CN105955932B - Timber density determination method based on iteration weight least square estimate method - Google Patents

Abstract

A timber density determination method based on an iteration weight least square estimate method relates to the timber density determination estimate method; the novel method solves the problems that a conventional regression analysis method has big errors in batch timber density determination; the method uses expected average estimation having unbiased estimate property under limited samples to serve as the initial mean value estimate value, uses the iteration weight least square method to carry out regression estimate fine tuning, and the product between a weight coefficient sampling density frequency and an error inverse distance can be randomly converged to a set threshold without having divergency phenomenon, thus relatively accurately estimating the total sample density. The timber density determination method based on the iteration weight least square estimate method is suitable for the batch timber density estimate field.

Description

A kind of Method for determination of the density of wood based on iteration weighted least square method

Technical field

The present invention relates to the method for estimation of Density Determination.

Background technology

During actual high-volume timber is applied, need the quality to load-bearing timber structure or the timber applied Estimated in advance, but each timber can not possibly be weighed during application.In actual load-bearing timber structure matter Amount is usually to be calculated according to the density of timber and volume during estimating, and can not in actual timber application process Density measure can be carried out to object all of in boarding, usually the density of boarding be estimated.

The mensure of density of wood is general to be implemented according to " GB/T 1933-2009 Method for determination of the density of wood ", but " GB/T 1933-2009 Method for determination of the density of wood " is not directed to boarding and how to estimate the total sample of this batch of timber by extracting sample density The method of density, and do not have pertinent literature to solve this problem.Due to density of wood discreteness is larger and measurement error very Little, the inapplicable the demand of traditional regression analysis, the quality estimation accuracy of timber structure is therefore made by certain batch of timber Can be greatly affected.

Content of the invention

The present invention carries out estimating exist for the density measurement of batch timber for methods such as traditional regression analysis The larger problem of error.

A kind of Method for determination of the density of wood based on iteration weighted least square method, comprises the following steps：

Step 1, randomly draw sample for boarding, calculate each sample density ρ_i；

Step 2, the expectation averaged power spectrum of sample density：

Step 2.1, the Frequency statistics of sample density：

From ρ_iIn determine maximum ρ_maxWith minimum ρ_min；

Take a=[ρ_min*10^l-1-0.5]/10^l-1, b=[ρ_max*10^l-1+0.5]/10^l-1；L is ρ_i、ρ_maxOr ρ_minArithmetic point Number of significant digit afterwards, [] computing represents round numbers；

Interval [a, b] is divided into m minizone, calculates sample rate number p falling into each minizone_j, this p_jIt is frequency Several, total sampling numbers

Step 2.2, expectation averaged power spectrum：

The expectation averaged power spectrum of sample density can be calculated by formula (2)

Each ρ_iCorresponding frequency is frequency p itself falling into corresponding to minizone_i；Although only m interval, also Only m p_jBut, by each ρ_iAll correspond to a p_i, ρ_iCorresponding p_iFor current ρ_iFall into frequency p corresponding to minizone_i, So n ρ_iN p can be corresponded to_i；

Step 3, the weighted least-squares iterative estimate of density：

Volume v of each sample_iConstitute matrix V=[v₁v₂… v_n]^T, quality g_iConstitute matrix G=[g₁g₂… g_n]^T, Density p_iConstitute matrix P=[ρ₁ρ₂… ρ_n]^T, by ρ₁To ρ_nN corresponding p_iConstitute diagonal matrix W；

Step 3.1：

A, generalSubstitute in formula (3), draw initial error apart from E₀：

B, by E₀Expand diagonally battle array, then initial weight coefficientDetermined by formula (4)；

C, calculate initial density with the weighted least-squares method of formula (5) and estimate

Step 3.2：

D, generalSubstitute in formula (6), draw the kth time error distance E of density estimation_k；

Wherein,It is the density of -1 estimation of kth；

E, by E_kExpand diagonally battle array, then kth time weight coefficientDetermined by formula (7)；

F, calculate kth time density estimation with the weighted least-squares method of formula (8)

Step 3.3：

G, calculate the kth time variance of density estimation with formula (9)

H, δ are the threshold value setting, and judgeThen proceed to step 3.2 continuation weighting minimum as being unsatisfactory for condition Two take advantage of iterative estimate to calculate, and such as meet condition and then terminate and output density estimated value

The present invention has the effect that：

This method has the expectation averaged power spectrum of unbiased esti-mator property as initial mean value estimated value using under finite sample, with Iteration weighted least-squares method does regression estimates fine setting, and weight coefficient takes the frequency of the sample rate product reciprocal with error distance (fully taken into account appearance probability and the estimation effect to sample of sample rate, meet density of wood dispersion between subjects larger but The less larger sample of density occurs that probability is little, the feature of finite sample capacity), can arbitrarily converge to the threshold value of setting Interior and occur without Divergent Phenomenon, can more accurately estimate to reflect the density of population sample.

Brief description

Fig. 1 is the iteration weighted least square operational flowchart of Method for determination of the density of wood；

Fig. 2 (a) is sample measurement scattergram；Fig. 2 (b) sample rate scattergram；

The frequency histogram of Fig. 3 sample density.

Specific embodiment

Specific embodiment one：

A kind of Method for determination of the density of wood based on iteration weighted least square method, comprises the following steps：

Step 1, randomly draw sample for boarding, calculate each sample density ρ_i；

Step 2, the expectation averaged power spectrum of sample density：

Step 2.1, the Frequency statistics of sample density：

From ρ_iIn determine maximum ρ_maxWith minimum ρ_min；

Take a=[ρ_min*10^l-1-0.5]/10^l-1, b=[ρ_max*10^l-1+0.5]/10^l-1；L is ρ_i、ρ_maxOr ρ_minArithmetic point Number of significant digit afterwards, [] computing represents round numbers；

Interval [a, b] is divided into m minizone, calculates sample rate number p falling into each minizone_j, this p_jIt is frequency Several, total sampling numbers

Step 2.2, expectation averaged power spectrum：

The expectation averaged power spectrum of sample density can be calculated by formula (2)

Each ρ_iCorresponding frequency is frequency p itself falling into corresponding to minizone_i；Although only m interval, also Only m p_jBut, by each ρ_iAll correspond to a p_i, ρ_iCorresponding p_iFor current ρ_iFall into frequency p corresponding to minizone_i, So n ρ_iN p can be corresponded to_i；

Step 3, the weighted least-squares iterative estimate of density：

Volume v of each sample_iConstitute matrix V=[v₁v₂… v_n]^T, quality g_iConstitute matrix G=[g₁g₂… g_n]^T, Density p_iConstitute matrix P=[ρ₁ρ₂… ρ_n]^T, by ρ₁To ρ_nN corresponding p_iConstitute diagonal matrix W；

Step 3.1：

A, generalSubstitute in formula (3), draw initial error apart from E₀：

B, by E₀Expand diagonally battle array, then initial weight coefficientDetermined by formula (4)；

C, calculate initial density with the weighted least-squares method of formula (5) and estimate

Step 3.2：

D, generalSubstitute in formula (6), draw the kth time error distance E of density estimation_k；

Wherein,It is the density of -1 estimation of kth；

E, by E_kExpand diagonally battle array, then kth time weight coefficientDetermined by formula (7)；

F, calculate kth time density estimation with the weighted least-squares method of formula (8)

Step 3.3：

G, calculate the kth time variance of density estimation with formula (9)

H, δ are the threshold value setting, and judgeThen proceed to step 3.2 continuation weighting minimum as being unsatisfactory for condition Two take advantage of iterative estimate to calculate, and such as meet condition and then terminate and output density estimated value

The expectation averaged power spectrum method that this method adopts and iteration weighted least-squares method all have extensively in measurement adjustment field Apply generally, but both approaches be both for measured value exist draw under equally accurate and unequal precision measurement error condition calibrated True estimated result, does not account for sample and there is larger dispersion and the less situation of measurement error.

In density estimation, arithmetic mean of instantaneous value estimator is entirely to receive pseudo- estimator.Arithmetic mean of instantaneous value estimator is namely A young waiter in a wineshop or an inn takes advantage of the maximum likelihood estimator of estimator or normal distribution, be built upon all observations containing only incidental error on the basis of A kind of method of estimation, if containing larger Discrete Distribution value in observation, this estimation is due to being by mean allocation error principle Carry out processing data, it will lead to estimated result to receive puppet.

Medion estimator and Crestor entitled " Sum Maximum Likelihood Estimate ", because it is according to stochastic variable probability event sum Derive unknown parameter estimator for maximum, there are thoroughly Robustness least squares, the larger Discrete Distribution of observation or rough error are not Impact can be produced on estimated result, but due to having thoroughly exclusiveness so that medion estimator eliminate all of unnecessary Observation, therefore brings and entirely abandons genuine essential behaviour, but thoroughly Robust filter method.

Under conditions of finite sample capacity, above two averaged power spectrum be biased estimation, medion estimator can gram Take the big situation of Density Distribution dispersion and arithmetic mean cannot overcome.Expect that averaged power spectrum occurs due to considering sample rate Frequency (the discrete data statistical result of probability density), estimate to have neither receive puppet, nor abandon genuine essential behaviour, that is, limited Under conditions of sample size, this averaged power spectrum is unbiased esti-mator.

The weight coefficient of iteration weighted least-squares method generally takes variance inverse to make mean square error and reach minimum, or according to right The feature of elephant takes signal to noise ratio, credibility etc. as weight coefficient, the following the example of of such weight coefficient be not all suitable for density of wood sample from Divergence is larger and measurement error is less, dispersion assumes quasi normal distribution feature；Iteration weighted least-squares method initially equal Value generally takes arithmetic average estimated value or Least Square Method value, the excess kurtosis estimated under finite sample due to estimated value Matter, iterative process is frequently present of Divergent Phenomenon.

This method has the expectation averaged power spectrum of unbiased esti-mator property as initial mean value estimated value using under finite sample, with Iteration weighted least-squares method does regression estimates fine setting, and weight coefficient takes the frequency of the sample rate product reciprocal with error distance (fully take into account appearance probability and the estimation effect to sample of sample, met that density of wood dispersion between subjects are larger but density Less larger sample occurs that probability is little, the feature of finite sample capacity), can arbitrarily converge in the threshold value of setting Occur without Divergent Phenomenon, can more accurately estimate to reflect the density of population sample.

Specific embodiment two：

The detailed process of present embodiment step 1 is as follows：

For boarding randomly draw sample (need representative, the quantity of sample should account for total sample four/ More than one, and it is no less than 20), the volume of each sample is measured according to " GB/T 1933-2009 Method for determination of the density of wood " v_i, quality g_i, i is the sequence number of sample, i=1,2...n；

Each sample density is calculated by formula (1)

Other steps and parameter are identical with specific embodiment one.

Specific embodiment three：

The determination method of the m described in present embodiment step 2.1 is as follows：

By empirical equation m ≈ [Δ (n-1)^0.4] determine the required interval m dividing；Δ value 1.80～1.90；

In formula, [] computing represents round numbers.

Other steps and parameter are identical with specific embodiment two.

Specific embodiment four：

Δ=1.90 described in present embodiment step 2.1.

Other steps and parameter are identical with specific embodiment three.

Embodiment

Using the present invention, a collection of Lignum paulowniae bar is tested：

Step 1, randomly draw sample for boarding, calculate each sample density；

The sample that boarding extracts is as shown in table 1.

Table 1 wood sample value

Sample measurement such as Fig. 2 (a) distribution in table 1, its density such as Fig. 2 (b) is distributed.

Step 2, the expectation averaged power spectrum of sample density：

1st, the Frequency statistics of sample density

Determined by the sample density values of table 1：

ρ_min=0.196, then take a=0.19；ρ_minSignificance bit l=3 after=0.196 arithmetic point, uses 10^l-1It is multiplied by this Number rounds, afterwards divided by 10 after deducting 0.5 again^l-1A, a=[0.196*100-0.5]/100；

ρ_max=0.306, then take b=0.31；ρ_maxSignificance bit l=3 after=0.306 arithmetic point, uses 10^l-1It is multiplied by this Number rounds, afterwards divided by 10 after adding 0.5^l-1B, b=[0.306*100+0.5]/100；

Can determine that density interval [0.19,0.31]；

Take Δ=1.9, m ≈ [Δ (n-1)^0.4]=[6.885]；M ≈ 6 is it is considered to the factor of [6.885] chooses m=7, really The fixed required interval dividing is 7；

Then fall into the sample rate number such as table 2 of each minizone, its rectangular histogram is as shown in Figure 3.

Table 2 sample rate frequency table

2nd, the expectation averaged power spectrum of sample density

The expectation averaged power spectrum of density can be calculated by formula (2)

Step 3, the iteration weighted least square of sample density

According to the iteration weighted least square operational flowchart of Fig. 1, an iteration weighting young waiter in a wineshop or an inn for density can be calculated Take advantage of estimation,

Wherein,

The interpretation of result of this method

According to the data of table 1, the density estimation result of conventional method of estimation is listed in table 3.

The density estimation result of method of estimation commonly used by table 3

As can be seen from Table 3, the density estimation variance minimum of this method is superior to other methods of estimation.

Claims (4)

1. a kind of Method for determination of the density of wood based on iteration weighted least square method is it is characterised in that include following walking Suddenly：

Step 1, randomly draw sample for boarding, calculate each sample density ρ_i；

Step 2, the expectation averaged power spectrum of sample density：

Step 2.1, the Frequency statistics of sample density：

From ρ_iIn determine maximum ρ_maxWith minimum ρ_min；

Take a=[ρ_min*10^l-1-0.5]/10^l-1, b=[ρ_max*10^l-1+0.5]/10^l-1；L is ρ i, ρ_maxOr ρ_minAfter arithmetic point Number of significant digit, [] computing represents round numbers；

Interval [a, b] is divided into m minizone, calculates sample density number p falling into each minizone_j, this p_jIt is frequency, Total sampling number

Step 2.2, expectation averaged power spectrum：

The expectation averaged power spectrum of sample density can be calculated by formula (2)

${\widehat{\mathrm{\ρ}}}_e=\frac{\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{p}_i\·{\mathrm{\ρ}}_i}{\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{p}_i}---\left(2\right)$

Each ρ_iCorresponding frequency is frequency p itself falling into corresponding to minizone_i；N ρ_iN p can be corresponded to_i；

Step 3, the weighted least-squares iterative estimate of density：

Volume v of each sample_iConstitute matrix V=[v₁v₂… v_n]^T, quality g_iConstitute matrix G=[g₁g₂… g_n]^T, density ρ_iConstitute matrix P=[ρ₁ρ₂… ρ_n]^T, by ρ₁To ρ_nN corresponding p_iConstitute diagonal matrix W；

Step 3.1：

A, generalSubstitute in formula (3), draw initial error apart from E₀：

$E_0=\left|{\widehat{\mathrm{\ρ}}}_{e}V-G\right|---\left(3\right)$

B, by E₀Expand diagonally battle array, then initial weight coefficientDetermined by formula (4)；

$W_{wls}^0={\mathrm{WE}}_0^{-1}---\left(4\right)$

C, calculate initial density with the weighted least-squares method of formula (5) and estimate

${\widehat{\mathrm{\ρ}}}_{wls}^0={\left(V^T{W}_{wls}^{0}V\right)}^{-1}{V}^T{W}_{wls}^{0}G---\left(5\right)$

Step 3.2：

D, generalSubstitute in formula (6), draw the kth time error distance E of density estimation_k；

$E_k=\left|{\widehat{\mathrm{\ρ}}}_{wls}^{k-1}V-G\right|---\left(6\right)$

Wherein,It is the density of -1 estimation of kth；

E, by E_kExpand diagonally battle array, then kth time weight coefficientDetermined by formula (7)；

$W_{wls}^k={\mathrm{WE}}_k^{-1}---\left(7\right)$

F, calculate kth time density estimation with the weighted least-squares method of formula (8)

${\widehat{\mathrm{\ρ}}}_{wls}^k={\left(V^T{W}_{wls}^{k}V\right)}^{-1}{V}^T{W}_{wls}^{k}G---\left(8\right)$

Step 3.3：

G, calculate the kth time variance of density estimation with formula (9)

${\mathrm{\σ}}_k^2=\frac{1}{n-1}\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{({\mathrm{\ρ}}_i-{\widehat{\mathrm{\ρ}}}_{wls}^k)}^2---\left(9\right)$

H, δ are the threshold value setting, and judgeThen proceed to step 3.2 continuation weighted least-squares as being unsatisfactory for condition Iterative estimate calculates, and such as meets condition and then terminates and output density estimated value

2. a kind of Method for determination of the density of wood based on iteration weighted least square method according to claim 1, its It is characterised by that the detailed process of step 1 is as follows：

Randomly draw sample for boarding, measure volume v of each sample_i, quality g_i, i is the sequence number of sample, i=1, 2...n；

Each sample density is calculated by formula (1)

${\mathrm{\ρ}}_i=\frac{g_i}{v_i}---\left(1\right).$

3. a kind of Method for determination of the density of wood based on iteration weighted least square method according to claim 1 and 2, It is characterized in that the determination method of m described in step 2.1 is as follows：

By m ≈ [Δ (n-1)^0.4] determine the required interval m dividing；Δ value 1.80～1.90；

In formula, [] computing represents round numbers.

4. a kind of Method for determination of the density of wood based on iteration weighted least square method according to claim 3, its It is characterised by Δ=1.90 described in step 2.1.