KR20090071219A - Method for determining mixing ratios in a mixture via non-negative independent component analysis - Google Patents

Abstract

A method for determining proportions of mixture through non-negative independent component analysis is provided to calculate the component proportions of a mixture sample from X-rays diffraction intensity data of the mixture when data to the pure substance forming the mixture sample is not able to be measured. A method for determining proportions of mixture through non-negative independent component analysis comprises: a step(S100) obtaining X rays diffraction intensity data toward a plurality of mixture samples; a step(S102) drawing the independent component of the mixture sample by using non-negative independent component analysis; and a step(S104) computing the mixture coefficient by using a non-negative independent least square method based on the extracted independent component. The mixture coefficient is the component proportions of the mixture sample.

Description

Method for Determining Mixing Ratios in a Mixture via Non-negative Independent Component Analysis}

The present invention relates to a method for determining the composition ratio of an unknown mixture, and more particularly, to determine a mixture composition ratio that can accurately predict the ratio of each component constituting the unknown mixture using non-negative independent component analysis and non-negative square method. It is about a method.

X-ray diffraction (XRD) analysis of the mixture sample can detect the components of the mixture and determine the component ratio of each component. Conventionally, in order to quantitatively analyze the components of a mixture by using powder X-ray diffraction, it is common to measure the XRD spectra of the pure components constituting the mixture sample and to compare them by measuring the XRD spectra of the mixture samples. Was used. However, this mixture analysis method has a problem that it cannot be used when XRD data of the pure substance constituting the sample cannot be obtained or cannot be measured. In addition, it takes a lot of time to measure and compare the pure substances as components, as well as mixture samples, and in some cases quantitative results cannot be obtained without some of the necessary data.

In order to overcome this problem, an independent component analysis (ICA) method is used to separate independent components based on the XRD intensity data of unknown mixtures, and the ratio of each separated component is estimated. Proposed. However, according to the quantitative analysis of the unknown mixture using the independent component analysis, the independent component derived by the independent component analysis is negative even though the components of the mixture are originally non-native and basal pure substances. .

If the independent components are nonnegative in nature, a range occurs in which the independent components derived by the existing independent component analysis are negative for two reasons. First, in the existing independent component analysis, a normal orthogonal matrix having a mean of 0 is generated by preprocessing mixture data composed of nonnegative independent components, and an independent component having the maximum nonnormality of the product of the weight vector is extracted. There arises a problem that the noise part of the independent component becomes negative in its entirety. Second, in the case of mixture data having a large dispersion between components in individual independent components and narrow widths of peaks, interference between independent components occurs, and a negative intensity peak of a large intensity occurs within a range of independent components instantaneously. In addition to the negative generation problem of the independent components described above, a problem may also occur in which the coefficient of each independent component derived by the independent component analysis is also negative. However, when calculating the proportion of independent components of a mixture, the coefficient of independent components representing that proportion should be a negative value.

One aspect of the present invention is proposed to solve the above-mentioned conventional problems, by applying a mixture consisting of independent non-negative independent components to non-negative independent component analysis to separate the independent and positive unknown components, and to separate the separated components It provides a method for predicting and determining accurate and fast by emptying a ratio.

Mixture component ratio determination method through non-negative independent component analysis according to an aspect of the present invention, the step of obtaining the X-ray diffraction intensity data for m mixture samples (m is an integer of 1 or more); Deriving independent components of the mixture sample from the mixture sample using non-negative independent component analysis based on the obtained X-ray diffraction intensity data; And calculating, based on the extracted independent components, a mixture coefficient representing the constituent ratio of the mixture sample using a nonnegative square method.

According to an embodiment of the present invention, the independent component derivation step,

A first step of randomly selecting an initial weight vector ( w ) from a standard normal distribution having a mean of 0 and a standard deviation of 1 with respect to the X-ray diffraction intensity data ( X ) of the m mixture samples;

w ← E [x g (w T x)] - wE [g (w T x)] the second step formula with a modifying said initial weight vector w with a new weight vectors (where x ^T is x ^T = (X ₁ , ..., X _m ) is a mixture variable vector, the expected E is estimated as the mean of the data, and the function g is the derivative of the non-quardratic function G, where g (u) = tanh (u)) ;

A third step of changing the current weight vector using a new weight vector w obtained in the second step through a formula of w ← w / ∥ w ∥;

A fourth step of non-negative processing the negative part such that w ^T x ? 0 with respect to the current weight vector w obtained in the third step;

A fifth step of determining whether a difference between the non-negative current weight vector and the previous weight vector converges to a predetermined value or less; And

And if the convergence result does not converge, proceeding to the second step and repeating subsequent steps; and if the convergence result converges, calculating the independent component S through S = w ^T x ; have.

The first to sixth steps may be repeated by the number of independent components of the mixture sample to obtain w and obtain each independent component therefrom. In the fifth step, the predetermined value may be set to 0.0001.

In the fourth step, the negative portion may be non-negative using at least one of three methods represented by the following equation.

.

.

According to an embodiment of the invention, the step of calculating the coefficient of mixture,

The mixture observation matrix X of (m × n) and the independent component matrix of (p × n) for the m mixture samples of which the measurement angle dimension of the x-ray diffraction intensity dimension is n (n is an integer of 1 or more) Calculating a mixture coefficient matrix B through equation B ^T = ( SS ^T ) ⁻¹ SX ^T by applying the least square method using S ) (p is an integer of 1 or more);

When the j th row vector of the mixture coefficient matrix B is b _j = (b _ji , ..., b _jp ), the minimum value of the elements of the row vector bj b _jmin = min (b _ji , ..., b _jp Obtaining;

Determining whether the minimum value b _jmin ≥ 0;

As a result of the determination, if the minimum value b _jmin ≥ 0, the coefficient (a _ij ) of the mixture is obtained by adjusting the sum of the elements of the row vector b _j to be 1 through the following equation, and if the minimum value b _jmin <0, the expression bji ← after adjusting the elements of the row vector bj through b _ji -b _jmin, and re-adjusting the coefficients of the mixture (a _ij ) by the following equation so that the sum of the elements of the adjusted row vector bj is 1; ,

;

;

It may include.

According to the present invention, it is possible to separate the independent components of the mixture data composed of the nonnegative and hypotonic independent components, and to quickly and accurately predict the component ratio of the mixture. Even when data on the pure substance constituting the mixture mixture is not available or cannot be measured, the ratio of the components can be obtained from the mixture's X-ray diffraction intensity data in a short time, and when the independent component analysis is used. The component ratio can be accurately determined by preventing the generation of negative independent components.

Hereinafter, embodiments of the present invention will be described with reference to the accompanying drawings. However, embodiments of the present invention may be modified in various other forms, and the scope of the present invention is not limited to the embodiments described below. Embodiments of the present invention are provided to more completely explain the present invention to those skilled in the art.

Non-negative independent component analysis is a statistical technique for extracting independent and positive independent components derived from solving mixture problem analysis problem under the condition that original independent component is non-negative. If it is assumed that there is an independent component vector s = [S1, S2, ..., Sp] consisting of P unknown components with real values, two conditions for the i-th independent component may be expressed as follows.

Nasal: Pr (Si <0) = 0, (i = 1, ..., p)

Basis: Pr (Si <δ)> 0, ∀δ (> 0), (i = 1, ..., p)

Where Pr represents the probability

In the present invention, by using the non-negative independent component analysis, the components are derived from the measured signal data of the mixture containing the purely nonnegative and hypo-negative substances, and applied to the non-negative square method to return the original mixture data values. We propose a method for predicting the composition ratio of independent components. As an application example, the components of the spectrum of the mixture powder composed of SiO 2, TiO 2, and MgO, which are the main components of the mold flux used as a catalyst in the production of iron, can be separated, and then the proportion of each component can be estimated.

BRIEF DESCRIPTION OF THE DRAWINGS It is a flowchart for demonstrating the mixture component ratio determination method which concerns on embodiment of this invention. Referring to FIG. 1, first, X-rays are irradiated to n mixture samples to obtain X-ray diffraction (XRD) intensity data diffracted from the sample (S100). The obtained XRD intensity data may be an XRD intensity value measured in n dimensions on the diffraction angle axis (x axis) in a specific angular range. For example, as XRD intensity data for 10 powder mixture samples, XRD intensity values on the y axis measured at 1000 diffraction angles (1000 dimensions) at 0.05 degree intervals in the 20 to 70 degree angle range on the x axis will be presented. Can be. In this case, m = 10 and n = 1000.

XRD data obtained in n-dimensional angles of measurement for m mixture samples may be expressed as (m × n) matrix X (mixture observation matrix) as follows.

For example, the diffraction intensity spectrum of FIG. 4 is a spectrum obtained by measuring a measurement angle of 1000 dimensions for 10 mixture samples. Here, each element (x _ij ) of the mixture observation matrix X represents an XRD intensity value.

Next, based on the obtained XRD data, an independent component is derived from the mixture sample using non-negative independent component analysis (S102). In this embodiment, instead of an independent component analysis method such as FastICA (Fast Independent Component Analysis) in order to extract an independent component from an unknown mixture sample, a modified non-negative component analysis method is proposed (see FIG. 2), and this is NNFICA. (Non-negative Fast Independent Component Analysis).

M mixture samples (e.g. j = 1, ..., 10), XRD measurement angles in n dimensions (e.g. k = 1, ..., 1000), p independent components S of the mixture ( Example: i = 1, 2, 3), the XRD intensity value (X) can be expressed by, for example, Equation 1 below.

Where a _ji represents the coefficient of the jth mixture with respect to the ith independent component.

When Equation 1 is expressed in a matrix form, when X = (X _jk ), S = (S _ik ), and A = (a _ji ), it may be expressed as X = AS .

A non-negative independence analysis method is used to derive the independent component S, which can be described below with reference to FIG. 2.

Derivation of Independent Ingredients

The random variable S representing one independent component may be expressed as S = w ^T x , and in order to obtain the independent component S, a weight vector w maximizing the nonnormality of w ^T x is obtained. Here, w is ^{T _{w T = (w 1, ...}} , w m) can be expressed as a transposed matrix of x is x ^T x ^T = (X _1, a transpose matrix of w ..., X _m ) as a mixture variable vector (superscript 'T' indicates a transpose). Thus, S = w ₁ X ₁ + ... + w _m X _m = w ^T x . The process of obtaining the weight vector w to be substituted into the equation S = w ^T x to find the independent component S is as follows.

(1), the first select the initial weight vector w with respect to the random X-ray diffraction intensity data (X) of the m sample mixture as shown in Figure 2 (S200). At this time, an initial weight vector w is randomly selected from a standard normal distribution having a mean of 0 and a standard deviation of 1, and a column vector of (m × 1) is extracted.

(2) Next, the initial weight vector w is modified to a new weight vector using the formula w ← E [ x g ( w ^T x )]-wE [g ( w ^T x )] (S202). Here, the expected value E is estimated as an average of the diffraction intensity data, and the function g is a derivative of the non-quardratic function G, where g (u) = tanh (u).

(3) Next, the new weight vector w obtained in the above (2) is changed into the current weight vector through the formula w ← w / ∥ w ∥ (S204).

(4) Next, the negative portion is non-negative and transformed so that w ^T x ≥ 0 with respect to the current weight vector w obtained in the third step (S206).

Here, three methods (method 1, method 2, method 3) are proposed as follows for the non-negative treatment of negative independent components.

.

.

The above three non-negative processing methods are proposed to converge to a constant weight vector while minimizing or compensating the negative part of the independent components derived for each iteration of NNFICA.

(5) Next, it is determined whether the difference between the magnitude of the current weight vector passed through the non-negative processing and the weight vector of the previous step converges below a predetermined value (S208). In the present embodiment, the convergence is set when the magnitude of the current w and the w in the previous step is 0.0001 or less.

(6) If the result of the determination does not converge, the process proceeds to (2) again and repeats subsequent steps, and if the result of the determination converges, the independent component S is calculated by using S = w ^T x using the final w . (S210).

Repeating steps (1) to (6) as many as the number of independent components of the mixture sample to obtain w for each independent component can be obtained from each independent component (see Figure 5).

As described above, after obtaining the independent component (S) by the non-negative independence analysis, in particular, the method according to the NNFICA proposed in this embodiment (see (1) to (6) described above), the independent component extracted as described above By calculating the mixture coefficients using non-negative least squares based on (S), the component proportions of the mixture samples are obtained (step S104 in FIG. 1). Specific mixture coefficient calculation method can be described as follows with reference to FIG.

Calculation of Mixture Factor

From the model of Equation 2 below, a _ji representing the mixture coefficient is obtained.

이며, ε_jk는 회귀모형관련 오차항을 나타낸다.only,

Ε _jk represents the error term related to the regression model.

Equation 2 is a formula representing a nonnegative minimum square method to obtain a mixture coefficient through the process described below. First, let the mixture observation matrix represented by the (m × n) matrix be X = (Xjk), and the independent component matrix represented by the (p × n) matrix is S = (Sik).

[1] The mixture coefficient matrix B is calculated through Equation 3 below by the least square method (S300).

B ^T = ( SS ^T ) ^-1 SX ^T

[2] Next, when the j th row vector of the mixture coefficient matrix B obtained in the above [1] is b _j = (b _ji , ..., b _jp ), the minimum value of the elements of the row vector bj b _jmin = min (b _ji , ..., b _jp ) is obtained (S302).

[3] Next, for non- _sound processing, it is determined whether b _jmin ≥ 0 (S304).

[4-1] As a result of the determination, if b _jmin ? 0, the process goes to the following [5].

[4-2] As a result of the determination, if b _jmin &lt; 0, the elements of the row vector bj are adjusted as shown in Equation 4 below (S306), and the routine goes to [5] below.

bji ← b _ji -b _jmin

[5] As shown in Equation 5 below, the sum of the elements of the row vector b _j is adjusted to 1 to obtain a _ji representing the mixture coefficient (S308).

Through [1] ~ [5], the mixture coefficients, ie the component ratios of the components of the mixture, are obtained.

(Example)

The present inventors have applied the above-described non-negative independent component analysis and non-negative minimum for powder mixture samples (including SiO ₂ , TiO ₂ , and MgO) of a mold flux used as a catalyst in the iron manufacturing process as an application example of the above-described mixture component ratio determination method. Square method was used to separate the independent components of the mixture and predict the proportion of each component.

4 shows the XRD intensity spectrum for 10 samples of the powder mixture. This spectrum represents the XRD intensity value (y-axis) measured in 1000 dimensions (x-axis) at 0.05-degree intervals from 20 degrees to 70 degrees. Figure 5 is based on the mixture XRD intensity data as shown in Figure 4, the three obtained through the inventors proposed NNFICA non-negative independence analysis (see (1) to (6) above) (SiO ₂ , TiO ₂ , MgO) is a graph showing the independent components. Referring to Figure 3, it was separated into three independent components, each independent component can be seen that shows a unique peak at a certain angle (SiO 2: 26.5 °, TiO 2: 25.5 °, MgO: 43 °).

FIG. 6 is a table showing the results of predicting each component ratio using the aforementioned non-negative square method (see the above-described processes of [1] to [5]) based on the independent components shown in FIG. 4. Mean absolute deviation (MAD) was used to confirm the accuracy of the mixture component ratio prediction value obtained by the method proposed by the present inventors and to compare the accuracy of the prediction value through the conventional independent component analysis method. The mean absolute error is expressed as

는 실제의 구성상분 비율을 나타낸다. Here, A _ji is a predicted value of the component ratio obtained through an analytical method using NNFICA or conventional ICA,

Represents the actual percentage of composition.

7 is a table showing the average absolute value error obtained by applying the mixed powder data of the mold flux to the existing FastICA, and the average absolute value error obtained by applying the NNFICA proposed by the present inventors. As shown in Figure 7, by applying the NNFICA proposed in the present specification it can be seen that the average absolute error is reduced to obtain a more stable result than when applying the conventional FastICA, thereby increasing the reliability of the accuracy of the prediction value have.

The present invention is not limited to the above-described embodiment and the accompanying drawings, but is intended to be limited by the appended claims, and that various modifications can be made without departing from the spirit of the invention described in the claims. It will be apparent to one of ordinary skill in the art.

BRIEF DESCRIPTION OF THE DRAWINGS It is a flowchart for demonstrating the mixture component ratio determination method which concerns on embodiment of this invention.

2 is a flowchart illustrating a process of obtaining a weight vector and independent components according to an embodiment of the present invention.

3 is a flowchart illustrating a mixture coefficient calculation process according to an embodiment of the present invention.

4 is a graph showing an X-ray diffraction (XRD) intensity spectrum of a mixed powder sample subjected to non-negative independent component analysis according to an embodiment of the present invention.

FIG. 5 is a graph illustrating independent component spectra of three kinds (SiO ₂ , TiO ₂ , MgO) obtained through non-negative independent component analysis according to an embodiment of the present invention.

6 is a table showing the results of predicting the ratio of three independent components by the mixture component ratio determining method according to an embodiment of the present invention.

7 is a table showing the average absolute value error obtained by applying FastICA and the average absolute value error obtained by applying NNFICA according to an embodiment of the present invention for the same mixed powder data.

Claims (6)

obtaining X-ray diffraction intensity data for m mixture samples (m is an integer of 1 or more); Deriving independent components of the mixture sample from the mixture sample using non-negative independent component analysis based on the obtained X-ray diffraction intensity data; And Calculating a mixture coefficient indicating a component ratio of the mixture sample by using a non-negative square method based on the extracted independent components. The method of claim 1, The independent component derivation step, A first step of randomly selecting an initial weight vector ( w ) from a standard normal distribution having an average of 0 and a standard deviation of 1 with respect to the X-ray diffraction intensity data of the m mixture samples; w ← E [x g (w T x)] - wE [g (w T x)] the second step formula with a modifying said initial weight vector w with a new weight vectors (where x ^T is x ^T = (X ₁ , ..., X _m ) is a mixture variable vector, the expected value E is estimated as the mean of the data, and the function g is the derivative of the non-quardratic function G, where g (u) = tanh (u)) ; A third step of changing the current weight vector using a new weight vector w obtained in the second step through a formula of w ← w / ∥ w ∥; A fourth step of non-negative processing the negative part such that w ^T x ? 0 with respect to the current weight vector w obtained in the third step; A fifth step of determining whether a difference between the non-negative current weight vector and the previous weight vector converges to a predetermined value or less; And And if the convergence result does not converge, proceeding to the second step and repeating subsequent steps; and if the convergence result converges, a sixth step of calculating the independent component S through S = w ^T x ; Mixture component ratio determination method through non-negative independent component characterized in that. The method of claim 2, Wherein the first to sixth step, mixture component ratio determination method through the non-negative independent component analysis, characterized in that it is repeatedly performed as the number of independent components of the mixture sample. The method of claim 2, In the fifth step, the predetermined value is a mixture component ratio determination method through the non-negative independent component analysis, characterized in that set to 0.0001. The method of claim 2, In the fourth step, the mixture component ratio determination method through the non-negative independent component analysis, characterized in that the negative portion is non-negative processing using at least one of the three methods represented by the following equation,

. The method of claim 1, The mixture coefficient calculating step, The mixture observation matrix X of (m × n) and the independent component matrix of (p × n) for the m mixture samples of which the measurement angle dimension of the x-ray diffraction intensity dimension is n (n is an integer of 1 or more) Using S ) (p is an integer of 1 or more), calculating a mixture coefficient matrix B through the equation B ^T = ( SS ^T ) ⁻¹ SX ^T by applying the least square method; When the j th row vector of the mixture coefficient matrix B is b _j = (b _ji , ..., b _jp ), the minimum value of the elements of the row vector bj b _jmin = min (b _ji , ..., b _jp Obtaining; Determining whether the minimum value b _jmin ≥ 0; And As a result of the determination, if the minimum value b _jmin ≥ 0, the coefficient (a _ij ) of the mixture is obtained by adjusting the sum of the elements of the row vector b _j to be 1 through the following equation, and if the minimum value b _jmin <0, the expression bji ← after adjusting the elements of the row vector bj through b _ji -b _jmin, and re-adjusting the coefficients of the mixture (a _ij ) by the following equation so that the sum of the elements of the adjusted row vector bj is 1; ,

; Mixture component ratio determination method through a non-negative independent component analysis comprising a.

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