CN112270229A - Landslide mass displacement prediction method based on singular spectrum analysis - Google Patents

Landslide mass displacement prediction method based on singular spectrum analysis Download PDF

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CN112270229A
CN112270229A CN202011110977.0A CN202011110977A CN112270229A CN 112270229 A CN112270229 A CN 112270229A CN 202011110977 A CN202011110977 A CN 202011110977A CN 112270229 A CN112270229 A CN 112270229A
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displacement
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李丽敏
张明岳
温宗周
郭伏
张俊
何洋
魏雄伟
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Abstract

The invention discloses a landslide mass displacement prediction method based on singular spectrum analysis, which is implemented as follows: preprocessing the time sequence by using a spectrum decomposition theory and an embedded reconstruction theory of singular spectrum analysis to obtain accumulated landslide displacement data; removing the trend item displacement from the accumulated displacement to obtain periodic item displacement; adopting Gaussian fitting to carry out fitting prediction on the trend item displacement; selecting an influence factor from the predicted trend item displacement by adopting a rapid multi-principal component parallel extraction algorithm, and selecting LSTM model related parameters by utilizing a Bayesian optimization algorithm; constructing a training set, a verification set and a prediction set, and establishing LSTM network model prediction period item displacement; and according to the time series decomposition principle, superposing the predicted values of the displacement subsequences to obtain the final predicted value of the displacement, and finishing the displacement prediction method of the landslide mass. The problem that in the prior art, multi-source heterogeneous influence factors are difficult to fuse for cooperation and dynamic prediction is solved.

Description

Landslide mass displacement prediction method based on singular spectrum analysis
Technical Field
The invention belongs to the technical field of geological disaster detection and forecast, and relates to a landslide mass displacement prediction method based on singular spectrum analysis.
Background
According to 2008 + 2018 national geological disaster report annual statement (not including hong Kong, Macau special administrative district and Taiwan province) released by China's national resource department (now natural resource department), in the last decade, the total number of geological disaster events occurring in each year is 15.2 ten thousand, wherein landslide disaster events account for 66% of collapse events and 22% of collapse events on average, debris flow geological disasters account for 6% of collapse events on average, 5853 people die altogether, and direct economic loss is more than 470 billion yuan. Landslide geological disasters are closely related to life and property safety of local people and national economic construction. Therefore, only through comprehensive and profound research on landslide cause types and failure mechanisms, the landslide deformation trend can be scientifically predicted and forecasted to reduce loss in all aspects.
The process from inoculation to disaster of the landslide body is mainly influenced by two major factors, namely internal evolution (trend term) and external induction (period term). In the past prediction case, the original accumulated displacement of the landslide is often directly analyzed and predicted, the evolution mechanism in the landslide body is ignored, and the prediction result is unreliable. Therefore, analyzing and predicting the deformation accumulated displacement of the sliding mass under the condition of considering both the external induction factor and the internal evolution mechanism of the sliding mass is still an important research subject.
Influence factors of different landslide displacement decomposition items are different, and if the influence factors of the decomposition items are not considered, the influence factors are directly input into a regular model for prediction, wherein the regular model cannot reflect the influence factors of the decomposition items, so that the operation efficiency and the prediction result of the model are seriously influenced. Therefore, it is particularly critical to adopt a suitable nonlinear model for prediction in combination with its corresponding influencing factors.
Singular Spectrum Analysis (SSA) is a nonparametric spectrum analysis method for a one-dimensional time sequence, and the method mainly comprises a spectrum decomposition theory and an embedded reconstruction theory. The singular spectrum analysis can identify different signals in an original sequence through decomposition and reconstruction of a one-dimensional time sequence, and extracts a trend component, a periodic component, a noise component and the like, so that each subsequence is predicted respectively. With the advent of the 90 s of the 20 th century, more and more learners apply an artificial intelligence learning method to an intelligent system for identifying time series and establishing a nonlinear model, and the nonlinear relation between set input and output is established through continuous learning to achieve the purpose of experiments. Such as Artificial Neural Networks (ANN), machine learning methods (SVM, etc.) but they are eventually static networks, while landslide is an evolutionary process with dynamic characteristics.
A Recurrent Neural Network (RNN) can process an arbitrary sequence of inputs using an internal storage unit, thereby enabling the RNN to learn a time sequence, however, when information or time intervals between nodes become long, it is difficult for the RNN to capture a long-term time correlation, which is called "gradient vanishing". As an improved RNN structure, the LSTM inherits good sequence learning characteristics of the RNN, and controls information transmission at different times by adding a state c and introducing a gate in each hidden layer unit and utilizing a memory mechanism, so that the integral scale at different moments is dynamically changed under the condition that model parameters are fixed, the problem of gradient disappearance is avoided, and the capacity of the RNN for processing long sequence data is effectively improved.
Disclosure of Invention
The invention aims to provide a landslide mass displacement prediction method based on singular spectrum analysis, and solves the problem that in the prior art, multi-source heterogeneous influence factors are difficult to fuse for cooperation and dynamic prediction.
The technical scheme adopted by the invention is that a landslide body displacement prediction method based on singular spectrum analysis is implemented according to the following steps:
step 1, carrying out data preprocessing on a time sequence by utilizing a spectrum decomposition theory and an embedded reconstruction theory of singular spectrum analysis to obtain accumulated landslide displacement data; removing the trend item displacement from the accumulated displacement to obtain periodic item displacement;
step 2, adopting Gaussian fitting to carry out fitting prediction on the trend item displacement;
step 3, selecting an influence factor from the predicted trend item displacement by adopting a rapid multi-principal component parallel extraction algorithm, and selecting LSTM model related parameters by utilizing a Bayesian optimization algorithm; constructing a training set, a verification set and a prediction set, and establishing LSTM network model prediction period item displacement;
and 4, superposing the predicted values of the displacement subsequences according to a time sequence decomposition principle to obtain a final predicted value of the displacement, and finishing the displacement prediction method of the landslide mass.
The invention is also characterized in that:
the step 1 is implemented according to the following steps:
step 1.1: constructing a track matrix;
singular spectral analysis is based on a one-dimensional, equally-spaced sampled time series X ═ X1,x2,...,xk) And the length is N, constructing a track matrix D, calculating a track matrix of a known time data sequence according to an embedding dimension K, and constructing an M X K order track matrix X, wherein K is N-M +1, M is the window length and is an integer, and the value range is
Figure BDA0002728589830000031
The trajectory matrix X is as shown in equation (1):
Figure BDA0002728589830000032
x is a Hankel matrix with each sub-diagonal value equal, where X isi＝(xi,...,xi+M-1)T,(1≤i≤K)；
Step 1.2: singular Value Decomposition (SVD);
defining matrix S-XXT，XTFor the transposed matrix of X, let λ1,...,λMIs a characteristic value of the matrix S, U1,...,UMAre each lambda1,...,λMCorresponding feature vector, where1≥...≥λMNot less than 0; let d ═ rank (x),
Figure BDA0002728589830000033
the trajectory matrix X can be expressed as shown in equation (2):
X＝X1+X2+...+Xd (2)
elementary matrix
Figure BDA0002728589830000041
rank(Xi)＝1，XiAnd X have the same matrix structure; u is the left singular vector of X, V is the right singular vector of X,
Figure BDA0002728589830000042
is the characteristic value of X and the characteristic value of X,
Figure BDA0002728589830000043
a singular spectrum called matrix X;
wherein
Figure BDA0002728589830000044
And | | | X | | non-conducting phosphor2＝λiAnd can thus define
Figure BDA0002728589830000045
Is a matrix XiThe rate of contribution of (a) to (b),
Figure BDA0002728589830000046
is the previous XiThe contribution rate of (c);
step 1.3: grouping;
partitioning the elementary matrix { 1. -, d } into m disjoint subsets I1,I2,...ImWherein I ═ { I ═ I1,...,ip}. The singular value decomposition of the trajectory matrix X can be expressed as:
Figure BDA00027285898300000411
the grouping being to determine I1,I2,...ImThe process of (2);
synthesizing matrix XIThe contribution ratio of (a) is expressed by equation (3):
Figure BDA0002728589830000047
step 1.4: carrying out diagonal averaging;
converting the matrix obtained by grouping into a series of new reconstruction components with the length of N; superposing all reconstruction components to obtain an original sequence, and defining that Z is XIk，z1,z2,...zNFor the sequence of Z obtained by diagonal averaging, let M ═ min (M, K), K ═ max (M, K) and N ═ M + K-1, if M is<K, then
Figure BDA0002728589830000048
Otherwise
Figure BDA0002728589830000049
The formula for diagonal averaging is expressed as equation (4):
Figure BDA00027285898300000410
specifically, data preprocessing is carried out on an original time sequence in matlab, the influence of random fluctuation items on an experimental result is eliminated, and then a landslide displacement nonlinear trend item sequence is extracted based on an SSA algorithm;
step 1.5: extracting based on SSA nonlinear trend;
time series X ═ X1,x2,......xNThe fourier expression of (a) is shown in equation (5):
Figure BDA0002728589830000051
wherein k belongs to N, N is more than or equal to 0 and less than or equal to N-1, and c is an odd number when N isN/20; periodic diagram of X at frequency ω k/NCan be defined as shown in formula (6):
Figure BDA0002728589830000052
for a given ω0E (0,0.5), interval [0, omega ∈0]Can be defined as shown in equation (7):
Figure BDA0002728589830000053
in the formula
Figure BDA0002728589830000054
Can be defined as the cumulative contribution of frequency 0, omega](ii) a Then selecting a characteristic vector meeting the following conditions for trend extraction, thereby selecting a characteristic function corresponding to low-frequency fluctuation;
C(Uj,ω0)≥C0 (8)
wherein C is0And ω0Is two parameters omega0∈(0,0.5)，C0∈[0,1]，UjIs the corresponding feature vector;
Figure BDA0002728589830000055
where X is a time series taking into account the length N,
Figure BDA0002728589830000056
is the median of the periodogram values of X. Low frequency contribution (C)0) The optimum value of (c) can be determined using equation (10):
Figure BDA0002728589830000057
Figure BDA0002728589830000058
T(ω0，C0) Is derived from the parameter C0And ω0The trend component extracted; Δ C is the search step, Δ R is the threshold; and selecting a characteristic function which meets the periodogram low-frequency contribution criterion and is larger than the optimal value of the low-frequency contribution as a trend component.
The step 2 is implemented according to the following steps:
adopting Gaussian fitting to perform function approximation fitting on the extracted values of the trend item displacement subsequence to obtain a time-displacement curve trend graph;
the calculation formula is as follows:
Figure BDA0002728589830000061
wherein y is0、xcW, A are 4 parameters, y0Represents a base line, A represents an area, w represents a full width at half maximum, and xcRepresents the peak position, PiRepresenting the circumferential ratio pi. Iterative calculation is carried out according to the formula 11, and fitting prediction is carried out on the trend term sequence obtained by extraction.
Step 3 is specifically implemented according to the following steps:
step 3.1: the method adopts a rapid multi-principal component parallel extraction algorithm to select the influence factors, and comprises the following specific steps:
for the linear neural network model, y (k) ═ WT(k)x(k) (12)
Wherein x (k) and y (k) are inputs and outputs of the neural network, respectively; wTA weight matrix that is a neural network;
wherein x (k) e Rn×1，y(k)＝Rr×1，W(k)∈Rn×rR is the autocorrelation matrix of the vector, n is the dimension of the input vector, and R is the dimension of the extracted principal component. Make the autocorrelation matrix of the input
Figure BDA0002728589830000062
The eigenvalue and eigenvector of R are respectively lambdaiAnd ui(i 1, 2.. times.n), since R is a symmetric positive definite matrix, the eigenvalue λ isiAnd (3) when the ratio is more than 0, decomposing the characteristic value of R to obtain:
R＝UΛUT (13)
wherein U is [ U ]1,u2,...,un]，Λ＝diag{λ1,λ2,...,λnAnd the characteristic values satisfy the following relationship:
λ1＞λ2＞...λr＞...＞λn＞0 (14)
the first R principal components of the matrix R are eigenvalues λiCorresponding eigenvector uiThe space generated by these principal components is called the principal subspace; the FMPCE seeks a suitable weight matrix iterative update equation so that the weight matrix can converge to the first R principal components of the matrix R. The algorithm form is as follows:
W(k+1)＝W(k)+η[W(k)C(k)+(E(k)A2-F(k)A)] (15)
wherein, the matrix A is an r × r diagonal matrix, and η is a learning rate. C (k) ═ w (k) ((w (k))TW(k))-1I) is a non-second-order matrix, and the introduction of C not only solves the problem of algorithm instability, but also improves the algorithm convergence speed;
E(k)＝RW(k)WTW(k)，F(k)＝W(k)AWT(k) RW (k) the autocorrelation matrix may be estimated by:
Figure BDA0002728589830000071
wherein the forgetting factor α ∈ (0,1), and when k → ∞ is satisfied, the matrix
Figure BDA0002728589830000072
Estimating the autocorrelation matrix by using the formula (16), and then extracting the input principal component by using the formula (13) and the formula (15);
step 3.2: selecting LSTM model relevant parameters by adopting a Bayesian optimization algorithm;
step 3.3: establishing an LSTM network model to predict periodic item displacement;
step 3.2 is specifically carried out as follows: and taking the parameters as independent variables, taking the accuracy of the test set as a target function, and setting the optimal result as the parameters.
Step 3.3 is specifically carried out as follows:
an LSTM landslide displacement dynamic prediction model is realized in Python by taking open-source machine learning library Tensorflow as a rear end and constructing a Keras package, wherein the structure diagram of the LSTM model is shown in FIG. 6:
beginning: load data [ ]
Initialization: each layer weight W (input layer, hidden layer, output layer), learning rate n, expected accuracy e, number of input layer nodes, number of hidden layer nodes
Given a target output h
Calculating the output value f of each neuron in forward directiont、it、ct、ot、ht
And reversely calculating the error term delta value of each neuron, wherein the error term delta value comprises two directions:
a. counter-propagation in time: i.e. the error term delta at time t-1 is calculatedt-1：
Figure BDA0002728589830000081
b. The error term is propagated up one layer: assuming that the current layer is the l-th layer, the error term defining the l-1 layer is the derivative of the error function to the weighted input of the l-1 layer, i.e.
Figure BDA0002728589830000082
And (3) updating the weight: calculating the gradient W of each weightfh、Wih、Wch、Woh、bf、bi、bc、bo、Wfx、Wix、Wcx、Wox。
And (3) final output: h ist。
The invention has the beneficial effects that: the invention discloses a landslide mass displacement prediction method based on singular spectrum analysis, which solves the problem that in the prior art, cooperation and dynamic prediction are difficult to perform by fusing multi-source heterogeneous influence factors. The time sequence is subjected to noise reduction treatment, so that the influence of random fluctuation of the time sequence on a prediction result can be effectively reduced, and the prediction accuracy is improved. Meanwhile, the long-short time memory network (LSTM) is used as a deep machine learning neural network, has the advantages of timing concept, solving the problems of easy falling into local minimum value, gradient deletion and the like inherent in the neural network, and has strong generalization capability and robust performance, and the final coupling value also shows that the combined model has higher precision.
Drawings
FIG. 1 is a technical route of a method for predicting displacement of a sliding mass based on singular spectrum analysis according to the present invention;
FIG. 2 is a comparison graph before and after SSA-based landslide accumulated displacement data reconstruction of the landslide body displacement prediction method based on singular spectrum analysis;
FIG. 3 is a graph of SSA-based trend term displacement extraction and Gaussian fitting trend term displacement prediction for the landslide mass displacement prediction method based on singular spectrum analysis;
FIG. 4 is a graph of the relationship between landslide influence factor dimensionality and prediction accuracy of a landslide mass displacement prediction method based on singular spectrum analysis;
FIG. 5 is a graph of accumulated displacement of a research area in the landslide body displacement prediction method based on singular spectrum analysis according to the present invention;
FIG. 6 is a diagram of an LSTM neural network structure involved in the landslide mass displacement prediction method based on singular spectral analysis according to the present invention;
FIG. 7 is a cycle term displacement extraction value of the landslide body displacement prediction method based on singular spectrum analysis according to the present invention;
FIG. 8 is a training error-iteration diagram of an LSTM landslide displacement dynamic prediction model in the landslide mass displacement prediction method based on singular spectrum analysis of the present invention;
FIG. 9 is a diagram showing the result of training the dynamic prediction model of LSTM landslide displacement in the method for predicting landslide body displacement based on singular spectrum analysis according to the present invention.
FIG. 10 is a diagram of the result of predicting the accumulated displacement of the landslide based on the SSA-LSTM combination in the method for predicting the displacement of the landslide body based on the singular spectrum analysis of the present invention.
Detailed Description
The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments.
The invention relates to a method for predicting displacement of a landslide mass based on singular spectrum analysis, which is implemented according to the following steps as shown in figure 1:
step 1, preprocessing time series by using a spectrum decomposition theory and an embedded reconstruction theory of Singular Spectrum Analysis (SSA) to obtain accumulated landslide displacement data; the method aims to eliminate noise in the sequence and reduce the influence of random fluctuation on the accuracy of a prediction analysis result. And selecting a characteristic function which meets the periodogram low-frequency contribution criterion and is larger than the optimal value of the low-frequency contribution as a trend component. Removing the trend item displacement from the accumulated displacement to obtain periodic item displacement;
the step 1 is implemented according to the following steps:
step 1.1: constructing a track matrix;
singular spectral analysis is based on a one-dimensional, equally-spaced sampled time series X ═ X1,x2,...,xk) And the length is N, constructing a track matrix D, calculating a track matrix of a known time data sequence according to an embedding dimension K, and constructing an M X K order track matrix X, wherein K is N-M +1, M is the window length and is an integer, and the value range is
Figure BDA0002728589830000101
The trajectory matrix X is as shown in equation (1):
Figure BDA0002728589830000102
x is a Hankel matrix with each sub-diagonal value equal, where X isi＝(xi,...,xi+M-1)T,(1≤i≤K)；
Step 1.2: singular Value Decomposition (SVD);
defining matrix S-XXT，XTFor the transposed matrix of X, let λ1,...,λMIs a characteristic value of the matrix S, U1,...,UMAre each lambda1,...,λMCorresponding feature vector, where1≥...≥λMNot less than 0; let d ═ rank (x),
Figure BDA0002728589830000103
the trajectory matrix X can be expressed as shown in equation (2):
X＝X1+X2+...+Xd (2)
elementary matrix
Figure BDA0002728589830000104
rank(Xi)＝1，XiAnd X have the same matrix structure; u is the left singular vector of X, V is the right singular vector of X,
Figure BDA0002728589830000105
is the characteristic value of X and the characteristic value of X,
Figure BDA0002728589830000106
a singular spectrum called matrix X;
wherein
Figure BDA0002728589830000107
And | | | X | | non-conducting phosphor2＝λiAnd can thus define
Figure BDA0002728589830000108
Is a matrix XiThe rate of contribution of (a) to (b),
Figure BDA0002728589830000109
is the previous XiThe contribution rate of (c);
step 1.3: grouping;
partitioning the elementary matrix { 1. -, d } into m disjoint subsets I1,I2,...ImWherein I ═ { I ═ I1,...,ip}. The singular value decomposition of the trajectory matrix X can be expressed as:
Figure BDA0002728589830000111
the grouping being to determine I1,I2,...ImThe process of (2);
synthesizing matrix XIThe contribution ratio of (a) is expressed by equation (3):
Figure BDA0002728589830000112
step 1.4: carrying out diagonal averaging;
converting the matrix obtained by grouping into a series of new reconstruction components with the length of N; superposing all reconstruction components to obtain an original sequence, and defining that Z is XIk，z1,z2,...zNFor the sequence of Z obtained by diagonal averaging, let M ═ min (M, K), K ═ max (M, K) and N ═ M + K-1, if M is<K, then
Figure BDA0002728589830000113
Otherwise
Figure BDA0002728589830000114
The formula for diagonal averaging is expressed as equation (4):
Figure BDA0002728589830000115
as shown in fig. 2, specifically, data preprocessing is performed on an original time sequence in matlab, the influence of random fluctuation terms on an experimental result is eliminated, and then a landslide displacement nonlinear trend term sequence is extracted based on an SSA algorithm;
step 1.5: extracting based on SSA nonlinear trend;
time series X ═ X1,x2,......xNThe fourier expression of (a) is shown in equation (5):
Figure BDA0002728589830000116
wherein k belongs to N, N is more than or equal to 0 and less than or equal to N-1, and c is an odd number when N isN/20; the periodogram of X at frequency ω ═ k/N can be defined as shown in equation (6):
Figure BDA0002728589830000121
for a given ω0E (0,0.5), interval [0, omega ∈0]Can be defined as shown in equation (7):
Figure BDA0002728589830000122
in the formula
Figure BDA0002728589830000123
Can be defined as the cumulative contribution of frequency 0, omega](ii) a Then selecting a characteristic vector meeting the following conditions for trend extraction, thereby selecting a characteristic function corresponding to low-frequency fluctuation;
C(Uj,ω0)≥C0 (8)
wherein C is0And ω0Is two parameters omega0∈(0,0.5)，C0∈[0,1]，UjIs the corresponding feature vector;
Figure BDA0002728589830000124
where X is a time series taking into account the length N,
Figure BDA0002728589830000125
is the median of the periodogram values of X. Low frequency contribution (C)0) The optimum value of (c) can be determined using equation (10):
Figure BDA0002728589830000126
Figure BDA0002728589830000127
T(ω0，C0) Is derived from the parameter C0And ω0The trend component extracted; Δ C is the search step, Δ R is the threshold; is selected to be fullAnd a characteristic function of the foot cycle chart with the low-frequency contribution criterion larger than the optimal value of the low-frequency contribution is taken as a trend component. The extracted value of the nonlinear trend term subsequence is shown in figure 3;
step 2, adopting Gaussian fitting to carry out fitting prediction on the trend item displacement; and obtaining a time-displacement curve trend graph.
The step 2 is implemented according to the following steps:
adopting Gaussian fitting to perform function approximation fitting on the extracted values of the trend item displacement subsequence to obtain a time-displacement curve trend graph;
the calculation formula is as follows:
Figure BDA0002728589830000131
wherein y is0、xcW, A are 4 parameters, y0Represents a base line, A represents an area, w represents a full width at half maximum, and xcRepresents the peak position, PiRepresenting the circumferential ratio pi. Iterative calculation is carried out according to the formula 11, and fitting prediction is carried out on the trend term sequence obtained by extraction. The time-displacement curve trend is shown in fig. 3.
Step 3, selecting an influence factor from the predicted trend item displacement by adopting a rapid multi-principal component parallel extraction algorithm (FMPCE), and selecting relevant parameters of the LSTM model by utilizing a Bayesian optimization algorithm; constructing a training set, a verification set and a prediction set, and establishing LSTM network model prediction period item displacement; and the multi-influence factor collaborative prediction is realized by a rapid multi-principal component parallel extraction algorithm.
Step 3 is specifically implemented according to the following steps:
step 3.1: the method adopts a rapid multi-principal component parallel extraction algorithm to select the influence factors, and comprises the following specific steps:
for the linear neural network model, y (k) ═ WT(k)x(k) (12)
Wherein x (k) and y (k) are inputs and outputs of the neural network, respectively; wTA weight matrix that is a neural network;
wherein x (k) e Rn×1，y(k)＝Rr×1，W(k)∈Rn×rR is the autocorrelation matrix of the vector, n is the dimension of the input vector, and R is the dimension of the extracted principal component. Make the autocorrelation matrix of the input
Figure BDA0002728589830000132
The eigenvalue and eigenvector of R are respectively lambdaiAnd ui(i 1, 2.. times.n), since R is a symmetric positive definite matrix, the eigenvalue λ isiAnd (3) when the ratio is more than 0, decomposing the characteristic value of R to obtain:
R＝UΛUT (13)
wherein U is [ U ]1,u2,...,un]，Λ＝diag{λ1,λ2,...,λnAnd the characteristic values satisfy the following relationship:
λ1＞λ2＞...λr＞...＞λn＞0 (14)
the first R principal components of the matrix R are eigenvalues λiCorresponding eigenvector uiThe space generated by these principal components is called the principal subspace; the FMPCE seeks a suitable weight matrix iterative update equation so that the weight matrix can converge to the first R principal components of the matrix R. The algorithm form is as follows:
W(k+1)＝W(k)+η[W(k)C(k)+(E(k)A2-F(k)A)] (15)
wherein, the matrix A is an r × r diagonal matrix, and η is a learning rate. C (k) ═ w (k) ((w (k))TW(k))-1I) is a non-second-order matrix, and the introduction of C not only solves the problem of algorithm instability, but also improves the algorithm convergence speed;
E(k)＝RW(k)WTW(k)，F(k)＝W(k)AWT(k) RW (k) the autocorrelation matrix may be estimated by:
Figure BDA0002728589830000141
wherein the forgetting factor α ∈ (0,1), and when k → ∞ is satisfied, the matrix
Figure BDA0002728589830000142
Use formula (16)Estimating an autocorrelation matrix, and then extracting the input principal component by using an equation (13) and an equation (15); the relationship between the influence factor dimension and the prediction accuracy is shown in fig. 4:
step 3.2: selecting LSTM model relevant parameters by adopting a Bayesian optimization algorithm;
step 3.2 is specifically carried out as follows: and taking the parameters as independent variables, taking the accuracy of the test set as a target function, and setting the optimal result as the parameters.
Step 3.3: establishing an LSTM network model to predict periodic item displacement;
step 3.3 is specifically carried out as follows:
an LSTM landslide displacement dynamic prediction model is realized in Python by taking open-source machine learning library Tensorflow as a rear end and constructing a Keras package, wherein the structure diagram of the LSTM model is shown in FIG. 6:
beginning: load data [ ]
Initialization: each layer weight W (input layer, hidden layer, output layer), learning rate n, expected accuracy e, number of input layer nodes, number of hidden layer nodes
Given a target output h
Calculating the output value f of each neuron in forward directiont、it、ct、ot、ht
And reversely calculating the error term delta value of each neuron, wherein the error term delta value comprises two directions:
a. counter-propagation in time: i.e. the error term delta at time t-1 is calculatedt-1：
Figure BDA0002728589830000151
b. The error term is propagated up one layer: assuming that the current layer is the l-th layer, the error term defining the l-1 layer is the derivative of the error function to the weighted input of the l-1 layer, i.e.
Figure BDA0002728589830000152
And (3) updating the weight: calculating the gradient W of each weightfh、Wih、Wch、Woh、bf、bi、bc、bo、Wfx、Wix、Wcx、Wox。
And (3) final output: h ist。
And 4, superposing the predicted values of the displacement subsequences according to a time sequence decomposition principle to obtain a final predicted value of the displacement, and finishing the displacement prediction method of the landslide mass.
Examples
1) A cumulative displacement map of the study area; as shown in fig. 5. The original displacement time sequence data set of the landslide research area is used as SSA algorithm input, the influence of random fluctuation items on an experiment result is eliminated, and the algorithm processes data in a front-back mode, such as the mode shown in figure 2. And extracting a trend item displacement subsequence based on an SSA algorithm to obtain a time-displacement curve trend graph.
2) And adopting Gaussian fitting to perform nonlinear curve approximation fitting on the trend term displacement extraction value, and converging the fitting when the iteration times reach 400 to achieve a Chi-sqr tolerance value of 1E-9, so that the accuracy is higher and the repeatability is good. Fitting the relevant parameter configuration and trend term time-displacement prediction curves as shown in FIG. 3, scaling the standard error using the Reduced Chi-Sqr root, and adjusting R20.98235. The experimental result shows that the average relative error of the prediction curve is 0.853%, and the root mean square error is 36.0632 mm. The predicted values are shown in Table 1.
The gaussian fit predictions (table 1) are as follows:
table 1 trend term displacement gaussian fit results
Figure BDA0002728589830000161
The fitting results plot (fig. 3) and fitting-related parameters (table 2) are as follows:
TABLE 1 Gaussian fitting correlation parameters
Figure BDA0002728589830000162
Figure BDA0002728589830000171
The standard error is scaled using the root of the Reduced Chi-Sqr, where sigma, FWHM, Height are the derived parameters.
3) A Fast multiple principal components parallel extraction algorithm (FMPCE) is adopted to screen and verify 10 multi-source heterogeneous evoked influence factors (table 3).
TABLE 3 list of influencing factors
Figure BDA0002728589830000172
4) As can be seen from FIG. 4, as the dimension of the influence factor increases, the prediction accuracy of the model gradually increases, and when the dimension reaches 6, the prediction accuracy of the model reaches 96.51%, and the model enters a stable stage. And then, the influence factors are continuously increased, and the accuracy of model prediction is not obviously changed. Finally, the rainfall x1(mm), ground water level x2(mm), soil moisture content x3(%), slope x of hillside4(
Figure BDA0002728589830000182
) Pore water pressure x5(kPa), displacement x of the period term6(mm), 6 influence factors are used as input vectors of the LSTM model, and the period term prediction displacement is output.
In order to prevent the influence of the inconsistency of the data type, the value range and the dimension of the influence factors on the network training speed, the original sequence is processed by normalization to obtain an interval [0, 1] as shown in the formula (17).
Figure BDA0002728589830000181
Wherein is provided with Pm*nOriginal landslide dataset, P, constructed for a landslide prediction modelm*nFor the normalized data set, max (P) is Pm*nMin (P) is Pm*nA medium to minimum value; min (T) and max (T) represent the maximum and minimum values, respectively, of the desired normalization interval target.
5) Optimizing parameters by adopting a Bayesian optimization algorithm to obtain the optimal hidden layer node number: and determining that the length of the optimal input sequence is 12, the number of nodes of the hidden layer is 18, and the learning rate is 0.001.
6) Removing the trend item displacement subsequence from the accumulated displacement sequence to obtain periodic item displacement, and inputting accumulated displacement data after removing the trend item into an LSTM landslide displacement dynamic prediction model for training; first step forward computation of each neuron ft、it、ct、ot、htThe output value of (d); the second step reversely calculates the error term value of each neuron, and the error term value comprises two directions: a. backward propagation along time (from the current t moment, calculating an error term at each moment); b. propagating the error term up one layer; and thirdly, calculating the gradient of each weight according to the corresponding error term and calculating the final displacement.
7) The samples were divided into the first 70% as training set and the last 30% as prediction set. In the training set, the first 60% of the data is used to train the model, and the last 40% of the data is used to verify the predictive reliability of the model. The periodic term displacement extraction value is shown in fig. 7, the training error-iteration of the LSTM neural network model is shown in fig. 8, the preset expected precision in the training is 0.0001, and the learning rate is 0.001. When the number of iteration steps is about 2000 steps, the error of the LSTM training result is about 0.0000732, and the preset desired accuracy is satisfied.
The accuracy and training results of the LSTM period dynamic displacement prediction model are shown in Table 4 and FIG. 9.
Figure BDA0002728589830000191
6) According to the time series decomposition principle, the predicted values of the trend item subsequence and the periodic item subsequence are superposed to obtain a predicted value curve of displacement, which is shown in fig. 10. The results show that: the prediction curve of the coupling model can have higher goodness of fit with the actual curve in the period that the displacement is greatly fluctuated by external influence factors in 5-9 months in rainy season, and therefore, the LSTM dynamic neural network can well represent the deformation evolution process of the landslide body along with the fluctuation of the external influence factors. In the longitudinal time sequence 1983, the overall average relative error of the model is 0.607%, the root mean square error RMSE is 26.317mm, and the change trend of the landslide displacement can be well predicted overall. The useful information of the original sequence can be well separated by the SSA decomposition algorithm, the obtained sequence can more easily extract the information for predicting the model learning rule, and the coupling model can be predicted more accurately. The coupling model predicted results are shown in table 5.
TABLE 5 results of displacement prediction based on singular spectral analysis and LSTM combined model
Figure BDA0002728589830000201
Aiming at the complex nonlinear evolution process of landslide, a Singular Spectrum Analysis (SSA) is adopted to perform noise reduction, decomposition and extraction processing on a displacement time sequence of newly shared landslide in China, and a deep learning dynamic prediction model (LSTM) is utilized to predict the displacement of a periodic item of the newly shared landslide, so that the combined model has the following advantages:
(a) and (3) decomposing the displacement time sequence by SSA, accurately extracting trend and periodic components from the univariate time sequence data, effectively reducing the influence of random fluctuation of the time sequence on a prediction result, and improving the accuracy of model prediction.
(b) The LSTM, as a novel multi-layer neural network learning algorithm, has a time sequence concept and can overcome the defects that a traditional machine learning model is easy to fall into local minimum values, gradient deficiency and the like to a certain extent. The LSTM as a deep learning model can bypass units so as to remember longer time steps, avoid endless continuous multiplication by a method of multiplication while adding, and solve the problem of disappearance of RNN gradient to a certain extent.

Claims (6)

1. A landslide mass displacement prediction method based on singular spectrum analysis is characterized by comprising the following steps:
step 1, carrying out data preprocessing on a time sequence by utilizing a spectrum decomposition theory and an embedded reconstruction theory of singular spectrum analysis to obtain accumulated landslide displacement data; removing the trend item displacement from the accumulated displacement to obtain periodic item displacement;
step 2, adopting Gaussian fitting to carry out fitting prediction on the trend item displacement;
step 3, selecting an influence factor from the predicted trend item displacement by adopting a rapid multi-principal component parallel extraction algorithm, and selecting LSTM model related parameters by utilizing a Bayesian optimization algorithm; constructing a training set, a verification set and a prediction set, and establishing LSTM network model prediction period item displacement;
and 4, superposing the predicted values of the displacement subsequences according to a time sequence decomposition principle to obtain a final predicted value of the displacement, and finishing the displacement prediction method of the landslide mass.
2. The singular spectrum analysis-based landslide body displacement prediction method according to claim 1, wherein step 1 is specifically implemented according to the following steps:
step 1.1: constructing a track matrix;
singular spectral analysis is based on a one-dimensional, equally-spaced sampled time series X ═ X1,x2,...,xk) And the length is N, constructing a track matrix D, calculating a track matrix of a known time data sequence according to an embedding dimension K, and constructing an M X K order track matrix X, wherein K is N-M +1, M is the window length and is an integer, and the value range is
Figure FDA0002728589820000011
The trajectory matrix X is as shown in equation (1):
Figure FDA0002728589820000012
x is a Hankel matrix with each sub-diagonal value equal, where X isi＝(xi,...,xi+M-1)T,(1≤i≤K)；
Step 1.2: singular Value Decomposition (SVD);
defining matrix S-XXT，XTFor the transposed matrix of X, let λ1,...,λMIs a characteristic value of the matrix S, U1,...,UMAre each lambda1,...,λMCorresponding feature vector, where1≥...≥λMNot less than 0; let d ═ rank (x),
Figure FDA0002728589820000021
the trajectory matrix X can be expressed as shown in equation (2):
X＝X1+X2+...+Xd (2)
elementary matrix
Figure FDA0002728589820000022
rank(Xi)＝1，XiAnd X have the same matrix structure; u is the left singular vector of X, V is the right singular vector of X,
Figure FDA0002728589820000023
is the characteristic value of X and the characteristic value of X,
Figure FDA0002728589820000024
a singular spectrum called matrix X;
wherein
Figure FDA0002728589820000025
And | | | X | | non-conducting phosphor2＝λiAnd can thus define
Figure FDA0002728589820000026
Is a matrix XiThe rate of contribution of (a) to (b),
Figure FDA0002728589820000027
is the previous XiThe contribution rate of (c);
step 1.3: grouping;
mapping the elementary matrix1, d is divided into m disjoint subsets I1,I2,...ImWherein I ═ { I ═ I1,...,ip}. The singular value decomposition of the trajectory matrix X can be expressed as:
Figure FDA0002728589820000028
the grouping being to determine I1,I2,...ImThe process of (2);
synthesizing matrix XIThe contribution ratio of (a) is expressed by equation (3):
Figure FDA0002728589820000029
step 1.4: carrying out diagonal averaging;
converting the matrix obtained by grouping into a series of new reconstruction components with the length of N; overlapping all the reconstructed components to obtain an original sequence defined as
Figure FDA00027285898200000210
z1,z2,...zNFor the sequence of Z obtained by diagonal averaging, let M ═ min (M, K), K ═ max (M, K) and N ═ M + K-1, if M is<K, then
Figure FDA0002728589820000031
Otherwise
Figure FDA0002728589820000032
The formula for diagonal averaging is expressed as equation (4):
Figure FDA0002728589820000033
specifically, data preprocessing is carried out on an original time sequence in matlab, the influence of random fluctuation items on an experimental result is eliminated, and then a landslide displacement nonlinear trend item sequence is extracted based on an SSA algorithm;
step 1.5: extracting based on SSA nonlinear trend;
time series X ═ X1,x2,......xNThe fourier expression of (a) is shown in equation (5):
Figure FDA0002728589820000034
wherein k belongs to N, N is more than or equal to 0 and less than or equal to N-1, and c is an odd number when N isN/20; the periodogram of X at frequency ω ═ k/N can be defined as shown in equation (6):
Figure FDA0002728589820000035
for a given ω0E (0,0.5), interval [0, omega ∈0]Can be defined as shown in equation (7):
Figure FDA0002728589820000036
in the formula
Figure FDA0002728589820000037
Can be defined as the cumulative contribution of frequency 0, omega](ii) a Then selecting a characteristic vector meeting the following conditions for trend extraction, thereby selecting a characteristic function corresponding to low-frequency fluctuation;
C(Uj,ω0)≥C0 (8)
wherein C is0And ω0Is two parameters omega0∈(0,0.5)，C0∈[0,1]，UjIs the corresponding feature vector;
Figure FDA0002728589820000041
where X is a time series taking into account the length N,
Figure FDA0002728589820000042
is the median of the periodogram values of X. Low frequency contribution (C)0) The optimum value of (c) can be determined using equation (10):
Figure FDA0002728589820000043
Figure FDA0002728589820000044
T(ω0，C0) Is derived from the parameter C0And ω0The trend component extracted; Δ C is the search step, Δ R is the threshold; and selecting a characteristic function which meets the periodogram low-frequency contribution criterion and is larger than the optimal value of the low-frequency contribution as a trend component.
3. The singular spectrum analysis-based landslide body displacement prediction method according to claim 1, wherein the step 2 is implemented specifically according to the following steps:
adopting Gaussian fitting to perform function approximation fitting on the extracted values of the trend item displacement subsequence to obtain a time-displacement curve trend graph;
the calculation formula is as follows:
Figure FDA0002728589820000045
wherein y is0、xcW, A are 4 parameters, y0Represents a base line, A represents an area, w represents a full width at half maximum, and xcRepresents the peak position, PiRepresenting the circumferential ratio pi. Iterative calculation is carried out according to the formula 11, and fitting prediction is carried out on the trend term sequence obtained by extraction.
4. The singular spectrum analysis-based landslide body displacement prediction method according to claim 1, wherein step 3 is implemented specifically according to the following steps:
step 3.1: the method adopts a rapid multi-principal component parallel extraction algorithm to select the influence factors, and comprises the following specific steps:
for the linear neural network model, y (k) ═ WT(k)x(k) (12)
Wherein x (k) and y (k) are inputs and outputs of the neural network, respectively; wTA weight matrix that is a neural network;
wherein x (k) e Rn×1，y(k)＝Rr×1，W(k)∈Rn×rR is the autocorrelation matrix of the vector, n is the dimension of the input vector, and R is the dimension of the extracted principal component. Make the autocorrelation matrix of the input
Figure FDA0002728589820000053
The eigenvalue and eigenvector of R are respectively lambdaiAnd ui(i 1, 2.. times.n), since R is a symmetric positive definite matrix, the eigenvalue λ isiAnd (3) when the ratio is more than 0, decomposing the characteristic value of R to obtain:
R＝UΛUT (13)
wherein U is [ U ]1,u2,...,un]，Λ＝diag{λ1,λ2,...,λnAnd the characteristic values satisfy the following relationship:
λ1＞λ2＞...λr＞...＞λn＞0 (14)
the first R principal components of the matrix R are eigenvalues λiCorresponding eigenvector uiThe space generated by these principal components is called the principal subspace; the FMPCE seeks a suitable weight matrix iterative update equation so that the weight matrix can converge to the first R principal components of the matrix R. The algorithm form is as follows:
W(k+1)＝W(k)+η[W(k)C(k)+(E(k)A2-F(k)A)] (15)
wherein, the matrix A is an r × r diagonal matrix, and η is a learning rate. C (k) ═ w (k) ((w (k))TW(k))-1I) is a non-second-order matrix, and the introduction of C not only solves the problem of algorithm instability, but also improves the algorithm convergence speed;
E(k)＝RW(k)WTW(k)，F(k)＝W(k)AWT(k) RW (k) the autocorrelation matrix may be estimated by:
Figure FDA0002728589820000051
wherein the forgetting factor α ∈ (0,1), and when k → ∞ is satisfied, the matrix
Figure FDA0002728589820000052
Estimating the autocorrelation matrix by using the formula (16), and then extracting the input principal component by using the formula (13) and the formula (15);
step 3.2: selecting LSTM model relevant parameters by adopting a Bayesian optimization algorithm;
step 3.3: and establishing an LSTM network model to predict the displacement of the period item.
5. The singular spectrum analysis-based landslide body displacement prediction method according to claim 4, wherein step 3.2 is specifically implemented as follows: and taking the parameters as independent variables, taking the accuracy of the test set as a target function, and setting the optimal result as the parameters.
6. The singular spectrum analysis-based landslide body displacement prediction method according to claim 4, wherein step 3.3 is specifically implemented as follows:
an LSTM landslide displacement dynamic prediction model is realized in Python by taking open-source machine learning library Tensorflow as a rear end and constructing a Keras package:
beginning: load data [ ]
Initialization: each layer weight W (input layer, hidden layer, output layer), learning rate n, expected accuracy e, number of input layer nodes, number of hidden layer nodes
Given a target output h
Calculating the output value f of each neuron in forward directiont、it、ct、ot、ht
And reversely calculating the error term delta value of each neuron, wherein the error term delta value comprises two directions:
a. counter-propagation in time: i.e. the error term delta at time t-1 is calculatedt-1：
Figure FDA0002728589820000061
b. The error term is propagated up one layer: assuming that the current layer is the l-th layer, the error term defining the l-1 layer is the derivative of the error function to the weighted input of the l-1 layer, i.e.
Figure FDA0002728589820000062
And (3) updating the weight: calculating the gradient W of each weightfh、Wih、Wch、Woh、bf、bi、bc、bo、Wfx、Wix、Wcx、Wox。
And (3) final output: h ist。
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