††thanks: These authors contributed equally to this work††thanks: These
authors contributed equally to this work

# Parity switching in a full-shell superconductor-semiconductor nanowire qubit

O. Erlandsson  Center for Quantum Devices, Niels Bohr Institute, University of
Copenhagen, 2100 Copenhagen, Denmark  D. Sabonis  Center for Quantum Devices,
Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark
Laboratory for Solid State Physics, ETH Zürich, CH-8093 Zürich, Switzerland
A. Kringhøj  Center for Quantum Devices, Niels Bohr Institute, University of
Copenhagen, 2100 Copenhagen, Denmark  T.W. Larsen  Center for Quantum Devices,
Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark  P.
Krogstrup  Center for Quantum Devices, Niels Bohr Institute, University of
Copenhagen, 2100 Copenhagen, Denmark  K. D. Petersson  Center for Quantum
Devices, Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen,
Denmark  C. M. Marcus  Center for Quantum Devices, Niels Bohr Institute,
University of Copenhagen, 2100 Copenhagen, Denmark

###### Abstract

The rate of charge-parity switching in a full-shell superconductor-
semiconductor nanowire qubit is measured by directly monitoring the dispersive
shift of a readout resonator. At zero magnetic field, the measured switching
time scale ${T_{P}}$ is on the order of $100\text{\,}\mathrm{ms}$. Two-tone
spectroscopy data post-selected on charge-parity is demonstrated. With
increasing temperature or magnetic field, ${T_{P}}$ is at first constant, then
exponentially suppressed, consistent with a model that includes both non-
equilibrium and thermally activated quasiparticles. As ${T_{P}}$ is
suppressed, qubit lifetime ${T_{1}}$ also decreases. The long
${T_{P}\sim$0.1\text{\,}\mathrm{s}$}$ at zero field is promising for future
development of qubits based on hybrid nanowires.

††preprint: AIP/123-QED

The coherent manipulation of any transmon qubit is threatened by the presence
of unpaired quasiparticles (QPs) in the superconductor, a problem referred to
as quasiparticle poisoning (QPP). Understanding the nature and rate of QPP is
therefore part of the challenge of using these superconducting circuits to
construct complex quantum information processing devices. In a transmon qubit
Koch  _et al._ (2007), a Josephson junction (JJ) with an associated energy
${E_{J}}$ separates two superconductors which are shunted by a large
capacitance with a charging energy ${E_{C}}$. As QPs tunnel across the JJ,
they contribute to relaxation if they carry energy matching the qubit
transition, and to dephasing in the case of non-vanishing charge dispersion.

Hybrid superconductor-semiconductor nanowire (NW) variants of the transmon
qubit de Lange  _et al._ (2015); Larsen  _et al._ (2015) provide an
alternative to conventional metallic systems. The semiconductor platform
offers advantages such as control of ${E_{J}}$ by electrostatic gating, and
reduced charge dispersion without loss of anharmonicity Bargerbos  _et al._
(2020); Kringhøj  _et al._ (2020). The system has been developed to be
compatible with large magnetic fields Luthi  _et al._ (2018); Kroll  _et al._
(2019); Kringhøj  _et al._ (2021a); Uilhoorn  _et al._ (2021) where it
potentially could be used for topological quantum computation Ginossar and
Grosfeld (2014); Vaitiekėnas  _et al._ (2020a). With a magnetic field applied,
NWs with a fully covering superconducting shell have been shown to exhibit
destructive Little-Parks effect, both in dc transport and qubit measurements
Vaitiekėnas  _et al._ (2020b); Sabonis  _et al._ (2020).

Figure 1:  (a) Parity branches of ground and excited states of the transmon
Hamiltonian. The dotted arrow symbolizes a charge-parity switch (poisoning) as
a quasiparticle tunnels across the Josephson junction. (b) False-color
scanning electron micrograph of device and diagram of surrounding cQED
circuit. An InAs (red) nanowire with an Al (yellow) shell is embedded in a NbN
(blue; rectangular contacts to nanowire are NbTiN) structure on a Si substrate
(black). A layer of HfO2 (green) separates the bottom gate ${V_{C}}$ from the
etched junction. (c) Time trace of demodulated transmission ${V_{H}}$ through
the readout resonator, showing discrete switching, interpreted as changes of
charge-parity state. Data taken at ${T=$30\text{\,}\mathrm{mK}$}$ and $B=0$.
The horizontal line between differently colored regions indicates data binning
threshold. (d) Average power spectrum density of 40 traces fitted to a
Lorentzian distribution, with ${T_{P}}$ as given by the fit.

In this Letter, the rate of QPP is measured in a full-shell superconductor-
semiconductor hybrid NW qubit. It is found that QPP occurs on a timescale of
${T_{P}\sim 0.1\,}$s, which is far from limiting the coherence time of the
qubit. As either the temperature or magnetic field is increased, ${T_{P}}$ is
first constant then exponentially suppressed, suggesting the existence of both
a non-equilibrium and equilibrium population of QPs in the qubit.
Additionally, while the qubit frequency recovers in the first Little-Parks
lobe, ${T_{P}}$ remains below our detectable range (not sufficiently exceeding
$30\text{\,}\mathrm{\SIUnitSymbolMicro s}$) after the initial suppression.

QPP in transmons was first measured directly by a Ramsey-type pulse sequences
sensitive to the charge-parity state of the qubit Ristè  _et al._ (2013). This
technique was subsequently used to study transitions across all four parity-
logical states Serniak  _et al._ (2018) and the effect of shielding and qubit
geometry on QPP levels Kurter  _et al._ (2021); Gordon  _et al._ (2021). A
technique based on direct dispersive monitoring of the charge-parity has also
been developed Serniak  _et al._ (2019). This was recently used to study QPP
in NWs similar to those investigated here but with the superconducting shell
only covering half the facets, where it was found that QPP occured on the
timescale of $100\text{\,}\mathrm{\SIUnitSymbolMicro s}$ and could have a non-
monotonic dependence on magnetic field Uilhoorn  _et al._ (2021). The effect
of geometry was also studied with the dispersive technique Pan  _et al._
(2022). Typical reported QPP time scales have ranged from
$100\text{\,}\mathrm{\SIUnitSymbolMicro s}$ up to $1\text{\,}\mathrm{s}$ Ristè
_et al._ (2013); Serniak  _et al._ (2018, 2019); Uilhoorn  _et al._ (2021);
Gordon  _et al._ (2021); Kurter  _et al._ (2021); Pan  _et al._ (2022).
Methods to mitigate unwanted quasiparticles in superconducting circuits have
included qubit pumping Gustavsson  _et al._ (2016) and trapping of
quasiparticles by vortices Vool  _et al._ (2014); Wang  _et al._ (2014) or
regions of normal metal Patel  _et al._ (2017).

QPP is measured as the rate of transitions between the two parity branches of
the transmon. How QPP leads to dephasing (for non-vanishing charge dispersion)
can be understood by considering the transmon Hamiltonian,
${\hat{H}=4E_{C}(\hat{n}-n_{g})^{2}-E_{J}\cos\hat{\phi}}$, where ${n}$ is the
number of Cooper pairs on the island, ${n_{g}}$ is the the effective offset
charge and ${\phi}$ is the phase difference across the JJ. As indicated in
Fig. 1(a), each time a poisoning event occurs ${n_{g}}$ is shifted by ${1e}$,
which except for at degeneracy points, changes the qubit transition frequency,
contributing to qubit dephasing and establishing two distinct charge-parity
branches (even and odd) of each logical state. The charge-parity state is
measured by monitoring the microwave transmission through the resonator,
sensitive to changes in qubit transition frequency via the dispersive shift.

We now describe the experimental set-up to realize a transmon using a NW. To
fabricate the sample, a 35 nm thick NbN film was sputtered onto a Si
substrate. The cQED architecture, consisting of a microwave feedline, readout
resonators, qubit islands, gate lines, and ground plane, was patterned in the
NbN film by reactive ion etching. The cQED circuit is shown in Fig. 1(b). For
the qubit studied here, the island was designed for a charging energy
${E_{C}/h=$500\text{\,}\mathrm{MHz}$}$ and the resonator frequency
${f_{\textrm{RO}}=$5.4\text{\,}\mathrm{GHz}$}$. The islands were connected to
the ground plane by micromanipulator deposited VLS-grown InAs NWs with a full
superconducting Al shell, placed on bottom gates covered by local HfO2
dielectric. The Al shell had a small segment removed by wet etching, forming
the semiconducting weak link of the JJ. Crossovers formed by crosslinked
resist on HfO2 tied the ground plane together across cQED features. NWs and
crossovers were contacted to the NbN film by ion milling followed by
sputtering of NbTiN. The sample was wire-bonded and placed in an In-sealed
CuBe box filled with Eccosorb, which was loaded into a dilution refrigerator
with a base temperature of ${\sim$30\text{\,}\mathrm{mK}$}$. After passing
through the sample, microwave signals were amplified by a traveling-wave
parametric amplifier at base temperature and a cryogenic amplifier at
$4\text{\,}\mathrm{K}$. A 6-1-1 T vector magnet was used to apply a magnetic
field in the plane of the substrate, along the axis of the NW.

Measurement of the charge-parity switching rate was first performed at base
temperature and zero magnetic field. In order for the dispersive QPP signal to
be visible, the qubit frequency ${f_{01}}$ must be in the vicinity of the
readout resonator frequency ${f_{\textrm{RO}}=$5.4\text{\,}\mathrm{GHz}$}$.
Setting ${V_{C}=$-0.71\text{\,}\mathrm{V}$}$ gave
${f_{01}=$4.1\text{\,}\mathrm{GHz}$}$, detected by two-tone spectroscopy. The
transmission at the resonator frequency was monitored for
$3.2\text{\,}\mathrm{s}$, yielding time traces as shown in Fig. 1(c). The
aggregated power spectral density ${P_{xx}}$ of ${40}$ binned time traces was
fit using a Lorentzian form

$S(f)=A\frac{T_{P}}{(\pi T_{P}f)^{2}+1}+B\,,$ (1)

with poisoning time ${T_{P}}$, amplitude $A$ and background $B$ as fit
parameters. Experimental data along with the resulting fit are shown in Fig.
1(d), yielding a low-temperature zero-field poisoning time of
${T_{P}\sim$0.1\text{\,}\mathrm{s}$}$. This is on the higher end of values
reported in similar systems.

Figure 2:  Parity-resolved spectroscopy of transmon charge dispersion. (a)
Gate voltage and microwave pulse scheme for measurement. ${V_{C}}$ is ramped,
while drive and readout tones are transmitted through the device. (b) All data
averages, showing two peaks at each gate value (even and odd charge-parity
branches). (c-d) Due to slow quasiparticle poisoning compared to the duration
of the spectroscopy measurement, the data can be post-selected as into
distinct parity branches. The small ${\Delta V_{C}}$ used here to control
${n_{g}}$ also varies ${E_{J}}$, resulting in the overall slope of features in
(b-d). Vertical white dashed lines are guides to the eye. Column averages are
subtracted for enhanced visibility.

Large ${T_{P}}$ compared to the measurement duration means that the charge-
parity branches of the qubit transition spectrum can be probed separately. In
order to demonstrate this, the charge dispersion is probed and the data is
post-selected according to charge parity. A ramped voltage is applied to the
gate ${V_{C}}$ and a two-tone spectroscopy measurement is performed, as shown
in Fig. 2(a). The ramped voltage was offset around $-0.33\text{\,}\mathrm{V}$,
yielding ${f_{01}\sim$4.4\text{\,}\mathrm{GHz}$}$. Around this offset, a small
ramp amplitude ${\Delta V_{C}}$ was used to control the effective charge-
offset ${n_{g}}$. Typically, this results in the pattern in Fig. 2(b) where
both charge-parity branches are visible Schreier  _et al._ (2008). However,
because the ramp signal and data acquisition can be applied considerably
faster than QPP, the two charge-parity branches can be separated by post-
selection sup , as shown in Figs. 2(c,d). A different technique for parity-
selective spectroscopy based on interspersed parity measurements was recently
demonstrated for Andreev bound states Wesdorp  _et al._ (2021).

Temperature dependence of QPP is shown in Fig. 3. Below
$80\text{\,}\mathrm{mK}$, QPP time ${T_{P}}$ was found to be independent of
temperature ${T}$. Above $80\text{\,}\mathrm{mK}$, ${T_{P}}$ is exponentially
suppressed. Assuming a QP density ${x_{\textrm{QP}}}$ with a fixed
contribution ${x_{\textrm{QP}}^{0}}$ from non-equilibrium QPs as well as a
thermally activated contribution, the total QP density, normalized by the
Cooper pair density, is given by Catelani  _et al._ (2011),

$x_{\textrm{QP}}=x_{\textrm{QP}}^{0}+\sqrt{2\pi
k_{B}T/\omega}\exp\left(-\omega/k_{B}T\right)$ (2)

where ${\omega}$ is the superconducting spectral gap. Assuming that
${1/T_{P}=Cx_{\textrm{QP}}}$ where ${C}$ is an unknown proportionality
constant, and fitting to the logarithm of data points yields the fit in Fig.
3(a). The resulting ${\omega=$139\pm 2\text{\,}\mathrm{\SIUnitSymbolMicro
eV}$}$ is in reasonable agreement with the bulk superconducting gap of Al
${\sim$200\text{\,}\mathrm{\SIUnitSymbolMicro eV}$}$. The density of non-
equilibrium QPs ${x_{\textrm{QP}}^{0}\sim 10^{-8}}$ is among the lower values
reported for similar systems Serniak  _et al._ (2018); Uilhoorn  _et al._
(2021); Pan  _et al._ (2022).

Figure 3:  Poisoning time ${T_{P}}$ as a function of temperature ${T}$. (a) As
${T}$ is increased, ${T_{P}}$ is constant then starts to decrease
exponentially around $80\text{\,}\mathrm{mK}$. This is interpreted as the
onset of significant contribution to QPP from thermally activated QPs. Model
(black curve) yields good decription of experimental data. (b, c) Examples of
individual ${V_{H}}$ time traces at different orders of magnitude of
${T_{P}}$, as indicated by the arrows in (a).

Next, we investigate qubit coherence and QPP as a function of axial magnetic
field $B$ relevant for potential applications in topological quantum
computation. The full-shell NW exhibits destructive Little-Parks effect
Sabonis  _et al._ (2020) which is reflected in the reentrant structure of
qubit frequency as a function of $B$ sup . We note, however, that the QP
spectral gap ${\omega}$ is different from the pairing energy ${\Delta}$, the
latter determining the qubit frequency via ${E_{J}=(\Delta/4)\sum\eta}$, where
${\eta}$ are the channel transmissions. Away from zero magnetic field,
${\omega}$ and ${\Delta}$ are distinct, as discussed in Sec. 10.2.2 of Ref.
Tinkham (1996). At each value of $B$, two-tone spectroscopy was performed and
a simple peak-finding algorithm was used to identify the qubit frequency. Rabi
and relaxation pulse sequences were then run at the qubit frequency. An
exponential fit to the relaxation-sequence data gave ${T_{1}}$, shown in Fig.
4.

Figure 4:  Qubit frequency ${f_{01}}$, poisoning time ${T_{P}}$, and qubit
relaxation time ${T_{1}}$ as functions of magnetic field $B$ along the NW
axis. (a) ${f_{01}}$ is suppressed as a function of ${B}$. The data are fit to
${\Delta(B)}$ as described by the Little-Parks effect (black curve). (b)
${T_{P}}$ is first constant, then diminishes above $20\text{\,}\mathrm{mT}$.
As ${T_{P}}$ is further suppressed, ${T_{1}}$ starts to reduce, suggesting
that ${T_{1}}$ becomes limited by QPP.

The ${f_{01}}$ data in Fig. 4(a) were fit using
${f_{01}=f_{0}\sqrt{\Delta(B)/\Delta(B=0)}}$, where ${\Delta(B)}$ is governed
by the destructive Little-Parks effect due to the full cylindrical
superconducting shell of the NW Sabonis  _et al._ (2020); sup . The fit yields
an NW radius ${R=$90\text{\,}\mathrm{nm}$}$, Al shell thickness
${t=$4\text{\,}\mathrm{nm}$}$ and superconducing coherence length
${\xi=$165\text{\,}\mathrm{nm}$}$. The physical dimensions are in rough
agreement with transmission electron micrographs of NWs from the same growth
batch sup , although ${t}$ is considerably small. In Fig. 4(b), measured
${T_{P}}$, ${T_{1}}$ are plotted along with a model for ${T_{P}}$ as
calculated by (2) using the parameters from the ${f_{01}}$ fit, and ${C}$,
${\omega(B=0)}$, ${x_{\textrm{QP}}^{0}}$ from the fit to temperature-dependent
data in Fig. 3(a) and ${T=$30\text{\,}\mathrm{mK}$}$ (base temperature). Here
${\omega(B)}$ is also governed by the Little-Parks effect sup . The onset of
increasing QPP occurs at a lower magnetic field than predicted by the model.
The reason for this discrepancy may be related to changes in the quasiparticle
density of states at finite field, where it is no longer exponential as
assumed in (2). The NW gap is softened at finite field Chang  _et al._ (2015)
and the coherence peaks are broadened Tinkham (1996). Furthermore, the
magnetic-field dependence of the non-equilibrium QP population is unknown.

Although QPP imposes a limit on qubit coherence, the ground state switching
measured in this work (${\ket{0e}\leftrightarrow\ket{0o}}$) does not itself
relax the qubit, since it is a transition between different charge-parities of
the qubit ground state. However, the rate of QP-induced transitions that
contribute to relaxation (${\ket{0e}\leftrightarrow\ket{1o}}$ and
${\ket{0o}\leftrightarrow\ket{1e}}$) are also related to the QP density, and
the similar magnetic fields at which ${T_{P}}$ and ${T_{1}}$ start to decrease
in Fig. 4(b) suggests that QPP becomes the limiting factor on the qubit
lifetime above $30\text{\,}\mathrm{mT}$. The full-shell NW exhibits the
Little-Parks effect, causing ${\Delta}$ and ${\omega}$ to revive as the field
is further increased, with a first reentrant lobe centered around
$100\text{\,}\mathrm{mT}$ sup . At $100\text{\,}\mathrm{mT}$, ${\omega}$ is
predicted to have sufficiently recovered such that ${T_{P}}$ is again
dominated by non-equilibrium QPs. At this magnetic field, we measured
${T_{1}=$0.4\text{\,}\mathrm{\SIUnitSymbolMicro s}$}$, however, no split
populations was observed in the dispersive charge-parity monitoring with
${f_{01}=$3.5\text{\,}\mathrm{GHz}$}$, which we interpret as ${T_{P}}$ not
sufficiently exceeding the measurement integration time
${\tau_{I}=$30\text{\,}\mathrm{\SIUnitSymbolMicro s}$}$. A likely cause of the
enhanced QPP in the first lobe is the increased density of subgap states in
that regime, as previously observed in the same NWs Kringhøj  _et al._
(2021b), increasing the QP density of states.

In conclusion, we have examined the rate of charge-parity switching in a full-
shell superconducting-semiconducting nanowire qubit. We found a long parity
switching time ${T_{P}}\sim$0.1\text{\,}\mathrm{s}$$ at zero field, and
exponential suppression of $T_{P}$ with field and temperature, in good
qualitative agreement with our model. We interpret the exponential onset as
associated with thermally activated QPs. Above $20\text{\,}\mathrm{mT}$, this
is observed as the rate of QPP rapidly increasing, coinciding with the
collapse of the gap in the zeroth lobe of the Little-Parks effect. In the
first lobe of reentrant superconductivity, ${T_{P}}$ does not recover along
with ${\Delta}$, consistent with previously observed supgap states in this
regime. The large value of ${T_{P}}$ at zero magnetic field indicates that QPP
is not a near-term barrier for the engineering of increased coherence times in
this type of qubit.

##  I Methods

Data in Figs. 1 c-d: A total of ${N_{T}=40}$ time traces of the in-phase and
quadrature components of transmission response are recorded by heterodyne
demodulation. Each trace consists of ${N_{P}=800}$ points separated by
${\tau_{C}=$4\text{\,}\mathrm{ms}$}$ and integrated over
${\tau_{I}=$100\text{\,}\mathrm{\SIUnitSymbolMicro s}$}$. For improved signal-
to-noise ratio, the data are then projected onto a rotation angle in the
phase-quadrature-plane, selected by eye, yielding a scalar ${V_{H}}$ sup . A
single time trace is shown in Fig. 1(c). Following the analysis in Ref. Ristè
_et al._ (2013), the data are binned such that points above (below) the mean
value are assigned to ${1}$ (${-1}$). The binned data are then fit to the
Lorentzian form in Eq. (2).

Data in Figs. 2 c-d: To control ${n_{g}}$, a ramp signal is applied to
${V_{C}}$ with a period $2\text{\,}\mathrm{ms}$, amplitude ${\Delta
V_{C}=$1\text{\,}\mathrm{mV}$}$ and offset $-0.33\text{\,}\mathrm{V}$. This
value of ${V_{C}}$ is sufficiently small so that the gate mainly controls
${n_{g}}$ and has a small effect on ${E_{J}}$ (visible as the overall slope of
features in Fig. 2). This is shown in Fig. 2(a), along with a variable qubit
drive tone ${f_{d}}$ and a readout tone at
${f_{\textrm{RO}}=$5.4\text{\,}\mathrm{GHz}$}$. For each value of ${f_{d}}$,
${V_{H}}$ is measured for ${15}$ consecutive ramps, with ${200}$ data points
separated by ${\tau_{C}=$10\text{\,}\mathrm{\SIUnitSymbolMicro s}$}$ taken per
ramp period. These ${15}$ ramp periods are averaged into one trace, plotted
horizontally in Fig. 2(b-d). Accordingly, one trace takes ${15\times
200\times\tau_{C}=$30\text{\,}\mathrm{ms}$}$ to acquire. The drive tone
${f_{d}}$ is stepped through a window of frequencies, and the entire process
is then repeated ${20}$ times. In Fig. 2(b), all ${20}$ traces are averaged
for each ${f_{d}}$, yielding a pearl-shaped pattern corresponding to the qubit
charge dispersion.

Data in Fig. 3: For ${T_{P}}$ to be probed over multiple orders of magnitude,
it was not possible to acquire all data at fixed ${N_{T}}$ and ${\tau_{C}}$.
This is because small ${\tau_{C}}$ is required to resolve small ${T_{P}}$, but
such a sampling rate could not be sustained for the duration needed to capture
${T_{P}\gg\tau_{C}}$ due to memory limitations in data acquisition. Therefore,
we repeated the temperature sweep trying different combinations of ${N_{T}}$,
${\tau_{C}}$ and ${\tau_{I}}$ at different ${T}$ until we could measure
${T_{P}}$ over a wide range of ${T}$. Furthermore, for simplicity, only
integration times ${\tau_{I}>T_{1}}$ are used, in order to predominantly
detect the ground states. Due to residual excited state population, all four
states in Fig. 1(a) would be resolved for lower ${\tau_{I}}$, and the
transition rates would need distinguishing. For ${\tau_{I}>T_{1}}$, the
visibility of the ground states dominates, and the analysis is
straightforward. At base temperature and ${B=0}$,
${T_{1}\sim$5\text{\,}\mathrm{\SIUnitSymbolMicro s}$}$ so we set
${\tau_{I}>$20\text{\,}\mathrm{\SIUnitSymbolMicro s}$}$ for all data in Fig. 3
and Fig. 4. This simplification defines the lower bound of detectable
${T_{P}}$ in those measurements
(${T_{P}\gg$20\text{\,}\mathrm{\SIUnitSymbolMicro s}$}$).

Data in Fig. 4: Measurements of ${T_{P}}$ are interleaved with measurements of
${f_{01}}$ and ${T_{1}}$. To achieve this, a measurement sequence was set up
as follows. At each setpoint of ${B}$, two-tone spectroscopy was performed
from which the qubit frequency was determined by a peak-finding routine. At
this frequency, standard Rabi and relaxation pulse schemes were used to
determine the $\pi$ pulse duration and ${T_{1}}$, respectively. Finally, a
measurement of ${T_{P}}$ was performed; as for the data in Fig. 3, different
combinations of ${N_{T}}$, ${\tau_{C}}$ and ${\tau_{I}}$ were used to resolve
switching events at different ${B}$. At $104\text{\,}\mathrm{mT}$, where no
split population was observed, the measurement was performed using
${N_{T}=5}$, ${N_{P}=2000}$,
${\tau_{C}=$100\text{\,}\mathrm{\SIUnitSymbolMicro s}$}$ and
${\tau_{I}=$30\text{\,}\mathrm{\SIUnitSymbolMicro s}$}$.

##  II Acknowledgments

We thank Arno Bargerbos, Gijs de Lange, Leonid Glazman, Torsten Karzig, Dmitry
Pikulin, Willemijn Uilhoorn, and Bernard van Heck for useful discussions. We
thank Will Oliver for providing the traveling-wave parametric amplifier used
in the experiment. We thank Martin Espiñeira for electron microscopy. We thank
Marina Hesselberg, Karthik Jambunathan, Robert McNeil, Karolis Parfeniukas,
Agnieszka Telecka, Shivendra Upadhyay, and Sachin Yadav at QDev, and Mahesh
Kumar, Rizwan Ali, Tommi Riekkinen, and Pasi Kostamo at Espoo for device
nanofabrication. Research is supported by the Danish National Research
Foundation, Microsoft, European Research Commission grant 716655, and a grant
(Project 43951) from VILLUM FONDEN.

## References

  * Koch  _et al._ (2007) J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schuster, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Phys. Rev. A 76, 042319 (2007).
  * de Lange  _et al._ (2015) G. de Lange, B. van Heck, A. Bruno, D. J. van Woerkom, A. Geresdi, S. R. Plissard, E. P. A. M. Bakkers, A. R. Akhmerov, and L. DiCarlo, Phys. Rev. Lett. 115, 127002 (2015).
  * Larsen  _et al._ (2015) T. W. Larsen, K. D. Petersson, F. Kuemmeth, T. S. Jespersen, P. Krogstrup, J. Nygård, and C. M. Marcus, Phys. Rev. Lett. 115, 127001 (2015).
  * Bargerbos  _et al._ (2020) A. Bargerbos, W. Uilhoorn, C.-K. Yang, P. Krogstrup, L. P. Kouwenhoven, G. de Lange, B. van Heck, and A. Kou, Phys. Rev. Lett. 124, 246802 (2020).
  * Kringhøj  _et al._ (2020) A. Kringhøj, B. van Heck, T. W. Larsen, O. Erlandsson, D. Sabonis, P. Krogstrup, L. Casparis, K. D. Petersson, and C. M. Marcus, Phys. Rev. Lett. 124, 246803 (2020).
  * Luthi  _et al._ (2018) F. Luthi, T. Stavenga, O. W. Enzing, A. Bruno, C. Dickel, N. K. Langford, M. A. Rol, T. S. Jespersen, J. Nygård, P. Krogstrup, and L. DiCarlo, Phys. Rev. Lett. 120, 100502 (2018).
  * Kroll  _et al._ (2019) J. Kroll, F. Borsoi, K. van der Enden, W. Uilhoorn, D. de Jong, M. Quintero-Pérez, D. van Woerkom, A. Bruno, S. Plissard, D. Car, E. Bakkers, M. Cassidy, and L. Kouwenhoven, Phys. Rev. Applied 11, 064053 (2019).
  * Kringhøj  _et al._ (2021a) A. Kringhøj, T. W. Larsen, O. Erlandsson, W. Uilhoorn, J. G. Kroll, M. Hesselberg, R. P. G. McNeil, P. Krogstrup, L. Casparis, C. M. Marcus, and K. D. Petersson, Phys. Rev. Applied 15, 054001 (2021a).
  * Uilhoorn  _et al._ (2021) W. Uilhoorn, J. G. Kroll, A. Bargerbos, S. D. Nabi, C.-K. Yang, P. Krogstrup, L. P. Kouwenhoven, A. Kou, and G. de Lange, arXiv:2105.11038 (2021).
  * Ginossar and Grosfeld (2014) E. Ginossar and E. Grosfeld, Nature communications 5, 4772 (2014).
  * Vaitiekėnas  _et al._ (2020a) S. Vaitiekėnas, G. Winkler, B. van Heck, T. Karzig, M.-T. Deng, K. Flensberg, L. Glazman, C. Nayak, P. Krogstrup, R. Lutchyn, _et al._ , Science 367, eaav3392 (2020a).
  * Vaitiekėnas  _et al._ (2020b) S. Vaitiekėnas, P. Krogstrup, and C. M. Marcus, Phys. Rev. B 101, 060507 (2020b).
  * Sabonis  _et al._ (2020) D. Sabonis, O. Erlandsson, A. Kringhøj, B. van Heck, T. W. Larsen, I. Petkovic, P. Krogstrup, K. D. Petersson, and C. M. Marcus, Phys. Rev. Lett. 125, 156804 (2020).
  * Ristè  _et al._ (2013) D. Ristè, C. C. Bultink, M. J. Tiggelman, R. N. Schouten, K. W. Lehnert, and L. DiCarlo, Nature Communications 4, 1913 (2013).
  * Serniak  _et al._ (2018) K. Serniak, M. Hays, G. de Lange, S. Diamond, S. Shankar, L. D. Burkhart, L. Frunzio, M. Houzet, and M. H. Devoret, Phys. Rev. Lett. 121, 157701 (2018).
  * Kurter  _et al._ (2021) C. Kurter, C. Murray, R. Gordon, B. Wymore, M. Sandberg, R. Shelby, A. Eddins, V. Adiga, A. Finck, E. Rivera, _et al._ , arXiv preprint arXiv:2106.11488 (2021).
  * Gordon  _et al._ (2021) R. Gordon, C. Murray, C. Kurter, M. Sandberg, S. Hall, K. Balakrishnan, R. Shelby, B. Wacaser, A. Stabile, J. Sleight, _et al._ , arXiv preprint arXiv:2105.14003 (2021).
  * Serniak  _et al._ (2019) K. Serniak, S. Diamond, M. Hays, V. Fatemi, S. Shankar, L. Frunzio, R. Schoelkopf, and M. Devoret, Phys. Rev. Applied 12, 014052 (2019).
  * Pan  _et al._ (2022) X. Pan, H. Yuan, Y. Zhou, L. Zhang, J. Li, S. Liu, Z. H. Jiang, G. Catelani, L. Hu, and F. Yan, arXiv preprint arXiv:2202.01435 (2022).
  * Gustavsson  _et al._ (2016) S. Gustavsson, F. Yan, G. Catelani, J. Bylander, A. Kamal, J. Birenbaum, D. Hover, D. Rosenberg, G. Samach, A. P. Sears, S. J. Weber, J. L. Yoder, J. Clarke, A. J. Kerman, F. Yoshihara, Y. Nakamura, T. P. Orlando, and W. D. Oliver, Science 354, 1573 (2016).
  * Vool  _et al._ (2014) U. Vool, I. M. Pop, K. Sliwa, B. Abdo, C. Wang, T. Brecht, Y. Y. Gao, S. Shankar, M. Hatridge, G. Catelani, M. Mirrahimi, L. Frunzio, R. J. Schoelkopf, L. I. Glazman, and M. H. Devoret, Phys. Rev. Lett. 113, 247001 (2014).
  * Wang  _et al._ (2014) C. Wang, Y. Y. Gao, I. M. Pop, U. Vool, C. Axline, T. Brecht, R. W. Heeres, L. Frunzio, M. H. Devoret, G. Catelani, L. I. Glazman, and R. J. Schoelkopf, Nature Communications 5, 5836 (2014).
  * Patel  _et al._ (2017) U. Patel, I. V. Pechenezhskiy, B. L. T. Plourde, M. G. Vavilov, and R. McDermott, Phys. Rev. B 96, 220501 (2017).
  * Schreier  _et al._ (2008) J. A. Schreier, A. A. Houck, J. Koch, D. I. Schuster, B. R. Johnson, J. M. Chow, J. M. Gambetta, J. Majer, L. Frunzio, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Phys. Rev. B 77, 180502 (2008).
  * (25) See Supplementary Material for details on data rotation in the phase-quadrature-plane, parity data post-selection, destructive Little-Parks effect, nanowire micrographs, and the experimenal setup.
  * Wesdorp  _et al._ (2021) J. Wesdorp, L. Grünhaupt, A. Vaartjes, M. Pita-Vidal, A. Bargerbos, L. Splitthoff, P. Krogstrup, B. van Heck, and G. de Lange, arXiv preprint arXiv:2112.01936 (2021).
  * Catelani  _et al._ (2011) G. Catelani, R. J. Schoelkopf, M. H. Devoret, and L. I. Glazman, Phys. Rev. B 84, 064517 (2011).
  * Tinkham (1996) M. Tinkham,  _Introduction to Superconductivity_ , 2nd ed., International Series in Pure and Applied Physics (McGraw Hill, New York, 1996).
  * Chang  _et al._ (2015) W. Chang, S. M. Albrecht, T. S. Jespersen, F. Kuemmeth, P. Krogstrup, J. Nygård, and C. M. Marcus, Nature Nanotechnology 10, 232 (2015).
  * Kringhøj  _et al._ (2021b) A. Kringhøj, G. W. Winkler, T. W. Larsen, D. Sabonis, O. Erlandsson, P. Krogstrup, B. van Heck, K. D. Petersson, and C. M. Marcus, Phys. Rev. Lett. 126, 047701 (2021b).
  * Sternfeld  _et al._ (2011) I. Sternfeld, E. Levy, M. Eshkol, A. Tsukernik, M. Karpovski, H. Shtrikman, A. Kretinin, and A. Palevski, Phys. Rev. Lett. 107, 037001 (2011).
  * Dao and Chibotaru (2009) V. H. Dao and L. F. Chibotaru, Phys. Rev. B 79, 134524 (2009).
  * Larkin (1965) A. I. Larkin, JETP 21, 153 (1965).

##  III Supplementary Material

###  III.1 Rotation of data in phase-quadrature-plane

Here we describe the procedure for determining a rotation angle ${\theta}$ in
the phase-quadrature-plane. For each measurement of ${T_{P}}$ in the main
text, a series of plots were generated for several different ${\theta}$, as
shown in Fig. S1 for the data set in Fig. 1(c-d) of the main text. The value
of ${\theta}$ was selected to emphasize the bimodality of the two charge-
parity populations, as visually apparent from histograms and time traces. Due
to instability of ${n_{g}}$ over time, the separation of the charge-parity
population varies. Data sets where two main populations were not observed were
not used to obtain values of ${T_{P}}$; this situation could result e.g. from
${n_{g}}$ on a degeneracy point, or a charge jump during the course of the
measurement. For many data sets, phase-quadrature-plane histogram contained
tail-like features, interpreted as partially resolved residual excited state
population. In these cases, ${\theta}$ and angle was selected in a compromise
between the axis connecting the charge-parity populations and the line
orthogonal to the one connecting the residual features. In general, this is
not the same as the line along the midpoints of the two populations. In the
case of Fig. S1, this meant that ${\theta=$1.369\text{\,}\mathrm{rad}$}$ was
selected (bottom row of plots). The particular criterion used in the selection
of ${\theta}$ is not expected to introduce a bias in ${T_{P}}$, as the
rotation does not directly alter time domain aspects of the data. For example,
in the case of the data in Fig. 1(c-d) of the main text, the standard
deviation of ${T_{P}}$ values resulting from the Lorentzian fit, across a
$1\text{\,}\mathrm{rad}$ interval around the selected angle, was found to be
${<5\%}$.

Figure S1:  Plots used to select phase-quadrature-plane rotation angle
${\theta}$. Two different ${\theta}$ for the data set in Fig. 1(c-d). In (a-d)
${\theta=$0.322\text{\,}\mathrm{rad}$}$. (a) Two-dimensional histogram with
the red line indicating ${\theta}$. The extension of each population are
interpreted to result from residual excited state population. (b) One-
dimensional histogram of the data rotated by ${\theta}$. (c) One of the
${N_{T}}$ traces; here ${N_{T}=40}$. (d) Average PSD of all ${N_{T}}$ data
sets. (e-h) as (a-d) but with ${\theta=$1.369\text{\,}\mathrm{rad}$}$, which
was selected for obtaining the ${T_{P}}$ value in the main text from this
data.

###  III.2 Parity data post-selection

Here we describe how the two-tone spectroscopy data in Fig. 2(b) were post-
selected into separate parity branches. As shown in Fig. S2, the rows are
divided into segments AB, BC, etc. Rows are post-selected into parities based
on whether the value at a specific point is higher or lower than the mean of
that row. The columns picked for the sorting are shown as veritcal lines in
Fig. S2. This particular procedure relies on a feature unique to one of the
parities existing in every row.

Figure S2:  Criterion points used in the parity post-selection for
spectroscopy. For a given interval of ${f_{d}}$, data are post-selected
conditioned on the value at a chosen ${\Delta V_{c}}$ (e.g. at AB for
$A>f_{d}>B$ ) exceeding the mean of that trace of ${V_{H}}$ as a function of
${V_{c}}$.

###  III.3 Destructive Little-Parks effect

Similar to Ref. 13, the NW for which we measure ${T_{P}}$ here exhibits a
destructive Little-Parks effect. This can be directly seen in the two-tone
spectroscopy measurement of qubit frequency ${f_{01}}$ as a function of
magnetic field, shown in Fig. S3. Following Ref. 13, the pair-breaking term
${\alpha}$ is given minimizing the function Sternfeld  _et al._ (2011);
Vaitiekėnas  _et al._ (2020b); Dao and Chibotaru (2009),

$\alpha(B)=\frac{4\xi^{2}k_{B}T_{C0}}{\pi
R^{2}}\left[\left(n-\frac{\Phi}{\Phi_{0}}\right)^{2}+\frac{t^{2}}{4R^{2}}\left(\frac{\Phi^{2}}{\Phi_{0}^{2}}+\frac{n^{2}}{3}\right)\right]$
(3)

in the winding number ${n}$ at each ${B}$, where ${\xi}$ is the
superconducting coherence length at zero field, ${T_{C0}}$ is the critical
temperature at zero field, ${\Phi}$ is the applied magnetic flux, ${R}$ is the
radius of the superconducing shell, and ${t}$ is the shell thickness. The
pairing energy ${\Delta}$ is then found by solving the implicit equation
Larkin (1965)

$\log\,\frac{\Delta_{0}}{\Delta}=\left\\{\begin{array}[]{ll}\frac{\pi\alpha}{4\Delta},&\mbox{if
$\alpha<\Delta$},\\\
\log\left(\frac{\alpha+\sqrt{\alpha^{2}-\Delta^{2}}}{\Delta}\right)-\frac{\sqrt{\alpha^{2}-\Delta^{2}}}{2\alpha}+\frac{\alpha}{2\Delta}\arcsin\frac{\Delta}{\alpha},&\mbox{if
${\Delta>\alpha}$}.\end{array}\right.$ (4)

We assume that ${\Delta}$ enters the transmon Hamiltonian as
${E_{J}=(\Delta/4)\sum\eta}$ where ${\eta}$ are the transmissions of the JJ
channels. In terms of ${\Delta}$, the spectral gap ${\omega}$ is given by
Larkin (1965)

$\omega=\left(\Delta^{2/3}-\alpha^{2/3}\right)^{3/2}\,.$ (5)

We assume that ${\omega}$ enters into Eq. (2).

Figure S3:  Destructive Little-Parks effect observed by measuring the qubit
frequency ${f_{01}}$, resulting in a lobe structure as a function of magnetic
field ${B}$.

###  III.4 Transmission electron micrographs of nanowires

Transmission electron micrographs of NWs from the same growth batch as the NW
used in the device described in the main text, are shown in Fig. S4.
Differences between NWs in micrographs and the one used for the device might
exist, due to aging, position on wafer, and individual wire-to-wire
variations.

Figure S4:  Transmission electron micrographs of two NWs from the same growth
batch as the NW discussed in the main text, with physical dimensions
indicated.

###  III.5 Experimental setup

A diagram of the experimental setup is shown in Fig. S5. The sample is mounted
on a printed circuit board in an In-sealed CuBe box, with added Eccosorb foam.

Figure S5:  Experimental setup.


# Building Autocorrelation-Aware Representations for Fine-Scale Spatiotemporal
Prediction

Yijun Lin  Department of Computer Science
University of Southern California
<EMAIL_ADDRESS>Yao-Yi Chiang  Spatial Sciences Institute
University of Southern California
<EMAIL_ADDRESS>Meredith Franklin  Sandrah P. Eckel  Department of Preventive
Medicine
University of Southern California
<EMAIL_ADDRESS>Department of Preventive Medicine
University of Southern California
<EMAIL_ADDRESS>José Luis Ambite  Information Sciences Institute
University of Southern California
<EMAIL_ADDRESS>
###### Abstract

Many scientific prediction problems have spatiotemporal data- and modeling-
related challenges in handling complex variations in space and time using only
sparse and unevenly distributed observations. This paper presents a novel deep
learning architecture, Deep learning predictions for LocATion-dependent Time-
sEries data (DeepLATTE), that explicitly incorporates theories of spatial
statistics into neural networks to addresses these challenges. In addition to
a feature selection module and a spatiotemporal learning module, DeepLATTE
contains an autocorrelation-guided semi-supervised learning strategy to
enforce both local autocorrelation patterns and global autocorrelation trends
of the predictions in the learned spatiotemporal embedding space to be
consistent with the observed data, overcoming the limitation of sparse and
unevenly distributed observations. During the training process, both
supervised and semi-supervised losses guide the updates of the entire network
to: 1) prevent overfitting, 2) refine feature selection, 3) learn useful
spatiotemporal representations, and 4) improve overall prediction. We conduct
a demonstration of DeepLATTE using publicly available data for an important
public health topic, air quality prediction, in a well-studied, complex
physical environment - Los Angeles. The experiment demonstrates that the
proposed approach provides accurate fine-spatial-scale air quality predictions
and reveals the critical environmental factors affecting the results.

###### Index Terms:

Fine-Scale Prediction, Spatiotemporal, Autocorrelation, Air Quality

##  I Introduction

Fine-scale spatiotemporal prediction is an important scientific problem
applicable to diverse phenomena, such as air quality, ambient noise, traffic
conditions, and meteorology.111Here, the prediction problem refers to
estimating measurements for unobserved locations at current or past times (in
contrast to forecasting future measurements). One of the primary motivations
for spatiotemporal prediction is that the observed data are only available at
a few unevenly distributed measurement locations (e.g., ground-based sensors)
[1, 2]. As an example, there is only a handful of Federal monitoring sites and
a few hundred low-cost sensors reporting air quality in Los Angeles, an area
that covers nearly 5,000 square miles and 15 million people. In many
epidemiological studies, the exposure assessment usually needs the air quality
at a fine spatial scale for analyzing the associations between inhalable
particles222 For example, fine particles with an aerodynamic diameter smaller
than 2.5$\mu$m, i.e., PM2.5. and a variety of health outcomes including
cardio-respiratory diseases [3]. However, sparse ground-based monitoring
networks do not provide the spatial resolution required to characterize
exposures where people spend their time: home, work, and school [4, 5].

Approaches for predicting spatiotemporal phenomena generally fall into the
categories of expert-based and data-driven methods. Expert-based air quality
prediction methods, such as atmospheric dispersion models, utilize
mathematical equations to simulate how pollutants travel in the physical
environment. They require significant expert knowledge and computational
resources (e.g., modeling chemical transformations in the atmosphere [2, 6]).
In contrast, data-driven methods aim to characterize spatiotemporal patterns
in the available measurement data and leverage the patterns for prediction.
One common data-driven approach is spatial interpolation, such as Inverse
Distance Weighting (IDW) and Ordinary Kriging (OK), which utilize spatial
distance-based relationships (in either non-geostatistical or geostatistical
manner) between observed and target locations to make predictions where there
are no observations [7]. Spatial interpolation methods do not consider
external environmental characteristics, such as meteorology or land use.
Hence, they have limited ability to produce reliable fine-scale predictions
(e.g., would have similar prediction values across a large area) [5].

Advanced data-driven methods take various contextual data about the
environment into account to improve the robustness of the prediction model and
outcomes. The main challenge is how to automatically discover meaningful
spatial and temporal relationships between rich contextual data and sparse
measurement data in an area. For example, land-use regression (LUR) models
generate explainable air quality predictions by including expert-selected
land-use predictors (e.g., traffic indicators, industrial facilities, and
population density) [8], which can vary from one geographic region to another
[9]. Machine learning models, such as neural networks and random forests, can
alleviate the requirement of expert-selected predictors in the contextual data
by algorithmically choosing the most useful features for prediction tasks
(e.g., [9, 10, 11, 5]). However, existing methods only take neighborhood-level
spatial or temporal dependencies into consideration (e.g., [11]), which could
suffer from overfitting when only a limited number of measurement locations
are available, or the measurement locations are sparse and not spatially
representative of the target region.

This paper proposes a novel data-driven approach, named DeepLATTE (Deep
learning prediction for LocATion-dependent Time-sEries data), that explicitly
incorporates theories of spatial statistics into neural networks for fine-
spatial-scale prediction of geographically-based spatiotemporal phenomena. The
goal333We demonstrate DeepLATTE within the air quality domain, but the overall
approach does not assume a specific application domain. is to estimate air
quality values at locations that do not have sensors over a fine spatial grid
by using the observed air quality measurements from a network of low-cost
sensors as well as the contextual data describing the environmental
characteristics. First, DeepLATTE automatically generates a condensed feature
embedding of the contextual data for both labeled (measurement) and unlabeled
(target) locations using a standard sparse layer and an auto-encoder. These
feature embeddings then go through a Convolutional Long Short-Term Memory
(ConvLSTM) module that incorporates the contextual information in local
spatial and temporal neighborhoods to generate spatiotemporal embeddings for
all locations. Next, DeepLATTE employs a novel autocorrelation-guided semi-
supervised learning strategy that 1) enforces the spatial and temporal
neighboring embeddings to be similar (i.e., the learned environmental
characteristics should change gradually or slightly over a small range of
space and time) and 2) enforces the distribution of the predicted values and
the distribution of the labels in the embedding space to be similar. The first
step helps propagate useful information within a local region in space and
time to refine the spatiotemporal embeddings of all locations. The second
process is inspired by Kriging using a semivariogram, which ensures the
predictions show similar graduated autocorrelation patterns in the learned
spatiotemporal embedding space as in the labeled data. This semi-supervised
learning strategy prevents the networks from focusing on only the labeled
locations and hence can fully utilize the contextual data for each location
and time to produce accurate prediction results.

In sum, the main contribution of this paper is a general network architecture
that exploits both local autocorrelation patterns and global autocorrelation
trends of the predictions and the labeled data in the learned spatiotemporal
embedding space for accurate fine-scale prediction task using limited labeled
data. Also, we show an end-to-end approach using this network architecture for
fine-scale air quality prediction.

##  II DeepLATTE Methodology

This section presents the proposed approach, DeepLATTE, for fine-spatial-scale
prediction of spatiotemporal data using air quality prediction as an example.
The inputs include the contextual data and the available measurements
represented in a grid structure (tensors). The contextual data are denoted as
$\boldsymbol{X}\in\mathbb{R}^{H\times W\times P}$, where $P$ is the number of
input features, and $H$ and $W$ are the height and width of the grid,
respectively. The measurements are denoted as
$\boldsymbol{Y}\in\mathbb{R}^{H\times W}$, which contain many missing values.
The prediction results are denoted as
$\boldsymbol{\hat{Y}}\in\mathbb{R}^{H\times W}$. Let $\boldsymbol{X}^{(t)}$
represent the input signal at time $t$, $T^{\prime}$ is the number of previous
hours (i.e., from $t-T^{\prime}+1$ to $t$). DeepLATTE aims to learn a function
$h$ that maps $T^{\prime}$ historical input signals to the output at time $t$:

$[\boldsymbol{X}^{(t-T^{\prime}+1)},\cdots,\boldsymbol{X}^{(t)}]\xrightarrow{\emph{h}}[\boldsymbol{\hat{Y}}^{(t)}]$

Figure 1 shows the overall architecture of DeepLATTE. After constructing the
input grid data $\boldsymbol{X}$ and labeled data $\boldsymbol{Y}$ (Section
II-A), DeepLATTE uses a sparse layer (Section II-B) and an encoder-decoder
module (Section II-C) to generate a condensed feature embedding for the
selected predictors. DeepLATTE then leverages multiple ConvLSTM layers with
varying kernel sizes to learn a spatiotemporal representation of embeddings
and enforces spatially and temporally nearby embeddings to be similar (Section
II-D). Afterwards, DeepLATTE constrains the global autocorrelation trends of
the predictions to be close to those of the labeled data in the learned space
of the spatiotemporal embeddings in a semi-supervised way (Section II-E). This
approach assumes that natural and built environments (i.e., contextual data)
contribute to the target spatiotemporal phenomenon (e.g., air quality), and
the predictions are highly correlated when their learned representations of
the environments are similar. The approach does not assume a specific
application domain.

Figure 1: DeepLATTE network architecture

###  II-A Generating Grid Data

The raw data (i.e., labeled air quality data from sensors and contextual data)
can have varying spatial and temporal resolutions. To unify and aggregate the
input data, DeepLATTE divides the target area into disjointed cells and
transforms the raw data into a grid representation. Each cell contains a set
of features, which represent an aggregation of the contextual data. We denote
$\boldsymbol{X}_{i}\in\mathbb{R}^{P}$ as the input vector of cell $i$ on the
grid. Let $\boldsymbol{D}_{i}\in\mathbb{R}^{P_{d}}$ represent the dynamic
(time-varying) features (e.g., weather) and
$\boldsymbol{S}_{i}\in\mathbb{R}^{P_{s}}$ represent the static (time-
invariant) features (e.g., built environment). $P_{d}$ and $P_{s}$ are the
number of dynamic features and static features, respectively. If the
measurements of a feature are sparser than the grid resolution but uniformly
distributed in space (e.g., weather data), DeepLATTE spatially up-scales the
feature with cubic interpolation; otherwise, DeepLATTE either directly adopts
the measurements for a cell (e.g., coordinate) or aggregates the measurements
within the cell (e.g., the area of parks) to construct the feature vector.
Figure 2(a) shows an example of one feature component, primary roads, in the
spatial grid covering Los Angeles. The cell value is the summation of the
lengths of individual primary roads within the cell.

This process assumes that the observations in a cell are uniform (i.e., one
value per cell). The system maps the available sensors to the corresponding
grid cells to generate labels. If one cell contains multiple sensors, the cell
takes the average of their sensor readings. Figure 2(b) shows an example grid
map of PM2.5 observations. The colored cells are the locations with ground
truth measurements (i.e., labeled locations), while the uncolored cells are
the target (unlabeled) locations for prediction.

(a) Grid primary roads

(b) Grid PM2.5 concentrations

Figure 2: An example grid map with 500m$\times$500m cells covering a
50km$\times$40km area in Los Angeles. (a) The aggregated length of primary
roads in the cells. (b) PM2.5 concentrations after mapping sensor data to the
grid. Missing values and zeros are not colored.

###  II-B Feature Selection

DeepLATTE first uses a sparse layer with $L_{1}$ regularization (Figure 1,
“Sparse Layer”) with the purpose of (1) removing irrelevant input features and
(2) balancing the contributions of dynamic and static features (e.g., in our
air quality case, there many more static features than dynamic ones).

The sparse layer is a linear layer containing the same number of nodes as the
input, and there exists only a single connection between the corresponding
nodes. Let $W_{sp}$ be the weight matrix of the sparse layer, which is a
diagonal matrix (zero off-diagonal weights). We add $L_{1}$ regularization as
the sparse constraint to this layer to force the sum of the absolute values of
the parameters to be small. Thus, the $L_{1}$ penalty term can cause many
weights to be close to zero. If the weight is a tiny value (less than a
predefined threshold), DeepLATTE sets the output of the sparse layer of the
corresponding feature to zero to achieve the purpose of feature selection. We
denote the output of the sparse layer as
$\boldsymbol{X}_{sp}\in\mathbb{R}^{H\times W\times P}$. The dimension of
$\boldsymbol{X}_{sp}$ is the same as $\boldsymbol{X}$ with irrelevant features
being zeros. The cost function of $L_{1}$ regularization is as follows:

$\mathcal{L}_{sp}={\sum_{w\in W_{sp}}{|w|}}$ (1)

###  II-C Learning Condensed Feature Embeddings

DeepLATTE pre-trains an auto-encoder to learn a condensed feature embedding
$\boldsymbol{E}_{i}$ for a cell $i$ from the selected dynamic
$\boldsymbol{D}_{i}$ and static $\boldsymbol{S}_{i}$ features (Figure 1
“Encoder” and “Decoder”). For example, suppose primary road is an indicator of
traffic volume, and its impact can vary with time, weather conditions, and
location. It is necessary to learn a latent embedding to condense these two
types of features instead of directly feeding them original features vector to
the network. The reconstruction loss for the auto-encoder is:

$\mathcal{L}_{ae}={\mathcal{\bar{L}}(\boldsymbol{X}_{sp},\boldsymbol{\hat{X}}_{sp})}$
(2)

where $\boldsymbol{X}_{sp}$ is the output of the sparse layer (the encoder
input), $\boldsymbol{\hat{X}}_{sp}$ is the reconstructed vector (the decoder
output), and $\mathcal{\bar{L}}$ is the loss function using the mean square
error. During the end-to-end training after pre-training, the network can
refine the condensed feature embedding $\boldsymbol{E}$ by updating the
encoder weights guided from the remainder of the network modules.

###  II-D Learning Spatiotemporal Feature Embeddings

Current air quality level at a location is highly correlated with the
environmental characteristics at the location and its neighbors at present and
previous time points. For example, suppose a South-West power plant emits
pollution at time $T$, the northeast cells can be significantly polluted at
$T+1$ with a northeast wind (Figure 3).

We define the spatiotemporal embedding of a cell $i$ at time $t$ as the joint
effects of the air quality-related factors in the condensed feature embeddings
from neighboring space and time, denoted as $\boldsymbol{R}^{(t)}_{i}$. To
jointly model the spatiotemporal impacts on air quality, DeepLATTE first
leverages the ConvLSTM operation [12] to learn useful information from the
combination of the condensed embeddings from spatial neighbors and the
previous hidden memory. For example, in Figure 3(c), when generating the
spatiotemporal embedding of the center cell (red box) using one-step neighbors
only (in the purple dotted box), DeepLATTE learns the interactive effects from
the northeast green areas, the northwest residential areas, the South-East
industrial areas, and the East commercial areas. To account for the influence
of environmental characteristics on air quality from varying distances,
DeepLATTE employs multiple ConvLSTM layers with various kernel sizes to learn
the impacts from the neighbors within increasing spatial distances. The
outputs of these ConvLSTM layers with varying kernel sizes are then
concatenated to form an initial spatiotemporal embedding for each cell.

Due to the limited number of sparse labeled data, the initial spatiotemporal
embeddings would be only learned from the labeled data. To overcome this
difficulty, DeepLATTE adds a representation constraint in the learning process
by enforcing the spatially and temporally nearby spatiotemporal embeddings to
be similar with the assumption that the embeddings would not change
significantly within a local spatial and temporal neighborhood. This
constraint guides the network to produce a graduated, continuous
spatiotemporal representation from all input data instead of focusing on the
labeled locations. The cost function is as follows:

$\displaystyle\mathcal{L}_{stc}=\lambda_{1}\times\sum_{i=1}^{N}\sum_{k=1}^{K_{S}}\sum_{j\in{\mathcal{N}^{(S_{k})}_{i}}}\frac{1}{k}\mathcal{\bar{L}}(\boldsymbol{R}_{i},\boldsymbol{R}_{j})$
(3)
$\displaystyle+\lambda_{2}\times\sum_{i=1}^{N}\sum_{k=1}^{K_{T}}\sum_{j\in{\mathcal{N}^{(T_{k})}_{i}}}\frac{1}{k}\mathcal{\bar{L}}(\boldsymbol{R}_{i},\boldsymbol{R}_{j})$

where $\boldsymbol{R}_{i}$ is the spatiotemporal embedding for cell $i$ at a
certain time. $N$ is the total number of spatiotemporal embeddings (the number
of target locations multiplied by the number of time points for prediction).
$K_{S}$ and $K_{T}$ are the number of neighborhoods to consider in space and
time, respectively, where $k=1$ implies the immediate, one-step neighborhood,
and $k=2$ is two-step neighborhood. ${\mathcal{N}^{(S_{k})}_{i}}$ is the
neighbors of the cell $i$ within $k$ spatial step. Similarly,
${\mathcal{N}^{(T_{k})}_{i}}$ represents the temporal neighbors.
$\mathcal{\bar{L}}$ is the loss function of mean square error. This cost
function ensures that nearby spatiotemporal embeddings in space and time are
similar, and the similarity gradually decreases when the distance increases.
Finally, the refined spatiotemporal embeddings are fed to fully connected
layers to infer air quality prediction for each cell.

(a) Time $T$

(b) Time $T+1$

(c) Spatial Effects

Figure 3: An example of learning spatiotemporal effects. The cell size is
500m$\times$500m. (a) A pollution emission from South-West power plant at time
$T$. (b) The northeast cells are polluted at time $T+1$ with the northeast
wind. (c) A process of learning environmental characteristics from one-step
neighbors (within purple dotted box).

###  II-E Autocorrelation-Guided Semi-Supervised Learning

To handle the fact that the number of unlabeled cells is much greater than the
labeled cells, typical semi-supervised learning methods could enforce nearby
cells in space and time to have a similar air quality value [11]. This is in
line with the assumption that air quality measurements often have spatial
autocorrelation that describes the tendency for areas or sites that are close
together to have similar values. Statistical methods, like Ordinary Kriging,
take advantage of spatial autocorrelation to generate reliable estimates for
the interpolated locations. However, it could be problematic when the observed
locations are not evenly distributed or far apart from each other, or there is
spatial non-stationarity in the data.

In addition to incorporating local spatial autocorrelation, DeepLATTE proposes
an autocorrelation-guided semi-supervised strategy that exploits the
autocorrelation in the representation space of the spatiotemporal embeddings
and enforces the predictions to embrace similar autocorrelation patterns to
the observations in the representation space. The assumption is that the air
quality values of two locations at the same or different times (or the same
location at different times) could be similar if their environmental
characteristics are similar (i.e., similar spatiotemporal embeddings) even
when their spatial or temporal distances are large. We define the
autocorrelation as the relationship between the air quality variability and
the similarity of spatiotemporal embeddings (the concept is close to the
spatial autocorrelation, but our method does not directly use the physical
distance). With this autocorrelation, we assume that similar spatiotemporal
embeddings will generate close air quality values within some “influence
range”, and this pattern should be applicable for both observations and
predictions.

First, DeepLATTE decides the autocorrelation influence range by constructing a
semivariogram to represent the autocorrelation in the real observations in the
representation space of spatiotemporal embeddings. Figure 4 shows an example
semivariogram, of which the x-axis represents the distance (lag) between two
spatiotemporal embeddings, $\boldsymbol{R}$. The distance lag is re-scaled to
$[0,1]$ and divided into multiple bins with a fixed lag size (e.g., 0.1). At a
lag size, $h$, the y-axis represents the semivariance, $\gamma(h)$, for all
the pairs of air quality values within that bin:

$\gamma(h)=\frac{1}{N(h)}\sum_{i}\sum_{j\neq
i}{(Y(\boldsymbol{R}_{i})-Y(\boldsymbol{R}_{j}))^{2}}$

where $Y(\boldsymbol{R})$ is the air quality value (label) of the
spatiotemporal embedding $\boldsymbol{R}$. The distance between
$\boldsymbol{R}_{i}$ and $\boldsymbol{R}_{j}$ must be in the corresponding
bin. $N(h)$ is the number of pairs of $\boldsymbol{R}$ in the bin.

In Figure 4, the red points represent the semivariances at the binned
distances. Then we estimate the relationship between semivariance and distance
lag with a curve function to describe the autocorrelation process. The blue
line is the estimated Gaussian process as an example:

$f(h)=n+s\times(1-\exp{(-\frac{h^{2}}{(r/2)^{2}})})$

where $h$ is the distance lag and $f(h)$ is the empirical semivariance. The
parameter $n$ refers to the “nugget”, which is the variability that cannot be
explained by distance (by default $n=0$). The parameter $s$ refers to the
“sill”, representing the maximum observed variability. The parameter $r$
refers to the “range” at which the semivariance stops increasing. For example,
the estimated influence range in Figure 4 is 0.6, meaning that air quality
values have strong autocorrelation at distances less than 0.6 while there is
no or little autocorrelation beyond this influence range that can be ignored.

Figure 4: An example semivariogram

After estimating the influence range from the semivariogram, DeepLATTE keeps
the bins within the range and computes the distribution of semivariances for
the labeled data and predictions separately in each of the kept bins.
Specifically, DeepLATTE computes a Gaussian distribution
$\mathcal{N}(\mu_{y_{h}},\sigma_{y_{h}})$ to describe the distribution of the
square errors between labels (i.e., $(y_{i}-y_{j})^{2}$) when
$\left\|\boldsymbol{R}_{i}-\boldsymbol{R}_{j}\right\|_{2}$ is within distance
lag $h$. Similarly, DeepLATTE computes
$\mathcal{N}(\mu_{\hat{y}_{h}},\sigma_{\hat{y}_{h}})$ for the predictions. The
Gaussian distribution represents a generative model describing the
autocorrelation pattern of the data at a given distance lag. Since the real
observations and predictions should share similar autocorrelation patterns,
DeepLATTE calculates the KL divergence to measure the distance between the two
Gaussian distributions at each bin to quantify the difference between the
autocorrelation of labels and predictions. Then DeepLATTE takes the summation
of the KL terms from all the valid bins as the autocorrelation loss:

$\begin{split}\mathcal{L}_{ac}&=\sum_{h}D_{\mathrm{KL}}(\mathcal{N}(\mu_{y_{h}},\sigma_{y_{h}})||\mathcal{N}(\mu_{\hat{y}_{h}},\sigma_{\hat{y}_{h}}))\\\
&=\frac{1}{2}\sum_{h}(log\frac{\sigma_{\hat{y}_{h}}^{2}}{\sigma_{y_{h}}^{2}}-1+\frac{\sigma_{y_{h}}^{2}+(\mu_{y_{h}}-\mu_{\hat{y}_{h}})^{2}}{\sigma_{\hat{y}_{h}}^{2}})\end{split}$
(4)

where $\mu_{y_{h}}$ and $\sigma_{y_{h}}$ are mean and std. for the labeled
pairs in lag distance $h$. $\mu_{\hat{y}_{h}}$ and $\sigma_{\hat{y}_{h}}$ are
for the prediction pairs in $h$.

###  II-F DeepLATTE Overall Architecture

The proposed network computes the supervised loss on labeled cells using the
following cost function:

$\mathcal{L}_{pred}={\frac{1}{m}\sum_{i=1}^{m}(y_{i}-\hat{y}_{i})^{2}}$ (5)

where $m$ is the number of labeled cells, $\hat{y}$ is the prediction value of
a cell, and $y$ is the real observation of the cell.

The overall cost function of the proposed architecture is the sum of equation
(5), (1), (2), (3), and (4):

$\mathcal{L}=\mathcal{L}_{pred}+\alpha\times\mathcal{L}_{sp}+\beta\times\mathcal{L}_{ae}+\lambda\times\mathcal{L}_{stc}+\eta\times\mathcal{L}_{ac}$
(6)

where $\mathcal{L}_{pred}$ is the loss over the training examples,
$\mathcal{L}_{sp}$ is the loss from the sparse layer, $\mathcal{L}_{ae}$ is
the reconstruction loss of the auto-encoder, $\mathcal{L}_{stc}$ is the loss
from the representation constraint, and $\mathcal{L}_{ac}$ is the loss from
applying the autocorrelation constraint; $\alpha$, $\beta$, $\lambda$, and
$\eta$ are the hyper-parameters. We train the model by updating the network
parameters using eq. (6).

##  III Experiments

We implemented and tested DeepLATTE for air quality prediction in Los Angeles.
444https://github.com/spatial-computing/deeplatte-fine-scale-prediction In
addition to its complex physical environment, Los Angeles is a well-studied
area in terms of air pollution so there is an abundant literature that can be
used to verify the prediction results (e.g., temporal and spatial
distributions and trends) even when ground observations are limited.

###  III-A Experimental Setup

Datasets. We conducted the experiments with the following open-source datasets
covering a 50km$\times$40km region in Los Angeles (Figure 5(a)). (1) Air
quality data: Our system collected hourly outdoor PM2.5 concentrations in 2018
and 2019 from PurpleAir sensors.555https://www2.purpleair.com/ The available
sensors increased over time, with the numbers ranging from 35 to 206. The
PM2.5 variations were generally higher during the winter (Nov., Dec., and
Jan.) than the summer period (May, Jun., and Jul.). (2) Meteorological data:
Our system collected hourly weather data in 2018 and 2019 from
DarkSky.666https://darksky.net/dev DarkSky reports worldwide fine-scale
weather data on various features, of which we extracted 10 features (e.g.,
temperature, humidity, and wind speed). The resolution of the requested
weather data was of 5km$\times$5km, and our system cubically interpolated them
to the target resolution. (3) Geographic data: Our system extracted geographic
information from OpenStreetMap to describe land uses, roads, traffic,
railways, and water areas in various spatial representations (polygons, lines,
and points). Each geographic feature contains various sub-types; for example,
the “roads” feature type contains sub-types such as motorways, primary roads,
and residential roads. We included a total of 80 geographic features. (4)
Other data: Our system included time information (i.e., hour of a day, day of
a week, and day of a year) and the geo-coordinates of the grid center (i.e.,
longitude and latitude) for each combination of time & location for the
prediction task.

Training Settings. Our system created a grid over the target region with a
500m$\times$500m cell size (total 6,992 cells). The sensor locations were
unevenly distributed over the target region. To maintain the original spatial
distribution of the sensors in both training and testing data, we divided the
target region evenly into four areas (Figure 5(b)), from which we randomly
picked 60% of the available locations as training, 20% as validation, and 20%
as testing.

For each month, we trained a predictive model using target-month sensor
observations and contextual data. Also, if historical data from previous
months were available, we leveraged the model trained from the previous months
and fine-tuned the model using the target-month data. For example, to predict
PM2.5 concentrations in Feb. 2018, we fine-tuned the model of Jan. 2018 with
Feb. data. We ensured that training and testing locations were mutually
exclusive over these months.

To predict PM2.5 concentrations at time $t$, we constructed the input features
with temporal lags of six hours (i.e., from $t-6$ to $t$). We set the latent
embedding of the auto-encoder as 32 neurons. We constructed each ConvLSTM
containing one layer with 64 hidden states. We applied three kernel sizes
(1$\times$1, 3$\times$3, and 5$\times$5), i.e., at most two-step neighbors
(distance range is 1,500m). Therefore, the output size of the ConvLSTM
component is 192. We set the threshold of the sparse layer as 0.0001, $K_{T}$
and $K_{S}$ as 1, and $\lambda_{1}=\lambda_{2}$. The initial learning rate was
0.001, with early stopping on the validation dataset. The hyper-parameters are
chosen by grid search on the validation set and we reported the results with
the set of hyper-parameters yielding the best average performance, that is
$\alpha=1$, $\beta=5$, $\lambda=5$, and $\eta=0.1$.

(a) Target area

(b) Four sub-areas

Figure 5: (a) shows the prediction region in Los Angeles; (b) shows the
equally-divided area as a gridded surface.

###  III-B Baseline Methods

We compared DeepLATTE with several state-of-the-art and commonly used models
as well as DeepLATTE variants:

Random Forest (RF). Yu et al. [13] demonstrated that RF outperformed other
approaches, including logistic regression, decision tree, and artificial
neural network for fine-spatial-scale air quality prediction.

Inverse Distance Weighting (IDW). IDW is a standard spatial interpolation
method that calculates predictions as the weighted average of the available
measured points based on their spatial distance. We set the power parameter as
2.

Ordinary Kriging (OK). OK is a spatial statistical interpolation method that
models the interpolated surface using a spatial process governed by the
spatial autocorrelation of the measured points. We set the kernel function as
Gaussian process and the number of bins as 10.

Lin et al. [5]. This is a machine learning method that mines important
PM2.5-related geographic features as “geo-context” for fine-grained air
quality prediction. We tested the approach with our datasets and picked the
number of clusters with the Elbow method.

Deep Air Learning (DAL) [11]. DAL is a semi-supervised learning approach that
leverages the information from the unlabeled data with spatiotemporal
smoothing for the prediction. We re-implemented the approach and tested with
our datasets. We did a similar way for hyper-parameter tuning and ended up
with the set $\lambda_{1}=1$, $\alpha=10$, and $\beta=10$.

DeepLATTE Variants. (1) DeepLATTE without feature selection module; (2)
DeepLATTE without the autocorrelation-guided constraint.

###  III-C Evaluation

Quantitative Evaluation. (1) We used the Root Mean Squared Error (RMSE) and
R-Squared (R2) scores to evaluate the model performance. (2) We quantified the
relationship between the variation in the fine-scale PM2.5 predictions and the
distance to critical environmental features, e.g., comparing the average
predictions near highways within 500m and 1,000m, and evaluated the results
with findings in the literature.

Qualitative Evaluation. (1) We presented both spatial and temporal
visualizations of the prediction results for verifying fine variations and
trends in spatial and temporal neighborhoods using studies in the literature.
(2) We verified the selected features with existing using studies in the
literature.

###  III-D Results and Discussion

Model Performance. Figure 6(a) and 6(b) show the RMSE and R2 for the monthly
models of DeepLATTE and baseline methods in 2018 and 2019. We observe that RF
has poor performance because it cannot handle complex relationships between
the predictors adequately. OK outperforms IDW since OK specifically models the
spatial autocorrelation instead of only relying on the spatial distance to
compute the weight of measured locations. However, OK does not consider
environmental characteristics and generates smoothing prediction surfaces (see
Section Visualizations). DAL has competitive performance but is worse than
DeepLATTE because DAL only enforces spatially and temporally nearby
predictions to be identical, while DeepLATTE considers the autocorrelation of
the air quality values in the representation space so that two locations or
times having similar spatiotemporal embeddings can still have similar air
quality values even when they are far away in space and time. Lin et al. [5]
generate predictions in terms of environmental characteristics (geo-context),
but the approach does not consider temporal dependencies. In general,
DeepLATTE achieved the best performance compared to other baseline methods.

We compare DeepLATTE to its variants by removing the feature selection module
and autocorrelation-guided constraint, respectively. We observe that the RMSE
increases 1.8-5.1% without the feature selection module (i.e., $\alpha$ is 0).
Also, the number of selected features decreases when $\alpha$ increases, but a
large $\alpha$ ($>1$) leads to worse performance. The feature selection module
aims to explicitly remove irrelevant features at the beginning of the network
to largely reduce the noise from these features during training. We manually
remove the irrelevant features to train the models without the feature
selection layer, which show little difference in the performance from
DeepLATTE. In comparison, if we only keep these irrelevant features for
prediction, the model performs much worse than DeepLATTE. The selected
features also offer insights to the PM2.5 contributors for the prediction
task. Moreover, we observe that the RMSE increases 4.1-8.3% without
autocorrelation-guided constraint (i.e., setting $\eta$ to be 0), indicating
that the constraint truly guides the network to generate more reliable
predictions that have similar autocorrelation to the labels in the
representation space. The model achieves worse performance when $\eta$ is
{0.01, 0.5} than $\eta$ is 0.1. We further examine the similarity of the
autocorrelation patterns between the labels and predictions with the
semivariograms, which results in a high correlation of 0.73 for the range
values and 0.85 for the sill values.

(a) RMSE

(b) R2

Figure 6: Performance of DeepLATTE and baseline methods in RMSE and R2

Variations in Predictions. Evaluating the model performance with limited
observed locations is not enough to demonstrate the prediction ability. Since
PM2.5 concentrations are highly correlated with environmental characteristics
[5], we examine the variations in predictions with varying distances to the
geographic features (e.g., the blue areas in Figure 7). Table I shows the
average predictions from DeepLATTE and OK (in Jan. 2019) with various
distances to motorways, light rail, and parks. We observe that the predictions
from DeepLATTE show high average values when the locations contain motorways
and light rail and decrease when the locations are further from them. In
contrast, the locations with parks have lower PM2.5 predictions than others.
Also, the predictions around parks are much better than motorways and light
rail, which is consistent with the existing finding [14]. Table I shows that
the predictions from OK have no apparent pattern regarding the environmental
characteristics. Since OK generates smoothing surfaces of PM2.5
concentrations, the average prediction values from OK are slightly larger than
those from DeepLATTE.

Figure 7: An example of the locations containing motorways (left), one-step neighbors away from motorways (middle), and two-step away from motorways (right) in blue. TABLE I: Variations of predictions towards geographic features | Motorways | Light Rail | Parks
---|---|---|---
LATTE | zero-step | 17.03 | 17.39 | 14.87
one-step | 16.65 | 16.55 | 15.79
two-step | 16.44 | 16.92 | 15.54
OK | zero-step | 17.35 | 17.57 | 16.73
one-step | 17.30 | 17.41 | 16.27
two-step | 17.25 | 18.01 | 16.38

Visualizations. Figures 8(a) and 8(c) spatially visualize the monthly average
of the predictions from DeepLATTE while Figures 8(b) and 8(d) are the
predictions from OK. We can see that the average predictions from OK are
smooth over the region, which only offers a general idea about the variation
of the PM2.5 concentrations at a coarse spatial scale. In comparison,
DeepLATTE predictions provide additional spatial details, consistent with some
of the selected features. For example, higher PM2.5 concentrations along
freeways (I-110, I-10, I-5, and I-405) are observed due to traffic-related
sources prevalent in Los Angeles [15]. PM2.5 values present an apparent
pattern along railways (light rail). Also, the predictions show quite
different patterns in February from October, i.e., Oct. has much worse air
quality than Feb. in 2019, probably due to the different weather conditions in
these two months. However, unlike the predictions from OK generating high
values everywhere, DeepLATTE can predict low PM2.5 concentrations in the north
of the region because of the green land. In general, our model successfully
generated predictions that present how the PM2.5 concentrations vary with
environmental characteristics over the space.

Figure 9 shows the hourly average of PM2.5 predictions in July and November
over three small regions, i.e., Downtown LA (in Figure 5(b) sub-area “B”),
Santa Monica (in sub-area “A”), and Long Beach (in sub-area “D”). We observe
that PM2.5 concentrations generally reach the peak at 8 and 9 a.m. and minimum
at 4 and 5 p.m.. This is because, in the morning, people start their
activities (e.g., commuting) that produce PM2.5, and in the afternoon, due to
sea-breezes and mountain-induced flows, the wind takes the particle matter
inland that leads to good air quality. Then the wind goes back from inland
with accumulated particle matter that worsen air quality during the night
[16]. Besides, the air quality during the winter (Nov.) is much worse than the
summer (Jul.), which is because winter has large temperature differences in a
day that creates a thick layer in the air preventing PM2.5 to escape [17].
Also, Downtown LA and Long Beach have higher PM2.5 concentrations than Santa
Monica during the morning because of heavy traffic and industrial emission
while Long Beach and Santa Monica are near the coastal areas, so PM2.5
decreases with the wind patterns in the afternoon.

(a) Feb. 2019, DeepLATTE

(b) Feb. 2019, Ordinary Kriging

(c) Oct. 2019, DeepLATTE

(d) Oct. 2019, Ordinary Kriging

Figure 8: Spatial visualizations of the predictions from DeepLATTE and OK
Figure 9: Hourly pattern of PM2.5 in Nov. and Jul. 2019 from DeepLATTE

Selected Features. Table II shows the most frequent 10 selected dynamic
features and 10 static features from all the DeepLATTE models ordered by the
number of appearing times. The strong relationship between PM2.5
concentrations and “visibility” is expected and has been seen previously [18].
Megaritis et al. [19] demonstrated PM2.5 concentrations are highly related to
“temperature”, “humidity”, “pressure”, and “wind speed”, which were
automatically selected by DeepLATTE. Time information including
weekday/weekend and day/nighttime is important since it indicates temporal
variation and population mobility.

In addition, most of the selected static features are consistent with existing
studies on PM2.5 in Los Angeles. Longitude and latitude are also important
indicators since PM2.5 concentrations are correlated with locations [20].
DeepLATTE identified light rail is important for predicting PM2.5, which has
been demonstrated in Kam et al. [21] that the light-rail lines are strongly
associated with ambient PM levels in Los Angeles (R2=0.61) by personally
monitoring the air quality at the stations. Also, Moore et al. [22] showed
that arterial roads and open green areas are statistically significantly
associated with PM2.5 in Los Angeles (R-value is approximately 0.4 to 0.6
respectively) using LUR approach. However, these existing studies require
long-term and costly investigations by the environmental scientists with
domain-specific expertise. In contrast, DeepLATTE automatically learns from a
variety of raw features, selecting those that are most pertinent to air
quality prediction. For other selected static features, such as “traffic fuel”
(referring to gas stations) and “traffic stop” (referring to the intersections
with stop signs), we do not find the existing work on analyzing the
relationship between them and PM2.5 concentrations to demonstrate our results,
but the “traffic fuel” and “traffic stop” features intuitively relate to
vehicle air pollution emissions.

Furthermore, the selected features could vary each month. For example, during
the summer time, DeepLATTE tends to select fewer features (around 20) for
predicting PM2.5 concentrations than the winter time (around 30). Visibility
and humidity stand out when predicting for the summer period while time
indicators are more likely to be selected for the winter period. This might be
because the temporal variation of PM2.5 in winter is higher than summer, the
models require time information to generate reliable estimations.

TABLE II: Selected dynamic features Dynamic Features |  visibility, humidity, day of week, wind speed, hour of day, temperature, pressure, day of year, cloud cover, wind direction
---|---
Static Features |  latitude, longitude, light rail, traffic fuel, motorway, traffic stop, primary road, secondary road, waterways river, park

##  IV Related Work

###  IV-A Spatiotemporal Modeling

Deep learning approaches have shown promising results in modeling
spatiotemporal relationships in data. Graph-based recurrent neural networks
are often used to deal with spatiotemporal data in a graph representation and
are flexible in modeling irregular graphs like road networks. Li et al. [23]
propose a diffusion convolutional recurrent neural network (DCRNN) to forecast
traffic volumes by combining the diffusion convolution operation with
recurrent neural networks (RNN) on a spatial distance-based graph. Lin et al.
[4] extends DCRNN by constructing a geo-context-based graph to model the
spatial relationship in the learned feature space for forecasting air quality
values. Hsieh et al. [10] build an affinity graph by connecting nearby sensor
measurements in space and time and apply a semi-supervised inference model to
predict air quality values at unobserved locations. However, the network
nature of graph-based approaches are not suitable for handling continuous
surfaces (as either input or output) and are not scalable for fine-spatial-
scale predictions that could have thousands of prediction locations.

In contrast, image processing methods, like the convolution operation, is more
suitable for spatiotemporal data represented in continuous surfaces (i.e.,
grids). ConvLSTM [12] embeds the convolution operation directly in the LSTM
network to capture spatiotemporal information simultaneously for weather
forecasting. Other work leverages the ConvLSTM network both for forecasting
(e.g., [24, 25] and for fine-scale prediction (e.g., [26]). In addition,
attention-based approaches also capture the spatial relationships by learning
the attention score for the observed locations to predict for the target
locations [27, 28]. However, these approaches usually require large amounts of
evenly distributed labeled data to achieve good performance, especially for
the prediction tasks.

###  IV-B Air Quality Prediction

Traditional spatial interpolation methods, like IDW and Ordinary Kriging, do
not explicitly include environmental characteristics and fail to generate
fine-scale predictions [5]. Classical dispersion models, such as Gaussian
Plume models [6], require expert knowledge and resources that are not
available everywhere, such as traffic data. Land-use regression models also
rely on expert-selected predictors, which cannot be easily generalized to
other geographic areas. In contrast, our data-driven approach can
automatically mine air quality-related predictors in contextual data from
multiple publicly available datasets. In addition, machine learning approaches
are widely used for air quality prediction. Lin et al. [5] propose a method to
identify significant air quality-related factors and apply them to generate
predictions. However, the proposed approach does not take temporal effects
into account. Zheng et al. [9] propose a co-training framework with separate
classifiers for spatial and temporal features, which does not jointly model
the spatiotemporal effects on air quality. Qi et al. [11] propose a semi-
supervised learning method with spatiotemporal smoothing. However, unlike
DeepLATTE, this proposed method ignores the potential spatial autocorrelation
patterns in air quality data and does not include predictors that describe the
environment.

##  V Conclusion and Future Work

This paper presented a novel approach, DeepLATTE, for fine-spatial-scale
prediction of location-dependent time series data and an application of air
quality prediction. The main contribution of the presented architecture is a
novel method that learns a spatiotemporal representation from contextual data
and labeled data and then uses the autocorrelation pattern in the
representation space to guide the predictions in a semi-supervised way. This
method allows for the integration of well-established spatial statistics tools
with neural networks and enables accurate prediction of location-dependent
time series data. Overall, the presented network architecture is flexible and
could apply to many scientific prediction problems dealing with spatiotemporal
phenomena with sparse ground truth measurements. We plan to test this approach
to other spatiotemporal phenomena, such as noise prediction.

##  VI Acknowledgement

This work is supported by the NIH grant 1U24EB021996- 01 and Nvidia
Corporation.

## References

  * Jiang [2018] Z. Jiang, “A survey on spatial prediction methods,” _IEEE TKDE_ , vol. 31, no. 9, pp. 1645–1664, 2018.
  * Karroum et al. [2020] K. Karroum, Y. Lin, Y.-Y. Chiang, Y. B. Maissa, M. El Haziti, A. Sokolov, and H. Delbarre, “A review of air quality modeling,” _MAPAN_ , pp. 1–14, 2020\.
  * Feng et al. [2016] S. Feng, D. Gao, F. Liao, F. Zhou, and X. Wang, “The health effects of ambient PM2.5 and potential mechanisms,” _Ecotoxicology and Environmental Safety_ , vol. 128, pp. 67–74, 2016.
  * Lin et al. [2018] Y. Lin, N. Mago, Y. Gao, Y. Li, Y.-Y. Chiang, C. Shahabi, and J. L. Ambite, “Exploiting spatiotemporal patterns for accurate air quality forecasting using deep learning,” in _ACM SIGSPATIAL_ , 2018, pp. 359–368.
  * Lin et al. [2017] Y. Lin, Y.-Y. Chiang, F. Pan, D. Stripelis, J. L. Ambite, S. P. Eckel, and R. Habre, “Mining public datasets for modeling intra-city PM2.5 concentrations at a fine spatial resolution,” in _ACM SIGSPATIAL_ , 2017, pp. 1–10.
  * Leelőssy et al. [2014] Á. Leelőssy, F. Molnár, F. Izsák, Á. Havasi, I. Lagzi, and R. Mészáros, “Dispersion modeling of air pollutants in the atmosphere: A review,” _Open Geosciences_ , vol. 6, no. 3, pp. 257–278, 2014\.
  * Li and Heap [2011] J. Li and A. D. Heap, “A review of comparative studies of spatial interpolation methods in environmental sciences: Performance and impact factors,” _Ecological Informatics_ , vol. 6, no. 3-4, pp. 228–241, 2011\.
  * Hoek et al. [2008] G. Hoek, R. Beelen, K. De Hoogh, D. Vienneau, J. Gulliver, P. Fischer, and D. Briggs, “A review of land-use regression models to assess spatial variation of outdoor air pollution,” _Atmospheric Environment_ , vol. 42, no. 33, pp. 7561–7578, 2008.
  * Zheng et al. [2013] Y. Zheng, F. Liu, and H.-P. Hsieh, “U-air: When urban air quality inference meets big data,” in _ACM SIGKDD_ , 2013, pp. 1436–1444.
  * Hsieh et al. [2015] H.-P. Hsieh, S.-D. Lin, and Y. Zheng, “Inferring air quality for station location recommendation based on urban big data,” in _ACM SIGKDD_ , 2015, pp. 437–446.
  * Qi et al. [2018] Z. Qi, T. Wang, G. Song, W. Hu, X. Li, and Z. Zhang, “Deep air learning: Interpolation, prediction, and feature analysis of fine-grained air quality,” _IEEE TKDE_ , vol. 30, no. 12, pp. 2285–2297, 2018.
  * Xingjian et al. [2015] S. Xingjian, Z. Chen, H. Wang, D.-Y. Yeung, W.-K. Wong, and W.-C. Woo, “Convolutional LSTM network: A machine learning approach for precipitation nowcasting,” in _NIPS_ , 2015, pp. 802–810.
  * Yu et al. [2016] R. Yu, Y. Yang, L. Yang, G. Han, and O. Move, “Raq–a Random Forest approach for predicting air quality in urban sensing systems,” _Sensors_ , vol. 16, no. 1, p. 86, 2016.
  * Nowak and Heisler [2010] D. J. Nowak and G. M. Heisler, _Air Quality Effects of Urban Trees and Parks: Executive Summary_. National Recreation and Park Association, 2010.
  * Westerdahl et al. [2005] D. Westerdahl, S. Fruin, T. Sax, P. M. Fine, and C. Sioutas, “Mobile platform measurements of ultrafine particles and associated pollutant concentrations on freeways and residential streets in Los Angeles,” _Atmospheric Environment_ , vol. 39, no. 20, pp. 3597–3610, 2005.
  * Lu and Turco [1995] R. Lu and R. P. Turco, “Air pollutant transport in a coastal environment—II. Three-dimensional simulations over Los Angeles basin,” _Atmospheric Environment_ , vol. 29, no. 13, pp. 1499–1518, 1995\.
  * Wallace and Kanaroglou [2009] J. Wallace and P. Kanaroglou, “The effect of temperature inversions on ground-level nitrogen dioxide (NO2) and fine particulate matter (PM2.5) using temperature profiles from the atmospheric infrared sounder (AIRS),” _Science of The Total Environment_ , vol. 407, no. 18, pp. 5085–5095, 2009\.
  * Pui et al. [2014] D. Y. Pui, S.-C. Chen, and Z. Zuo, “PM2.5 in China: Measurements, sources, visibility and health effects, and mitigation,” _Particuology_ , vol. 13, pp. 1–26, 2014.
  * Megaritis et al. [2014] A. Megaritis, C. Fountoukis, P. Charalampidis, H. Denier Van Der Gon, C. Pilinis, and S. Pandis, “Linking climate and air quality over Europe: Effects of meteorology on PM2.5 concentrations,” _Atmospheric Chemistry and Physics_ , vol. 14, no. 18, pp. 10 283–10 298, 2014.
  * Zamani Joharestani et al. [2019] M. Zamani Joharestani, C. Cao, X. Ni, B. Bashir, and S. Talebiesfandarani, “PM2.5 prediction based on Random Forest, XGBoost, and deep learning using multisource remote sensing data,” _Atmosphere_ , vol. 10, no. 7, p. 373, 2019.
  * Kam et al. [2011] W. Kam, K. Cheung, N. Daher, and C. Sioutas, “Particulate matter (PM) concentrations in underground and ground-level rail systems of the Los Angeles Metro,” _Atmospheric Environment_ , vol. 45, no. 8, pp. 1506–1516, 2011.
  * Moore et al. [2007] D. Moore, M. Jerrett, W. Mack, and N. Künzli, “A land use regression model for predicting ambient fine particulate matter across Los Angeles, CA,” _Journal of Environmental Monitoring_ , vol. 9, no. 3, pp. 246–252, 2007.
  * Li et al. [2018] Y. Li, R. Yu, C. Shahabi, and Y. Liu, “Diffusion convolutional recurrent neural network: Data-driven traffic forecasting,” in _ICLR_ , 2018.
  * Guo et al. [2019] C. Guo, W. Guo, C.-H. Chen, X. Wang, and G. Liu, “The air quality prediction based on a convolutional LSTM network,” in _International Conference on Web Information Systems and Applications_. Springer, 2019, pp. 98–109.
  * Wang et al. [2018] B. Wang, Z. Yan, J. Lu, G. Zhang, and T. Li, “Deep multi-task learning for air quality prediction,” in _NIPS_. Springer, 2018, pp. 93–103.
  * Le et al. [2020] V.-D. Le, T.-C. Bui, and S.-K. Cha, “Spatiotemporal deep learning model for citywide air pollution interpolation and prediction,” in _IEEE BigComp_ , 2020, pp. 55–62.
  * Pan et al. [2020] Z. Pan, W. Zhang, Y. Liang, W. Zhang, Y. Yu, J. Zhang, and Y. Zheng, “Spatio-temporal meta learning for urban traffic prediction,” _IEEE TKDE_ , 2020.
  * Cheng et al. [2018] W. Cheng, Y. Shen, Y. Zhu, and L. Huang, “A neural attention model for urban air quality inference: Learning the weights of monitoring stations,” in _AAAI_ , 2018.