recently coined term for denominating this multisource and
rich stream of information [36]. Such markers are acknowledge to be objective,
quantifiable, physiological and behavioral measures collected by portable,
wearable, implantable or digestible devices [2].

Digital biomarkers have the potential to provide a more accurate
characterization of physiological states of subjects than traditional
diagnostic tests. Current medical care is based on periodic, or sporadic,
blood tests at fasting condition, physical examinations, or questionnaires
which assess the patient’s health status at a specific point in time.
Alternatively, digital biomarkers can provide real-time information about a
patient’s behavior and physiological well-being by tracking them over extended
periods. Therefore such markers allows for the detection of variations in
physiological and biochemical measures only detectable under real life
conditions. Moreover, it provides more objective measures about patients
lifestyles than self-reported ones which are likely to be influenced by poor
memory or cognitive biases [15, 37]. Research examples where digital
biomarkers proves their clinical value can be found in cardiovascular disease
[9], gastrointestinal complications [16], mental health [21] or diabetes [43]
among others.

A major challenge for the medical usefulness of digital biomarkers is the
complex clinical interpretation of this vast amount of data. From a
statistical point of view, these novel markers are far more complex than
traditional ones, as each patient provides a set of values recorded over time
that exhibit auto-correlation and a functional behavior. Additionally, each
recorded track exhibits high variability among individuals and even for the
same individual when measured on different days.

Cutting-edge researchers on statistical learning from digital biomarkers data
relies on a data pre-processing of such information in order to identify
relevant features inside these statistical objects. In some cases, feature
extraction is previously defined by the wearable manufacturer using their own,
usually unknown, algorithms [27]. Alternatively, generally accepted metrics
may be defined for a given condition (e.g., the sum of squared magnitudes of
acceleration in mobility studies). Once the features have been defined,
statistical learning models, either supervised or unsupervised, are applied to
identify useful information. These procedures carry the risk of overlooking
useful information that is not contained in the designed or identified
features.

Automatic feature selection is also possible by applying deep learning
methodologies such as Long Short-Term Memory (LSTM) neural networks. Deep
learning automatic featurization poses an interpretation problem, as it
provides few clues about why a subject is classified as healthy or not. For
instance, Sükey et al., (2023)[48] used an LSTM pipeline to analyze digital
biomarkers data and identified five “hidden components” that captured the
underlying patterns. However, these non-identifiable components are unlikely
to provide any clinical explanation about the etiology and evolution of a
disease. Additionally, assigning subjects to a given category without
understanding the underlying rationale raises both practical and ethical
concerns.

In the forthcoming sections, focusing on the statistical modeling of
continuous glucose monitoring data, we base our analysis on a distributional
representation of functional digital biomarkers. Statistically, this
representation effectively describes the full spectrum of variability within
the recorded data. Additionally, these representations provide valuable
clinical insights, serving as a foundation for the medical classification of
subjects.

###  1.1 Challenges on continuous glucose monitoring data analysis

Continuous glucose monitoring devices are called to be a benchmark in the
diagnosis and treatment of diabetes mellitus disease. This chronic condition
is characterized by sustained high glucose levels in blood. In order to
prevent the health consequences of the disease (e.g., retinopathy,
nephropathy, stroke) we need an early diagnosis and strict control of blood
glucose levels. CGM data may improve classic diagnostic procedures based on
sporadic blood tests measuring glucose levels at fasting or surrogate markers
of glycemia as glycated hemoglobin. Regarding disease control it also upgrades
manual recordings of glycemia done by patients in order to self control
treatments administration. Indeed, Spanish health system had introduced such
devices as regular care of type I diabetes cases in 2021.

Besides the great technological and medical improvement CGM supposes, data
provided by these devices are not easy to interpret by physicians or data
scientist. A CGM device offers a set of curves for each patient with peaks
after food or soft drinks intake and valleys due to starving periods and
physical activity. Free will of subjects places these peaks and valleys at a
random pattern along the function domain. Such data feature makes inter-, but
also intra-, subject comparisons a challenge. To date physicians relies on
summary measurements, such as the mean value of all glucose levels recorded
during a given time period, their standard deviation or different indexes.
More popular indexes includes the mean glucose (MG), the standard deviation of
glucose (SD), CONGA (continuous overall net glycemia action), J-Index, LI
(Liability Index), LBGI (Low Blood Glucose Index), HBGI (High Blood Glucose
Index), GRADE (glycaemic risk assessment diabetes equation), MAGE (mean
aplitude of glycaemic excursions), M value, MAG (mean absolute glucose) [38].
Moreover, time inside, above or below range, are also widely used in diabetes
literature [6, 3]. Even though CGM-based indexes are widely used, they provide
a limited view of the glucose homeostasis. Indeed, subjects showing similar
values on the aforementioned indexes may have huge difference on CGM records.

To overcome the limitations of CGM-indexes Matabuena et al., (2021) [35]
introduced the glucodensity framework. A methodology which is gained attention
in medical research [12, 23, 32] due to the advatangues to incorporate more
information than traditional CGM biomarkers. In a nutshell, a glucodensity is
a density function that captures the proportion of time spent at each glucose
value. It can be used to summarize all the information captured by a
continuous glucose monitoring (CGM) tracker. Each subject provides one of
these functions, which can be analyzed as functional data working with
distributional representation. This biostatistical concept can be seen as a
generalization of the time above range idea, but for the entire range of
glucose levels. The clinical interpretability of the glucodensity and the
toolbox of statistical concepts designed to handle this data [33, 34] make it
a promising framework for predicting several conditions based on biosensors in
diabetes research.

Figure 1: Toprow plots represent the transformation from Continuous Glucose
Monitoring (CGM) raw data into glucodensities and its distributional
representation. This process is illustrated for two subjects; ID = 498 who was
diagnosed of diabetes during follow-up, and a healthy control ID = 68. In the
bottom plot we display the distributional representation of CGM data for our
sample of subjects differentiating by baseline or follow-up diabetes status.

Figure 1 presents raw continuous glucose monitoring (CGM) data alongside the
corresponding glucodensity and its distributional representation for two
subjects. Subject 68 serves as a healthy control, while subject 496 developed
diabetes in the medium term. While the raw data reveals minimal discernible
differences between the individuals, the glucodensity and its distributional
representation effectively capture variations in both the mean value and
variability of glucose recordings. As we extend our analysis to a larger
sample (depicted in the bottom panels of Figure 1), it becomes apparent that
the distributional representation of glucodensity may facilitate
differentiation between two groups of subjects.

###  1.2 Optimal cut-points and functional data

The research field of optimal cut points is primarily focused on determining
the best values to categorize continuous variables into discrete groups,
particularly in the context of diagnostic studies. The clinical relevance of
this area of statistical research is evident as it influences how diseases are
diagnosed and treated. The estimation of optimal cut points in diagnostic
studies encompasses several methodological approaches, each with its
theoretical foundations and practical implications. In brief, optimal cut
points choice relies on accurately identifying positive and negative cases,
which corresponds to sensitivity and specificity. One strategy to obtain a cut
point might be based on the maximization of either test’s specificity or
sensitivity. However, this procedure proves more extreme, as the choice of an
optimal cut point should generally imply an equilibrium between both measures.
Notably, Youden’s Index, a widely used method, is referenced in statistical
and clinical studies [20, 41], due to its efficacy in maximizing the sum of
both sensitivity and specificity to determine the optimal threshold. Receiver
Operating Characteristic (ROC) Curve Analysis, extensively discussed in [18],
offers a graphical representation of the trade-off between sensitivity and
specificity across various cut points, aiding in the selection of the most
suitable threshold. Additionally, methods like diagnostic likelihood ratio [7]
or cost-effectiveness analysis [39] add up to the statistical toolbox of
optimal cut points estimation. To gain a comprehensive understanding and
practical implementation of the existing methods for estimating optimal cut
points of continuous markers, we recommend consulting Lopez et al.’s work
(2014) on this topic [30].

On the other hand the task of categorizing patients using functional data
encounters a significant challenge: traditional multivariate analysis
techniques, including logistic regression and discriminant analysis, are not
directly applicable to functional data due to its infinite-dimensional nature.
As a result, a common solution is to modify these multivariate methods for use
in functional scenarios, leading to adaptations such as functional logistic
regression [1] or functional discriminant analysis [28, 44, 47]. Additional,
strategies include the use of kernel methods [19] and support vector machines
[46], principal components analysis [26, 22] or functional depth classifiers
[11] among others.

In theoretical works, it was proven that perfect classification is feasible
for Gaussian functional processes [14, 10] under certain conditions. However,
in biomedical problems the classification problems are challenging and there
is overlap beetween groups that difficult the perfect clasifications or some
close notion to them. This is the case of diabetes, that is heteregoneous and
multifactorial disease, that can have different accepted definition according
to the medical guidelines that we consider, and universal consensus is not
possible about this metabolic disorder.

In this work we extend the field of optimal cut points estimation to the
functional data arena. Specifically, we propose a statistical learning model
to estimate a functional cut point for classifying subjects based on
functional digital biomarkers. The method relies on the distributional
representation of the recorded information on which we can estimate a curve
discriminating healthy and diseased individuals. Our proposal proved to be
reliable with simulated and real world clinical data.

##  2 Mathematical models

The objective of this section is to introduce a comprehensive framework for
determining optimal cut points within statistical objects that take values in
arbitrary separable Hilbert spaces, denoted as $\mathcal{H}$. Given our
ultimate aim of applying such models to the distributional representation of
continuous glucose monitoring (CGM) data, we initiate our discussion by
formally defining suchs representations. Subsequently, we present a
generalized formulation of an optimal-cut algorithm designed for application
in any separable Hilbert spaces. Finally, to exemplify the algorithm’s
application, we focus on its implementation within the context of
distributional representations derived from CGM data.

###  2.1 Definition of distributional representation of glucose values:
glucodensity

We start with the formal definition of distributional representation of
glucose values. For a given patient $i$, we denote the glucose monitoring data
by pairs $(t_{ij},X_{ij})$, $j=1,\ldots,m_{i}$, where $t_{ij}$ represents the
recording times that are typically equally spaced across the observation
interval, while $X_{ij}\in[a,b]$ is the glucose level at time
$t_{ij}\in[0,T_{i}]$, where $a$ and $b$ represent the minimum and maximum
range of CGM monitor, that in our case $a=40$ mg/dL, and $b=400$ mg/dL. The
number of records $m_{i}$, and thus the spacing between them, and the overall
observation length $T_{i}$, are allowed to vary across patients. One can see
these data as discrete observations of a continuous latent process $Y_{i}(t)$
with $X_{ij}=Y_{i}(t_{ij})$. The glucodensity is defined as
$f_{i}(x)=F_{i}^{\prime}(x)$, where $F_{i}^{\prime}(x)$ represents the
derivative of the cumulative distribution function $F_{i}(x)$. This
distribution function is given by:

$F_{i}(x)=\frac{1}{T_{i}}\int_{0}^{T_{i}}\mathbb{I}(Y_{i}(t)\leq x)dt$

for $\underset{t\in[0,T_{i}]}{\inf}Y_{i}(t)\leq
x\leq\underset{t\in[0,T_{i}]}{\sup}Y_{i}(t)$.

This expression refers to the proportion of the observational interval in
which the glucose level remains below the value $x$. $F_{i}$ are a set of
increasing functions from $0$ to $1$. While, for $x\in[a,b]$, $f_{i}(x)$,
roughly speaking, measures the proportion of individuals spending time with
continuous glucose concentration $x$ over the continuous domain $[a,b]$.
Density function modeling is a natural extension of traditional CGM data
analysis methods called time-in-range metrics, that measure the proportion of
time individuals spend over some range, e.g. hypoglycemia
$\widehat{Hipo}_{i}=\frac{1}{n_{i}}\sum_{j=1}^{n_{i}}\mathbf{I}\\{X_{ij}\leq
54\\}$, but from a functional perspective. From a technical perspective, the
use of probability distribution or density functions of the individual CGM
time series introduces technical considerations in the statistical analysis
because such representations are defined in a metric space without the linear
structure of vectorial space. In practice, to overcome such technical
difficulties, it is common to consider as representation the quantile function
of observed time series, $\widehat{Q}_{i}(\rho)$, that is defined as
$\widehat{Q}_{i}(\rho)=\arg\min\\{t\in\mathbb{R}:\widehat{F}_{i}(t)=\frac{1}{n_{i}}\sum_{j=1}^{n_{i}}\mathbf{I}\\{X_{ij}\leq
t\\}\geq\rho\\}$. Mathematically, in some modeling tasks, it is equivalent to
consider the space of univariate probability distribution and density
functions equipped with the $2$-Wasserstein distance (see for example [35]).

###  2.2 Defining Optimal Cut-Points in Separable Hilbert Spaces for
Statistical Objects

This section outlines a methodology to determine optimal cut-points for
categorizing a response variable $Y$ situated within a separable Hilbert space
$\mathcal{H}$. For each subject $i$, let $Z_{i}\in\\{0,1\\}$ represent a
binary indicator of group membership, where $0$ signifies the control (non-
disease) group and $1$ signifies the case (disease) group. We introduce a
continuous parameter $c$ within a specified range $[e,f]$, to partition the
space $\mathcal{H}$ into two regions: $\mathcal{A}^{c+}$ for subjects with a
positive condition and $\mathcal{A}^{c-}$ for those without. The
classification of an observation $Y_{i}$ into $\mathcal{A}^{c+}$ or
$\mathcal{A}^{c-}$ depends on the indicator function $I^{c}_{i}$, which
compares against the actual group membership $Z_{i}$. We aim to optimize $c$
by maximizing the following metrics:

Sensitivity $\displaystyle:\max_{c\in[e,f]}\mathbb{P}(I^{c}_{i}=1|Z_{i}=1),$
Specificity $\displaystyle:\max_{c\in[e,f]}\mathbb{P}(I^{c}_{i}=0|Z_{i}=0),$
Youden Index
$\displaystyle:\max_{c\in[e,f]}\left(\mathbb{P}(I^{c}_{i}=1|Z_{i}=1)+\mathbb{P}(I^{c}_{i}=0|Z_{i}=0)-1\right).$

To facilitate the partitioning into $\mathcal{A}^{c+}$ and $\mathcal{A}^{c-}$,
we define a function $h^{\sigma}_{c}(\rho)$, with $c$ as a real-valued
parameter and $\rho$ belonging to the domain $\mathcal{S}$ of $Y$. This
function is given by:

$h^{\sigma}_{c}(\rho)=\mu(\rho)+c\sigma(\rho),\quad\forall\rho\in\mathcal{S},$

where $\mu=\mathbb{E}(Y)$ denotes the expected mean (or other centrality
measures like median or mode could be used), and $\sigma$ represents a scaling
function, which, for simplicity, we assume to be constant ($\sigma(\rho)=1$)
across all $\rho$ in $\mathcal{S}$ for clinical applications. With a selected
$c$, $h^{\sigma}_{c}(\rho)$ helps in defining the regions $\mathcal{A}^{c+}$
and $\mathcal{A}^{c-}$ as follows:

$\mathcal{A}^{c+}=\\{f\in\mathcal{H}:f(\rho)\geq
h^{\sigma}_{c}(\rho),\forall\rho\in\mathcal{S}\\},\quad\mathcal{A}^{c-}=\\{f\in\mathcal{H}:f(\rho)<h^{\sigma}_{c}(\rho),\forall\rho\in\mathcal{S}\\}.$

###  2.3 Practical Calculation of Optimal Cut Points for CGM Time Series Data

In contexts where $\mathcal{H}$ denotes the space of increasing quantile
functions, ranging from $[40,400]$, to reflect the measurement spectrum of
Continuous Glucose Monitoring (CGM) devices, we represent the random response
for patient $i$ as $Y_{i}(\rho)=\widehat{Q}_{i}(\rho)$, with $\rho\in[0,1]$
signifying the quantile probabilities.

Considering a i.i.d sample $\mathcal{D}_{n}=\\{(Y_{i},Z_{i})\sim
P\\}_{i=1}^{n}$, the function $h^{\sigma}_{c}(\rho)$ for any $c$ value is
defined as:

$h^{\sigma}_{c}(\rho)=\frac{1}{n}\sum_{i=1}^{n}Y_{i}(\rho)+c,$

enabling the categorization into $\mathcal{A}^{c+}$ and $\mathcal{A}^{c-}$,
thereby calculating $I^{c}_{i}$. This facilitates the empirical estimation of
sensitivity, specificity, and the Youden Index:

$\displaystyle\widehat{\text{Sen}}_{c}=\frac{\sum_{i=1}^{n}\mathbb{I}(I^{c}_{i}=1\land
Z_{i}=1)}{\sum_{i=1}^{n}\mathbb{I}(Z_{i}=1)},$
$\displaystyle\widehat{\text{Spe}}_{c}=\frac{\sum_{i=1}^{n}\mathbb{I}(I^{c}_{i}=0\land
Z_{i}=0)}{\sum_{i=1}^{n}\mathbb{I}(Z_{i}=0)},$
$\displaystyle\widehat{\text{You}}_{c}=\widehat{\text{Sen}}_{c}+\widehat{\text{Spe}}_{c}-1.$

By assessing these metrics over a range of $c$ values,
$\mathcal{A}=\\{c_{1},\ldots,c_{m}\\}$, we identify the $c$ that optimizes
these criteria based on empirical measures.

###  2.4 Bootstrapping for Uncertainty Quantification in Optimal Cut-off
Determination

Utilizing the dataset $\mathcal{D}_{n}=\\{(Y_{i},Z_{i})\\}_{i=1}^{n}$, we
identify the optimal cut-point $\widehat{c}$, optimizing for sensitivity,
specificity, and the Youden Index through a one-dimensional $c$-process,
integrating a family of functions:

$h^{\sigma}_{c}(\rho)=\mu(\rho)+c\sigma(\rho),\quad c\in[e,f].$

Bootstrap resampling is conducted $B$ times, generating samples
$\mathcal{D}^{b}_{n}=\\{(Y_{i},Z_{i})\\}_{i\in\mathbb{I}_{b}}$ by drawing with
replacement from the index set $\mathcal{I}=\\{1,\dots,n\\}$. This approach
yields a series of optimal cut-points $\\{\widehat{c}_{b}\\}_{b=1}^{B}$. We
employ empirical quantiles $\left[\alpha/2,1-\alpha/2\right]$ to calculate
confidence intervals, establishing confidence regions for these optimal cut
points. This method facilitates inferential analysis on the optimal cut-points
within a functional space using bootstrapping.

######  Theorem 1.

The asymptotic distribution of the estimated optimal cut-point $\widehat{c}$,
concerning sensitivity, specificity, and the Youden Index criteria, follows a
Gaussian distribution. Consequently, the application of the bootstrap
resampling technique for inference is justified.

###### Proof.

Assuming the functions $\mu$ and $\sigma$ are fixed for simplicity, the
regions $\mathcal{A}^{c+}$ exhibit a finite V.C. dimension. This
characteristic ensures that the empirical process associated with the
$M$-estimator, which maximizes sensitivity, specificity, and the Youden Index,
complies with the Donsker principle. Therefore, employing a bootstrapping
algorithm with Efron multipliers to construct confidence bands around the
estimator $\widehat{c}$ achieves universal consistency due to the guaranteed
asymptotic normality. In instances where the functions $\mu$ and $\sigma$ are
estimated using the sample data, a data splitting strategy can be applied to
simplify the analysis, preserving the bootstrap’s reliability and consistency.
∎

###  2.5 Computational details

The implementation of this methodology involves the functionalities of two R
packages for optimal cutpoint estimation within functional data. First, the
fda [45] package is employed to calculate the functional mean, representing
the average curve across the provided data. Subsequently, we estimate the
maximum value of c at which a given curve is above the functional cutpoint.
Subsequently, the optimal.cutpoints package [30] facilitates the core
analysis. Specifically, this package calculates a chosen performance metric
(e.g., Youden’s Index) over the distribution of the obtained c values. This
cutpoint corresponds to the value of c which offer the best performance
metric, signifying the ideal separation between classes within the functional
data.

##  3 Simulation study

In this section, we assess the efficacy of our proposal using simulated
functional distributional data. The data generating process is based on the
following formulation. Given a binary variable $Z\in Ber(0.5)$, and the
quantile function $Q_{0}(\rho)$ from a truncate normal distribution
$\text{TN}(\mu=1,\sigma^{2}=1,a=-5,b=5)$ the quantile function of the
functional distributional records are given by

$Q(\rho)=aZ+U_{1}+U_{2}v+(5+b)ZU_{3}Q_{0}(\rho)$ (1)

where $U_{1}\sim U(-1,1)$, $U_{2}\sim U(-1,1)$ and $U_{3}\sim U(0.8,1.2)$. The
parameter $v$ was fixed to $v=2$ and it controls the degree of variability
along $\rho$ of the simulated curves. The parameter $a$ governs the degree of
difference between both groups regarding the location term of distributional
functional data. Finally, $b$ is a parameter modulating the degree of
difference between both groups due to the variability of the distributional
functional data. A sketch of the simulated data is shown in Figure 2 for
different values of these parameters.

Figure 2: Example of the simulated distributional data for different values of
parameters $a$ and $b$, with $n=100$.

In this simulation scenario, we generated data according to equation (1) using
various values for parameters $a$ and $b$. Then, we assessed the
discrimination capability of the proposed method between both groups. The
disparities between the $Z=0$ and $Z=1$ groups are due to variations in either
the location parameter (represented by varying $a$) or the scale parameter
(represented by varying $b$).

We estimated the functional cutpoint under three criteria: 1) Youden index
criteria, 2) maximum sensitivity estimator , and 3) maximum specificity
estimator. Sample sizes were set to $n=100$ and $n=1000$. The assessment was
carried out across $R=1000$ replicates, with empirical sensitivity and
specificity calculated for each classifier in every replicate.

In Figure 3, we present the sensitivity and specificity distribution across
different values of $a$ and $b$. The Youden criteria display better
performance improvements with increasing values of $a$ or $b$. Notably, the
model exhibits the ability to accurately classify subjects regardless of
whether the difference stems from location or scale parameters. As sample size
increases the variability of the obtained sensitivities and specificities
decreases.

On the other hand, the maximum sensitivity criteria consistently yield the
highest sensitivity values regardless of the parameter values, while
specificity gradually increases with greater disparities between groups. For
values of the parameters $a$ and $b$ higher than $3.5$ we observe fair values
for specificity. As in the Youden case, a higher sample size reduces the
variability of the estimated specificities. Conversely, the maximum
specificity criteria exhibit an inverse trend, showing a fair sensitivity for
$a$ or $b$ values higher than 3.5. Finally, the sensitivity variability
decreases as sample size becomes higher. Both maximum sensitivity and
specificity criteria works well with independence if the groups show
differences on location or scale parameters.

Figure 3: Simulation results: estimated sensitivities and specificities for
optimal cutoff curves based on Youden’s, maximum specificity, and maximum
sensitivity criteria across varying sample sizes and differences in location
and scale parameters between groups.

##  4 Clinical validation of the optimal functional cutpoint

In this section, we apply the proposed methodology to classify individuals
into two groups based on their diabetes status, using continuous glucose
monitoring (CGM) records as the primary data source. Our first objective is to
identify individuals currently experiencing anomalies in glycemic regulation.
Our second objective is to identify individuals who face an increased risk of
developing diabetes in the medium term.

The data source for this study is the A-Estrada Glycation and Inflammation
Study, a population-based research project in which 580 participants underwent
CGM during a week ten years ago. Currently, health events are being recorded
based on electronic health records (EHRs).

###  4.1 AEGIS project

The present study comprises a subsample of subjects from the A Estrada
Glycation and Inflammation Study (AEGIS; trial NCT01796184 at
www.clinicaltrials.gov). AEGIS is a cross-sectional population study of adults
(aged $\geq$ 18) from a Spanish municipality. A detailed description of AEGIS
can be found in Gude et al., (2017)[25].

In AEGIS, participants were periodically examined at their primary care center
for one year, beginning in March. During these visits, participants completed
an interviewer-administered structured questionnaire, provided a lifestyle
description, underwent biochemical measurements, and were prepared for
continuous glucose monitoring (CGM).

From 1516 subjects enrolled in the AEGIS project, 622 participants consented
to undergo a CGM. Moreover, those subjects accepted to writting down their
meal intake into a diary. From this sample, 5 did not fulfilled the study
protocol and 37 experienced a disconnection of the biosensor during the
monitoring period. Therefore, we count with 580 participants (360 women, 220
men) who completed at least two days of CGM. Of those, 65 subjects suffered
from diabetes.

At the start of each monitoring period, a research nurse inserted an EnliteTM
continuous glucose monitoring (CGM) sensor (Medtronic, Inc., Northridge, CA,
USA) subcutaneously into the subject’s abdomen. The sensor continuously
measured the interstitial glucose level (40-400 mg/dL) of the subcutaneous
tissue, recording values every 5 minutes. Participants were also provided with
a conventional OneTouchR VerioR Pro glucometer (LifeScan, Milpitas, CA, USA)
as well as compatible lancets and test strips for calibrating the CGM. All
subjects were asked to make at least three capillary blood glucose
measurements (usually before the main meals) without checking the current CGM
reading. The sensor was removed on the seventh day, and the data was
downloaded and stored for further analysis. If the total number of data
acquisition skips per day exceeded 2 hours, all of the data for that day was
discarded.

During the following years monitored individuals were followed using their
electronic health records. Based on periodic blood test ordered by their
primary care doctor we could identify which subjects developed the disease. A
participant was considered to have or developed diabetes if they met either of
the following criteria (as given by the American Diabetes Association):

  1. 1.

A glycated hemoglobin level above 6.5% and a fasting plasma glucose level of
126 mg/dL or higher in a single blood test.

  2. 2.

One of these criteria above the diagnostic cut points in two separate blood
tests.

###  4.2 Optimal functional cutpoint estimation

From the 580 subjects 65 was previously diagnosed with diabetes which
corresponds to a prevalence of $11.2\%$ $($CI95$\%$: $3.8\%-14.1\%))$. By
August 2022, the diabetes status of all participants was checked based on
their electronic health records. From 515 subjects showing no diabetes at
baseline, $26$ (incidence rate = $5.0\%$ (CI$95\%$: $3.4\%-7.4\%$) developed
the disease during the follow-up.

Optimal cut-off curves for maximum sensitivity, Youden index and maximum
specificity criteria are depicted in Figure 4 for predicting cases of diabetes
prevalence and incidence. Alongside the estimated curve we present their
corresponding $95\%$ pointwise confidence bands.

Firstly, for predicting diabetes prevalence maximum sensitivity criteria offer
poor specificity $0.15$ (CI$95\%:0.12-0.68$). The median glucose value for
this cutoff is $108.2$ mg/dL, spending $\geq 4.0\%$ time above the 140 mg/dL
threshold. The Youden criteria shows a good balance between sensitivity and
specificity being $0.89$ (CI$95\%:0.85-0.92$) for both quantities. The median
value of this cutoff curve rises until $135.9$ mg/dL, so does time above 140
mg/dL which increases until $\geq 38.2\%$. Finally, maximum specificity cut-
off curve show a low sensitivity of $0.51$ (CI$95\%:0.43-0.60$). The median
value rises until $201.8$ mg/dL, and the curve is above 140 mg/dL along the
domain function. This curve identify subjects who reach very high glucose
excursion a low proportion of time.

Figure 4: Optimal cut-off curves for predicting diabetes prevalence and
incidence alongside with 95% pointwise confidence band. The confidence band
was obtained using 1000 uniform bootstrap replicates.

On the other hand, when predicting diabetes incidence, criteria based on
maximum sensitivity exhibit poor specificity ($0.19$, 95% CI: $0.14$ \-
$0.46$). The median glucose value for this cutoff curve is $106.2$ mg/dL, with
$\geq 3.1\%$ of time spent above the 140 mg/dL threshold. In contrast, the
Youden criteria demonstrates a balanced trade-off between sensitivity and
specificity, yielding $0.69$ (95% CI: $0.62$ \- $0.73$) for both metrics. The
median value of this cutoff curve increases until reaching $118.8$ mg/dL, with
time spent above 140 mg/dL also increasing to $\geq 9.5$%. Conversely, the
curve for maximum specificity cut-off demonstrates a very low sensitivity
($0.06$, 95% CI: $0.04$ \- $0.08$). Its median value rises until $174.7$
mg/dL, remaining above the 140 mg/dL threshold throughout.

As depicted in Figure 5 the estimated cutoff curves show higher discrimination
for diabetes prevalence than for diabetes incidence. When we considered the
specificities and (1-sensitivities) for the entire range of $c$ we obtain the
depicted ROC curves. As can be seen, while the prediction of diabetes
prevalence shows an AUC of $0.94$ (CI$95\%:0.90-0.95$), the prediction of
future development of the disease show a lower area under the curve of $0.78$
(CI$95\%:0.68-0.80$).

Figure 5: ROC curves, depicting specificities vs. (1 - sensitivities),
obtained for a varying value of $c$ in the functional cut-point. Left plot
shows the discrimination capability of the functional-cutpoint for diabetes
prevalence, while right plot focus on predicting diabetes incidence.

##  5 Discussion

This paper proposes a new methodology for defining optimal cut points for
statistical objects within a separable Hilbert space. The method’s reliability
is demonstrated using both simulated and real-world data. We applied the
method to a clinical problem: classifying patients based on digital biomarkers
obtained from continuous glucose monitoring. This allowed us to identify
prevalent cases of diabetes and to predict the incidence of the disease. The
method exhibited excellent discrimination capability, and for the first time,
we were able to determine various functional cut-points that balanced
sensitivity and specificity. Additionally, we estimated a 95% confidence band
to quantify the uncertainty associated with the cut-point estimation.

Our findings hold significant relevance for diabetes mellitus research.
Continuous glucose monitoring (CGM) data is increasingly playing a key role in
diagnosis and disease management, potentially leading to improved diagnostic
criteria and treatment protocols [4]. A statistical model that get advantage
of CGM data to formulate a classification rule for diabetes would serve as a
valuable practical resource. Existing research has identified individuals at
higher risk of developing diabetes based on metrics like time spent above a
certain blood sugar range [31]. Additionally, studies have shown changes in
CGM glucodensity with aging [40]. Notably, our work presents the first
evidence that distributional representations of glucodensity can be
instrumental in early diagnosis of diabetes. Integrating this interpretable
graphical rule into CGM software would be highly beneficial for both
clinicians and the CGM industry.

Finally, the applications of this statistical methodology may be extended
broadly across the biomedical field. It can be used to define interpretable
decision rules in clinical trials involving continuous subject monitoring [5],
or in the context of digital health [49]. As wearable devices become
increasingly prevalent for disease characterization, our methodology has the
potential to be instrumental in defining new metabolic states of patients or
diseases. Furthermore, this framework can be potentially applied to other
domains, such as neuroimaging, where data can be represented by Laplacian
graphs embedded in a separable Hilbert space.

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