IMPUTING PRICING CRITERIA FOR PARKING USING A COMBINATION OF PAYMENT DATA AND INCOMPLETE SENSOR DATA
A method for estimating average occupancy of a set of stalls based on incomplete occupancy data includes acquiring payment data for the set of stalls for each of a set of times and acquiring occupancy data for the set of stalls for each of the set of times, the occupancy data being acquired from a reporting subset consisting of fewer than all of the stalls. An estimate of a time integral of a function of occupancy is computed, based on the payment data and occupancy data and information is output based on the estimate.
1 . A method for inferring occupancy information for a set of stalls based on incomplete occupancy data, the method comprising:
for each of a series of times:
acquiring payment data for a set of stalls; and
acquiring occupancy data for the set of stalls, the set of stalls including a reporting subset of the stalls which report occupancy and a non-reporting subset of stalls which do not report occupancy, the occupancy data being acquired from the reporting subset;
with a processor, computing an estimate of a time integral of a function of occupancy over the series of times based on the acquired payment data and occupancy data; and
outputting information based on the estimate.
2 . The method of claim 1 , wherein the stalls comprise parking spaces.
3 . The method of claim 1 , wherein the occupancy data includes a count of the stalls in the reporting subset that are occupied at each of the times in the set of times.
4 . The method of claim 1 , wherein the estimating of the time integral includes, for each of the times, computing a set of observations Y(t) based on the acquired payment data and occupancy data, the observations being selected from:
a first count Z m p (t), of the non-reporting stalls that are paid for, at time t;
a second count Z s op (t), of the reporting stalls that are occupied and paid for, at time t;
a third count Z s o p (t), of the reporting stalls that are occupied and not paid for, at time t;
a fourth count Z s ōp (t), of the reporting stalls that are unoccupied and paid for, at time t.
5 . The method of claim 4 , further comprising, extracting a fraction of time F(y) that the set of stalls spends at each state y in a set of states from the observations, each of the states in the set of states representing different values for each of the first, second, third, and fourth counts.
6 . The method of claim 5 , wherein the estimate of the time integral of the function of occupancy is computed as a function of:
∑
Z
o
=
0
n
∑
Y
∈
c
(
Z
o
)
P
(
Z
o
Y
,
θ
^
)
F
(
Y
)
(
1
)
for each time t in the set of times,
where Z o (t) represents a count of the stalls which are occupied at time t,
c(Z o (t)), is the real-valued function of occupancy;
is a set of possible values of the set of observations Y(t),
n is the total number of stalls in the set of stalls, and
P(Z o |Y,{circumflex over (θ)}) is a probability distribution function which estimates Z o (t) given Y(t) and an estimated set of parameters B which predict observations for non-reporting stalls based on predefined distributions.
7 . The method of claim 6 , wherein the probability distribution function
P
(
Z
o
Y
,
θ
)
=
P
(
Z
o
,
Y
θ
)
∑
Z
o
=
0
n
P
(
Z
o
,
Y
θ
)
.
8 . The method of claim 6 , wherein the predefined distributions are selected from beta-binomial, hypergeometric, and binomial distributions.
9 . The method of claim 7 , wherein the joint distribution P(Z o ,Y|θ) is expressed as a joint distribution P(Z o ,Z s o ,Z m p |θ)=P(Z o |θ)P(Z s o |Z o ,θ)P(Z m p |Z o ,Z s o ,θ),
where Z m p is a count of non-reporting stalls for which payment has been made.
10 . The method of claim 7 , wherein the distributions include:
Z o |θ is treated as a beta-binomial (n, θ α , θ β ),
Z s o |Z o treated as a hypergeometric,
Z m op |Z m o ,θ is treated as a binomial (Z m o ,θ p|o ), and
Z m ōp |Z m ō ,θ is treated as a binomial (Z m ō ,θ p|ō ).
11 . The method of claim 10 , wherein for the beta-binomial, parameter estimates are:
θ MOM α :=γp, θ MOM β :=γ−α,
where
p
:=
∑
z
=
0
n
s
F
Z
s
o
(
z
)
z
n
s
,
γ
:=
n
-
p
ρ
-
1
,
and where F Z s o (z) is the fraction of time T for which the occupancy of the reporting stalls is z and p is an over-dispersion parameter and wherein for the binomials the parameter estimates are:
θ
MOM
p
o
=
∑
z
=
0
n
s
zF
Z
o
op
(
z
)
∑
z
=
0
n
os
zF
Z
o
o
(
z
)
and
θ
MOM
p
o
_
=
∑
z
=
0
n
s
zF
Z
s
o
_
p
(
z
)
∑
z
=
0
n
s
zF
Z
s
o
_
(
z
)
.
12 . The method of claim 6 , wherein the real-valued function of occupancy c(Z o (t)) is a fraction-high-minus-fraction-low function which is defined as:
=
{
-
1
,
Z
=
0
(
n
-
1
)
∫
Z
-
1
n
-
1
Z
n
-
1
(
1
x
>
h
-
1
x
<
l
)
x
,
0
<
Z
<
n
1
,
Z
=
n
where n is the capacity of the set of stalls,
1. is the indicator function,
Z is a value of the occupancy,
and parameters l and h are predefined low and high fractional occupancies.
13 . The method of claim 1 , wherein the outputting information comprises outputting the estimate.
14 . The method of claim 1 , wherein the method further comprises generating a pricing policy for the stalls based on the estimate and the outputting information comprises outputting the pricing policy.
15 . The method of claim 1 , wherein a ratio of non-reporting stalls to the total number of stalls in the set of stalls is at least 1:5.
16 . The method of claim 1 , further comprising computing the estimate of a time integral of a function of occupancy for each of a plurality of sets of stalls.
17 . The method of claim 1 , wherein the payment data comprises, for each stall in the set of stalls, whether a payment covering time t has been made.
18 . A computer program product comprising a non-transitory computer-readable storage medium storing instructions, which when executed by a computer, perform the method of claim 1 .
19 . A system comprising memory which stores instructions for performing the method of claim 1 and a processor in communication with the memory for executing the instructions.
20 . A system for estimating average occupancy of a set of stalls based on incomplete occupancy data, the system comprising:
memory which stores:
for each of a set of times:
payment data received for a set of stalls; and
occupancy data received for the set of stalls, the set of stalls including a reporting subset of the stalls which report occupancy and a non-reporting subset of stalls which do not report occupancy, the occupancy data being acquired from the reporting subset; and
instructions for computing an estimate of a time integral of a function of occupancy based on the payment data and occupancy data for each of the set of times; and
a processor in communication with the memory for executing the instructions.
21 . A method for inferring occupancy information for a set of parking spaces based on incomplete occupancy data, the method comprising:
for each of a set of times in a series of times:
acquiring payment data for each parking space in a set of parking spaces; and
acquiring occupancy data from only a subset of parking spaces which include occupancy sensors, the set of parking spaces including a subset of parking spaces which do not include occupancy sensors;
with a processor, computing an estimate of occupancy of the set of parking spaces over the series of times based on the payment data and occupancy data for each of the set of times; and
outputting information based on the estimate.