CN105354644A - Financial time series prediction method based on integrated empirical mode decomposition and 1-norm support vector machine quantile regression - Google Patents

Abstract

The invention belongs to the field of financial risk management, in particular to a time series probability distribution prediction method based on integrated empirical mode decomposition and nonlinear quantile regression. The method comprises the following steps: firstly, carrying out the integrated empirical mode decomposition on a financial price time series to obtain components with high regularity under different scales; secondly, independently predicting each component by 1-norm support vector machine quantile regression to obtain all quantile prediction results of each component; and thirdly, taking each quantile as a statistics target, independently adding the prediction result of each qnantile of each component, and integrating all prediction results to obtain the preduction result of each quantile so as to obtain the financial time series probability distribution prediction. The method provided by the invention can effectively predict a change probability of financial price, and can be applied to financial risk management and investment practice.

Description

A kind of Financial Time Series Forecasting method based on integrated empirical mode decomposition and 1-norm support vector machine quantile estimate

Technical field

The invention belongs to financial field, be specifically related to the Probability distribution prediction method of the price of various securities, earning rate.

Background technology

Financial time series, mainly comprises the data such as the price of various securities, earning rate.Although price movement has very strong uncertainty, among its change, always contain certain law.Such as, some experienced portfolio investors can rely on technical indicator, realize making prediction to the ups and downs of price, and therefrom make a profit.Find these rules, and these rules are used for the variation of forecast price, for the managing risk helping financial institution or common investor's science, making investment decision has great meaning.

Financial time series often has non-stationary and nonlinear feature, and traditional linear model is difficult to the rule caught wherein.At present conventional Forecast of Nonlinear Time Series model has the model such as neural network and support vector machine, and these models can the nonlinear characteristic of fit time sequence well, and then makes prediction.Although the precision of prediction of these models of mind is than the height of conventional model, the result that predicted is applied to decision-making and also has a certain distance.And these predictions are made prediction to the average of price, and lacked the description of the various possibilities to price movement, this is not easy to investor and carries out risk control.Quantile estimate model exactly can the distribution situation of estimated price variation, and it, by calculating a kind of absolute value residual error minimization function of unsymmetric form, estimates the regression function of each fractile respectively.Quantile estimate has lot of advantages, but is most importantly the overall picture that it more comprehensively can describe the distribution of explained variable condition, instead of only analyzes the average of explained variable.

The present invention, in conjunction with integrated empirical mode decomposition and 1-norm support vector machine quantile estimate model, predicts the variation distribution situation of financial time series.Wherein, integrated empirical mode decomposition is used for original time series being decomposed into regular strong component, to improve precision of prediction; Support vector machine quantile estimate model is used for estimating the predicted value of Nonlinear Time Series each fractile.

Summary of the invention

The object of the invention is to the deficiency existed in the Probability distribution prediction method for original financial time series, the Forecasting Methodology that a kind of precision is higher is provided, carries out risk management and investment decision with ancillary investment person.

The essence of time series forecasting modeling is the relation of history value by analytical sequence and its future value, sets up funtcional relationship.Use x _trepresent known raw financial time series, wherein subscript t represents the time, total T phase a: t=1,2 ..., T; As set up anticipation function x _t=f (x _t-1, x _t-2..., x _t-l+1, x _t-l), (l<T), wherein l represents the lag period of prediction, by autocorrelation analysis, and is determined by Schwarz minimization principle.Be such as 6 by analysis time series lag period l, then use its t-1, t-2 ..., the value of t-6 phase, as independent variable, predicts the value of t phase.Use y _irepresent the dependent variable x of i-th training sample _l+i, x _irepresent the independent variable vector [x of i-th training sample _i, x _i+1..., x _i+l-2, x _i+l-1], then original time series { x ₁, x ₂..., x _ttraining set (the x that one group of sample size is T-l can be formed _i, y _i), i=1,2 ..., T-l+1, T-l.For given training set (x _i, y _i), (i=1,2 ..., T-l+1, T-l), use 1-norm support vector machine quantile estimate model to set up x _i→ y _iat fractile τ, the anticipation function f of (τ ∈ (0,1)) _τ(x _i)=f _τ(x _i, x _i+1..., x _i+l-2, x _i+l-1).

This invention first step carries out integrated empirical mode decomposition to financial Time Series of Random Macro-price, under obtaining different scale, and regular N+1 stronger Decomposition Sequence; Second step, carries out the model prediction of 1-norm support vector machine quantile estimate to each Decomposition Sequence respectively, obtains 9 fractile τ=0.1 of each sequence, 0.2 ..., 0.9 predict the outcome.3rd step, in units of each decomposed component, based on the following value under each fractile of each Decomposition Sequence of model outside forecast trained.4th step, predicts the outcome each fractile of each sequence and sums up respectively, integrate each fractile and predict the outcome.

The concrete steps of described Forecasting Methodology are as follows:

(1) integrated empirical mode decomposition is carried out to original time series;

Use x _trepresent raw financial time series { x ₁, x ₂..., x _t, wherein subscript t represents the time, total T phase a: t=1,2 ..., T; Use integrated empirical mode decomposition algorithm by x _tdecompose, obtain N number of intrinsic mode function sequence c _j,t, (j=1,2 ..., N) and a residual value sequence r _t, decomposition result is formulated as:

$x_t=\underset{j=1}{\overset{N}{\&Sigma;}}{c}_{j,t}+r_t$

(2) to each Decomposition Sequence difference sequence prediction function Time Created;

The essence of time series forecasting modeling is the relation of history value by analytical sequence and its future value, sets up funtcional relationship.Use x _trepresent raw financial time series { x ₁, x ₂..., x _t, as set up anticipation function x _t=f (x _t-1, x _t-2..., x _t-l+1, x _t-l), (l<T), wherein l represents the lag period of prediction, by autocorrelation analysis, and is determined by Schwarz minimization principle.Be such as 6 by analysis time series lag period l, then use its t-1, t-2 ..., the value of t-6 phase, as independent variable, predicts the value of t phase.Use y _irepresent the dependent variable x of i-th training sample _l+i, x _irepresent the independent variable vector [x of i-th training sample _i, x _i+1..., x _i+l-2, x _i+l-1], then original time series { x ₁, x ₂..., x _ttraining set (the x that one group of sample size is T-l can be formed _i, y _i), i=1,2 ..., T-l+1, T-l.For given training set (x _i, y _i), (i=1,2 ..., T-l+1, T-l), use 1-norm support vector machine quantile estimate model to set up x _i→ y _iat fractile τ, the anticipation function f of (τ ∈ (0,1)) _τ(x _i)=f _τ(x _i, x _i+1..., x _i+l-2, x _i+l-1), its concrete grammar is:

For fractile τ ∈ (0,1), the essence of training 1-norm support vector machine quantile estimate model is by Optimal Parameters α _i, α _i ^*, (i=1,2 ..., T-l+1, T-l), wherein, α _iwith α _i ^*for the Lagrange factor of constraint condition corresponding to i-th sample point predicted value is less than actual value or is greater than actual value.Make following the minimization of object function:

$\underset{{\mathrm{\&alpha;}}_i,{{\mathrm{\&alpha;}}_i}^{*},b}{min}\underset{i=1}{\overset{T-l}{\&Sigma;}}({\mathrm{\&alpha;}}_i+{{\mathrm{\&alpha;}}_i}^{*})+C\underset{i=1}{\overset{T-l}{\&Sigma;}}({\mathrm{\&tau;\&xi;}}_i+\left(1-\mathrm{\&tau;}\right){\mathrm{\&xi;}}_i^{*})$

And meet constraint condition:

$\left\{\begin{array}{c}\underset{s=1}{\overset{T-l}{\&Sigma;}}({\mathrm{\&alpha;}}_s-{{\mathrm{\&alpha;}}_s}^{*})K(x_s,x_i)+b-y_i\&le;{\mathrm{\&xi;}}_i^{*},i=1,2,...,T-l;s=1,2,...,T-l,s\&NotEqual;i;\\ -\underset{s=1}{\overset{T-l}{\&Sigma;}}({\mathrm{\&alpha;}}_s-{{\mathrm{\&alpha;}}_s}^{*})K(x_s,x_i)-b+y_i\&le;{\mathrm{\&xi;}}_i,i=1,2,...,T-l;s=1,2,...,T-l,s\&NotEqual;i;\\ {\mathrm{\&xi;}}_i,{\mathrm{\&xi;}}_i^{*}\&GreaterEqual;0\end{array}\right.$

Wherein, dummy variable ξ _i, represent model predication value respectively be less than actual value y _ibe greater than actual value y _iresidual error, b is intercept to be estimated, C for punishment parameter.K () is Radial basis kernel function, for any one group (t ₁, t ₂∈ [1,2 ..., T-l+1, T-l]):

$K(x_{t_1},x_{t_2})=\exp (-\frac{\left|\right|x_{t_1}-x_{t_2}|{|}^2}{2{\mathrm{\&sigma;}}^2})$

σ is the core width of Radial basis kernel function, t ₁, t ₂represent the subscript of any two independents variable.

Parameter C and σ in model has material impact to prediction effect, and different time serieses may have different optimal parameter combinations.Optimal parameter is combined through gridding method and chooses: in given range C, σ ∈ [1,2,3,4,5,6,7,8,9,10,20,30,40,50,60,70,80,90,100], have 19 × 19 groups of parameters.To often organizing parameter, using 1-norm Nonlinear Support Vector Machines quantile estimate model to set up anticipation function respectively, and according to anticipation function, the data on training set being predicted.Predicated error between computational prediction value and actual value, chooses one group of minimum parameter combinations of predicated error as final argument.Wherein, for fractile τ, its predicated error is:

$E_{\mathrm{\&tau;}}=\frac{1}{T-l}\underset{i=1}{\overset{T-l}{\&Sigma;}}({\mathrm{\&tau;\&xi;}}_{i,train}+(1-\mathrm{\&tau;}\left){\mathrm{\&xi;}}_{i,train}^{*}\right)$

Herein, T-l represents training sample number, ξ _{i, train}represent the actual value y of training set i-th sample _ibe greater than predicted value residual error, represent the actual value y of training set i-th sample _ibe less than predicted value residual error:

$\left\{\begin{array}{c}{\mathrm{\&xi;}}_{i,train}=max\{y_i-f_{\mathrm{\&tau;}}(x_i),0\}\\ {\mathrm{\&xi;}}_{i,train}^{*}=max\left\{f_{\mathrm{\&tau;}}\right(x_i)-y_i,0\}\end{array}\right.$

After using gridding method to determine optimal parameter C and σ, this model is the linear programming problem of a band Linear Constraints in essence, classical simplicial method can be used to solve, obtain decision variable α _i, α _i ^*, (i=1,2 ..., T-l+1, T-l) and intercept b.For fractile τ, (τ ∈ (0,1)), its final anticipation function is as follows:

$f_{\mathrm{\&tau;}}\left(x\right)=\underset{i=1}{\overset{T-l}{\&Sigma;}}({\mathrm{\&alpha;}}_i-{{\mathrm{\&alpha;}}_i}^{*})K(x_i,x)+b$

Use 1-norm support vector machine quantile estimate model to decomposing N+1 the sequence obtained, each sequence sets up 9 component figure place anticipation functions, i.e. τ=0.1, and 0.2 ..., 0.9 totally 9 fractiles, obtain 9 × (N+1) individual anticipation functions altogether.Use f _{τ, j}, (j=1,2 ..., N) represent the jth intrinsic mode function sequence c of fractile τ _j,tanticipation function, f _{τ, r}represent residual error r _tanticipation function.

(3) outside forecast is carried out based on the model trained.

Use training set to train this model, obtain the anticipation function of 9 fractiles of N+1 Decomposition Sequence respectively, totally 9 × (N+1) is individual.Based on the history value of these anticipation functions and each Decomposition Sequence, can obtain each intrinsic mode function sequence and the residual sequence predicted value in the T+1 phase, its anticipation function is as follows:

${\hat{c}}_{\mathrm{\&tau;},j,T+1}=f_{\mathrm{\&tau;},j}(c_{\mathrm{\&tau;},j,T},c_{\mathrm{\&tau;},j,T-1},c_{\mathrm{\&tau;},j,T-2},...,c_{\mathrm{\&tau;},j,T-l+1}),j=\mathrm{1,2},...,N,\mathrm{\&tau;}=\mathrm{0.1,0.2},...,0.9$

${\hat{r}}_{\mathrm{\&tau;},T+1}=f_{\mathrm{\&tau;},r}(r_{\mathrm{\&tau;},T},r_{\mathrm{\&tau;},T-1},r_{\mathrm{\&tau;},T-2},...,r_{\mathrm{\&tau;},T-l+1}),\mathrm{\&tau;}=\mathrm{0.1,0.2},...,0.9$

(4) finally integrated to predicting the outcome of Decomposition Sequence.

Finally, being predicted the outcome by each component of each fractile, it is integrated to sum up, and obtains final the predicting the outcome of all fractiles:

${\hat{x}}_{\mathrm{\&tau;},T+1}=\underset{j=1}{\overset{N}{\mathrm{\&Sigma;}}}{\hat{c}}_{\mathrm{\&tau;},j,T+1}+{\hat{r}}_{\mathrm{\&tau;},T+1},j=\mathrm{1,2},...,N,\mathrm{\&tau;}=\mathrm{0.1,0.2},...,0.9$

Relative to traditional Time Series Forecasting Methods, integrated Empirical mode decomposition is being introduced in the forecast of distribution of Nonlinear Time Series by advantage of the present invention.First data are decomposed, obtain the component of the better high and low frequency of predictability; So each component is carried out respectively the prediction of 1-norm support vector machine quantile estimate, obtain the forecast of distribution result of each component; Finally, then according to fractile, important predicting the outcome is carried out integrated, obtain the forecast of distribution result of original time series.

Forecasting Methodology provided by the invention can predict the situation of the complete probability distribution of time series, and has higher precision of prediction, has a wide range of applications at financial risk management and investment decision field.

Accompanying drawing explanation

Fig. 1 is this patent Forecasting Methodology implementation framework figure

Fig. 2 is that Index of Shanghai Stock Exchange monthly set of prices in March ,-2014 in January, 2000 becomes empirical mode decomposition component

Fig. 3 be the present invention to Index of Shanghai Stock Exchange from totally 30 phases in March ,-2014 in October, 2011,10%, 20% ..., 90% fractile predicts the outcome

Embodiment

The present invention is the financial time series distribution forecasting method based on integrated empirical mode decomposition and support vector machine quantile estimate, and its framework as shown in Figure 1.In order to the use of the method is described, for the actual effect of monthly closing price data (the totally 171 phases) test model in Index of Shanghai Stock Exchange in January, 2000 in March, 2014.Wherein, front 141 issues are according to being used for training pattern parameter (known sequential, maximum phase T=141), and rear 30 phases are for testing actual outside forecast effect (test set quantity T ^test=30).

Concrete steps are:

The first step: given original time series x _toriginal series is carried out integrated empirical mode decomposition, is specifically implemented as follows.

Because white noise is random, be repeated below (1)-the decomposable process of (6) 100 times, use k=1 each time, 2,3 ..., 100 represent.

(1) at primordial time series data x _ton to add a standard deviation be the white noise of 0.2, obtain and make $r_t^k=x_t^k;$

(2) sequence is found in all local maximums and local minimum;

(3) by these local maximums and minimum value, its coenvelope line of Cubic Spline Functions Fitting and lower envelope line is used respectively with

(4) the average packet winding thread of upper and lower envelope is calculated

(5) from sequence middle separation obtain if meet two conditions: the number of extreme point is equal with the number of zero crossing or to differ be 1; At any some place, the average envelope generated by local maximum envelope and local minimum envelope is 0, then become new mode sequence, and make if do not satisfy condition, then as the residual error of decomposing;

(6) repeat (2)-(6) process, until find out all N number of intrinsic mode function sequences and residual error, decomposition result can be expressed as

(7) Decomposition Sequence 100 times obtained and residual error average respectively, obtain with final integrated empirical mode decomposition result is expressed as:

$x_t=\underset{j=1}{\overset{N}{\&Sigma;}}{c}_{j,t}+r_t.$

The last decomposition result of Index of Shanghai Stock Exchange as shown in Figure 2.

Second step: to using 1-norm support vector machine quantile estimate model to decomposing N+1 the sequence obtained, each sequence sets up 9 component figure place anticipation functions, i.e. τ=0.1,0.2,, 0.9 totally 9 fractiles, obtain 9 × (N+1) individual anticipation functions altogether.Use f _{τ, j}, (j=1,2 ..., N) represent the jth intrinsic mode function sequence c of fractile τ _j,tanticipation function, f _{τ, r}represent the anticipation function of residual error.Its specific implementation method is as follows:

(1) according to seasonal effect in time series autocorrelation analysis, lag period l (l<T) is determined by Schwarz minimization principle (Bayesian Information amount).

With original time series x _tfor example, set up its p rank autoregressive model (AR (p)):

x _t＝β ₀+β ₁x _t-1+β ₂x _t-2+...+β _px _t-p+ε _t

Try to achieve its maximum likelihood function value L, then the Bayesian Information amount (BIC) of this model is:

BIC _p＝－2ln(L)+ln(T-p)*(p+1)

In order to determine best lag period l, first set up p=1,2 ..., 20 totally 20 rank autoregressive models, and calculate its corresponding BIC _pvalue; Then, { BIC is found out ₁, BIC ₂..., BIC ₂₀minimum one of intermediate value, its exponent number p is as the best lag period:

l＝arcmin{BIC ₁,BIC ₂,...,BIC ₂₀}

Such as, in Index of Shanghai Stock Exchange prediction case, the best lag period is set as 6.

(2) in order to improve the training speed of model, need training sample normalization.For decomposing N+1 the sequence obtained, using following formula respectively, making it normalize to [0,1] interval.

$c_{j,t}^{scale}=\frac{c_{j,t}-min\{c_{j,1},c_{j,2},...,c_{j,T}\}}{max\{c_{j,1},c_{j,2},...,c_{j,T}\}-\min \{c_{j,1},c_{j,2},...,c_{j,T}\}},j=\mathrm{1,2},...,N$

$r_t^{scale}=\frac{r_t-min\{r_1,r_2,...,r_T\}}{max\{r_1,r_2,...,r_T\}-\min \{r_1,r_2,...,r_T\}},j=\mathrm{1,2},...,N$

(3) training data generates.

The essence of time series forecasting modeling is the relation of history value by analytical sequence and its future value, sets up funtcional relationship.In this example, need N+1 component, 9 fractiles set up 9 × (N+1) individual anticipation functions.And for the different fractile functions of same Decomposition Sequence, its training set used is identical, therefore need to generate N+1 group training set.

Below with original series x _tfor example illustrates the generation method of training sample: use y _irepresent the dependent variable x of i-th training sample _l+i, x _irepresent the independent variable vector [x of i-th training sample _i, x _i+1..., x _i+l-2, x _i+l-1], then original time series { x ₁, x ₂..., x _ttraining set (the x that one group of sample size is T-l can be formed _i, y _i), i=1,2 ..., T-l+1, T-l.For given training set (x _i, y _i), (i=1,2 ..., T-l+1, T-l), use 1-norm support vector machine quantile estimate model to set up x _i→ y _iat fractile τ, the anticipation function f of (τ ∈ (0,1)) _τ(x _i)=f _τ(x _i, x _i+1..., x _i+l-2, x _i+l-1).

In the case of Index of Shanghai Stock Exchange time series forecasting, the data of front 141 phases are used as training data, can generate 141-6=135 training sample, as follows:

x ₁([x ₁,x ₂,x ₃,x ₄,x ₅,x ₆])→y ₁(x ₇)

x ₂([x ₂,x ₃,x ₄,x ₅,x ₆,x ₇])→y ₂(x ₈)

x ₃([x ₃,x ₄,x ₅,x ₆,x ₇,x ₈])→y ₃(x ₉)

x ₁₃₄([x ₁₃₄,x ₁₃₅,x ₁₃₆,x ₁₃₇,x ₁₃₈,x ₁₃₉])→y ₁₃₄(x ₁₄₀)

x ₁₃₅([x ₁₃₅,x ₁₃₆,x ₁₃₇,x ₁₃₈,x ₁₃₉,x ₁₄₀])→y ₁₃₅(x ₁₄₁)

In like manner, test set 30 samples are as follows:

x ₁₃₆([x ₁₃₆,x ₁₃₇,x ₁₃₈,x ₁₃₉,x ₁₄₀,x ₁₄₁])→y ₁₃₆(x ₁₄₂)

x ₁₃₇([x ₁₃₇,x ₁₃₈,x ₁₃₉,x ₁₄₀,x ₁₄₁,x ₁₄₂])→y ₁₃₇(x ₁₄₃)

x ₁₃₈([x ₁₃₈,x ₁₃₉,x ₁₄₀,x ₁₄₁,x ₁₄₂,x ₁₄₃])→y ₁₃₈(x ₁₄₄)

x ₁₆₄([x ₁₆₄,x ₁₆₅,x ₁₆₆,x ₁₆₇,x ₁₆₈,x ₁₆₉])→y ₁₆₄(x ₁₇₀)

x ₁₆₅([x ₁₆₅,x ₁₆₆,x ₁₆₇,x ₁₆₈,x ₁₆₉,x ₁₇₀])→y ₁₆₅(x ₁₇₁)

(4), after using said method to generate respective training set (altogether N+1) and test set (altogether N+1) to each Decomposition Sequence, each fractile of 1-norm Nonlinear Support Vector Machines quantile estimate model to each Decomposition Sequence is used to set up 9 × (N+1) individual anticipation functions.Below with original series x _tfor example illustrates the generation method of anticipation function:

For fractile τ ∈ (0,1) and training set (x _i, y _i), i=1,2 ..., T-l+1, T-l, the essence of training 1-norm support vector machine quantile estimate model is by Optimal Parameters α _i, α _i ^*, (i=1,2 ..., T-l+1, T-l), wherein, α _iwith α _i ^*for the Lagrange factor of constraint condition corresponding to i-th sample point predicted value is less than actual value or is greater than actual value.Make following the minimization of object function:

$\underset{{\mathrm{\&alpha;}}_i,{{\mathrm{\&alpha;}}_i}^{*},b}{min}\underset{i=1}{\overset{T-l}{\&Sigma;}}({\mathrm{\&alpha;}}_i+{{\mathrm{\&alpha;}}_i}^{*})+C\underset{i=1}{\overset{T-l}{\&Sigma;}}({\mathrm{\&tau;\&xi;}}_i+(1-\mathrm{\&tau;}\left){\mathrm{\&xi;}}_i^{*}\right)$

And meet constraint condition:

$\left\{\begin{array}{c}\underset{s=1}{\overset{T-l}{\&Sigma;}}({\mathrm{\&alpha;}}_s-{{\mathrm{\&alpha;}}_s}^{*})K(x_s,x_i)+b-y_i\&le;{\mathrm{\&xi;}}_i^{*},i=1,2,...,T-l;s=1,2,...,T-l,s\&NotEqual;i;\\ -\underset{s=1}{\overset{T-l}{\&Sigma;}}({\mathrm{\&alpha;}}_s-{{\mathrm{\&alpha;}}_s}^{*})K(x_s,x_i)-b+y_i\&le;{\mathrm{\&xi;}}_i,i=1,2,...,T-l;s=1,2,...,T-l,s\&NotEqual;i;\\ {\mathrm{\&xi;}}_i,{\mathrm{\&xi;}}_i^{*}\&GreaterEqual;0\end{array}\right.$

Wherein, dummy variable ξ _i, represent model predication value respectively be less than actual value y _ibe greater than actual value y _iresidual error, b is intercept to be estimated, C for punishment parameter.K () is Radial basis kernel function, for any one (t ₁, t ₂∈ [1,2 ..., T-l+1, T-l]):

$K(x_{t_1},x_{t_2})=\exp (-\frac{\left|\right|x_{t_1}-x_{t_2}|{|}^2}{2{\mathrm{\&sigma;}}^2})$

σ is the core width of Radial basis kernel function, t ₁, t ₂represent the subscript of any two independents variable.In the case of Index of Shanghai Stock Exchange time series forecasting, such as, for two independent variable x ₇with x ₁₀, σ=1, the value of its Radial basis kernel function is:

$\begin{array}{c}K(x_7,x_{10})=\exp (-\frac{\left|\right|x_7-x_{10}|{|}^2}{2\&times;1})\\ =\exp (-\frac{1}{2}\&times;{}^{}{}^{}{}^{}|\left|\begin{array}{c}\&lsqb;0.0969;0.1336;0.1511;0.1585;0.1704;0.1772]-\\ [0.1585;0.1704;0.1772;0.1967;0.1963;0.1736\&rsqb;\end{array}\right|{|}^2)\\ =\exp \{-\frac{1}{2}\&times;\&lsqb;{(0.0969-0.1585)}^2+{(0.1336-0.1704)}^2+\mathrm{...}+{(0.1772-0.1736)}^2\&rsqb;\}\\ =\exp \{-\frac{1}{2}\&times;0.0080\}\\ =0.9960\end{array}$

Parameter C and σ in model has material impact to prediction effect, and different time serieses may have different optimal parameter combinations.Optimal parameter is combined through gridding method and chooses: in given range C, σ ∈ [1,2,3,4,5,6,7,8,9,10,20,30,40,50,60,70,80,90,100], have 19 × 19 groups of parameters.To often organizing parameter, using 1-norm Nonlinear Support Vector Machines quantile estimate model to set up anticipation function respectively, and according to anticipation function, the data on training set being predicted.Predicated error between computational prediction value and actual value, chooses one group of minimum parameter combinations of predicated error as final argument.Wherein, for fractile τ, its predicated error is:

$E_{\mathrm{\&tau;}}=\frac{1}{T-l}\underset{i=1}{\overset{T-l}{\&Sigma;}}({\mathrm{\&tau;\&xi;}}_{i,train}+(1-\mathrm{\&tau;}\left){\mathrm{\&xi;}}_{i,train}^{*}\right)$

Herein, T-l represents training sample number, ξ _{i, train}represent the actual value y of training set i-th sample _ibe greater than predicted value residual error, represent the actual value y of training set i-th sample _ibe less than predicted value residual error:

$\left\{\begin{array}{c}{\mathrm{\&xi;}}_{i,train}=max\{y_i-f_{\mathrm{\&tau;}}(x_i),0\}\\ {\mathrm{\&xi;}}_{i,train}^{*}=max\left\{f_{\mathrm{\&tau;}}\right(x_i)-y_i,0\}\end{array}\right.$

Illustrate, for fractile τ=0.5, the final predicated error E of each combination of C and σ _τrow are in Table 1 (for original Index of Shanghai Stock Exchange sequence x _tthe example that is predicted as be described)

The original Index of Shanghai Stock Exchange sequence x of table 1 _ttime prediction error value of each parameter combinations on training set in fractile τ=0.5

Wherein, when parameter C and σ is 90 and 1, obtain minimum predicated error, therefore for fractile τ=0.5, get C=90, σ=1.

This model is the linear programming problem of a band Linear Constraints in essence, classical simplicial method can be used to solve, obtain decision variable α _i, α _i ^*, (i=1,2 ..., T-l+1, T-l) and intercept b.For fractile τ, (τ ∈ (0,1)), its final anticipation function is as follows:

$f_{\mathrm{\&tau;}}\left(x\right)=\underset{i=1}{\overset{T-l}{\&Sigma;}}({\mathrm{\&alpha;}}_i-{{\mathrm{\&alpha;}}_i}^{*})K(x_i,x)+b$

Such as, the predicted value of outside forecast testing period first sample in fractile τ=0.5, substitutes into following formula:

${\hat{x}}_{\mathrm{\&tau;}=0.5,T+1}=f_{\mathrm{\&tau;}=0.5}\left(x_{T-l+1}\right)=\underset{i=1}{\overset{T-l}{\&Sigma;}}(\mathrm{\&alpha;}-{{\mathrm{\&alpha;}}_i}^{*})K(x_i,x_{T-l+1})+b$

In order to test the forecast result of this method, the precision of prediction of testing model in test set 30 samples.Choose and do not carry out integrated empirical mode decomposition, only for original series x _tcarry out respectively 1-norm support vector machine quantile estimate prediction and linear quantile estimate predict that two kinds of methods are tested in contrast.

In contrast method, the prediction of linear quantile estimate and 1-norm support vector machine quantile estimate forecast model carry out predicting the process of modeling and this method similar, all want the autocorrelation analysis of elapsed time sequence, samples normalization and training pattern to set up anticipation function.Be that prediction modeling is carried out to original Index of Shanghai Stock Exchange sequence unlike contrast method, instead of modeling is carried out to each decomposed component through integrated empirical mode decomposition.In addition, the setting of independent 1-norm support vector machine quantile estimate forecast model is identical with choice of parameters and this method.

Because this method can decomposite multiple decomposed component, using during 1-norm support vector machine quantile estimate model modeling each decomposed component and will carry out as the screening in table 1 parameter C, σ in normal situation.For the purpose of simplifying the description, when testing this method below, no longer screen parameter C, σ, the 1-norm support vector machine quantile estimate prediction model parameters of each decomposed component is unified is set to C=10, σ=2.As can be seen from last result, even if without Model Parameter Optimization, the result that this method draws all significantly will be better than contrast method.

The standard evaluated is the asymmetric tolerances function of fractile, selects the evaluation method used during parameter similar to gridding method before:

$E_{\mathrm{\&tau;},test}=\frac{1}{T_{test}}\underset{i=1}{\overset{T_{test}}{\&Sigma;}}({\mathrm{\&tau;\&xi;}}_{T-l+i,test}+(1-\mathrm{\&tau;}\left){\mathrm{\&xi;}}_{T-l+i,test}^{*}\right)$

$\left\{\begin{array}{c}{\mathrm{\&xi;}}_{T-l+i,test}=max\{y_{T-l+i}-f_{\mathrm{\&tau;}}(x_{T-l+i}),0\}\\ {\mathrm{\&xi;}}_{T-l+i,test}^{*}=max\left\{f_{\mathrm{\&tau;}}\right(x_{T-l+i})-y_{T-l+i},0\}\end{array}\right.$

Wherein, ξ _{t-l+i, test}for test set i-th sample actual value y _t-l+ilower than predicted value f _τ(x _t-l+i) residual error, test set i-th sample is actual value y _t-l+ihigher than predicted value f _τ(x _t-l+i) residual error.T _testrepresent test set sample size, in the prediction case of Index of Shanghai Stock Exchange, T _test=30.

Such as, during τ=0.3,

$\begin{array}{c}{E}_{\mathrm{\&tau;}=0.3,test}=\frac{1}{30}\underset{t=142}{\overset{171}{\&Sigma;}}(0.3\&times;{\mathrm{\&xi;}}_{t,real}+(1-0.3\left){\mathrm{\&xi;}}_{t,real}^{*}\right)=\\ \frac{1}{30}\&times;0.3\&times;\max \{2468.3-2270.6,0\}+\frac{1}{30}\&times;0.7\&times;\max \{2270.6-2468.3,0\}\\ +\frac{1}{30}\&times;0.3\&times;\max \{2333.4-2295.5,0\}+\frac{1}{30}\&times;0.7\&times;\max \{2295.5-2333.4,0\}+\mathrm{...}\\ +\frac{1}{30}\&times;0.3\&times;\max \{2056.3-1982.2,0\}+\frac{1}{30}\&times;0.7\&times;\max \{1982.2-2056.3,0\}+\\ \frac{1}{30}\&times;0.3\&times;\max \{2033.3-1970.8,0\}+\frac{1}{30}\&times;0.7\&times;\max \{1970.8-2033.3,0\}\\ =23.53\end{array}$

Again such as, during τ=0.6,

$\begin{array}{c}{E}_{\mathrm{\&tau;}=0.6,test}=\frac{1}{30}\underset{t=142}{\overset{171}{\&Sigma;}}(0.6\&times;{\mathrm{\&xi;}}_{t,real}+(1-0.6\left){\mathrm{\&xi;}}_{t,real}^{*}\right)=\\ \frac{1}{30}\&times;0.6\&times;\max \{2468.3-2381.4,0\}+\frac{1}{30}\&times;0.4\&times;\max \{2381.4-2468.3,0\}\\ +\frac{1}{30}\&times;0.6\&times;\max \{2333.4-2275.5,0\}+\frac{1}{30}\&times;0.4\&times;\max \{2275.5-2333.4,0\}+\mathrm{...}\\ +\frac{1}{30}\&times;0.6\&times;\max \{2056.3-2069.4,0\}+\frac{1}{30}\&times;0.4\&times;\max \{2069.4-2056.3,0\}+\\ \frac{1}{30}\&times;0.6\&times;\max \{2033.3-2113.3,0\}+\frac{1}{30}\&times;0.4\&times;\max \{2113.3-2033.3,0\}\\ =24.78\end{array}$

Table 2 the inventive method and other two kinds of classic methods each fractile predicated error in test set contrast

As can be seen from Table 2, these three kinds of models have obvious good and bad level.Wherein, the effect of 1-norm support vector machine Quantile Regression is better than linear quantile estimate model, and this shows that the method can the nonlinear characteristic of matching Index of Shanghai Stock Exchange.The most important thing is, no matter from deviation or the mean deviation of each fractile, the measurement error based on integrated empirical mode decomposition 1-norm support vector machine Quantile Regression of the present invention is all better than single model, this demonstrates superiority of the present invention.

Claims (1)

1., based on a Financial Time Series Forecasting method for integrated empirical mode decomposition and 1-norm support vector machine quantile estimate, it is characterized in that, the method comprises following operation steps:

(1) Time Series;

Use x _trepresent raw financial time series { x ₁, x ₂..., x _t, wherein subscript t represents the time, total T phase a: t=1,2 ..., T; Use integrated empirical mode decomposition algorithm by x _tdecompose, obtain N number of intrinsic mode function sequence c _j,t, (j=1,2 ..., N) and a residual value sequence r _t, decomposition result is formulated as:

$x_t=\underset{j=1}{\overset{N}{\&Sigma;}}{c}_{j,t}+r_t$

(2) to each Decomposition Sequence difference sequence prediction function Time Created;

The essence of time series forecasting modeling is the relation of history value by analytical sequence and its future value, sets up funtcional relationship; Use x _trepresent raw financial time series { x ₁, x ₂..., x _t, wherein subscript t represents the time, total T phase a: t=1,2 ..., T; As set up anticipation function x _t=f (x _t-1, x _t-2..., x _t-l+1, x _t-l), (l<T), wherein l represents the lag period of prediction, by auto-correlation and partial Correlation Analysis, and is determined by Schwarz minimization principle; Use y _irepresent the dependent variable x of i-th training sample _l+i, x _irepresent the independent variable vector [x of i-th training sample _i, x _i+1..., x _i+l-2, x _i+l-1], then original time series { x ₁, x ₂..., x _tform the training set (x that one group of sample size is T-l _i, y _i), i=1,2 ..., T-l+1, T-l; For given training set (x _i, y _i), (i=1,2 ..., T-l+1, T-l), use 1-norm support vector machine quantile estimate model to set up x _i→ y _iat fractile τ, the anticipation function f of (τ ∈ (0,1)) _τ(x _i)=f _τ(x _i, x _i+1..., x _i+l-2, x _i+l-1), its concrete grammar is:

For fractile τ ∈ (0,1), the essence of training 1-norm support vector machine quantile estimate model is by Optimal Parameters α _i, α _i ^*, (i=1,2 ..., T-l+1, T-l), wherein, α _iwith α _i ^*for the Lagrange factor of constraint condition corresponding to i-th sample point predicted value is less than actual value or is greater than actual value; Make following the minimization of object function:

$\underset{{\mathrm{\&alpha;}}_i,{{\mathrm{\&alpha;}}_i}^{*},b}{min}\underset{i=1}{\overset{T-l}{\&Sigma;}}({\mathrm{\&alpha;}}_i+{{\mathrm{\&alpha;}}_i}^{*})+C\underset{i=1}{\overset{T-l}{\&Sigma;}}({\mathrm{\&tau;\&xi;}}_i+(1-\mathrm{\&tau;}\left){\mathrm{\&xi;}}_i^{*}\right)$

And meet constraint condition:

$\left\{\begin{array}{c}\underset{s=1}{\overset{T-l}{\&Sigma;}}({\mathrm{\&alpha;}}_s-{{\mathrm{\&alpha;}}_s}^{*})K(x_s,x_i)+b-y_i\&le;{\mathrm{\&xi;}}_i^{*},i=1,2,...,T-l;s=1,2,...,T-l,s\&NotEqual;i;\\ -\underset{s=1}{\overset{T-l}{\&Sigma;}}({\mathrm{\&alpha;}}_s-{{\mathrm{\&alpha;}}_s}^{*})K(x_s,x_i)-b+y_i\&le;{\mathrm{\&xi;}}_i,i=1,2,...,T-l;s=1,2,...,T-l,s\&NotEqual;i;\\ {\mathrm{\&xi;}}_i,{\mathrm{\&xi;}}_i^{*}\&GreaterEqual;0\end{array}\right.$

Wherein, dummy variable ξ _i, ξ _i ^*represent model predication value respectively be less than actual value y _ibe greater than actual value y _iresidual error, b is intercept to be estimated, C for punishment parameter; K () is Radial basis kernel function, for any one (t ₁, t ₂∈ [1,2 ..., T-l+1, T-l]):

$K(x_{t_1},x_{t_2})=\exp (-\frac{\left|\right|x_{t_1}-x_{t_2}|{|}^2}{2{\mathrm{\&sigma;}}^2})$

σ is the core width of Radial basis kernel function, t ₁, t ₂represent the subscript of any two independents variable;

Parameter C and σ in model has material impact to prediction effect, and different time serieses may have different optimal parameter combinations; Optimal parameter is combined through gridding method and chooses: in given range C, σ ∈ [1,2,3,4,5,6,7,8,9,10,20,30,40,50,60,70,80,90,100], have 19 × 19 groups of parameters; To often organizing parameter, using 1-norm Nonlinear Support Vector Machines quantile estimate model to set up anticipation function respectively, and according to anticipation function, the data on training set being predicted; Predicated error between computational prediction value and actual value, chooses one group of minimum parameter combinations of predicated error as final argument; Wherein, for fractile τ, its predicated error is:

$E_{\mathrm{\&tau;}}=\frac{1}{T-l}\underset{i=1}{\overset{T-l}{\&Sigma;}}({\mathrm{\&tau;\&xi;}}_{i,train}+(1-\mathrm{\&tau;}\left){\mathrm{\&xi;}}_{i,train}^{*}\right)$

Herein, T-l represents training sample number, ξ _{i, train}represent the actual value y of training set i-th sample _ibe greater than predicted value residual error, represent the actual value y of training set i-th sample _ibe less than predicted value residual error:

$\left\{\begin{array}{c}{\mathrm{\&xi;}}_{i,train}=max\{y_i-f_{\mathrm{\&tau;}}\left(x_i\right),0\}\\ {\mathrm{\&xi;}}_{i,train}^{*}=max\left\{f_{\mathrm{\&tau;}}\right(x_i)-y_i,0\}\end{array}\right.$

After using gridding method to determine optimal parameter C and σ, this model is the linear programming problem of a band Linear Constraints in essence, uses classical simplicial method to solve, obtains decision variable α _i, α _i ^*, (i=1,2 ..., T-l+1, T-l) and intercept b; For fractile τ, (τ ∈ (0,1)), its final anticipation function is as follows:

$f_{\mathrm{\&tau;}}\left(x\right)=\underset{i=1}{\overset{T-l}{\&Sigma;}}({\mathrm{\&alpha;}}_i-{{\mathrm{\&alpha;}}_i}^{*})K(x_i,x)+b$

Use 1-norm support vector machine quantile estimate model to decomposing N+1 the sequence obtained, each sequence sets up 9 component figure place anticipation functions, i.e. τ=0.1, and 0.2 ..., 0.9 totally 9 fractiles, obtain 9 × (N+1) individual anticipation functions altogether; Use f _{τ, j}, (j=1,2 ..., N) represent the jth intrinsic mode sequence c of fractile τ _j,tanticipation function, f _{τ, r}represent residual error r _tanticipation function;

(3) outside forecast is carried out based on the model trained;

Use training set to train this model, obtain the anticipation function of 9 fractiles of N+1 Decomposition Sequence respectively, totally 9 × (N+1) is individual; Based on the history value of these anticipation functions and each Decomposition Sequence, obtain each intrinsic mode function sequence and the residual sequence predicted value in the T+1 phase, its anticipation function is as follows:

${\hat{c}}_{\mathrm{\&tau;},j,T+1}=f_{\mathrm{\&tau;},j}(c_{\mathrm{\&tau;},j,T},c_{\mathrm{\&tau;},j,T-1},c_{\mathrm{\&tau;},j,T-2},...,c_{\mathrm{\&tau;},j,T-l+1}),j=\mathrm{1,2},...,N,\mathrm{\&tau;}=\mathrm{0.1,0.2},...,0.9$

${\hat{r}}_{\mathrm{\&tau;},T+1}=f_{\mathrm{\&tau;},r}(r_{\mathrm{\&tau;},T},r_{\mathrm{\&tau;},T-1},r_{\mathrm{\&tau;},T-2},...,r_{\mathrm{\&tau;},T-l+1}),\mathrm{\&tau;}=\mathrm{0.1,0.2},...,0.9$

(4) finally integrated to predicting the outcome of Decomposition Sequence;

Finally, being predicted the outcome by each component of each fractile, it is integrated to sum up, and obtains final the predicting the outcome of all fractiles:

${\hat{x}}_{\mathrm{\&tau;},T+1}=\underset{j=1}{\overset{N}{\mathrm{\&Sigma;}}}{\hat{c}}_{\mathrm{\&tau;},j,T+1}+{\hat{r}}_{\mathrm{\&tau;},T+1},j=\mathrm{1,2},...,N,\mathrm{\&tau;}=\mathrm{0.1,0.2},...,0.9.$