TELKOM
NIKA
, Vol. 11, No. 12, Decem
ber 20
13, pp.
7379
~73
8
5
e-ISSN: 2087
-278X
7379
Re
cei
v
ed Ma
y 3, 2013; Re
vised July 4,
2013; Accept
ed Augu
st 23
, 2013
Wavelet
Neural Network-based
Short-Term Passenger
Flow Forecastin
g on Urban Rail Transit
Xiaojie Zhang*, Baohu
a Mao, Yongliang Wa
ng, J
i
a Feng, Minggao Li
MOE Ke
y
La
bo
rator
y
for Urb
a
n
T
r
ansportatio
n
Comp
le
x S
y
s
t
ems
T
heor
y
a
nd T
e
chnol
og
y, Beijin
g
Jiaoto
ng U
n
ive
r
sit
y
(BJT
U),
The 8
th
Buil
din
g
, No. 3 Shan
g Yuan C
un, Ha
i Dian D
i
strict Beiji
ng, Ch
ina,
100
04
4 Ph. /Fax: +
8
6
10-5
1
6
8
226
4
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: zhang
xia
o
ji
e
@
bjtu.e
du.cn
A
b
st
r
a
ct
Accurate forec
a
sting of short-
term pass
e
n
g
e
r
fl
ow
has been one of the most
imp
o
rtant i
ssues i
n
urba
n rail tra
n
s
i
t plan
nin
g
a
nd
oper
ation. C
o
n
s
ideri
ng
th
e sh
ortcomin
gs of traditi
ona
l forec
a
sting
meth
ods
,
and
in or
der t
o
i
m
prov
e fore
casting
accura
cy of passe
ng
er flow
, this p
aper
prese
n
ts a w
a
velet n
eur
a
l
netw
o
rk (W
NN) for short-term p
a
sse
nger fl
ow
forecastin
g
.
One real ur
b
an rai
l
transit
station w
i
th lar
g
e
and s
i
gn
ifica
n
tl
y chan
ge
d pas
seng
er flow
is
chose
n
to b
e
the ex
a
m
pl
e. T
he pr
op
osed
meth
od
an
d B
P
neur
al
netw
o
r
k
hav
e b
e
e
n
compar
ed w
i
th
the r
e
sult
s,
w
h
ich sh
ow
that the W
N
N
mo
de
l h
a
s
mo
re
adva
n
tag
e
s. T
he W
N
N
mo
d
e
l fe
atures
hi
g
her
lear
nin
g
s
pee
d
and
dr
a
s
tically
less
c
onver
genc
e ti
me,
show
ing th
at it is me
an
ingf
ul i
n
pr
actica
l ap
pl
icatio
n. F
u
rthermor
e
, the ca
lc
ulate
d
rel
a
tive
errors usi
ng B
P
neur
al n
e
tw
ork are n
ear
ly in
the ra
ng
e
[-0.4,
0.3] an
d the c
a
lculat
ed re
lati
v
e
errors
are
in t
he ra
ng
e [-0.25
,
0.1] using th
e W
NN, w
h
ich
demonstrate th
e
superi
o
r accur
a
cy of using th
i
s
appro
a
ch.
Ke
y
w
ords
:
ur
ban ra
il transit,
short-term p
a
s
s
eng
er flow
forecastin
g, w
a
velet neur
al n
e
tw
ork
Copy
right
©
2013 Un
ive
r
sita
s Ah
mad
Dah
l
an
. All rig
h
t
s r
ese
rved
.
1. Introduc
tion
Urb
an
rail transit
sho
r
t-t
e
rm p
a
sse
n
ger fl
o
w
foreca
sting i
s
one of the
essential
element
s in t
r
an
spo
r
tation
system
s whi
c
h
can b
e
u
s
ed to fine-tu
n
e
travel beh
a
v
iors, redu
c
e
passe
nge
r co
nge
stion, an
d
enha
nce service qu
ality
of tran
spo
r
tatio
n
syste
m
s. T
he forecastin
g
results
of short-te
rm p
a
s
seng
er flo
w
can
be
ap
plied to
sup
port tra
n
spo
r
tation syste
m
manag
eme
n
t su
ch a
s
ope
ration plan
nin
g
, and st
ation
passen
ger
crowd regul
atio
n planni
ng [1].
In rece
nt years, a nu
mb
er of techni
q
ues a
r
e u
s
e
d
in sho
r
t-te
rm pa
sseng
e
r
flow
forecastin
g. Kalman filterin
g model is si
mple in ca
l
c
ul
ation and fast
in speed; ho
wever, it fails to
reflect
un
cert
ainty and
no
nlinea
rity in traffic flo
w
p
r
oce
s
s an
d i
s
unabl
e to h
andle th
e ra
pid
variation an
d
compli
cated
process cha
nge
s unde
r
l
y
ing of traffic flow [2-3]. Suppo
rt vector
reg
r
e
ssi
on
(SVR)
ha
s be
e
n
succe
s
sfull
y
use
d
to
predic
t traffic pa
rameters
s
u
c
h
as hourly flow,
and travel time. However, when the n
e
w traffi
c dat
a beco
m
e a
v
ailable in e
v
ery coupl
e of
minutes o
r
seco
nd
s, the t
r
adition
al SV
R m
e
thod
is not
a practi
cal
optio
n b
e
cause it
req
u
ires
compl
e
te mo
del traini
ng wheneve
r
a ne
w data p
o
int i
s
ad
ded [4]. Geneti
c
algo
rithm is abl
e to
reserve
a fe
w be
st fitted membe
r
s
of the wh
ole p
opulatio
n for the next ge
neratio
n in t
he
operation
proce
s
s, ho
we
ver, after
some g
ene
rat
i
ons
gen
etic algo
rithm
may lead to
a
prem
ature
co
nverge
nce to
a local optim
um in t
he
sea
r
chi
ng the
sui
t
able pa
ramet
e
rs
of a mod
e
l
[5-6]. Simulated an
nealin
g
(SA) is
a st
och
a
sti
c
ba
sed ge
neral search tool th
at mimics th
e
anne
aling p
r
oce
s
s of ma
terial phy
sics: howeve
r
, it cost
s mo
re
comp
utation
time [7]. Th
e
cla
ssi
cal
rep
r
esentative i
s
a
r
tificial n
eural
network (A
NN) m
odel d
ue to
its su
pe
rio
r
perfo
rman
ce
to app
roximat
e
any d
egree
of co
mplexity and
without
prio
r
kno
w
le
dge of
pro
b
le
m
solving [8].
ANN m
odel
is ba
se
d on
a mod
e
l
of
emulating
the p
r
ocesse
s of the
hu
man
neurologi
cal
system to det
ermin
e
the n
u
mbe
r
s of
ve
hicle a
nd tem
poral
cha
r
a
c
t
e
risti
cs from the
histori
c
al traffic flow patterns, espe
cially
for nonline
a
r
and dynami
c
evolution
s
[9].
Wavelet
anal
ysis i
s
b
e
tter meth
od
which
is
appli
ed to the
n
on-stationa
ry
sig
nal
analysi
s
[10-11]. So this pape
r co
mbi
nes t
he
wave
let transfo
rm and BP neural netwo
rk, a
n
d
pre
s
ent
s the
wavelet
neu
ral net
work. It
descri
b
e
s
the
wavel
e
t ne
ural net
work th
at is
develo
p
ed
to pre
d
ict th
e urban
rail
transit
sh
ort-t
e
rm p
a
ss
e
n
g
e
r flo
w
. The
feasibility of
this meth
od i
s
demon
strated
throug
h the
basi
s
of time
se
ries of
pa
sseng
er flo
w
. The results
reporte
d in thi
s
Evaluation Warning : The document was created with Spire.PDF for Python.
e-ISSN: 2
087-278X
TELKOM
NIKA
Vol. 11, No
. 12, Dece
mb
er 201
3: 737
9 – 7385
7380
pape
r cl
early
sho
w
the a
d
v
antage
s of impleme
n
ting
this ap
pro
a
ch
for urb
an
rai
l
transit
sho
r
t-
term pa
sseng
er flow p
r
edi
ction.
This p
ape
r i
s
orga
nized a
s
follo
ws: Se
ction 2
pre
s
e
n
ts the
wave
let neural ne
twork
model
s; Se
ction 3 int
r
od
uce
s
th
e da
ta pro
c
e
s
sin
g
; Section
4
illustrates the forecastin
g
perfo
rman
ce
and ma
ke
s some discu
s
si
ons; Se
ction 5 gives the concl
u
si
on
s.
2. Wav
e
let Neural Ne
t
w
o
r
k
2.1. Wav
e
let Trans
f
orm
Wavelet is a t
y
pe of transfo
rmation that reta
ins b
o
th time and fre
q
u
ency informat
ion of
the s
i
gnal [12]. In wavelet
trans
form, all bas
ic
func
tion
)
(
,
x
b
a
can b
e
deri
v
ed from a m
o
ther
wav
e
let
)
(
x
through the follo
wing dil
a
tion and tran
slatio
n pro
c
e
s
ses:
,
1
()
ab
tb
a
a
a
,
bR
and
0
a
(1)
Whe
r
e
a
an
d
b
are the dilat
i
on and tra
n
sl
ation paramet
ers, respe
c
tively.
Given a
time-varying
sign
a
l
)
(
t
f
, then, the Continuous Wavelet Trans
f
orm is
defined
as
follows
:
dt
t
t
f
b
a
t
f
CWT
b
a
)
(
)
(
)
,
:
)
(
(
,
(2)
Whe
r
e “*
” de
notes the co
mplex conj
ug
ation. When
j
a
2
,
j
k
b
2
,
Z
k
j
,
(
Z
is
the set of integers), it can b
e
written a
s
:
)
(
,
k
t
j
j
k
j
2
2
2
(3)
The fast
algo
rithm of DWT
(Di
s
crete
Wavelet Tra
n
sf
orm)
ca
n be
written a
s
foll
owin
g
[13]:
1
1
22
22
(2
)
(2
)
jj
jj
dd
k
d
k
A
fh
k
n
A
f
D
fg
k
n
A
f
(4)
Whe
r
e
1
()
(
1
)
(
1
)
n
gn
n
h
n
,
nZ
. In
(4)
,
g
(
n
) and
h
(
n
) a
r
e the
high-pa
ss an
d
low-
pa
ss f
ilt
e
r
s,
re
spe
c
t
i
v
e
l
y
,
f
is the
discrete sig
nal. A sign
al or fu
nction
)
(
t
f
de
comp
o
s
ed by
wavelet tran
sform is exp
r
e
s
sed finitely as
follows
:
1
0
1
0
0
0
0
)
(
)
(
)
(
N
jk
N
jk
jk
j
k
k
k
j
d
C
t
w
t
f
t
f
(5)
Whe
r
e
0
()
f
t
rep
r
e
s
ents the lo
west freq
uen
cy compo
nent
,
j
w
rep
r
e
s
ent
s differe
nt
freque
ncy
co
mpone
nt a
n
d
N
re
pre
s
e
n
ts d
e
comp
osit
ion level;
k
j
d
is
the wavelet coefficient at
scale
j
.
2.2. Wav
e
let Neur
al Net
w
ork
The wavelet neural
net
wo
rk (WNN) co
nsi
s
ts
of
th
re
e layers: inp
u
t layer, hid
d
en laye
r
and
output l
a
yer. Unlike
a
traditio
nal
b
a
ck-p
ro
pag
ation n
eural n
e
twork that a
p
p
lies a
c
tivation
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
e-ISSN:
2087
-278X
Wa
velet Neural Network-b
a
se
d
Short-T
e
rm
Passe
ng
er Flo
w
Forecasting
…
(Xia
ojie Zhan
g)
7381
function
s to
both hid
den
and o
u
tput l
a
yers,
WNN employs m
o
ther
wavele
ts (o
r wavel
e
t
function
s) in t
he hidd
en lay
e
r only [14]. The c
onne
cti
ons b
e
twe
en
input units a
n
d
hidde
n unit
s
,
and b
e
twe
e
n
hidde
n unit
s
and o
u
tput
units a
r
e
call
ed weight
s
ti
w
and
t
W
, res
p
ec
tively. The
netwo
rk
stru
cture is
sho
w
n
in Figure 1.
)
(
1
t
x
)
(
2
t
x
)
(
t
x
m
)
(
2
t
y
ti
w
t
W
1
2
n
Figure 1. Wa
velet Neural Ne
twork Stru
ctural
Diag
ra
m
In this WNN, the training p
r
oce
dure is de
scribe
d as foll
ows:
Initializing the dilation parameter
t
a
, trans
lation parameter
t
b
and node
con
n
e
c
tion
weig
hts
ti
w
,
t
W
to some
ra
ndom
values. All th
ose
ran
dom v
a
lue
s
are limi
t
ed in the inte
rval (0,
1).
Input data
)
(
i
X
n
and the corre
s
pondi
ng outp
u
t values
T
n
V
, where
i
varie
s
from 1 t
o
S
, representin
g the nu
mbe
r
of the input
node
s,
n
repre
s
ent
s the nth
data sampl
e
of trainin
g
set, and
T
represe
n
ts the target output sta
t
e.
The output va
lue of the sa
mple
n
V
is calcul
ated with the
followin
g
formula:
T
t
t
S
i
t
n
ti
t
n
a
b
i
x
w
W
V
1
1
)
)
(
(
(6)
Whe
r
e
is co
nsid
ere
d
a mother wavelet
,
such
a
s
the
Morlet wave
let filter, and is
r
e
pr
es
e
n
t
ed
b
y
:
)
5
.
0
exp(
)
cos(
)
(
2
0
t
t
t
(7)
To
redu
ce
th
e e
rro
r,
t
W
,
ti
w
,
t
a
,
t
b
are
adj
uste
d
usin
g
t
W
,
ti
w
,
t
a
,
t
b
. In the
WNN, the gra
d
ient de
scen
d
algorith
m
is employed, throug
h the followin
g
equati
ons:
)
(
)
(
)
1
(
j
W
j
W
E
j
W
t
t
t
(8)
)
(
)
(
)
1
(
j
w
j
w
E
j
w
t
t
ti
(9)
)
(
)
(
)
1
(
j
a
j
a
E
j
a
t
t
t
(10
)
)
(
)
(
)
1
(
j
b
j
b
E
j
b
t
t
t
(11
)
Whe
r
e the e
r
ror functio
n
E
is tak
en as
:
Evaluation Warning : The document was created with Spire.PDF for Python.
e-ISSN: 2
087-278X
TELKOM
NIKA
Vol. 11, No
. 12, Dece
mb
er 201
3: 737
9 – 7385
7382
N
n
n
T
n
V
V
E
1
2
)
(
2
1
(12
)
And,
N
sta
ndin
g
for th
e data
numb
e
r
of training
set,
,
being the
lea
r
n
i
ng rate an
d
the momentum term, respec
tively.
The p
r
o
c
e
s
s
contin
ue
s u
n
til
E
satisfie
s th
e given
erro
r
crite
r
ia, a
nd t
he
whol
e trai
ning
of the WNN is co
mpleted [
15].
3. Data
Colle
ction and Pr
e-pro
ces
sing
The ente
r
ing
passen
ger fl
ow data
s
et o
f
one urba
n rail tran
sit st
ation is colle
cted to
investigate
the
viability of the prop
ose
d
WN
N
approa
ch for foreca
sting
the short-t
e
rm
passe
nge
r flo
w
. The
data
s
et wa
s
colle
cted du
ring
6:
00 AM to
9:0
5
AM, on five
wo
rki
ng
day
s,
2012, a
nd th
e sa
mpling
p
e
riod
wa
s
2
minutes.
One
day’s p
a
sse
nger flow
dat
a are sho
w
n
in
Figure 2. In orde
r to exami
ne wh
ethe
r WNN give
s
better res
u
lt
s
or not, the datas
e
t is
divided
into two part
s
to be used fo
r trainin
g
and
testing.
One
part on the first four days a
r
e used a
s
the
training
samp
le, for d
e
term
ining th
e
WNN p
a
ra
meters; the othe
r p
a
r
t on t
he fifth
day are u
s
e
d
as the testin
g
sample, for v
a
lidating
the perfo
rman
ce of
the
trained model.
Figure 2. One
day’s Passe
nger Fl
ow
In ord
e
r to
redu
ce th
e inf
l
uen
ce of th
e
pre
d
ictio
n
p
e
rform
a
n
c
e
d
ue to the
different
dimen
s
ion
s
o
f
sample data
,
the sample
data are n
o
rmalize
d
acco
rding to the fo
llowing fo
rmul
a:
min
max
min
x
x
x
x
x
(13
)
Whe
r
e
x
,
is the data before
norm
a
lization
,
x
is the data after norm
a
lization,
min
x
and
max
x
are the minim
u
m and maxi
mum value
s
of the raw dat
a.
After traini
ng
the
WNN a
nd o
b
tainin
g
the forecasti
ng o
u
tput, th
e forecasting
mod
e
l
sho
u
ld re
no
rmalize the o
u
t
put data as the followi
ng:
min
min
max
)
(
*
x
x
x
u
y
(14
)
Whe
r
e
y
is the output of the netwo
rk,
u
is the norm
a
lized
output.
4 Foreca
stin
g Performan
ce and Dis
c
u
ssion
4.1. Structu
r
e of WNN
The wavel
e
t neural network used for p
r
edictin
g sh
ort
-
term p
a
ssen
ger flow
con
s
ists of
three layers.
It is develo
ped usi
ng th
e 4 neuro
n
s
as input laye
r. The output
layer has o
n
e
neuron that
predi
cts
sh
ort-term pa
sse
nger flo
w
by the model. T
he num
ber o
f
neuro
n
in the
Training series
Passenger flow
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Short-T
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rm
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7383
hidde
n layer
is un
kno
w
n
a
nd nee
ds to
be optimi
z
ed.
In addition t
o
the numb
e
r of neurons i
n
the hid
den
la
yer, the
WNN p
a
ramete
rs
con
s
i
s
t of t
he le
arni
ng
rate, the m
o
m
entum
and
th
e
numbe
r of iteration
s
sho
u
ld also be
optimize
d
. In this pap
er, the num
ber
o
f
neuro
n
s in
the
hidde
n laye
r and
othe
r paramete
r
s, except
th
e num
be
r o
f
iteration
s
, are
optimi
z
e
d
simultan
eou
sl
y.
De
cidin
g
the
numbe
r of neuron
s in the hidd
en la
yers i
s
a very importa
nt part of
deci
d
ing you
r
overall neu
ral netwo
rk a
r
chite
c
ture. T
here a
r
e ma
ny rule-of-thu
m
b method
s
for
determi
ning t
he co
rrect nu
mber of ne
urons to u
s
e
in
the hidde
n
la
yers, such as the following:
l
n
m
(15
)
n
m
2
log
(16
)
nl
m
(17
)
Whe
r
e
m
,
is th
e
numb
e
r of n
euro
n
s in t
he
hidde
n laye
r,
n
is
the
nu
mber
o
f
n
e
ur
o
n
s
in
the input layer,
l
is the number of neu
ron
s
in the outp
u
t layer,
is the con
s
tant b
e
twee
n 1-
10.
In
th
is
pa
p
e
r
,
th
e
fo
r
m
u
l
a
(1
5
)
is
s
e
lec
t
ed
, a
fte
r
man
y
time
s
o
f
e
x
p
e
r
imen
ts
, th
e
h
i
d
den
layer ad
opts
6 neu
ron
s
with faster
sp
ee
d and
bette
r l
earni
ng effe
ct, so the
stru
cture of
WNN
i
s
4-6
-
1. The le
arnin
g
rate of
all those neu
ral net
work is determine
d as 0.02, the momentum te
rm
as 0.6. The training p
r
o
c
ed
ure is d
e
scrib
ed in se
ction
2.2.
4.2. Forecas
ting Results
After the urb
an rail tran
sit sho
r
t-term p
a
ssen
ger flo
w
fore
ca
sting
model i
s
est
ablished
,
the traini
ng
d
a
taset i
s
use
d
for traini
ng
the BP
ne
ural network
and
WNN
by MA
TLAB. (Th
e
B
P
neural n
e
two
r
k ha
s th
e
same p
a
ramet
e
rs with
the
WNN.) Rece
ntly, some
re
sea
r
che
r
s ha
ve
tried to
devel
op the
BP n
eural
net
wo
rk a
p
p
r
oa
ch
fo
r the
c
i
ty traffic
flow forec
a
s
t
ing [16]. In
orde
r to eva
l
uate the foreca
sting a
ccura
cy
and
stability, this study comp
ares BP neu
ra
l
netwo
rk with
WN
N.
The
fo
re
ca
sting results i
ndicate that
t
he p
r
op
osed
WNN m
odel i
s
fea
s
i
b
le an
d
effec
t
ivefor the s
h
ort-term
passe
nge
r flow fore
ca
sting
.
The re
sult
s are an
alyzie
d
as follow:
(1) T
he
WNN neu
ral n
e
t
work mod
e
l
can g
e
t higher l
earnin
g
spe
e
d a
n
d
less
conve
r
ge
nce
time that a
r
e
used
for sh
ort-te
rm
p
a
ssenge
r flo
w
predictio
n
. Fo
r
the BP n
eura
l
netwo
rk, a ve
ry satisfa
c
tory result is obt
ained
after a
bout 239 trai
ning epo
ch
s, but the WNN is
about
106
tra
i
ning
epo
ch
s. The t
r
aini
ng
time of BP
ne
ural
network i
s
3.9
25m
s, a
nd the
traini
n
g
time of WNN is 1.256m
s.
Figure 3. Foreca
sting
Curv
es of BP Neu
r
al
Netw
or
k
Figure 4. Foreca
sting
Curv
es of WNN
F
o
r
e
casting ser
i
es
F
o
r
e
casting ser
i
es
Passenger flow
Passenger flow
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Vol. 11, No
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mb
er 201
3: 737
9 – 7385
7384
(2)
The i
m
proved p
r
edi
ction results of
t
he WNN a
l
gorithm
co
rresp
ond to t
h
e re
al
passe
nge
r flo
w
better
.
Figure 3, Figure 4 illustrate t
hat the predi
c
tion value
curve of WNN fits
the real val
u
e cu
rve bett
e
r than
the
BP neural
n
e
twork d
o
e
s
. To better
a
ppre
c
iate th
e
perfo
rman
ce
of the WNN, the relative e
rro
r betw
een
the actual a
n
d
the forecasting flow se
ri
es
is co
nsi
dered
as app
rop
r
ia
te and used here.
F
r
om t
he Figu
re 5, we can see that the relati
ve
errors of BP neural net
wo
rk a
r
e n
e
a
r
ly in the
ran
g
e
[-0.4, 0.3], and that of WNN
are i
n
the
rang
e [-0.25,
0.1].
Figure 5. The
Relative Erro
r Cu
rve of BP Neural Net
w
ork a
nd
WNN
The BP algori
t
hm has seve
ral drawb
a
cks, for
exampl
e, the perform
ance of the netwo
rk
learni
ng i
s
strictly depe
nde
nt on the
sha
pe of the
erro
r surfa
c
e, val
ues of
the i
n
itial co
nne
ctio
n
weig
hts, and
the conve
r
ge
nce to the gl
obal opt
imum
is not guara
n
teed. The
WNN is a no
vel
approa
ch to
wards th
e le
arnin
g
fun
c
tion. It combi
nes the
wav
e
let theory a
nd feed
-forward
neural net
wo
rks, and
utilizes
wavelets
as the b
a
si
s function
to construct
a ne
twork.
Wavel
e
t
function i
s
a l
o
cal fu
nctio
n
and influe
nce
s
the n
e
two
r
ks’ outp
u
t only
in som
e
lo
ca
l rang
es. T
he
wavelet ne
ural netwo
rk sh
ows su
rp
risi
n
g
effect
iveness in
solving the co
nventio
nal problem
s
of
poor
conve
r
g
ence or even
diverge
n
ce e
n
co
unt
ered in
other kin
d
s o
f
neural net
works.
5. Conclusio
n
The tradition
al predi
ction
model
s
ha
ve so
m
e
weakne
sses; t
herefo
r
e
this pap
er
establi
s
h
ed a
wavelet n
e
u
r
al net
wo
rk-b
ase
d
short
-
te
rm pa
ssen
ge
r flow fo
re
ca
sting mo
del f
o
r
urba
n rail tra
n
sit station, combinin
g the theory of
wav
e
let transfo
rm
with the BP
neural network.
Thro
ugh a
nal
yzing in the p
aper, the follo
wing
con
c
lu
si
ons may be i
n
ferred:
(1) T
he WNN model features a hig
h
e
r
le
arnin
g
sp
eed,
redu
ce
d co
n
v
ergen
ce tim
e
, and
being a
p
p
r
op
riate to practi
cal a
pplicatio
n, so th
i
s
me
thod ha
s fair
pro
s
pe
cts
of appli
c
ation fo
r
the s
h
ort-term pass
e
nger flow forec
a
s
t
ing.
(2)
Fro
m
the
relative e
r
ror
curve
of BP
neur
al n
e
two
r
k an
d
WNN,
we
can
see t
hat the
testing pe
rformance of
the WNN is foun
d to be better than BP neural network in
accura
cy an
d
robu
stne
ss.
Therefore, th
e favorable
re
sults o
b
taine
d
in
this wo
rk reveal that th
e prop
osed m
odel is
a valid alte
rn
ative for the
short-te
rm
passen
ger fl
ow f
o
re
ca
sting. In
addition,
eve
n
the p
r
op
ose
d
WNN mod
e
l
is one of the hybrid fo
reca
sti
ng mo
dels; some
other adva
n
ced optimization
algorith
m
s
ca
n be ap
plied
for the WNN model to
imp
r
ove the a
c
cura
cy
of the neural net
work,
and this
woul
d be valuabl
e
future wo
rk.
Ackn
o
w
l
e
dg
ement
The resea
r
ch
described
in
this p
ape
r
wa
s sub
s
tant
ially sup
porte
d by a g
r
ant
from
Nation
al Basi
c Research
Prog
ram of
Chi
na (2012
CB7
2540
6) a
nd t
he National
Natural S
c
ien
c
e
Found
ation of
China (P
roje
ct No. 711
31
001; Proje
c
t No. 709
710
1
0
).
F
o
r
e
casting ser
i
es
Relative error
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-278X
Wa
velet Neural Network-b
a
se
d
Short-T
e
rm
Passe
ng
er Flo
w
Forecasting
…
(Xia
ojie Zhan
g)
7385
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