CN103745079A - Curve fitting method based on abstract convex estimations - Google Patents

Abstract

A curve fitting method based on abstract convex estimations includes the following steps that supporting vectors begin to be built from two boundary points to form an initial supporting matrix, and the initial supporting matrix is used as a root to built a binary tree; supporting vectors are built for known data points, and the binary tress is updated according to a certain condition; a sampling step length is set, sampling is conducted within a definitional domain range of x, tree leaf child nodes to which sampling points belong are found out, and lower bound low estimators of a tree leaf child node zone to which the sampling points belong are calculated; all the sampling points are connected, and then lower bound saw tooth estimations of a curve to be fitted can be obtained; upper bound saw tooth estimations of the curve to be fitted can be obtained through the same method, and the curve which is obtained by averaging the upper bound saw tooth estimations and the lower bound saw tooth estimations is used as the curve to be fitted. The curve fitting method based on the abstract convex estimations has the advantages that calculated amount is small, the original data points can be reserved, and fitting efficiency and the accuracy rate are high.

Description

A kind of curve-fitting method of estimating based on abstract convex

Technical field

The present invention relates to a kind of mathematical modeling, computer application field, in particular, a kind of curve-fitting method of estimating based on abstract convex.

Background technology

Science and engineering problem can obtain some discrete data by methods such as sampling, experiment, and according to these data, we often wish to obtain a continuous curve and given data matches, and this process is just called curve.Curve application is very extensive, occupies very consequence in computational science.For example, the field of people to a certain the unknown, in order to explore its inherent rule, holds its developing direction, will use curve.Therefore, comprehensive research of curve-fitting method is had to positive realistic meaning.At present, curve fitting technique has all obtained wide application in the fields such as processes and displays of image processing, reverse-engineering, computer-aided design (CAD) and test data.The method of curve has a lot, and conventional method has least square method and method of interpolation etc.

The least square method of curve is according to known data (x ^k, y ^k) (k=1,2 ..., K) (x wherein ^kfor measured quantity, y ^kfor its corresponding functional value, the quantity size size that K is data point) choose an approximate function f (x), make E ₂(f) (root-mean-square error) minimum; Least square fitting great advantage is that fitting precision can regulate and control, can select different fitting precisions according to actual service condition, but in actual applications, some occasion need to retain former data point, at this moment use the common methods such as least square method just improper.The method of interpolation of curve is according to interpolation principle structure polynomial of degree n P _n(x), make P _n(x) in the data of each test point, just in time pass through eyeball; Although method of interpolation can retain former data point,, in the ordinary course of things, we have gathered a lot of sampling points in order to reflect actual conditions as far as possible, have caused interpolation polynomial P _n(x) number of times is very high, and this has not only increased calculated amount, and has affected fitting precision.

Therefore, existing technology exists defect aspect curve, needs to improve.

Summary of the invention

In order to overcome existing method, can not retain former data point, calculated amount is large, fitting precision is lower deficiency, the invention provides a kind of when guaranteeing fitting precision, can retain former data point, calculated amount is little, fitting precision the is higher curve-fitting method based on abstract convex estimation.

The technical solution adopted for the present invention to solve the technical problems is:

A curve-fitting method of estimating based on abstract convex, described curve-fitting method comprises the following steps:

1) parameter initialization: constant M is set, input data (x ^k, y ^k), k=1,2 ..., K, the quantity size size that wherein K is data point;

2) boundary condition initialization, process is as follows;

2.1) according to existing data point, determine the field of definition [a, b] of variable x, wherein a is the lower bound of variable x, and b is the upper bound;

2.2) model conversion: former variable x is done to following linear transformation:

X wherein ₁', x ₂' be linear transformation variable;

Formula (1) is converted, obtains:

x＝x ₁′(b-a)+a （2）

2.2.1) establish

wherein,

the linear transformation variable 1 that represents the 1st data,

the linear transformation variable 1 of the 2nd data;

2.2.2) right according to formula (2)

be converted to former variable x ¹, x ², x wherein ¹=b, x ²=a;

2.3) calculate initial support vector sum proppant matrix, and set up tree:

2.3.1) according to formula (4) respectively calculation level (x ' ¹, y _b) and (x ' ², y _a) support vector,

$l^k=(\frac{x_1^k}{f\left(x^k\right)},\frac{x_2^k}{f\left(x^k\right)},...,\frac{x_{N+1}^k}{f\left(x^k\right)})---\left(4\right)$

Wherein, y _brepresent functional value corresponding to upper bound b, y _arepresent functional value corresponding to lower bound a, the x in formula ^kwith x ' replacement, f () replaces with y;

2.3.2) two support vectors of gained are formed to proppant matrix L, as shown in Equation (7):

2.3.3) take proppant matrix L as root foundation tree;

3) calculate the support vector of each data point and upgrade tree, each data point is handled as follows:

3.1) according to formula (1) to x ^kas linear transformation, be

3.2) according to formula (4) calculate point (x ' ^k, y ^k) support vector l ^k, the x in formula ^kwith x ' ^kreplace, f () uses y ^kreplace;

3.3) according to conditional relationship formula (5) and (6), upgrade tree:

$\∀i,j\∈I,i\≠j:l_i^{k_i}>l_i^{k_j}---\left(5\right)$

$\∀v\∈{\mathrm{\Λ}}^k\backslash L,\∃i\∈I:l_i^{k_i}\≤v_i---\left(6\right)$

V ∈ Λ wherein ^kl represent that v belongs to Λ ^kbut do not belong to L,

represent to exist;

3.3.1) find for step 3.2) the support vector l that produces ^kthe leaf node of (6), i.e. v do not satisfy condition _i=l ^k;

3.3.2) use l ^kreplacement step 3.3.1) i support vector in the leaf node matrix finding in

thereby form new leaf node;

3.3.3) the new leaf node producing the determining step 3.3.2) relational expression (5) that whether satisfies condition, if met, retains, otherwise deletes;

4) calculate Lower Bound Estimation value;

4.1) sampling step length is set, and sampling in field of definition [a, b] scope;

4.2) to step 4.1) each sampled point x _cbe handled as follows:

4.2.1) according to formula (1), do linear transformation, obtain new vector x _c'=(x _c1', x _c2');

4.2.2) according to formula (10) from step 3.3) tree find and comprise vector x _c' leaf node, in formula

use x _c' replace, if leaf node meets formula (10), comprise, otherwise do not comprise;

$x_j^{k_j}{\hat{x}}_i^r>x_i^{k_j}{\hat{x}}_j^r,i,j\∈I,i\≠j---\left(10\right)$

Wherein,

for the element in looked for leaf node matrix;

4.2.3) according to formula (3), obtain vector x _cthe Lower Bound Estimation value y of the leaf node corresponding region at ' place _l';

4.2.5) due in step 2.3) in added a constant M while being converted into IPH, in the hope of Lower Bound Estimation value should deduct M, i.e. y _l=y _l'-M;

4.2.6) output Lower Bound Estimation value y _land corresponding sampled point x _c;

5) calculate Estimation of Upper-Bound value, process is as follows:

5.1) the y in data point ^kget negative;

5.2) by step 2) to 4) calculating Lower Bound Estimation value;

5.3) to step 5.2) the Lower Bound Estimation value of gained gets negatively, is Estimation of Upper-Bound value y _u;

5.4) output Estimation of Upper-Bound value y _uand corresponding sampled point x _c;

6) curve, process is as follows:

6.1) connect each point (x _c, y _l) obtain treating the Lower Bound Estimation of matched curve;

6.2) connect each point (x _c, y _u) obtain treating the Estimation of Upper-Bound of matched curve;

6.3) Estimation of Upper-Bound and Lower Bound Estimation being averaged to the curve obtaining is and treats matched curve.

Further, the constant M in step 1), if function f meets following two conditions:

$i)\∀x,y\∈R_{+}^n,x\≥y\⇒f\left(x\right)\≥f\left(y\right)$

$\mathrm{ii})\∀x\∈R_{+}^n,\∀\mathrm{\λ}\∈R_{++}:f\left(\mathrm{\λx}\right)=\mathrm{\λf}\left(x\right)$

Wherein,

represent arbitrarily,

claim function f for just homogeneous increasing function IPH;

Add one enough large constant M can will treat that function corresponding to matched curve is converted to IPH function, adds M to each y in data.

Further, the method for calculating support vector and proppant matrix step 2.3) is:

Suppose that the field of definition of N dimension objective function utilized model conversion formula (1) unit's of being mapped to simplex space

suppose that it has K support function, the point (x based on given ^k, f (x ^k)) (k=1,2 ..., K), the Lower Bound Estimation of objective function f is provided by following formula:

$H^K\left(x\right)=\underset{k\≤K}{\max }\underset{i=1,...N+1}{\min }\frac{x_i}{l_i^k}---\left(3\right)$

Wherein max represents maximum, and min represents minimum, x _ifor the vector in unit simplex space;

K support function is:

$h^k\left(x\right)=\underset{i=1,...N+1}{\min }\frac{x_i}{l_i^k}=f\left(x^k\right)\min \{\frac{x_1}{x_1^k},...,\frac{x_{N+1}}{x_{N+1}^k}\}$

Wherein, l ^kbe called support vector, as the formula (4).Consider a set that contains K support vector

make I={1,2 ..., N+1}, N+1 the support vector of satisfy condition (5) and (6)

be called an effective proppant matrix;

Suppose an effective proppant matrix as the formula (7):

According to this proppant matrix, can calculate corresponding lower bound support function extreme value x _minthe Lower Bound Estimation value d corresponding with it:

x _min(L)＝diag(L)/Trace(L) （8）

d(L)＝H ^k(x _min)＝1/Trace(L) ^-1 （9）

Wherein, diag is the positive diagonal entry of matrix, and Trace is matrix trace, i.e. the positive diagonal entry sum of matrix.

Further again, step 4.2.2) in, find out and comprise vector x _c' the method for leaf node be:

Suppose

for H ^k(x) r local minimum, the relational expression that satisfies condition (5) of its correspondence and the proppant matrix of (6) are L ^r, by formula (9), had lower Bound Estimation value

$H^k\left({\hat{x}}^r\right)=1/\mathrm{Trace}{\left(L^r\right)}^{-1}$

i element be

${\hat{x}}^r=\frac{l_i^{k_i}}{\mathrm{Trace}\left(L^r\right)}=\frac{x_i^{k_i}}{f\left(x^{k_i}\right)\mathrm{Trace}\left(L^r\right)}$

By conditional relationship formula (5):

and formula (4) has

$\frac{x_i^{k_i}}{f\left(x^{k_i}\right)}>\frac{x_i^{k_j}}{f\left(x^{k_j}\right)}$

Be multiplied by simultaneously on above formula both sides

divided by Trace (L ^r):

$\frac{1}{\mathrm{Trace}\left(L^r\right)}{x}_j^{k_j}\frac{x_i^{k_i}}{f\left(x^{k_i}\right)}>\frac{1}{\mathrm{Trace}\left(L^r\right)}\frac{x_j^{k_j}}{f\left(x^{k_j}\right)}{x}_i^{k_j}$

Above formula is equivalent to (10), if institute the leaf node of looking for for

meet formula 10), this leaf node district inclusion is vectorial

Technical conceive of the present invention is: with two frontier points, start to set up support vector, form initial support matrix, take this initial matrix to set up binary tree as root; Then known data point is set up to support vector, and upgrade this binary tree according to condition; A sampling step length is set again, within the scope of the field of definition of x, samples, find out the affiliated leaf node of sampled point, calculate the low valuation of lower bound of the affiliated leaf node region of sampled point; Connect the lower bound sawtooth estimation that all sampled points just can obtain treating matched curve; With same method, try to achieve the upper bound sawtooth estimation for the treatment of matched curve, the curve on average obtaining of getting the upper bound and the estimation of lower bound sawtooth is treats matched curve.The abstract convex estimation curve matching of application based on tree, has not only improved efficiency and the accuracy rate of matching, and has retained raw data points, has reduced calculated amount.

Accompanying drawing explanation

Fig. 1 is the unit's of being converted to simplex space, common definition territory schematic diagram, and wherein, left side subgraph is common definition territory, and the right subgraph is unit simplex space.

Fig. 2 is the schematic diagram that the lower bound sawtooth of one dimension Shubert function is estimated.

Fig. 3 is the schematic diagram that one dimension Shubert function upper bound sawtooth is estimated.

Fig. 4 is the schematic diagram of the matched curve of one dimension Shubert function.

Embodiment

Below in conjunction with accompanying drawing, the invention will be further described.

With reference to Fig. 1～Fig. 4, a kind of curve-fitting method of estimating based on abstract convex, comprises the following steps:

1) parameter initialization: constant M is set, input data (x ^k, y ^k) (k=1,2 ..., K), the quantity size size that wherein K is data point.

2) boundary condition initialization.

2.1) according to existing data point, determine the field of definition [a, b] of variable x, wherein a is the lower bound of variable x, and b is the upper bound;

2.2) model conversion;

If x is ∈ [a, b], wherein, x is the variable on boundary constraint feasible zone [a, b], and former variable x is done to following linear transformation:

X wherein ₁', x ₂' be linear transformation variable.

Obviously, (b-a) > 0, and x is easily known in (x-a)>=0 ₁'>=0, x ₂'>=0, $x^{\′}={[x_1^{\′},x_2^{\′}]}^T\∈S=\{x^{\′}\∈R_{+}^2:{\mathrm{\Σ}}_{i=1}^2{x}_i^{\′}=1\},$ Wherein $R_{+}^2=\{x^{\′}=(x_1^{\′},x_2^{\′})|x_i^{\′}\≥0,i=\mathrm{1,2}\},$ S representation unit simplex region; Fig. 1 has provided the schematic diagram in the unit's of being converted to simplex space, common definition territory.

Formula (1) is converted, can obtain:

x＝x ₁′(b-a)+a （2）

2.2.1) establish

wherein,

the linear transformation variable 1 that represents the 1st data,

the linear transformation variable 1 of the 2nd data;

2.2.2) right according to formula (2)

be converted to former variable x ¹, x ², x wherein ¹=b, x ²=a;

2.3) calculate initial support vector sum proppant matrix, and set up tree;

Definition 1: if function f meets following two conditions

$i)\∀x,y\∈R_{+}^n,x\≥y\⇒f\left(x\right)\≥f\left(y\right)$

$\mathrm{ii})\∀x\∈R_{+}^n,\∀\mathrm{\λ}\∈R_{++}:f\left(\mathrm{\λx}\right)=\mathrm{\λf}\left(x\right)$

Wherein,

represent arbitrarily,

claim function f for just homogeneous increasing function (Increasing Positively Homogeneous of degree one, IPH).

Objective function also can by add one enough large constant M be converted to IPH function.Suppose that the field of definition of N dimension objective function utilized model conversion formula (1) unit's of being mapped to simplex space

suppose that it has K support function, the point (x based on given ^k, f (x ^k)) (k=1,2 ..., K), the Lower Bound Estimation of objective function f can be provided by following formula:

$H^K\left(x\right)=\underset{k\≤K}{\max }\underset{i=1,...N+1}{\min }\frac{x_i}{l_i^k}---\left(3\right)$

Wherein max represents maximum, and min represents minimum, x _ifor the vector in unit simplex space;

K support function is:

$h^k\left(x\right)=\underset{i=1,...N+1}{\min }\frac{x_i}{l_i^k}=f\left(x^k\right)\min \{\frac{x_1}{x_1^k},...,\frac{x_{N+1}}{x_{N+1}^k}\}$

Wherein

$l^k=(\frac{x_1^k}{f\left(x^k\right)},\frac{x_2^k}{f\left(x^k\right)},...,\frac{x_{N+1}^k}{f\left(x^k\right)})---\left(4\right)$

Be called support vector.Consider a set that contains K support vector

make I={1,2 ..., N+1}, N+1 support vector of two conditions (5) and (6) below meeting

be called an effective proppant matrix;

$\∀i,j\∈I,i\≠j:l_i^{k_i}>l_i^{k_j}---\left(5\right)$

$\∀v\∈{\mathrm{\Λ}}^k\backslash L,\∃i\∈I:l_i^{k_i}\≤v_i---\left(6\right)$

V ∈ Λ wherein ^kl represent that v belongs to Λ ^kbut do not belong to L,

represent to exist;

Explain for convenience principle, suppose an effective proppant matrix as the formula (7):

According to this proppant matrix, can calculate corresponding lower bound support function extreme value x _minthe Lower Bound Estimation value d corresponding with it:

x _min(L)＝diag(L)/Trace(L) （8）

d(L)＝H ^k(x _min)＝1/Trace(L) ^-1 （9）

Wherein, diag is the positive diagonal entry of matrix, and Trace is matrix trace, i.e. the positive diagonal entry sum of matrix.

2.3.1) according to formula (4) respectively calculation level (x ' ¹, y _b) and (x ' ², y _a) support vector, wherein, y _brepresent functional value corresponding to upper bound b, y _arepresent functional value corresponding to lower bound a, the x in formula ^kwith x ' replacement, f () replaces with y;

2.3.2) 2 support vectors of gained are formed to proppant matrix L, as shown in Equation (7);

2.3.3) take proppant matrix L as root foundation tree;

3) calculate the support vector of each data point and upgrade tree, each data point is handled as follows:

3.1) according to formula (1) to x ^kas linear transformation, be

3.2) according to formula (4) calculate point (x ' ^k, y ^k) support vector l ^k, the x in formula ^kwith x ' ^kreplace, f () uses y ^kreplace;

3.3) according to conditional relationship formula (5) and (6), upgrade tree;

3.3.1) find for step 3.2) the support vector l that produces ^k(be v _i=l ^k) leaf node of do not satisfy condition (6);

3.3.2) use l ^kreplacement step 3.3.1) i support vector in the leaf node matrix finding in

thereby form new leaf node;

3.3.3) the new leaf node producing the determining step 3.3.2) relational expression (5) that whether satisfies condition, if met, retains, otherwise deletes;

4) calculate Lower Bound Estimation value.

Suppose

for H ^k(x) r local minimum, the relational expression that satisfies condition (5) of its correspondence and the proppant matrix of (6) are L ^r.By formula (9), had

lower Bound Estimation value

$H^k\left({\hat{x}}^r\right)=1/\mathrm{Trace}{\left(L^r\right)}^{-1}$

i element be

${\hat{x}}^r=\frac{l_i^{k_i}}{\mathrm{Trace}\left(L^r\right)}=\frac{x_i^{k_i}}{f\left(x^{k_i}\right)\mathrm{Trace}\left(L^r\right)}$

By conditional relationship formula (5):

and formula (4) has

$\frac{x_i^{k_i}}{f\left(x^{k_i}\right)}>\frac{x_i^{k_j}}{f\left(x^{k_j}\right)}$

Be multiplied by simultaneously on above formula both sides divided by Trace (L ^r):

$\frac{1}{\mathrm{Trace}\left(L^r\right)}{x}_j^{k_j}\frac{x_i^{k_i}}{f\left(x^{k_i}\right)}>\frac{1}{\mathrm{Trace}\left(L^r\right)}\frac{x_j^{k_j}}{f\left(x^{k_j}\right)}{x}_i^{k_j}$

Above formula is equivalent to:

$x_j^{k_j}{\hat{x}}_i^r>x_i^{k_j}{\hat{x}}_j^r,i,j\∈I,i\≠j---\left(10\right)$

Wherein

for the element in looked for leaf node matrix.

4.1) sampling step length is set, and sampling in field of definition [a, b] scope;

4.2) to step 4.1) each sampled point x _cbe handled as follows:

4.2.1) according to formula (1), do linear transformation, obtain new vector x _c'=(x _c1', x _c2');

4.2.2) according to formula (10) from step 3.3) tree find and comprise vector x _c' leaf node, in formula

use x _c' replace, if leaf node meets formula (10), comprise, otherwise do not comprise;

4.2.3) according to formula (3), obtain vector x _cthe Lower Bound Estimation value y of the corresponding subregion of leaf node at ' place _l';

4.2.5) due in step 2.3) in added a constant M while being converted into IPH, in the hope of Lower Bound Estimation value should deduct M, i.e. y _l=y _l'-M;

4.2.6) output Lower Bound Estimation value y _land corresponding sampled point x _c;

5) calculate Estimation of Upper-Bound value.

5.1) the y in data point ^kget negative;

5.2) by step 2) to 4) calculating Lower Bound Estimation value;

5.3) to step 5.2) the Lower Bound Estimation value of gained gets negatively, is Estimation of Upper-Bound value y _u;

5.4) output Estimation of Upper-Bound value y _uand corresponding sampled point x _c;

6) curve.

6.1) connect each point (x _c, y _l) obtain treating the Lower Bound Estimation of matched curve;

6.2) connect each point (x _c, y _u) obtain treating the Estimation of Upper-Bound of matched curve;

6.3) Estimation of Upper-Bound and Lower Bound Estimation being averaged to the curve obtaining is and treats matched curve.

It is embodiment that the present embodiment be take one dimension Shubert function, and a kind of curve-fitting method of estimating based on abstract convex, wherein comprises following steps:

1) parameter initialization: constant M=300 is set, 100 data point (x of random generation in field of definition [0,5] ^k, y ^k), k=1,2 ..., 100.

2) boundary condition initialization.

2.1) field of definition of determining variable x is [0,5];

2.2) model conversion;

If x is ∈ [a, b], wherein, x is the variable on boundary constraint feasible zone [a, b], and former variable x is done to following linear transformation:

X wherein ₁', x ₂' be linear transformation variable.

Obviously, (b-a) > 0, and x is easily known in (x-a)>=0 ₁'>=0, x ₂'>=0, $x^{\′}={[x_1^{\′},x_2^{\′}]}^T\∈S=\{x^{\′}\∈R_{+}^2:{\mathrm{\Σ}}_{i=1}^2{x}_i^{\′}=1\},$ Wherein $R_{+}^2=\{x^{\′}=(x_1^{\′},x_2^{\′})|x_i^{\′}\≥0,i=\mathrm{1,2}\},$ S representation unit simplex region; Fig. 1 has provided the schematic diagram in the unit's of being converted to simplex space, common definition territory.

Formula (1) is converted, can obtain:

x＝x ₁′(b-a)+a （2）

2.2.1) establish

wherein,

the linear transformation variable 1 that represents the 1st data,

the linear transformation variable 1 of the 2nd data;

2.2.2) right according to formula (2)

be converted to former variable x ¹, x ², x wherein ¹=5, x ²=0;

2.3) calculate initial support vector sum proppant matrix, and set up tree;

Definition 1: if function f meets following two conditions

$i)\∀x,y\∈R_{+}^n,x\≥y\⇒f\left(x\right)\≥f\left(y\right)$

$\mathrm{ii})\∀x\∈R_{+}^n,\∀\mathrm{\λ}\∈R_{++}:f\left(\mathrm{\λx}\right)=\mathrm{\λf}\left(x\right)$

Wherein,

represent arbitrarily,

claim function f for just homogeneous increasing function (Increasing Positively Homogeneous of degree one, IPH).

Objective function also can add one enough large constant M be converted to IPH function.Suppose that the field of definition of N dimension objective function utilized model conversion formula (1) unit's of being mapped to simplex space

suppose that it has K support function, the point (x based on given ^k, f (x ^k)) (k=1,2 ..., K), the Lower Bound Estimation of objective function f can be provided by following formula:

$H^K\left(x\right)=\underset{k\≤K}{\max }\underset{i=1,...N+1}{\min }\frac{x_i}{l_i^k}---\left(3\right)$

Wherein max represents maximum, and min represents minimum, x _ifor the vector in unit simplex space;

K support function is:

$h^k\left(x\right)=\underset{i=1,...N+1}{\min }\frac{x_i}{l_i^k}=f\left(x^k\right)\min \{\frac{x_1}{x_1^k},...,\frac{x_{N+1}}{x_{N+1}^k}\}$

Wherein

$l^k=(\frac{x_1^k}{f\left(x^k\right)},\frac{x_2^k}{f\left(x^k\right)},...,\frac{x_{N+1}^k}{f\left(x^k\right)})---\left(4\right)$

Be called support vector.Consider a set that contains K support vector

make I={1,2 ..., N+1}, N+1 support vector of two conditions (5) and (6) below meeting

be called an effective proppant matrix.

$\∀i,j\∈I,i\≠j:l_i^{k_i}>l_i^{k_j}---\left(5\right)$

$\∀v\∈{\mathrm{\Λ}}^k\backslash L,\∃i\∈I:l_i^{k_i}\≤v_i---\left(6\right)$

Explain for convenience principle, suppose an effective proppant matrix as the formula (7):

According to this proppant matrix, can calculate corresponding lower bound support function extreme value x _minthe Lower Bound Estimation value d corresponding with it:

x _min(L)＝diag(L)/Trace(L) （8）

d(L)＝H ^k(x _min)＝1/Trace(L) ^-1 （9）

Wherein, diag is the positive diagonal entry of matrix, and Trace is matrix trace, i.e. the positive diagonal entry sum of matrix.

2.3.1) according to formula (4) respectively calculation level (x ' ¹, y ₅) and (x ' ², y ₀) support vector, wherein, y ₅the functional value that represents the upper bound 5 correspondences, y ₀the functional value that represents lower bound 0 correspondence, the x in formula ^kwith x ' replacement, f () replaces with y;

2.3.2) 2 support vectors of gained are formed to proppant matrix L, as shown in Equation (7);

2.3.3) take proppant matrix L as root foundation tree;

3) calculate the support vector of each data point and upgrade tree, each data point is handled as follows:

3.1) according to formula (1) to x ^kas linear transformation, be

3.2) according to formula (4) calculate point (x ' ^k, y ^k) support vector l ^k, the x in formula ^kwith x ' ^kreplace, f () uses y ^kreplace;

3.3) according to conditional relationship formula (5) and (6), upgrade tree;

3.3.1) find for step 3.2) the support vector l that produces ^k(be v _i=l ^k) leaf node of do not satisfy condition (6);

3.3.2) use l ^kreplacement step 3.3.1) i support vector in the leaf node matrix finding in

thereby form new leaf node;

3.3.3) the new leaf node producing the determining step 3.3.2) relational expression (5) that whether satisfies condition, if met, retains, otherwise deletes;

4) calculate Lower Bound Estimation value.

Suppose

for H ^k(x) r local minimum, the relational expression that satisfies condition (5) of its correspondence and the proppant matrix of (6) are L ^r.By formula (9), had

lower Bound Estimation value

$H^k\left({\hat{x}}^r\right)=1/\mathrm{Trace}{\left(L^r\right)}^{-1}$

i element be

${\hat{x}}^r=\frac{l_i^{k_i}}{\mathrm{Trace}\left(L^r\right)}=\frac{x_i^{k_i}}{f\left(x^{k_i}\right)\mathrm{Trace}\left(L^r\right)}$

By conditional relationship formula (5): and formula (4) has

$\frac{x_i^{k_i}}{f\left(x^{k_i}\right)}>\frac{x_i^{k_j}}{f\left(x^{k_j}\right)}$

Be multiplied by simultaneously on above formula both sides

divided by Trace (L ^r):

$\frac{1}{\mathrm{Trace}\left(L^r\right)}{x}_j^{k_j}\frac{x_i^{k_i}}{f\left(x^{k_i}\right)}>\frac{1}{\mathrm{Trace}\left(L^r\right)}\frac{x_j^{k_j}}{f\left(x^{k_j}\right)}{x}_i^{k_j}$

Above formula is equivalent to:

$x_j^{k_j}{\hat{x}}_i^r>x_i^{k_j}{\hat{x}}_j^r,i,j\∈I,i\≠j---\left(10\right)$

Wherein

for the element in looked for leaf node matrix.

4.1) sampling step length being set is 0.001, and sampling in field of definition [0,5] scope;

4.2) to step 4.1) each sampled point x _cbe handled as follows:

4.2.1) according to formula (1), do linear transformation, obtain new vector x _c'=(x _c1', x _c2');

4.2.2) according to formula (10) from step 3.3) tree find and comprise vector x _c' leaf node, in formula use x _c' replace, if leaf node meets formula (10), comprise, otherwise do not comprise;

4.2.3) according to formula (3), obtain vector x _cthe Lower Bound Estimation value y of the leaf node corresponding region at ' place _l';

4.2.5) due in step 2.3) in added a constant M while being converted into IPH, in the hope of Lower Bound Estimation value should deduct M, i.e. y _l=y _l'-M;

4.2.6) output Lower Bound Estimation value y _land corresponding sampled point x _c;

5) calculate Estimation of Upper-Bound value.

5.1) the y in data point ^kget negative;

5.2) by step 2) to 4) calculating Lower Bound Estimation value;

5.3) to step 5.2) the Lower Bound Estimation value of gained gets negatively, is Estimation of Upper-Bound value y _u;

5.4) output Estimation of Upper-Bound value y _uand corresponding sampled point x _c;

6) curve.

6.1) connect each point (x _c, y _l) the lower bound sawtooth that obtains one dimension Shubert function curve estimates, as shown in Figure 2;

6.2) connect each point (x _c, y _u) upper bound sawtooth that obtains one dimension Shubert function curve estimates, as shown in Figure 3;

6.3) Estimation of Upper-Bound and Lower Bound Estimation are averaged to the matched curve that the curve obtaining is one dimension Shubert function, as shown in Figure 4.

The one dimension Shubert function of take is embodiment, uses above method to obtain its matched curve as shown in Figure 4, and in figure, " * " point is known data point, and sawtooth curve is the matched curve of one dimension Shubert function.As shown in Figure 2, in figure, " * " point is known data point in the lower bound sawtooth estimation of one dimension Shubert function, and jagged equity curve is that the lower bound sawtooth of one dimension Shubert function is estimated.As shown in Figure 3, in figure, " * " point is known data point in upper bound sawtooth estimation, and jagged equity curve is that the upper bound sawtooth of one dimension Shubert function is estimated.

What more than set forth is the excellent results that embodiment shows that the present invention provides, obviously the present invention is not only applicable to above-described embodiment, can do many variations to it and is implemented not departing from essence spirit of the present invention and do not exceed under the prerequisite of the related content of flesh and blood of the present invention.

Claims (4)

1. a curve-fitting method of estimating based on abstract convex, is characterized in that: described curve-fitting method comprises the following steps:

1) parameter initialization: constant M is set, input data (x ^k, y ^k), k=1,2 ..., K, the quantity size size that wherein K is data point;

2) boundary condition initialization, process is as follows;

2.1) according to existing data point, determine the field of definition [a, b] of variable x, wherein a is the lower bound of variable x, and b is the upper bound;

2.2) model conversion: former variable x is done to following linear transformation:

X wherein ₁', x ₂' be linear transformation variable;

Formula (1) is converted, obtains:

x＝x ₁′(b-a)+a （2）

2.2.1) establish wherein,

the linear transformation variable 1 that represents the 1st data,

the linear transformation variable 1 of the 2nd data;

2.2.2) right according to formula (2) be converted to former variable x ¹, x ², x wherein ¹=b, x ²=a;

2.3) calculate initial support vector sum proppant matrix, and set up tree:

2.3.1) according to formula (4) respectively calculation level (x ' ¹, y _b) and (x ' ², y _a) support vector,

Wherein, y _brepresent functional value corresponding to upper bound b, y _arepresent functional value corresponding to lower bound a, the x in formula ^kwith x ' replacement, f () replaces with y;

2.3.2) two support vectors of gained are formed to proppant matrix L, as shown in Equation (7):

2.3.3) take proppant matrix L as root foundation tree;

3) calculate the support vector of each data point and upgrade tree, each data point is handled as follows:

3.1) according to formula (1) to x ^kas linear transformation, be

3.2) according to formula (4) calculate point (x ' ^k, y ^k) support vector l ^k, the x in formula ^kwith x ' ^kreplace, f () uses y ^kreplace;

3.3) according to conditional relationship formula (5) and (6), upgrade tree:

V ∈ Λ wherein ^kl represent that v belongs to Λ ^kbut do not belong to L, represent to exist;

3.3.1) find for step 3.2) the support vector l that produces ^kthe leaf node of (6), i.e. v do not satisfy condition _i=l ^k;

3.3.2) use l ^kreplacement step 3.3.1) i support vector in the leaf node matrix finding in

thereby form new leaf node;

3.3.3) the new leaf node producing the determining step 3.3.2) relational expression (5) that whether satisfies condition, if met, retains, otherwise deletes;

4) calculate Lower Bound Estimation value;

4.1) sampling step length is set, and sampling in field of definition [a, b] scope;

4.2) to step 4.1) each sampled point x _cbe handled as follows:

4.2.1) according to formula (1), do linear transformation, obtain new vector x _c'=(x _c' ₁, x _c' ₂);

4.2.2) according to formula (10) from step 3.3) tree find and comprise vector x _c' leaf node, in formula

use x _c' replace, if leaf node meets formula (10), comprise, otherwise do not comprise;

Wherein, for the element in looked for leaf node matrix;

4.2.3) according to formula (3), obtain vector x _cthe Lower Bound Estimation value y of the leaf node corresponding region at ' place _l';

4.2.4) due in step 2.3) in added a constant M while being converted into IPH, in the hope of Lower Bound Estimation value should deduct M, i.e. y _l=y _l'-M;

4.2.5) output Lower Bound Estimation value y _land corresponding sampled point x _c;

5) calculate Estimation of Upper-Bound value, process is as follows:

5.1) the y in data point ^kget negative;

5.2) by step 2) to 4) calculating Lower Bound Estimation value;

5.3) to step 5.2) the Lower Bound Estimation value of gained gets negatively, is Estimation of Upper-Bound value y _u;

5.4) output Estimation of Upper-Bound value y _uand corresponding sampled point x _c;

6) curve, process is as follows:

6.1) connect each point (x _c, y _l) obtain treating the Lower Bound Estimation of matched curve;

6.2) connect each point (x _c, y _u) obtain treating the Estimation of Upper-Bound of matched curve;

6.3) Estimation of Upper-Bound and Lower Bound Estimation being averaged to the curve obtaining is and treats matched curve.

2. a kind of curve-fitting method of estimating based on abstract convex as claimed in claim 1, is characterized in that: the constant M in step 1), if function f meets following two conditions:

Wherein,

represent arbitrarily,

claim function f for just homogeneous increasing function IPH;

Add one enough large constant M can will treat that function corresponding to matched curve is converted to IPH function, adds M to each y in data.

3. a kind of curve-fitting method of estimating based on abstract convex as claimed in claim 1 or 2, is characterized in that: step 2.3) in calculate support vector and proppant matrix method be:

Suppose that the field of definition of N dimension objective function utilized model conversion formula (1) unit's of being mapped to simplex space

suppose that it has K support function, the point (x based on given ^k, f (x ^k)) (k=1,2 ..., K), the Lower Bound Estimation of objective function f is provided by following formula:

Wherein max represents maximum, and min represents minimum, x _ifor the vector in unit simplex space;

K support function is:

Wherein, l ^kbe called support vector, as the formula (4), consider a set that contains K support vector

make I={1,2 ..., N+1}, N+1 the support vector of satisfy condition (5) and (6)

be called an effective proppant matrix;

Suppose an effective proppant matrix as the formula (7):

According to this proppant matrix, can calculate corresponding lower bound support function extreme value x _minthe Lower Bound Estimation value d corresponding with it:

x _min(L)＝diag(L)/Trace(L) （8）

d(L)＝H ^k(x _min)＝1/Trace(L) ^-1 （9）

Wherein, diag is the positive diagonal entry of matrix, and Trace is matrix trace, i.e. the positive diagonal entry sum of matrix.

4. a kind of curve-fitting method of estimating based on abstract convex as claimed in claim 3, is characterized in that: step 4.2.2), find out and comprise vector x _c' the method for leaf node be:

Suppose

for H ^k(x) r local minimum, the relational expression that satisfies condition (5) of its correspondence and the proppant matrix of (6) are L ^r, by formula (9), had lower Bound Estimation value

i element be

By conditional relationship formula (5):

and formula (4) has

Be multiplied by simultaneously on above formula both sides

divided by Trace (L ^r):

Above formula is equivalent to formula (10), if institute the leaf node of looking for for

meet formula 10), this leaf node district inclusion is vectorial

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