中南大学（本、专）科生考试试卷.doc

啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊9, 简述ITTFA方法的基本思路,并说明该方法是怎样来实现其迭代逼近的 (5)本题的要点在于:1,说明ITTFA (Iterative Target Transformation Factor Analysis) 方法的基本.啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊中南大学（本、专）科生考试试卷啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊啊Examination Paper in Central South UniversityCourse：Chemometrics Time：May 31 2005Student No.： Name: Score: 1. What is the meaning of the linearly dependent and the linearly independent among vectors? For a chemical signal matrix, what is the relationship between the rank of the matrix and the number of the chemical components in the sample? Which hypothesis does this relationship base?2、What is singular value decomposition (SVD)? Perform SVD on a chemical signal matrix, which kinds of three matrices can be obtained, try to explain the mathematical and physical meanings of these three matrices.3. y=c1x1+c2x2+c3x3+eWhere y indicates observing spectrum of mixed sample with three components, xi(i=1,2,.,n) signify observing spectrum of pure component and e symbolizes observing error vector. Try to explain how to turn the above formula to the following one in the means of matrix operation.y = Xc +ewhere matrix X=x1, x2, x3 is so-called sensitivity matrix, while c=c1, c2, c3t is unknown concentration vector. After matrix operation, we get c= (XtX)-1XtyThis is the result of Multiple Linear Regression (MLR), try to explain that MLR is essentially Least Squares method.4. What is Chemometrics? And what does Chemometrics mainly study?5. Briefly explain why it is necessary for chemists to perform an experimental design. And what are the major ideas of different experimental designs?6. How to construct normal plot? What are the functions of effect and residual normal plots, what kind of conclusions can we achieve when using the two plots? 7. Try to explain the main idea of Savizky-Golay smoother and compare this method with moving window average smoother.8、What is the selective information in the two-dimensional hyphenated data and how can the selective information be detected?9、Describe the basic idea of Iterative Target transformation Factor analysis (ITTFA) in brief and further elucidate how ITTFA can approach the feasible solution by iteration?10、Describe briefly the basic rationale of window factor analysis (WFA). Why can the chromatographic profile of the nth component be calculated through the eqaution Xp0n=gnncn in WFA?Examination Paper in Central South UniversityCourse：Chemometrics Time：May 31 2005Student No.： Name: Score: Answer and grade rule:1、What is the meaning of the linearly dependent and the linearly independent among vectors ? For a chemical signals matrix, what is the relationship between the rank of the matrix and the number of the chemical components in the sample? Which hypothesis does this relationship base?The outlines of this question:What is the rank of the data matrix, the rank of the data matrix equals the number of chemical components within the sample. (3 points)The corresponding relationship between the data matrix and the chemical components within the sample. (4 points)This hypothesis is based on the spectrum of every absorbing chemical component is different from all the other spectra. (3 points)2、What is SVD? Perform SVD on a chemical signals matrix, which three matrices can we obtain, explain the mathematical and the physical meaning of the three matrices.The outlines of this question:The formula of SVD ( 2 points)The names of the three matrices(scores, eigenvalues and loadings), actually the eigenvalues and the eigenvectors are obtained. (4 points)Mathematical meaning: all columns of scores are orthogonal to each other, and all rows of loadings are orthogonal to each other. Physical meaning: the scores are the abstract spectrum and the loadings are the abstract chromatogram. (4 points)3. y=c1x1+c2x2+c3x3+eWhere y indicates observing spectrum of mixed sample with three components, xi(i=1,2,.,n) signify observing spectrum of pure component and e symbolizes observing error vector. Try to explain how to turn the above formula to the following one in the means of matrix operation.y = Xc +ewhere X=x1, x2, x3 is so-called sensitivity matrix, while c=c1, c2, c3t is unknown concentration vector. After matrix operation, we get C= (XtX)-1XtyThis is the result of Multiple Linear Regression (MLR), try to explain that MLR is essential to Least Squares.The outlines of this question:1. show the process of the y=c1x1+c2x2+c3x3+e to y = Xc +e on the basis of matrix operation (4 points)2. this part is a key of this issue. The students should explain the essential of MLR is Least Squares. The mainly processes are as follows:f(c) =(-y)t (-y) =(-Xc)t(-Xc) =ete =S ei2f(c) =(y-Xc)t(y-Xc)=yty - yt(Xc) - (Xc)ty +(Xc)t(Xc) =yty - ytXc - ctXy +ctXtXc =yty - 2ytXc +ctXtXcWhere ytXc is scalar, so ytXc=ctXy， d f(c)/dc = -2Xty +2XtXcif suppose d f(c)/dc=0, XtXc =XtyAnd then,=(XtX)-1Xty (10 points)4. what is Chemometrics? And what do Chemometrics mainly study?The outlines of this question:Show the definition of Chemometrics (4 points); give connotation and the main parts of Chemometrics.(6 points).5. Briefly explain why it is necessary to perform an experimental design? And introduce what the basic mentality through the experimental design.The outlines of this question:1) Explain the importance of the experimental design, especially in the study of chemistry.2) The Introduction of the mentality of experimental design should contain how to design an experiment, such as the basic mentality of factorial designs (factorial design, fractional factorial design and orthogonal design) and uniform design and simplex design. It will be enough to take one or two aspects mentioned above to answer.6.How to construct normal plot? What are the functions of effect and residual normal plots, and the conclusions can we learn when use the two plots? The outlines of this question:1. Explain how to construct normal plot. 2. What are the functions effect and residual of normal plots. 3. What are the conclusions can we learn when use the plots.7. try to explain the main idea of Savizky-Golay smoother and compare this method with moving window average smoother.The outlines of this question:1. show the the main idea of Savizky-Golay smoother is a weighted average smoother(2 points).2. explain how to gain the weighted coefficient by Polynomial Least Squares by means of performance of mathmatical calculations (6 points).Compare this method with moving window average smoother by means of principle, mathmatical calculations and physical meaning. (2 points).8、What is the selective information in the two-dimensional hyphenated data and how can the selective information be detected?The outlines of this question:Due to the separating ability of chromatography, the analytical signal of a given mixture by hyphenated instrument can be separated to some extent. Thus the resulted two-way data usually contains the regions where only one component elutes. The information in such one-component region is called selective information. In the evolving latent projection graph (ELPG) of HELP, the straight line whose prolongation passes through the origin suggests that there is a selective chromatographic region. In addition, the selective information can be detected by evolving factor analysis (EFA) or fixed size moving window factor analysis (FSMWFA). 9、Describe the basic idea of ITTFA in brief and further elucidate how ITTFA can approach the feasible solution by iteration?The outlines of this question:For a two-way data X, it can be written asX = CSt + E = TPt + E (1)Thus the equation below can be easily derived: CSt = TRR-1Pt (2)Then we have: C= TR (3)St =R-1Pt (4)Here, R is called the rotation matrix which can linearly transform the abstract chromatography/spectra profiles into real chromatography/spectra ones. Lets focus on the resolution of chromatography and rewrite Equation (2) into the vectorial form.ci=Tri (5)By least squares,ri=(TtT)-1Ttci (6)In fact, ci is unkonw to us. The key point of ITTFA is to approach ci by iteration. This procedure works in the following steps:1) Initialize ,and calculate the corresponding by Equation (6).2) In the kth step, suppose we have got and calculate a new .Pay attention that the negative element in should be set to zero because the chromatography profile cannot include negative value.3) Calculate the error e=. If e is bigger than the predefined allowed error then go to step 2), otherwise terminate the iteration and set as the final solution.10、Describe briefly the basic rationale of WFA. Why can the chromatography profile of the nth component be calculated as Xp0n=gnncn in WFA?The outlines of this question:The key point of WFA is that the chromatography profile of the nth component can be derived using the window containing the information of the other n-1 components. First, the space expanded from the n-1 dimensional subspace pojt (j=1, ., n) and the original n-dimensional space can be linearly expressed by each other because that they are indeed the same space. So the spectrum of the ith component can be written as: sit= gijpojt (1)Insert this equation into X = Si=1n cisit , we have X = Si=1n ci ( gij pojt ) = Si=1n gijcipojt (2)Then right-multiply Equation (2) by pon, thenX pon = Si=1n ( gijcipojt ) pon (3)Because pon is orthorgonal to poj,(j=1,2,n-1),so X pon = ( ginci )pontpon=ginci (4)Here pontpon=1 as pon is orthonormal vector. 中南大学（本、专）科生考试试卷课程名称：化学计量学 考试时间：2005/05/31专业年级 学号 姓名 成绩 1、 矢量间的线性相关与线性无关是什么意义？对于一个化学量测矩阵，其秩与组分数有什么关系？这一关系的基本假设是什么？（10）2、 什么叫SVD? 通过对任意化学量测矩阵进行SVD运算我们可得到三个什么样的矩阵，试解释这三个矩阵数学物理意义。（10）3、 如果有y=c1x1+c2x2+c3x3+e式中y表示三组分混合物样本的量测光谱，xi(i=1,2,.,n)为纯物质的量测纯光谱，e为量测误差矢量，试说明这一矢量表达式为什么可用矩阵式表达为y = Xc +e 其中X=x1, x2, x3，可称为敏感度矩阵。c=c1, c2, c3t，为未知待估浓度矢量。对上述矩阵表达式运用矩阵运算易得C= (XtX)-1Xty 这是多元线性回归（MLR）所得的解，试说明MLR实质上是最小二乘方法。（14）4、 什么叫化学计量学？化学计量学的研究内容包括那几个方面？（10）5、 简述为什么有必要进行试验设计？试验设计的基本思路是什么？（10）6、 怎样构建正态图？因子设计的效应正态图和残差正态图各有什么用处，用它可作出什么样的决断？（12）7、 试说明窗口移动多项式最小二乘平滑法的基本思路，并在此的基础上，比较它与窗口移动平均法的异同。（12）8、 什么是联用色谱的二维量测数据中的信息？怎样确定选择性信息？（12）9、 简述ITTFA方法的基本思路，并说明该方法是怎样来实现其迭代逼近的？（5）10、 简述WFA是怎样来实现重叠色谱峰的分辨的，为什么它可以由一个简单的矩乘法即Xp0n=gnncn 来求得第n个组分的色谱曲线？（5）中南大学（本、专）科生考试试卷答案课程名称：化学计量学 考试时间：2005/05/311、 矢量间的线性相关与线性无关是什么意义？对于一个化学量测矩阵，其秩与组分数有什么关系？这一关系的基本假设是什么？（10）本题的要点在于：什么叫矩阵的秩，即线性无关的矢量的数目（3分），而对于一个化学量测矩阵，其秩与组分数有着一一对应的关系（4分），这一关系的基本假设为不同物种的波谱是不同的（3分）。2、 什么叫SVD? 通过对任意化学量测矩阵进行SVD运算我们可得到三个什么样的矩阵，试解释这三个矩阵数学物理意义。（10）本题的要点在于：什么是SVD的数学表达式（2分），应写出，得到三个矩阵的名称（scores, eigenvalues and loadings），实际所得到的是特征值与特征矢量，（4分）,此外，它们的数学意义，即特征矢量的列正交与行正交，特征值的方差特征，物理意义：抽象波谱与抽象色谱（4分）。3、 如果有y=c1x1+c2x2+c3x3+e式中y表示三组分混合物样本的量测光谱，xi(i=1,2,.,n)为纯物质的量测纯光谱，e为量测误差矢量，试说明这一矢量表达式为什么可用矩阵式表达为y = Xc +e 其中X=x1, x2, x3，可称为敏感度矩阵。c=c1, c2, c3t，为未知待估浓度矢量。对上述矩阵表达式运用矩阵运算易得C= (XtX)-1Xty 这是多元线性回归（MLR）所得的解，试说明MLR实质上是最小二乘方法。（14）本题的要点在于：1、怎样从y=c1x1+c2x2+c3x3+e变为y = Xc +e应该从矩阵运算的角度讲清楚变化数学过程（4分）；2这一步是此题的关键，主要需说明为什么MLR实质上是最小二乘方法。在这里首先需要从eTe出发，然后，通过求极小而得出 C= (XtX)-1Xty多元线性回归（MLR）所得的解实质上是最小二乘解， 必须应有数学推导过程说明 （10分）。4、 什么叫化学计量学？化学计量学的研究内容包括那几个方面？（10）本题的要点在于：什么是化学计量学的定义？（4分）；化学计量学的内涵：它包括了具体哪些内容（6分）。5、 简述为什么有必要进行试验设计？试验设计的基本思路是什么？（10）本题的要点在于：1、说明进行试验设计的重要性，尤其要说明它们在化学研究中的重要性（4分）；2、试验设计的基本思路应包括如何进行试验设计，比如因子设计（包括因子设计、部分因子设计正交设计）的基本思路，均匀设计的思路，以及单纯形设计的基本思路，其中举12个例子说明就行（6分）。6、 怎样构建正态图？因子设计的效应正态图和残差正态图各有什么用处，用它可作出什么样的决断？（12）本题的要点在于：1、说明怎样构建正态图（4分）；2、因子设计的效应正态图和残差正态图各有什么用处（4分）3、用它可作出什么样的决断？（2分）7、 试说明窗口移动多项式最小二乘平滑法的基本思路，并在此的基础上，比较它与窗口移动平均法的异同。（12）本题的要点在于：1、说明窗口移动多项式最小二乘平滑法的基本思路在于其是一种加权平均的平滑方法（2分）；2、而其加权系数是通过多项式最小二乘而求出的，此步说明需要有数学式推导（6分）；比较它与窗口移动平均法的异同须从原理和数学式和物理意义上加以说明（2分）。8、 什么是联用色谱的二维量测数据中的信息？怎样确定选择性信息？（12）本题的要点在于：1、说明联用色谱的二维量测数据中不但包含了色谱的分离特性，又包含了光谱的定性信息，所以可用来解决复杂体系的分析难题（4分）；2、分别可采用特征投影图中过零点的直线获得单组分区来确定选择性信息，即用渐进因子分析法或固定窗口演进因子分析方法就可以确定选择性信息（6分）。9、 简述ITTFA方法的基本思路，并说明该方法是怎样来实现其迭代逼近的？（5）本题的要点在于：1、说明ITTFA (Iterative Target Transformation Factor Analysis) 方法的基本思路在于如何精心构建一个迭代步骤以逼近可能解，需要数学意义的说明（4分）；2、ITTFA方法是通过两个约束，及浓度非负和波谱非负，来不断逼近可能解。（4分）10、 简述WFA是怎样来实现重叠色谱峰的分辨的，为什么它可以由一个简单的矩乘法即Xp0n=gnncn 来求得第n个组分的色谱曲线？（5）主要思路是利用一个只有(n-1)个物种的“窗口”，从而求出那第n个物种的纯色谱来（2分）由(n-1)维子空间拓广而得的正交空间pojt (j=1, ., n)，因其与原始的n维正交空间pjt (j=1, ., n)可以相互线性表出，所以它们实际上张成同一空间。于是，无论是采用pojt (j=1, ., n)还是pjt (j=1, ., n)来表示该体系的纯物种光谱都是可行的。sit= gijpojt将上式代入式 X = Si=1n cisit 可得X = ci ( gij pojt ) = gijcipojt注意到上式中将量测误差矩阵项已经忽略。对上式两边左乘pon，则有X pon = ( gijcipojt ) pon由于pon与poj( j=1, ., n-1) 都正交，即pojtpon= 0 ( j=1, ., n-1)，所以有X pon = ( ginci )pontpon因为当sit中的i小于n时，即纯物种被包括在那(n-1)个之中时，sit都可以由前(n-1)个pojt来线性表出，此时就有gin对除n外的所有i来说都等于零，只有snt需要pon的线性组合得到，同时，注意到是可以归一化的，这样X pon= gnncn （3分） Examination Paper in Central South UniversityCourse：Chemometrics Time： 2006/6/22Student No.： Name: Score: 1、 Try to explain the meaning of the following words:Experiment design, Factor level, Experiment domain, Self-modeling curve resolution, orthogonal vector. 2. What is the meaning of the linearly dependent and the linearly independent among vectors? For a chemical signal matrix, what is the relationship between the rank of the matrix and the number of the chemical components in the sample? Which hypothesis does this relationship base?3、What is singular value decomposition (SVD)? Perform SVD on a chemical signal matrix, which kinds of three matrices can be obtained, try to explain the mathematical and physical meanings of these three matrices.4. y=c1x1+c2x2+c3x3+eWhere y indicates observing spectrum of mixed sample with three components, xi(i=1,2,.,n) signify observing spectrum of pure component and e symbolizes observing error vector. Try to explain how to turn the above formula to the following one in the means of matrix operation.y = Xc +ewhere matrix X=x1, x2, x3 is so-called sensitivity matrix, while c=c1, c2, c3t is unknown concentration vector. After matrix operation, we get c= (XtX)-1XtyThis is the result of Multiple Linear Regression (MLR), try to explain that MLR is essentially Least Squares method.5. Why we say that principal component regression (PCR) and partial least squares (PLS) could give significant improvement for P-matrix method? 6. Briefly describe what are factorial design, Fractional factorial design, orthogonal design and uniform design, and analyze their advantages and disadvantages, respectively. 7. How to construct normal plot? What are the functions of effect and residual normal plots, what kind of conclusions can we achieve when using the two plots? 8. Try to explain the main idea of Savizky-Golay smoother and compare this method with moving window average smoother.9. How to check the purity of a chromatographic peak? What is the principle of checking peak purity? 10. Describe the basic idea of Iterative Target transformation Factor analysis (ITTFA) in brief and further elucidate how ITTFA can approach the feasible solution by iteration?Examination Paper in Central South UniversityCourse：Chemometrics Time：2006Student No.： Name: Score: Answer and grade rules:1、Try to explain the means of the following words:Experiment design：This is a kind of method to systematically arrangement of the experiments in order to get as much as possible information from the experiment. The examples are, factorial experiment design, orthogonal design and etc. (2 points)Factor level：In experiment design, there are always several factor levels, in which the experiment variables are of different values. (2 points)Experiment domain： In experiment design, there are always several factor levels and also different factors, the domain reflects the change range of these levels and these factors is called as Experiment domain. (2 points)Self-modeling curve resolution：The model is unknown. There are only two constraints, that is, nonnegative concentrations and non-negative absorbance in it. The aim is to resolve the real spectral curve from the overlapping peaks. (2 points)Orthogonal vector：if the inner product of any two vectors is equal to zero, these two vectors are said to be orthogonal with each other. (2 points)2 What is the meaning of the linearly dependent and the linearly independent among vectors ? For a chemical signals matrix, what is the relationship between the rank of the matrix and the number of the chemical components in the sample? Which hypothesis does this relationship base?The outlines of this question:What is the rank of t