阅读说明：本技术 多重自适应快速弱敏容积卡尔曼滤波方法及其在金融市场数据处理中的应用 (Multiple self-adaptive fast weak-sensitive volume Kalman filtering method and application thereof in financial market data processing ) 是由 陈沛 方璞 张云玲 马贵珍 张昊 娄泰山 李原厂 余红奎 于 2021-06-02 设计创作，主要内容包括：本发明公开了一种多重自适应快速弱敏容积卡尔曼滤波方法及其在金融市场模型中的应用。本发明将多重渐消因子引入到快速弱敏容积卡尔曼滤波中,有效地解决了金融市场微结构模型价格预测过程中受模型参数不确定的影响导致预测精度下降的问题,不但提高了金融市场微结构模型价格预测的精度,而且减少了运算时间,提升了预测结果的时效性。(The invention discloses a multiple self-adaptive fast weak-sensitive volume Kalman filtering method and application thereof in a financial market model. According to the method, multiple fading factors are introduced into the rapid weak-sensitive volume Kalman filtering, so that the problem of reduction of prediction precision caused by uncertain influence of model parameters in the price prediction process of the microstructure model of the financial market is effectively solved, the price prediction precision of the microstructure model of the financial market is improved, the operation time is reduced, and the timeliness of a prediction result is improved.)

1. A multiple self-adaptive fast weak-sensitive volume Kalman filtering method is characterized in that multiple fading factors are introduced in a step of calculating filter gain of fast weak-sensitive volume Kalman filtering:

wherein l is the number of uncertain parameters, lambda_kdiag[μ₁,μ₂,…,μ₈]Is a multiple fading factor, w_i,kCorresponding to the ith uncertain parameter c_iA sensitivity weight of; w is a_i,kThe following method was used:

ac_i＜c_i＜bc_i

in the above formula, bc_iAnd ac_iRespectively correspond to c_iThe upper and lower limits of (a) are,is c_iThe mean value of (a);

the fading factor value is as follows:

wherein the content of the first and second substances,in the formula, the compound is shown in the specification,

in the formula V_kAs a residual matrix, estimated by:

2. the multiple adaptive fast weak-sensitive volume kalman filtering method according to claim 1, comprising the steps of:

s1: performing time updates

Let t be_k-1The state estimation value and the error variance matrix at the moment are respectivelyAndthen t is_kThe volume points at the time are:

where the superscript "+" indicates its posterior estimate,is an error variance matrixSquare root of (b), satisfiesξ_jThe j value of 2n volume points is as follows:

wherein e is_j(i-1, 2, …, n) is a unit vector with the jth value of 1 and the remaining elements are zero, i.e., e_j＝[0,…,0,1,0,…,0]^T；

Calculating the sensitivity of the rank sampling point of the step k-1:

in the formula (I), the compound is shown in the specification,sensitivity for step k-1;

volume points after nonlinear transmission are as follows:

then their corresponding prior state estimates and variances are:

in the formula, the superscript "-" represents a prior estimate of the variable;

updating a rank sampling point set:

calculating the prior state estimate and the sensitivity of the prior covariance matrix:

in the formula (I), the compound is shown in the specification,estimating the sensitivity prior for the k step;

s2: performing measurement update

Estimating from a priori stateSum varianceRecalculating the volume points:

secondly, calculating volume point positions transferred by the nonlinear measurement function:

calculating the sensitivity of the re-rank sampling point set and the sensitivity of the measurement rank sampling point

Fourthly, calculating the measurement prior estimation value, the variance of the prior estimation value and the covariance of the prior estimation value:

calculating the measured sensitivity:

calculating the sensitivity of the state and the metrology covariance and the sensitivity of the metrology variance:

s3: calculating a weak-sensitive filter gain matrix

Where l-8 is the number of uncertain parameters, λ_kdiag[μ₁,μ₂,…,μ₈]Is a multiple fading factor, w_i,kCorresponding to the ith uncertain parameter c_iA sensitivity weight of; w is a_i,kThe following method was used:

wherein bc is_iAnd ac_iRespectively correspond to c_iThe upper and lower limits of (a) are,is c_iThe mean value of (a);

the fading factors are:

wherein the content of the first and second substances,in the formula, the compound is shown in the specification,

in the formula (24), V_kAs a residual matrix, estimated by:

s4: estimating state and variance

Calculating state estimation values and corresponding variance matrixes:

the variance matrix is:

calculating the sensitivity of state estimation and the sensitivity of state error variance matrix:

wherein the content of the first and second substances,

wherein the content of the first and second substances,is an oblique symmetrical matrix, and the oblique symmetrical matrix,satisfy the Gamma^TAll of ═ Γ, Ψ, and Θ are nonsingular matrices, and satisfy

The loop iterates the steps S1-S3.

3. A financial market microstructure model price prediction method based on multiple self-adaptive fast weak-sensitive volume Kalman filtering comprises the following steps:

step one, establishing a microstructure model of a financial market

Wherein, P_kIs the asset price; phi is a_kThe demand for surplus in the market;λ_kis the reciprocal of market liquidity; zeta_v,kIs Bernoulli random variable and satisfies P { zeta_v,k＝1}＝v_tΔt；α₁，β₁，α₂，β₂，γ₁，γ₂，γ₃Parameters to be identified are provided with uncertainty; p is a radical of_kIs the jump amplitude, and obeys mean value of zero and variance ofNormal distribution of (2); xi_1,k,ξ_2,k,ξ_3,kAll follow a standard normal distribution and are independent of each other, Zeta_v,kThe two are mutually independent;

step two, establishing a measuring model of the financial market microstructure model with price as a measuring value

z_k＝P_k+w_k ②，

Wherein, w_kObeying a mean of zero and a variance of R_kIs turning toState distribution;

step three: state equation and measurement equation of financial market microstructure model

Selecting a state variable x_k＝[P_k,φ_k,σ_k]^TThen, there are:

z_k＝h(x_k)+w_k＝P_k+w_k ④，

wherein c ═ α₁,β₁,α₂,β₂,γ₁,γ₂,γ₃,η_p]^TIs an uncertain parameter vector; xi_k＝[ξ_1,k,ξ_2,k,ξ_3,k]^TObedience mean value is zero and variance is Q_kIs normally distributed, and

step four: estimating an uncertain parameter c in the model (3) by using the known data, and determining a parameter reference value of the model

Wherein the content of the first and second substances,and P_zz,kThe filtered measurement error and the measurement prediction variance, respectively, may be obtained by CKF filtering;

step five: initializing states of financial market micro-architectural modelAndsensitivity weight W₀＝P_cc，P_ccFor uncertainty of the variance of the parameter, error sensitivity S₀0 and error sensitivity varianceEmpirical value of parameter c

Step six: the multi-adaptive fast and weakly sensitive cubature Kalman filtering method of claim 1 is used for filtering processing to obtain the real-time asset price and market excess demand estimation value of the financial market microstructure model.

4. The method for forecasting prices according to the financial market micro-structural model of claim 3, wherein in the fourth step, when estimating the parameters, the non-uniform jump of the financial market is detected by using the following non-parameter jump method:

detecting statisticsIf the statistic is larger than the threshold value of 4.6, the jump is considered to occur, otherwise, no jump occurs;

in the above formula, the first and second carbon atoms are,wherein r is_k+1The logarithmic yield estimated from the known data is calculated by the formula:

Technical Field

The invention relates to the technical field of modeling data processing, in particular to a multiple self-adaptive fast weak sensitive volume Kalman filtering method and application thereof in financial market data processing.

Background

The Kalman filtering is a calculation method for carrying out optimal estimation on the system state by inputting and outputting observation data through the system, has important significance, and is better applied to various fields such as communication, navigation, guidance and control. In a linear steady system containing known noise, general linear Kalman filtering can be effectively used, but in a nonlinear system, because a state transition matrix cannot be linearly represented, a plurality of improved Kalman filtering techniques are generated; the application field of the method is also expanded continuously.

On the other hand, with the rapid advance of the global economy integration process, the financial market is continuously opened, which also leads to the increasingly complex change process of the financial market, the more complex the related data processing and calculation amount, the long calculation time and the difficult guarantee of the prediction timeliness; therefore, a more suitable financial market prediction model is needed to predict the occurrence of financial crisis in time and to alert the financial market to take timely counter-measures. Analyzing and researching the influence of the surplus demand and market liquidity on the asset price behavior from the microscopic view is one of important means for effectively and reasonably predicting the financial market, guiding the effective allocation of the assets, preventing the financial risk and improving the risk control capability. To realize the above means, a computer system is used for modeling data such as financial time series of a financial market (such as filter state estimation and the like); but the financial time series has the typical characteristics of nonlinearity, non-Gaussian, fluctuation time variation, model parameter uncertainty and the like. The classical extended kalman filtering method is to estimate the state by time update and measurement update, but the extended kalman filtering method is to use a first-order linear approximation nonlinear system equation; therefore, a large error may be generated during the filtering process, especially when the model has strong nonlinearity or the model parameters are not accurate, which may even cause the filtering system to diverge, thereby causing a huge amount of data calculation and a long calculation time.

In recent years, Arasaratnam proposes a volume Kalman filter, which uses a group of equal-weight volume points to propagate the mean value and variance of the system state, has higher filtering precision (Arasaratnam i., Haykin s.foundation Kalman filters. automatic Control, IEEE Transactions on,2009,54(6): 1254 + 1269.), and provides a new implementation way for the state estimation problem of a nonlinear system model; however, the volumetric kalman filter cannot exhibit a good state estimation effect even when faced with the problems of uncertain model parameters and time variation of fluctuation, and thus is difficult to be applied to processing such nonlinear system model data.

Disclosure of Invention

The invention aims to provide a multiple self-adaptive fast weak-sensitive volume Kalman filtering method to solve the technical problem that the volume Kalman filtering cannot show a better state estimation effect aiming at the problems of uncertain parameters and time-varying fluctuation of a nonlinear system model.

The invention also aims to provide a financial market microstructure model price prediction method based on multiple self-adaptive fast and weak sensitive volume Kalman filtering, so as to solve the technical problems that the existing prediction method is long in data calculation and processing time consumption, cannot give a prediction result timely and effectively, and is poor in prediction precision and satisfactory.

In order to solve the technical problems, the invention adopts the following technical scheme:

a multiple self-adaptive fast weak sensitive volume Kalman filtering method is characterized in that multiple fading factors are introduced in a step of calculating filter gain of fast weak sensitive volume Kalman filtering:

wherein l is the number of uncertain parameters, lambda_kdiag[μ₁,μ₂,…,μ₈]Is a multiple fading factor, w_i,kCorresponding to the ith uncertain parameter c_iA sensitivity weight of; w is a_i,kThe following method was used:

ac_i＜c_i＜bc_i

in the above formula, bc_iAnd ac_iRespectively correspond to c_iThe upper and lower limits of (a) are,is c_iThe mean value of (a);

the fading factor value is as follows:

wherein the content of the first and second substances,in the formula, the compound is shown in the specification,

in the formula V_kAs a residual matrix, estimated by:

the method specifically comprises the following steps:

s1: performing time updates

Let t be_k-1The state estimation value and the error variance matrix at the moment are respectivelyAndthen t is_kThe volume points at the moment are:

where the superscript "+" indicates its posterior estimate,is an error variance matrixSquare root of (1), satisfyξ_jThe j value of 2n volume points is as follows:

wherein e is_j(i-1, 2, …, n) is a unit vector with the jth value of 1 and the remaining elements are zero, i.e., e_j＝[0,…,0,1,0,…,0]^T；

Calculating the sensitivity of the rank sampling point of the step k-1:

in the formula (I), the compound is shown in the specification,sensitivity for step k-1;

volume points after nonlinear transmission are as follows:

then their corresponding prior state estimates and variances are:

in the formula, the superscript "-" represents a prior estimate of the variable;

updating a rank sampling point set:

calculating the prior state estimate and the sensitivity of the prior covariance matrix:

in the formula (I), the compound is shown in the specification,estimating the sensitivity prior for the k step;

s2: performing measurement update

Is given by priorState estimationSum varianceRecalculating the volume points:

secondly, calculating volume point positions transferred by the nonlinear measurement function:

calculating the sensitivity of the re-rank sampling point set and the sensitivity of the measurement rank sampling point

Fourthly, calculating the measurement prior estimation value, the variance of the prior estimation value and the covariance of the prior estimation value:

calculating the measured sensitivity:

calculating the sensitivity of the state and the metrology covariance and the sensitivity of the metrology variance:

s3: calculating a weak-sensitive filter gain matrix

Where l-8 is the number of uncertain parameters, λ_kdiag[μ₁,μ₂,…,μ₈]Is a multiple fading factor, w_i,kCorresponding to the ith uncertainty parameter c_iA sensitivity weight of; w is a_i,kThe following method was used:

wherein bc is_iAnd ac_iRespectively correspond to c_iThe upper and lower limits of (a) are,is c_iThe mean value of (a);

the fading factors are:

wherein the content of the first and second substances,in the formula, the compound is shown in the specification,

in the formula (24), V_kAs a residual matrix, estimated by:

s4: estimating state and variance

Calculating state estimation values and corresponding variance matrixes:

the variance matrix is:

calculating the sensitivity of state estimation and the sensitivity of state error variance matrix:

wherein the content of the first and second substances,

wherein the content of the first and second substances,is an oblique symmetric matrix satisfying gamma^TAll of ═ Γ, Ψ, and Θ are nonsingular matrices, and satisfy

The loop iterates steps S1 to S3.

The financial market microstructure model price prediction method based on the multiple self-adaptive fast weak-sensitive volume Kalman filtering comprises the following steps:

step one, establishing a microstructure model of a financial market

Wherein, P_kIs the asset price; phi is a_kThe demand for surplus in the market;λ_kis the reciprocal of market liquidity; zeta_v,kIs Bernoulli random variable and satisfies P { zeta_v,k＝1}＝v_tΔt；α₁，β₁，α₂，β₂，γ₁，γ₂，γ₃Parameters to be identified are provided with uncertainty; p is a radical of_kIs the jump amplitude, and obeys mean value of zero and variance ofNormal distribution of (2); xi_1,k,ξ_2,k,ξ_3,kAll follow a standard normal distribution and are independent of each other, Zeta_v,kThe two are mutually independent;

step two, establishing a measuring model of the financial market microstructure model with price as a measuring value

z_k＝P_k+w_k ②，

Wherein, w_kObeying a mean of zero and a variance of R_kNormal distribution of (2);

step three: state equation and measurement equation of financial market microstructure model

Selecting a state variable x_k＝[P_k,φ_k,σ_k]^TThen, there are:

z_k＝h(x_k)+w_k＝P_k+w_k ④，

wherein c ═ α₁,β₁,α₂,β₂,γ₁,γ₂,γ₃,η_p]^TIs an uncertain parameter vector; xi_k＝[ξ_1,k,ξ_2,k,ξ_3,k]^TObedience mean value is zero and variance is Q_kIs normally distributed, and

step four: estimating an uncertain parameter c in the model (3) by using the known data, and determining a parameter reference value of the model

Wherein the content of the first and second substances,and P_zz,kThe filtered measurement error and the measurement prediction variance, respectively, may be obtained by CKF filtering;

step five: initializationState of financial market micro-architectural modelAndsensitivity weight W₀＝P_cc，P_ccFor uncertainty of variance of parameter, error sensitivity S₀0 and error sensitivity varianceEmpirical value of parameter c

Step six: the multi-adaptive fast and weak sensitive volume Kalman filtering method of claim 1 is used for filtering processing to obtain the real-time asset price of the financial market micro-structure model and the estimated value of the market excess demand.

In the fourth step, when estimating the parameters, the unequal intensity jump of the financial market is detected by using the following non-parameter jump method:

detecting statisticsIf the statistic is larger than the threshold value of 4.6, the jump is considered to occur, otherwise, no jump occurs;

in the above formula, the first and second carbon atoms are,wherein r is_k+1The logarithmic yield estimated from the known data is calculated by the formula:

compared with the prior art, the invention has the main beneficial technical effects that:

1. compared with the volume Kalman filtering, the filtering method creatively introduces multiple fading factors into the fast weak sensitive volume Kalman filtering to establish the multiple self-adaptive fast weak sensitive volume Kalman filtering, so that the method can obtain better state estimation effect when facing the problems of uncertain model parameters and time variation of fluctuation; and the filtering system is easy to converge, the data processing amount is reduced, and the computing power and the memory resource occupation amount are small.

2. The filtering method is applied to the financial field (price estimation and prediction of related financial markets such as stock market, futures market and the like), can effectively solve the problems of large calculation amount, large occupied system resources and prediction precision reduction caused by uncertain influence of model parameters in the price prediction process of the microstructure model of the financial market, not only improves the price prediction precision of the microstructure model of the financial market, but also reduces the calculation time and improves the timeliness of the prediction result; the method has important guiding significance for stably promoting the continuous and healthy development of the financial market in China, better performing asset combination configuration, effectively preventing financial risks and improving the risk control and handling capacity.

Drawings

FIG. 1 is a flow chart of the multiple adaptive fast weak-sensitive volume Kalman filtering of the present invention.

FIG. 2 is a root mean square error plot of financial market prices obtained in accordance with an embodiment of the present invention.

FIG. 3 is a root mean square error plot of financial market excess demand obtained in accordance with an embodiment of the present invention.

FIG. 4 is a root mean square error plot of the liquidity of the financial market in accordance with an embodiment of the present invention.

Detailed Description

The following examples are intended to illustrate the present invention in detail and are not intended to limit the scope of the present invention in any way.

The measurement or calculation methods mentioned in the following examples are conventional methods unless otherwise specified, and the letters and symbols mentioned are conventional meanings unless otherwise specified.

Example 1: a financial market microstructure model price prediction method based on multiple self-adaptive fast weak-sensitive volume Kalman filtering comprises the following steps:

step one, establishing a microstructure model of a financial market

Consider a discrete financial market micro-structure model with an inhomogeneous poisson jumping process:

wherein, P_kFor the price of the asset, phi_kIn order to meet the excessive demand of the market,(λ_kis the inverse of market liquidity); zeta_v,kIs Bernoulli random variable and satisfies P { zeta_v,k＝1}＝v_tΔt；α₁，β₁，α₂，β₂，γ₁，γ₂，γ₃Parameters to be identified are provided with uncertainty; p is a radical of_kIs a jump amplitude and obeys a mean of zero variance ofA positive state distribution of; xi_1,k,ξ_2,k,ξ_3,kAll follow a standard normal distribution and are independent of each other, Zeta_v,kTwo are independent of each other.

Step two, establishing a measuring model of the financial market microstructure model

The price of the financial market micro-structure model is observable, so the price is selected as the measurement value, and the measurement model is:

z_k＝P_k+w_k (2)，

wherein, w_kObeying a mean of zero and a variance of R_kIs normally distributed.

Step three: state equation and measurement equation of financial market microstructure model

Selecting a state variable x_k＝[P_k,φ_k,σ_k]^TThen, there are:

z_k＝h(x_k)+w_k＝P_k+w_k (4)，

wherein c ═ α₁,β₁,α₂,β₂,γ₁,γ₂,γ₃,η_p]^TFor uncertain parameter vectors, xi_k＝[ξ_1,k,ξ_2,k,ξ_3,k]^TObedience mean value is zero and variance is Q_kIs normally distributed, and

step four: and estimating an uncertain parameter c in the model (3) by using the known data, and determining a parameter reference value of the model. In this example, the maximum likelihood function is used to estimate the maximum likelihood function, and the corresponding formula is:

wherein the content of the first and second substances,and P_zz,kThe filtered measurement error and the measurement prediction variance, respectively, may be obtained by CKF filtering.

In estimating the parameters, it is necessary to check the financial market for unequal strength jumps. The present example uses a nonparametric jump method for detection, i.e. using the detection statistics(wherein ) Whether the value exceeds a corresponding threshold value (4.6 is selected under the 1% significance level) or not is judged to judge whether jumping occurs or not; if the statistic is greater than the threshold value, the jump is considered to occur, otherwise, no jump occurs.

In the above formula r_k+1The logarithmic yield estimated from the known data is given by the formula:

step five: initializing states of financial market micro-architectural modelAndsensitivity weight W₀＝P_cc(P_ccVariance of uncertain parameter), error sensitivity S₀0 and error sensitivity varianceEmpirical value of parameter c

Performing multiple adaptive fast weak-sensitive volume kalman filtering (see fig. 1):

step six: performing time updates

Let t be_k-1The state estimation value and the error variance matrix at the moment are respectivelyAndthen t is_kThe volume points at the moment are:

where the superscript "+" indicates its posterior estimate,is an error variance matrixSquare root of (1), satisfyξ_jThe j value of 2n volume points is the value set of

Wherein e is_j(i-1, 2, …, n) is a unit vector with the jth value of 1 and the remaining elements are zero, i.e., e_j＝[0,…,0,1,0,…,0]^T；

Calculating the sensitivity of the rank sampling point of the step k-1:

in the formula (I), the compound is shown in the specification,sensitivity at step k-1.

Volume points after nonlinear transmission are as follows:

then their corresponding prior state estimates and variances are:

in the formula, the superscript "-" represents a prior estimate of the variable;

updating a rank sampling point set:

calculating the sensitivity of the prior state estimate and the prior covariance matrix

In the formula (I), the compound is shown in the specification,is the k-th step sensitivity prior estimation.

Step seven: and (3) carrying out measurement updating:

estimating from a priori stateSum varianceRecalculating the volume points:

secondly, calculating volume point positions transferred by the nonlinear measurement function:

calculating the sensitivity of the re-rank sampling point set and the sensitivity of the measurement rank sampling point

Fourthly, calculating the measurement prior estimation value, the variance of the prior estimation value and the covariance of the prior estimation value:

calculating the measured sensitivity:

calculating the sensitivity of the state and the metrology covariance and the sensitivity of the metrology variance:

step eight: calculating a weak-sensitive filter gain matrix

Where l-8 is the number of uncertain parameters, λ_kdiag[μ₁,μ₂,…,μ₈]Is a multiple fading factor, w_i,kCorresponding to the ith uncertain parameter c_iA sensitivity weight of; w is a_i,kThe following method was used:

wherein bc is_iAnd ac_iRespectively correspond to c_iThe upper and lower limits of (a) are,is c_iThe mean value of (a);

the fading factors are:

wherein the content of the first and second substances,in the formula (I), the compound is shown in the specification,

v in formula (30)_kAs a residual matrix, estimated by:

step nine: estimating state and variance

Calculating state estimation values and corresponding variance matrixes:

the variance matrix is:

calculating the sensitivity of state estimation and the sensitivity of state error variance matrix:

wherein the content of the first and second substances,

wherein the content of the first and second substances,is an oblique symmetric matrix satisfying gamma^TAll of ═ Γ, Ψ, and Θ are nonsingular matrices, and satisfy

And circularly iterating the sixth step to the ninth step to obtain the real-time asset price of the financial market microstructure model and the estimation value of the market excess demand.

Test examples

Model prediction was performed according to the procedure described in example 1

The sampling time in this experimental example was 1 day for a total of 1000 days.

The initial values of the financial market microstructure models are respectively P₀＝199.3,φ₀＝0.1489,σ₀0.1024, the initial estimates are respectivelyInitial variance matrix P₀＝diag[7.324e-4,1.055e-4,4.914e-4]And the measurement variance is R-0.22²(ii) a The parameters are respectively:

α₁＝0.0013,β₁＝-0.02,α₂＝-0.035,β₂＝-0.03,γ₁＝0.047,γ₂＝0.032,γ₃＝1,η_p＝10。

the sensitivity weights of the MAFDCKF algorithm are:

W₀＝diag[2.7040e-07,6.4000e-05,1.9600e-04,1.4400e-04]。

the test prediction results are shown in fig. 2 to 4, which are respectively the Root Mean Square Error (RMSE) of the price, the surplus demand and the market liquidity of the financial market obtained by the multiple adaptive fast and weak sensitive volumetric kalman filter algorithm (MAFDCKF) and the CKF algorithm. It can be seen that the algorithm provided by the invention has smaller RMSE, namely, the algorithm provided by the invention has higher financial market prediction precision.

According to the invention, by improving the definition of the sensitivity matrix, the analytic solution of the gain matrix is obtained, the problem of solving the matrix algebraic equation is solved, and the calculation amount of a filtering algorithm can be greatly reduced; the invention introduces multiple fading factors, provides multiple self-adaptive fast weak-sensitive volume Kalman filtering with analytical gain, and further improves the precision and robustness of the state estimation of the financial market microstructure model with uncertain parameters.

While the invention has been described in detail with reference to the drawings and examples, it will be understood by those skilled in the art that various changes in the details of the embodiments may be made without departing from the spirit of the invention, and equivalents of the related components, structures, and materials may be substituted for elements, structures, and materials to form multiple embodiments, which are common variations of the invention and will not be described in detail herein.