CN106875314B - Dynamic estimation method for passenger flow OD of urban rail transit network - Google Patents

Dynamic estimation method for passenger flow OD of urban rail transit network Download PDF

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CN106875314B
CN106875314B CN201710062955.3A CN201710062955A CN106875314B CN 106875314 B CN106875314 B CN 106875314B CN 201710062955 A CN201710062955 A CN 201710062955A CN 106875314 B CN106875314 B CN 106875314B
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张宁
王健
刘洋
何铁军
石庄彬
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Southeast University
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Abstract

The invention discloses a dynamic estimation method for passenger flow OD of an urban rail transit network, which comprises the following steps: firstly, based on historical passenger flow data, a one-way ticket passenger flow OD matrix with poor passenger flow space-time distribution stability is improved by a moving average method through setting time intervals to generate an improved passenger flow OD distribution matrix, on the basis, a passenger flow distribution rate matrix is calculated, a dynamic flow relation between the OD flow and the passenger flow entering and exiting stations is established by combining a travel time distribution rule of the OD flow, then an OD dynamic estimation state space model is established according to the dynamic flow relation and the passenger flow information of the entering and exiting stations uploaded in real time, a Kalman filtering method is used for solving the model, an OD estimation result is corrected by a standardization method to obtain an optimal estimation value, and the effectiveness of the method is checked. According to the method, statistics is carried out on the business data and the historical passenger flow data uploaded in real time, a state space model based on Kalman filtering is established, the real-time passenger flow demand distribution structure information can be estimated, and data support is provided for passenger flow dynamic management of rail transit enterprises.

Description

Dynamic estimation method for passenger flow OD of urban rail transit network
Technical Field
The invention relates to a dynamic estimation method for passenger flow OD of an urban rail transit network.
Background
With the rapid development of urban rail transit, large and medium cities gradually cross into networked operation. The complex of the wire mesh structure leads to the enhancement of the randomness of the travel behaviors of passengers, presents the characteristics of dynamic and complex on a passenger flow demand distribution structure, brings great challenges to the transportation organization of rail transit, and urgently needs to utilize a proper demand estimation model to research the distribution rule of the passenger flow demand in time and space in a short time range so as to improve the dynamic operation management level and the system strain capacity of the rail transit.
Since the 80 s of the last century, scholars at home and abroad carry out a great deal of research on OD dynamic estimation and form a series of OD dynamic estimation models. However, the application of the existing OD estimation method in rail transit mainly has the following disadvantages: the existing research is mainly focused on the field of road traffic, the structure and passenger flow characteristics of a rail traffic network are not considered, and the defects of low precision and low operation efficiency exist in the aspect of dynamic estimation of the passenger flow OD of the rail traffic network; secondly, the flow relation between the OD flow and the collected information flow in the existing model is mostly established on the basis that the section flow is easy to obtain, but the real-time section passenger flow information in the rail transit is difficult to obtain, and only the passenger flow information of the station can be obtained, so that the dynamic flow equation based on the section collected flow is difficult to be applied to the rail transit network. Therefore, a new method is needed to realize the real-time OD estimation of the passenger flow of the rail transit network by combining the passenger flow characteristics of the rail transit network and the collected data information.
Disclosure of Invention
The purpose of the invention is as follows: aiming at the defects in the prior art, the invention provides a dynamic estimation method for the OD of the passenger flow of an urban rail transit network, which is used for estimating the current real-time OD distribution structure information of the passenger flow based on historical passenger flow travel data and transaction data uploaded by terminal equipment of an automatic rail transit fare collection system in real time.
The technical scheme is as follows: in order to achieve the purpose, the invention adopts the technical scheme that:
a dynamic estimation method for passenger flow OD of an urban rail transit network comprises the following steps:
(1-1) setting a time interval delta t, and segmenting the historical passenger flow data of each day according to the time interval delta t; counting the passenger flow data of the stored value tickets and the passenger flow data of the one-way tickets in each time period every day;
(1-2) improving the passenger flow data of the one-way ticket by adopting a moving average method, wherein the improved passenger flow data of the one-way ticket is as follows:
Figure BDA0001219871050000021
in the formula, i is not equal to j; q. q.s0ij(t) represents the number of one-way ticket passengers that will eventually exit from station j among the passengers that entered station i during the modified tth time period;
Figure BDA0001219871050000022
representing history of compositionCounting the number of passengers of one-way tickets which finally exit from a station j in the passengers which enter from the station i in the t-a time period obtained by passenger flow data; r represents the number of time periods of moving average, and R is less than t;
(1-3) calculation of
Figure BDA0001219871050000023
In the formula,
Figure BDA0001219871050000024
represents the number of passengers who enter the station from the station i and finally exit the station from the station j in the t time period obtained by the statistics of historical passenger flow travel data, qij(t) represents the total number of passengers arriving at station j from the passenger flow arriving at station i during the t-th time period;
according to qij(t) constructing an OD distribution matrix A (t) of inbound passenger flow in the t time period and a passenger flow distribution rate matrix B (t) in the t time period of the whole network:
Figure BDA0001219871050000025
Figure BDA0001219871050000026
wherein n is the total number of sites; bij(t) is a passenger flow distribution rate which represents the proportion of the passenger flow to j station in the passengers arriving at the station from i station in the time period t to the total passenger flow of the station from i station,
Figure BDA0001219871050000027
and is
Figure BDA0001219871050000028
Converting the passenger flow distribution ratio matrix B (t) into a column vector form:
B(t)＝[b12(t)，b13(t)，…，b1n(t)，…，b21(t)，…，b2n(t)，…，bn(n-1)(t)]T (1.4)
(1-4) constructing a passenger outbound arrival coefficient as follows:
Figure BDA0001219871050000031
wherein,
Figure BDA0001219871050000032
the coefficient is the passenger flow outbound arrival coefficient and represents the proportion of OD passenger flows which start from a station i in the t-m time period and take the station j as the destination to a destination station j in the time period t, and t is more than or equal to m
Figure BDA0001219871050000033
uij(t) represents the average travel time of passengers departing from station i to station j in the t time period,
Figure BDA0001219871050000034
the standard deviation of the average travel time of passengers departing from the station i to the station j in the t time period; f. ofij(x) Is a probability density function representing the probability that a passenger flow departing from station i and destined to station j arrives at station j at time x;
(1-5) establishing a constraint equation between an OD passenger flow distribution proportion and station inlet and outlet flow based on real-time passenger flow data:
Figure BDA0001219871050000035
qij(t-m)＝Ii(t-m)·bij(t-m) (1.7)
Figure BDA0001219871050000036
in the formula Ii(t-m) is the total number of inbound passengers standing at i during the t-m time period; q. q.sij(t-m) represents the total number of passengers arriving at station j by the passenger flow arriving at station i in t-m time period; o isj(t) line representing the number M of passengers at stop j in the t-th time periodThe maximum number of the time periods of the passenger travel time between any two time periods in the network; vij(t) is an outbound quantity error generated when a flow constraint equation is established;
(1-6) constructing a passenger flow OD dynamic estimation state space model by taking the passenger flow split flow rate as a state variable, wherein the state space model comprises a state transition equation 1.9 and an observation equation 1.10:
Figure BDA0001219871050000041
Figure BDA0001219871050000042
in the formula (1.9), B (t) is the actual passenger flow distribution rate bij(t) R of compositionodX 1 dimensional matrix, RodDenotes the total number of OD pairs, Rod＝n×(n-1)；Bk(t) is the forward k-th week history passenger flow split rate under the condition of the same passenger flow characteristic day
Figure BDA0001219871050000043
Of composition RodX 1 dimensional matrix; f (t) and Gk(t) all are state transition matrixes, represent the state evolution characteristics of the system and are represented by weight coefficients gammakObtained Rod×RodA dimension constant matrix; w (t) error W generated by establishing system state transition equationij(t) a white noise matrix;
in the formula (1.10), Oj(t) and Ii(t-m) is real-time station entering and exiting passenger flow data; o (t) is an n × 1 dimension outbound passenger flow matrix; h (t) is a passenger outbound arrival matrix which dynamically changes with time and represents the correlation between the state variable B (t) and the observation variable O (t), and is nxRodA dimension matrix;
Figure BDA0001219871050000044
is formed by
Figure BDA0001219871050000045
Constructed RodA matrix of dimension x 1 is formed,
Figure BDA0001219871050000046
the passenger flow distribution rate is the mean value of the passenger flow distribution rate containing the current time period and M forward continuous time periods; v (t) is the error v generated by establishing the system observation equationij(t) a white noise matrix;
(1-7) solving the passenger flow OD dynamic estimation state space model by adopting a Kalman filtering method, and correcting an OD estimation result by adopting a standardized method; establishing an index according to the corrected OD estimation result, and checking whether the constructed passenger flow OD dynamic estimation state space model is correct or not by using the index; if the check result is correct, judging that the passenger flow OD dynamic estimation state space model is correct, and outputting the estimation result of the passenger flow OD dynamic estimation state space model; if the checking result is incorrect, resetting the parameter value of the passenger flow OD dynamic estimation state space model, and returning to the step (1-6); the reset parameters include: number of periods R of moving average and weight coefficient gammak。
Further, the step of constructing the passenger flow OD dynamic estimation state space model in the step (1-6) is as follows:
(2-1) establishing a passenger flow split flow rate relation between adjacent time periods:
Figure BDA0001219871050000051
in the formula,
Figure BDA0001219871050000052
the passenger flow distribution rate of the t time period is obtained by statistics of passenger flow data of a forward k week history under the same passenger flow characteristic day; gamma raykIs a weight coefficient, gamma is more than or equal to 0kLess than or equal to 1, and is used for measuring the reliability of the passenger flow information of the forward k-th week history; w is aij(t) is a normally distributed white Gaussian noise variable used for representing a state transition error generated when a state transition equation is constructed;
(2-2) converting the passenger flow distribution ratio relation among the adjacent time periods into a standard matrix form to obtain a state transition equation as follows:
Figure BDA0001219871050000053
wherein W (t) is the error w generated by establishing the system state transition equationij(t) and W (t) -N (0, Q (t)), Q (t) is a state transition error variance, and represents an error variance generated when a state transition equation is established, wherein an unbiased estimation expression of Q (t) is as follows:
Figure BDA0001219871050000054
in the formula, Wk(t) represents the historical state transition error for the t-th period of the k-th week forward on the same traffic signature day,
Figure BDA0001219871050000055
the average value of the historical state transfer errors of p days;
(2-3) approximately replacing the average value of the passenger flow distribution rate in the adjacent time periods with the passenger flow distribution rate of each time period, and converting the expression of the formula (1-8) into the following form:
Figure BDA0001219871050000061
the observation equation for the state space model from the above equation is:
Figure BDA0001219871050000062
wherein, V (t) is an error matrix of the observation equation, and V (t) -N (0, R (t)), R (t) is an error variance matrix of the outbound quantity, which represents the error variance generated when the observation equation is established, and the unbiased estimation expression of R (t) is as follows:
Figure BDA0001219871050000063
in the formula,Vk(t) represents the historical error matrix of the observed equation for the t-th time period on day k,
Figure BDA0001219871050000064
the average value of historical observation errors of n consecutive days is shown.
Further, in the step (1-7), the method for solving the passenger flow OD dynamic estimation state space model by using the kalman filtering method and correcting the OD estimation result by using the normalization method to obtain the optimal estimation value comprises the steps of:
(3-1) defining the covariance matrix as P (t); initializing t ═ 1; definition of
Figure BDA0001219871050000065
P(1)＝[1]n×n；
Wherein, Bk(1) The forward k-th week history passenger flow split rate is the forward k-th week history passenger flow split rate under the condition of the same passenger flow characteristic day
Figure BDA0001219871050000066
Of composition RodX 1 dimensional matrix;
(3-2) performing prior estimation according to a state transition equation:
Figure BDA0001219871050000067
in the formula,
Figure BDA0001219871050000068
an a priori estimate of the state variable b (t) representing the t-th period,
Figure BDA0001219871050000069
a posteriori estimate of a state variable B (t-1) representing the t-1 th epoch;
(3-3) calculating an a priori estimated covariance matrix;
Figure BDA00012198710500000610
wherein,
Figure BDA00012198710500000611
a priori estimated covariance matrix representing the t-th time period;
Figure BDA00012198710500000612
representing the posterior estimated covariance matrix of the t-1 th time period;
(3-4) calculating a Kalman filtering gain;
Figure BDA0001219871050000071
(3-5) correcting the prior estimated value according to the Kalman filtering gain and the residual error between the estimated value and the observed value
Figure BDA0001219871050000072
Obtaining the posterior estimated value
Figure BDA0001219871050000073
Namely, the solution of the standard kalman filtering method based on the state space model:
Figure BDA0001219871050000074
(3-6) constraint correction of the estimated value by a standard Kalman filtering method: in the OD dynamic estimation process, the state variable b (t) should satisfy the constraint of equation (1.21), where equation (1.21) is:
Figure BDA0001219871050000075
with the mean square error minimization as the objective function, we can get:
Figure BDA0001219871050000076
wherein | | | | represents the two-norm of the vector;
Figure BDA0001219871050000077
to correct the passenger flow split rate
Figure BDA0001219871050000078
Of composition RodX 1 dimensional matrix, which is the estimated value obtained by the standard Kalman filtering step
Figure BDA0001219871050000079
On the basis, a vector formed by the corrected estimation value after being adjusted by a mean square error method; equation of
Figure BDA00012198710500000710
For the corrected state vector
Figure BDA00012198710500000711
The equation constraint equation to be satisfied, Y is an n multiplied by 1 dimension constant matrix, and the element values are all 1; x is nxRodA dimension matrix;
constructing a lagrangian condition function for the constraint problem represented by equation (1.22) can obtain:
Figure BDA00012198710500000712
wherein Z is the constructed Lagrangian condition function; beta is a Lagrangian multiplier vector; p (B (t) O (t)) is a conditional probability density function;
assuming that initial system state variables B (1), W (t), V (t) are joint Gaussian variables, combining with the property of Kalman filtering: when B (1), W (t), V (t) are joint Gaussian variables, Kalman filtering estimation value
Figure BDA0001219871050000081
The conditional mean value of B (t) under the condition of O (t) can obtain:
Figure BDA0001219871050000082
then respectively align the formula (1.2)3) In (1)
Figure BDA0001219871050000083
And β is first order derived, which can be solved:
Figure BDA0001219871050000084
(3-7) updating the posteriori estimation covariance matrix;
Figure BDA0001219871050000085
further, the step (1-7) of establishing an index according to the corrected OD estimation result, and checking whether the constructed passenger flow OD dynamic estimation state space model is correct by using the index includes:
(4-1) constructing a normalized root mean square error index:
Figure BDA0001219871050000086
the lower the RMSN value is, the more accurate the estimation model is;
(4-2) judging whether the value of RMSN is less than a preset threshold value RMSNminIf RMSN < RMSN is satisfiedminJudging that the passenger flow OD dynamic estimation state space model is correct; otherwise, judging that the passenger flow OD dynamic estimation state space model is incorrect.
Further, in the step (1-7), the method for resetting the parameters includes:
and (3) calculating:
Figure BDA0001219871050000087
in the formula,
Figure BDA0001219871050000088
iterating the step size for a preset number of time periods, an
Figure BDA0001219871050000089
Is an integer; tau is a preset weight coefficient iteration step, and tau is less than 1.
It should be noted that when k is smaller, it indicates that γ is closer to the actual passenger flow datakWhen reset, the value of (A) can be increased, whereas when k is larger, the value of (Y) is increasedkMay be reduced when reset.
If the evaluation passes the inspection, the established method for dynamically estimating the OD of the passenger flow of the line network is determined to be effective and can be used for actual rail transit operation management.
Has the advantages that: the dynamic estimation method for the passenger flow OD of the urban rail transit network combines the OD distribution rule of historical passenger flow, processes the passenger flow of the one-way ticket with larger volatility by adopting a moving average method, extracts an improved historical passenger flow split rate matrix, and can improve the estimation precision of a model; the method has the advantages that a passenger flow outbound arrival coefficient is constructed, a double integral calculation method is provided for establishing a dynamic flow relation between OD flow and inbound and outbound passenger flow, and the problem that the flow relation is greatly reduced due to the fact that the rail transit section passenger flow is difficult to collect is solved; a state space model based on a Kalman filtering algorithm is constructed, the OD of the line network passenger flow is dynamically estimated, and the OD estimation result is corrected by adopting a standardized method, so that the OD estimation precision is further improved; and finally, the effectiveness of the model is checked by adopting a standardized weighted root mean square error method, so that the reliability of the estimation result is ensured. The method can effectively carry out real-time OD estimation on the passenger flow of the urban rail transit network, and provides data support for urban rail transit operation management decisions.
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FIG. 1 is a schematic flow chart of the method operation of the present invention.
Detailed Description
Fig. 1 is a schematic diagram illustrating an operation flow of a dynamic estimation method for passenger flow OD of an urban railway network, and the invention is further described with reference to an implementation process.
In this embodiment, a time interval Δ t is first set, the operation time is segmented according to the time interval Δ t, and since the real-time uploading of the passenger flow data is generally at 15min intervals, in order to meet the requirement of operation management, the time interval Δ t may be set to 15 min. Dividing historical passenger flow travel data into stored value ticket passenger flow data and single-pass ticket passenger flow data, counting time-interval OD distribution passenger flow in the stored value ticket passenger flow data and the single-pass ticket passenger flow data, considering that the space distribution stability of the single-pass ticket passenger flow is poor and the difference of arrival time of passengers is large, and considering that a moving average method is adopted to improve the single-pass ticket OD passenger flow. At each time period, the following steps are performed:
step 1: calculating a passenger flow OD distribution matrix:
Figure BDA0001219871050000091
wherein q is0ijSubscript 0 of (t) denotes a one-way ticket, subscripts i and j are station numbers, q is a station number0ij(t) represents the number of one-way ticket passengers that finally exit from station j among passengers that entered station i within the t-th time period improved by the moving average method;
Figure BDA0001219871050000101
representing the number of passengers of the one-way ticket which finally exit from the station j in the t-a time period acquired by the original passenger flow data statistics; r represents the number of time periods of moving average, and R < t.
Accumulating the passenger flow of the single ticket improved by the moving average method and the passenger flow of the stored value ticket to obtain an OD distribution matrix A (t) of the passenger flow entering the station in the t time period on the same day of the global network, wherein the OD distribution matrix A (t) is as follows:
Figure BDA0001219871050000102
wherein n is the total number of sites; q. q.sij(t) represents the total number of passengers arriving at station j from the flow of passengers arriving at station i during the t-th time period, and i ≠ j;
the value of the moving average time period number R is determined according to the passenger flow distribution fluctuation of the same characteristic day calculated by the historical single-pass ticket passenger flow data.
Step 2: calculating a passenger flow distribution rate matrix: based on the OD distribution matrix a (t), the passenger flow distribution rate matrix b (t) in the t-th time period can be obtained as follows:
Figure BDA0001219871050000103
wherein, bij(t) is a passenger flow distribution rate which represents the proportion of the passenger flow going to j station in the passengers arriving at the station i in the time period t to the total passenger flow arriving at the station i, and the passenger flow distribution rate can be known
Figure BDA0001219871050000104
To facilitate the construction of the subsequent model, the passenger flow partial flow rate matrix b (t) is converted into a column vector form as follows:
B(t)＝[b12(t)，b13(t)，…，b1n(t)，…，b21(t)，…，b2n(t)，…，bn(n-1)(t)]T
step 3: constructing a passenger outbound arrival coefficient: the rail transit AFC system records passenger travel information, and meanwhile, the rail transit passenger travel time has the characteristic of high reliability (the travel time of the subway section is basically fixed). Therefore, the traffic relationship between inbound and outbound traffic and OD traffic can be characterized by the time distribution of passenger travel between the respective ODs. Suppose that the travel time of the passenger flow from the station i to the station j at the xth minute within any time period t obeys normal distribution, namely x epsilon N (u)ij(t)，
Figure BDA0001219871050000111
Wherein u isij(t) represents the average travel time of a passenger from station i to station j during time period t;
Figure BDA0001219871050000112
standard deviation, u, representing the average travel time for a passenger to travel from station i to station j during time period kij(t) and
Figure BDA0001219871050000113
the method can be obtained by performing statistical analysis on historical passenger flow OD data.
On the basis, assuming that the passenger flow arriving from the station i to the station j in any time period t uniformly arrives at the station i in the time period, the passenger travel time probability density function f between the OD can be passedij(x) And integrating the probability that the passenger flow which is sent from the station i to the station j in the t-m time period reaches the station j in the time period t, thereby calculating the passenger flow outbound arrival coefficient
Figure BDA0001219871050000114
The flow relation between the incoming and outgoing passenger flow and the OD flow is described, and the expression is as follows:
Figure BDA0001219871050000115
wherein,
Figure BDA0001219871050000116
the outbound arrival coefficient is a passenger flow outbound arrival coefficient which represents the proportion of OD passenger flows which start from a station i in the t-m th time period and take the station j as a destination to arrive at a destination station j in the time period t (t is more than or equal to m) (the outbound passenger flow information collected in a certain time period is the result of arriving and gathering of inbound passenger flows from other stations in a plurality of time periods in the early stage);
step 4: constructing a dynamic flow relation: because the rail transit section passenger flow is difficult to collect, the dynamic flow relation is difficult to establish by taking the road traffic field as reference and collecting the section passenger flow of the road section in real time as the basis, therefore, the method establishes the OD passenger flow distribution proportion and the constraint equation between the inlet and outlet flow on the basis of the time-varying inlet and outlet station collected passenger flow:
Figure BDA0001219871050000117
qij(t-m)＝Ii(t-m)·bij(t-m)
Figure BDA0001219871050000118
wherein n represents the total number of the wire net stations; i isi(t-m) is the total station entering passenger flow of the station i in the t-m time period; q. q.sij(t-m) represents the total number of passengers arriving at station j from the passenger flow arriving at station i during the t-m time period; bij(t-m) is the passenger flow separation rate in the t-m time period, and represents the proportion of the passenger flow from the station i to the station j to the total passenger flow entering the station i in the t-m time period; o isj(t) represents the outbound amount of the station j in the time period t; m is the maximum crossing time period number of the passenger travel time between any OD in the road network, and the value of the maximum crossing time period number M depends on the line network scale of local urban rail transit. (ii) a
Figure BDA0001219871050000121
An outbound arrival coefficient for the passenger; vij(t) is the outbound error that results when the observation equation is established.
Step 5: establishing a state space model: and selecting the station passenger flow split ratio as an estimation variable, and establishing a network passenger flow OD dynamic estimation state space model. In view of the fact that the passenger flow fluctuation in the short time is not too large, the relationship between the passenger flow split ratios in adjacent time periods can be obtained to satisfy:
Figure BDA0001219871050000122
in the formula,
Figure BDA0001219871050000123
the passenger flow distribution rate of the t time period is obtained by statistics of passenger flow data of a forward k week history under the same passenger flow characteristic day; gamma raykIs a weight coefficient, gamma is more than or equal to 0kLess than or equal to 1, and is used for measuring the reliability of the passenger flow information of the forward k-th week history; w is aijAnd (t) is a normally distributed white Gaussian noise variable used for representing a state transition error generated when the state transition equation is constructed.
Converting the passenger flow distribution rate relational expression between the adjacent time periods into a standard matrix form to obtain a state transition equation as follows:
Figure BDA0001219871050000124
wherein W (t) is the error w generated by establishing the system state transition equationij(t) and W (t) -N (0, Q (t)), Q (t) is a state transition error variance, and represents an error variance generated when a state transition equation is established, wherein an unbiased estimation expression of Q (t) is as follows:
Figure BDA0001219871050000125
in the formula, Wk(t) represents the historical state transition error for the t-th period of the k-th week forward on the same traffic signature day,
Figure BDA0001219871050000126
the mean value of the historical state transition errors for p days.
According to the relation between the passenger flow split ratios in the adjacent time periods, the estimation process needs to contain state variable information of a plurality of time periods, and the passenger flow split ratio b in the t-th time period is estimatedij(t) and bij(t-1)，…，bij(t-m) then, it is necessary to integrate variable information of a plurality of periods into one period, thereby facilitating model construction. For this reason, assuming that the variation of the passenger flow fluctuation is small in a certain time range, the mean value of the passenger flow distribution rate in the adjacent time period is adopted
Figure BDA0001219871050000131
Will be b of a plurality of periodsij(t) integrating into a time interval, and converting the relation between the passenger flow split ratios in the adjacent time intervals into the following form:
Figure BDA0001219871050000132
the above equation is expressed as a standard matrix form, and the obtained system observation equation is:
Figure BDA0001219871050000133
wherein O (t) is an n × 1 dimensional outbound quantity matrix; h (t) is a compound of
Figure BDA0001219871050000134
A determined passenger outbound arrival matrix which dynamically changes with time, characterizes the interrelation between state variables and observation variables, is nxRodA dimension matrix;
Figure BDA0001219871050000135
is based on the mean value of the passenger flow distribution rate in continuous time periods
Figure BDA0001219871050000136
Constructed RodX 1 dimensional matrix; v (t) is an error matrix of the outbound quantity equation system, V (t) -N (0, R (t)), R (t) is an outbound quantity error variance matrix which represents error variances generated when the observation equation is established, and the error variances can be obtained by carrying out statistical analysis according to observation error variance sample values V (t) in historical data, wherein the unbiased estimation expression is as follows:
Figure BDA0001219871050000137
Vk(t) represents the historical observation error for the t-th time period on the k-th day,
Figure BDA0001219871050000138
the average value of historical observation errors of n consecutive days is shown.
Step 6: solving by a Kalman filtering algorithm: the Kalman filtering algorithm is a classical solving method of a state space model, is actually an optimized autoregressive data processing algorithm, and assumes that two estimation values exist in a state vector B (t) of any time period t, namely a priori estimation value
Figure BDA0001219871050000139
And a posteriori estimate
Figure BDA00012198710500001310
The basic idea of the kalman filter algorithm is: OD estimation value of any time interval t
Figure BDA00012198710500001311
Are all prior estimates
Figure BDA00012198710500001312
Is obtained by further correcting the system observed value O (t), and the prior estimated value of the time interval t
Figure BDA00012198710500001313
Always a posteriori estimate over a period of t-1
Figure BDA0001219871050000141
The improvement is made on the basis. The iteration recursion comprises the following specific steps:
1) initializing a system: defining a covariance matrix as P (t); initializing t ═ 1; definition of
Figure BDA0001219871050000142
P(1)＝[1]n×n(ii) a Where the initial period T is 1 st estimated time period of the operation day, i.e., [ T ═ 1 { (T) }0，T0+Δt]，T0Representing the starting time of the subway operation day; the initial OD passenger flow split-flow rate B (1) can be understood as the posterior passenger flow split-flow rate of the 1 st estimation period, and is the basis of the Kalman filtering iterative algorithm, but B (1) is difficult to obtain by collecting data in real time and is usually replaced by the average value of the passenger flow split-flow rate matrix of the initial period of continuous k weeks in historical data; b isk(1) The forward k-th week history passenger flow split rate is the forward k-th week history passenger flow split rate under the condition of the same passenger flow characteristic day
Figure BDA0001219871050000143
Of composition RodX 1 dimensional matrix;
the initial covariance matrix P (1) can be understood as a posterior covariance matrix of the 1 st estimation period, and since P (1) is difficult to obtain by real-time data collection or statistical analysis of historical data, P (1) can be set as an identity matrix.
2) Carrying out prior estimation according to a state transition equation;
Figure BDA0001219871050000144
wherein,
Figure BDA0001219871050000145
an a priori estimate of the state variable b (t) representing the time period t,
Figure BDA0001219871050000146
a posterior estimate of the state variable B (t-1) representing time period t-1;
3) calculating a prior estimation covariance matrix;
Figure BDA0001219871050000147
wherein,
Figure BDA0001219871050000148
a priori estimation error variance representing time period t;
Figure BDA0001219871050000149
representing the posterior estimation error variance representing time period t-1;
4) calculating Kalman filtering gain;
Figure BDA00012198710500001410
5) correcting the prior estimated value according to the Kalman filter gain and the residual error between the estimated value and the observed value
Figure BDA00012198710500001411
Obtaining the posterior estimated value
Figure BDA00012198710500001412
Namely, the solution of the standard kalman filtering method based on the state space model:
Figure BDA0001219871050000151
6) updating a posterior estimation covariance matrix;
Figure BDA0001219871050000152
the above steps are standard steps of the kalman filtering method. It should be noted that, in the process of dynamically estimating the passenger flow OD of the rail transit line network, the estimated value of the state variable b (t) needs to satisfy the equality constraint
Figure BDA0001219871050000153
For this purpose, before step 6), the OD estimation result obtained in step 5) may be corrected by using a normalization method:
Figure BDA0001219871050000154
in the formula,
Figure BDA0001219871050000155
as state variable passenger flow split bij(t) a posterior estimate of the value of,
Figure BDA0001219871050000156
the passenger flow split flow rate estimation value is corrected by adopting a standardization method.
The specific steps for obtaining the correction result are as follows:
with the mean square error minimization as the objective function, we can get:
Figure BDA0001219871050000157
therein without| | | represents the two-norm of the vector;
Figure BDA0001219871050000158
to correct the passenger flow split rate
Figure BDA0001219871050000159
Of composition RodX 1 dimensional matrix, which is the estimated value obtained by the standard Kalman filtering step
Figure BDA00012198710500001510
On the basis, a vector formed by the corrected estimation value after being adjusted by a mean square error method; equation of
Figure BDA00012198710500001511
For the corrected state vector
Figure BDA00012198710500001512
The equation constraint equation to be satisfied, Y is an n multiplied by 1 dimension constant matrix, and the element values are all 1; x is nxRodThe dimension matrix has the following elements:
Figure BDA00012198710500001513
in the above formula, the zeros function is to generate a matrix with all 0 element values.
Constructing a lagrangian condition function for the objective function can obtain:
Figure BDA0001219871050000161
wherein Z is the constructed Lagrangian condition function; beta is a Lagrangian multiplier vector; p (B (t) O (t)) is a conditional probability density function.
Here, we assume that the initial system state variables B (1), w (t), v (t) are joint gaussian variables, combined with the properties of kalman filtering: when B (1), W (t), V (t) are joint Gaussian variables, then the Kalman filter estimate
Figure BDA0001219871050000162
The conditional mean value of B (t) under the condition of O (t) can obtain:
Figure BDA0001219871050000163
then respectively align with those in the formula (1.24)
Figure BDA0001219871050000164
And β is first order derived, which can be solved:
Figure BDA0001219871050000165
therefore, the correction of the estimation result of the standard Kalman filtering algorithm is realized through the correction steps. Passenger flow split flow rate estimated value of each calculated certain time period t
Figure BDA0001219871050000166
And then, correcting the passenger flow distribution ratio by using the method so as to ensure that the passenger flow distribution ratio estimated value meets the equality constraint.
Step 7: the estimation method comprises the following steps: using the sample data to check the estimated value obtained at Step6, if the estimated value fails to pass the check, returning to the previous Step, and resetting the values of parameters such as the time period number R, the weight coefficient alpha and the like of the moving average; if the time series prediction model passes the inspection, the established time series prediction model is determined to be effective and can be used for actual rail transit operation management.
And (3) evaluating the effectiveness of the estimation model by adopting a Weighted Root Mean Square Error Normalized (WRMSN), wherein the WRMSN index expression is as follows:
Figure BDA0001219871050000167
wherein R isodRepresents the total number of network ODs; n is the total number of stations; bij(t)And
Figure BDA0001219871050000168
respectively obtaining a real value of the net passenger flow distribution rate and an estimated average value of the net passenger flow distribution rate in the time period t; lower RMSN values indicate more accurate estimation models.
When the RMSN index data calculated according to the actual value and the estimated value of the sample data is in an allowed range (for example, the RMSN is less than or equal to 20 percent), the method is considered to be feasible and can be applied to actual rail transit operation management; if the RMSN value is too large, the time period number R and the weight coefficient gamma of the moving average are resetkUntil the model passes the validity check.
The method for resetting the parameters comprises the following steps:
and (3) calculating:
Figure BDA0001219871050000171
in the formula,
Figure BDA0001219871050000172
iterating the step size for a preset number of time periods, an
Figure BDA0001219871050000173
Is an integer; tau is a preset weight coefficient iteration step, and tau is less than 1.
It should be noted that when k is smaller, it indicates that γ is closer to the actual passenger flow datakWhen reset, the value of (A) can be increased, whereas when k is larger, the value of (Y) is increasedkMay be reduced when reset.
The above description is only of the preferred embodiments of the present invention, and it should be noted that: it will be apparent to those skilled in the art that various modifications and adaptations can be made without departing from the principles of the invention and these are intended to be within the scope of the invention.

Claims (5)

1. A dynamic estimation method for passenger flow OD of an urban rail transit network is characterized by comprising the following steps: the method comprises the following steps:
(1-1) setting a time interval delta t, and segmenting the historical passenger flow data of each day according to the time interval delta t; counting the passenger flow data of the stored value tickets and the passenger flow data of the one-way tickets in each time period every day;
(1-2) improving the passenger flow data of the one-way ticket by adopting a moving average method, wherein the improved passenger flow data of the one-way ticket is as follows:
Figure FDA0002687873720000011
in the formula, i is not equal to j; q. q.s0ij(t) represents the number of one-way ticket passengers that will eventually exit from station j among the passengers that entered station i during the modified tth time period;
Figure FDA0002687873720000012
representing the number of passengers of the one-way ticket which finally exit from the station j in the t-a time period acquired by the historical passenger flow data statistics; r represents the number of time periods of moving average, and R is less than t;
(1-3) calculation of
Figure FDA0002687873720000013
In the formula,
Figure FDA0002687873720000014
represents the number of passengers who enter the station from the station i and finally exit the station from the station j in the t time period obtained by the statistics of historical passenger flow travel data, qij(t) represents the total number of passengers arriving at station j from the passenger flow arriving at station i during the t-th time period;
according to qij(t) constructing an OD distribution matrix A (t) of inbound passenger flow in the t time period and a passenger flow distribution rate matrix B (t) in the t time period of the whole network:
Figure FDA0002687873720000015
Figure FDA0002687873720000016
wherein n is the total number of sites; bij(t) is a passenger flow distribution rate which represents the proportion of the passenger flow to j station in the passengers arriving at the station from i station in the time period t to the total passenger flow of the station from i station,
Figure FDA0002687873720000021
and is
Figure FDA0002687873720000022
Converting the passenger flow distribution ratio matrix B (t) into a column vector form:
B(t)＝[b12(t)，b13(t)，…，b1n(t)，…，b21(t)，…，b2n(t)，…，bn(n-1)(t)]T (1.4)
(1-4) constructing a passenger outbound arrival coefficient as follows:
Figure FDA0002687873720000023
wherein,
Figure FDA0002687873720000024
the ratio of OD passenger flow which starts from a station i in the t-m time period and takes the station j as a destination to arrive at a target station j in the time period t is represented, and t is larger than or equal to m;
Figure FDA0002687873720000025
uij(t) represents the average travel time of passengers departing from station i to station j in the t time period,
Figure FDA0002687873720000026
the standard deviation of the average travel time of passengers departing from the station i to the station j in the t time period; f. ofij(x) Is a probability density function representing the probability that a passenger flow departing from station i and destined to station j arrives at station j at time x;
(1-5) establishing a constraint equation between an OD passenger flow distribution proportion and station inlet and outlet flow based on real-time passenger flow data:
Figure FDA0002687873720000027
qij(t-m)＝Ii(t-m)·bij(t-m) (1.7)
Figure FDA0002687873720000028
in the formula Ii(t-m) is the total number of inbound passengers standing at i during the t-m time period; q. q.sij(t-m) represents the total number of passengers arriving at station j by the passenger flow arriving at station i in t-m time period; o isj(t) represents the number of passengers outbound from station j during the t-th time period; m is the maximum crossing time period number of the passenger travel time between any two time periods in the line network; vij(t) is an outbound quantity error generated when a flow constraint equation is established;
(1-6) constructing a passenger flow OD dynamic estimation state space model by taking the passenger flow split flow rate as a state variable, wherein the state space model comprises a state transition equation 1.9 and an observation equation 1.10:
Figure FDA0002687873720000031
Figure FDA0002687873720000032
in the formula (1.9), B (t) is the actual passenger flow distribution rate bij(t) R of compositionodX 1 dimensional matrix, RodDenotes the total number of OD pairs, Rod＝n×(n-1)；Bk(t) is the forward k-th week history passenger flow split rate under the condition of the same passenger flow characteristic day
Figure FDA0002687873720000033
Of composition RodX 1 dimensional matrix; f (t) and Gk(t) all are state transition matrixes, represent the state evolution characteristics of the system and are represented by weight coefficients gammakObtained Rod×RodA dimension constant matrix; w (t) error w generated by establishing system state transition equationij(t) a white noise matrix;
in the formula (1.10), Oj(t) and Ii(t-m) is real-time station entering and exiting passenger flow data; o (t) is an n × 1 dimension outbound passenger flow matrix; h (t) is a passenger outbound arrival matrix which dynamically changes with time and represents the correlation between the state variable B (t) and the observation variable O (t), and is nxRodA dimension matrix;
Figure FDA0002687873720000034
is formed by
Figure FDA0002687873720000035
Constructed RodA matrix of dimension x 1 is formed,
Figure FDA0002687873720000036
the passenger flow distribution rate is the mean value of the passenger flow distribution rate containing the current time period and M forward continuous time periods; v (t) is the error v generated by establishing the system observation equationij(t) a white noise matrix;
(1-7) solving the passenger flow OD dynamic estimation state space model by adopting a Kalman filtering method, and correcting an OD estimation result by adopting a standardized method; establishing an index according to the corrected OD estimation result, and checking whether the constructed passenger flow OD dynamic estimation state space model is correct or not by using the index; if the check result is correct, judging that the passenger flow OD dynamic estimation state space model is correct, and outputting the estimation result of the passenger flow OD dynamic estimation state space model; if the test result is not correct, thenResetting the parameter value of the passenger flow OD dynamic estimation state space model, and returning to the step (1-6); the reset parameters include: number of periods R of moving average and weight coefficient gammak。
2. The method for dynamically estimating the OD of the urban rail transit network passenger flow according to claim 1, wherein the step of constructing the dynamic estimation state space model of the OD of the passenger flow in the steps (1-6) comprises:
(2-1) establishing a passenger flow split flow rate relation between adjacent time periods:
Figure FDA0002687873720000041
in the formula,
Figure FDA0002687873720000042
the passenger flow distribution rate of the t time period is obtained by statistics of passenger flow data of a forward k week history under the same passenger flow characteristic day; gamma raykIs a weight coefficient, gamma is more than or equal to 0kLess than or equal to 1, and is used for measuring the reliability of the passenger flow information of the forward k-th week history; w is aij(t) is a normally distributed white Gaussian noise variable used for representing a state transition error generated when a state transition equation is constructed;
(2-2) converting the passenger flow distribution ratio relation among the adjacent time periods into a standard matrix form to obtain a state transition equation as follows:
Figure FDA0002687873720000043
wherein W (t) is the error w generated by establishing the system state transition equationij(t) and W (t) -N (0, Q (t)), Q (t) is a state transition error variance, and represents an error variance generated when a state transition equation is established, wherein an unbiased estimation expression of Q (t) is as follows:
Figure FDA0002687873720000044
in the formula, Wk(t) represents the historical state transition error for the t-th period of the k-th week forward on the same traffic signature day,
Figure FDA0002687873720000045
the average value of the historical state transfer errors of p days;
(2-3) replacing the average value of the passenger flow distribution rate in the adjacent time periods with the passenger flow distribution rate of each time period, and converting the expression of the formula (1.8) into the following form:
Figure FDA0002687873720000051
the observation equation for the state space model from the above equation is:
Figure FDA0002687873720000052
wherein, V (t) is an error matrix of the observation equation, and V (t) -N (0, R (t)), R (t) is an error variance matrix of the outbound quantity, which represents the error variance generated when the observation equation is established, and the unbiased estimation expression of R (t) is as follows:
Figure FDA0002687873720000053
in the formula, Vk(t) represents the historical error matrix of the observed equation for the t-th time period on day k,
Figure FDA0002687873720000054
the average value of historical observation errors of p consecutive days is shown.
3. The method for dynamically estimating the OD of the passenger flow of the urban rail transit network according to claim 2, wherein the step (1-7) of solving a state space model of the dynamic estimation of the OD of the passenger flow by using a Kalman filtering method and correcting an OD estimation result by using a normalization method to obtain an optimal estimation value comprises the following steps:
(3-1) defining the covariance matrix as P (t); initializing t ═ 1; definition of
Figure FDA0002687873720000055
P(1)＝[1]n×n；
Wherein, Bk(1) The forward k-th week history passenger flow split rate is the forward k-th week history passenger flow split rate under the condition of the same passenger flow characteristic day
Figure FDA0002687873720000056
Of composition RodX 1 dimensional matrix;
(3-2) performing prior estimation according to a state transition equation:
Figure FDA0002687873720000057
in the formula,
Figure FDA0002687873720000058
an a priori estimate of the state variable b (t) representing the t-th period,
Figure FDA0002687873720000059
a posteriori estimate of a state variable B (t-1) representing the t-1 th epoch;
(3-3) calculating an a priori estimated covariance matrix;
Figure FDA00026878737200000510
wherein,
Figure FDA0002687873720000061
a priori estimated covariance matrix representing the t-th time period;
Figure FDA0002687873720000062
representing the posterior estimated covariance matrix of the t-1 th time period;
(3-4) calculating a Kalman filtering gain;
Figure FDA0002687873720000063
(3-5) correcting the prior estimated value according to the Kalman filtering gain and the residual error between the estimated value and the observed value
Figure FDA0002687873720000064
Obtaining the posterior estimated value
Figure FDA0002687873720000065
Figure FDA0002687873720000066
I.e. the solution of the standard kalman filtering method based on the state space model:
Figure FDA0002687873720000067
(3-6) constraint correction of the estimated value by a standard Kalman filtering method: in the OD dynamic estimation process, the state variable b (t) should satisfy the constraint of equation (1.21), where equation (1.21) is:
Figure FDA0002687873720000068
with the mean square error minimization as the objective function, we can get:
Figure FDA0002687873720000069
wherein | | | | represents the two-norm of the vector;
Figure FDA00026878737200000610
to correct the passenger flow split rate
Figure FDA00026878737200000611
Of composition RodX 1 dimensional matrix, which is the estimated value obtained by the standard Kalman filtering step
Figure FDA00026878737200000612
On the basis, a vector formed by the corrected estimation value after being adjusted by a mean square error method; equation of
Figure FDA00026878737200000613
For the corrected state vector
Figure FDA00026878737200000614
The equation constraint equation to be satisfied, Y is an n multiplied by 1 dimension constant matrix, and the element values are all 1; x is nxRodA dimension matrix;
constructing a lagrangian condition function for the constraint problem represented by equation (1.22) can obtain:
Figure FDA00026878737200000615
wherein Z is the constructed Lagrangian condition function; beta is a Lagrangian multiplier vector; p (B (t) O (t)) is a conditional probability density function;
assuming that initial system state variables B (1), W (t), V (t) are joint Gaussian variables, combining with the property of Kalman filtering: when B (1), W (t), V (t) are joint Gaussian variables, Kalman filtering estimation value
Figure FDA0002687873720000071
The conditional mean value of B (t) under the condition of O (t) can obtain:
Figure FDA0002687873720000072
then respectively align with those in the formula (1.23)
Figure FDA0002687873720000073
And β is first order derived, which can be solved:
Figure FDA0002687873720000074
(3-7) updating the posteriori estimation covariance matrix;
Figure FDA0002687873720000075
4. the method for dynamically estimating OD (origin-destination) of passenger flow of urban rail transit network according to claim 3, wherein the step (1-7) of establishing an index according to the corrected OD estimation result and using the index to check whether the constructed dynamic OD state space model of passenger flow is correct comprises the steps of:
(4-1) constructing a normalized root mean square error index:
Figure FDA0002687873720000076
the lower the RMSN value is, the more accurate the estimation model is;
(4-2) judging whether the value of RMSN is less than a preset threshold value RMSNminIf RMSN < RMSN is satisfiedminJudging that the passenger flow OD dynamic estimation state space model is correct; otherwise, judging that the passenger flow OD dynamic estimation state space model is incorrect.
5. The method for dynamically estimating OD of urban rail transit network passenger flow according to claim 4, wherein in the steps (1-7), the method for resetting parameters comprises:
and (3) calculating:
Figure FDA0002687873720000077
in the formula,
Figure FDA0002687873720000078
iterating the step size for a preset number of time periods, an
Figure FDA0002687873720000079
Is an integer; tau is a preset weight coefficient iteration step, and tau is less than 1.
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