重金属元素浓度数据的插值函数数学专业英语论文.doc

四 符号说明：坐标的空间位置；The space coordinates;：重金属的空间分布浓度； The concentration of heavy metal spatial distribution;：空间某点的属性值；A point of attribute values;：点到点的水平距离； the horizontal distance of Point (x, y) to (Xi, Yi);：反距离加权插值函数的加权幂指数； Weighted exponential inverse distance weighted interpolation function;：反距离加权插值函数中各点浓度的权重； The weight of each point concentration of inverse distance weighted interpolation function;：相关系数矩阵；Correlation matrix.：特征根；Characteristic root;：特征向量；The feature vector;：主成分载荷；Principal component loadings;：主成分贡献率；Principal component contribution rate;：最优参考序列；The optimal reference sequence;：两级最大差；two Level maximum difference：两级最小差；Two level smallest difference;：关联度；Correlation degree;：综合评价系数；Comprehensive evaluation coefficient;隶属度系数；Membership degree coefficient;： 第个评价因子污染强度最大的超标值（以级准则值为界）；the largest pollution exceeding the standards value of the ith evaluation factor intensity ( grade criterion value for the sector);：第个评价因子级准则限值；The ith evaluation factor intensity of P grade criterion limit;：超标因子个数；The number of Exceed the standard factors;：评价因子总数，当时，取； The total number of evaluation factor, when , ;：各区与之间的相对距离；The relative distance between the and ;：各级标准隶属度向量与之间的相对距离。 The relative distance between levels of standard membership vector and .A. 样品采集点的分布与分析 the distribution and analysis of the sample patch我们使用Excel工具对319个取样点按照其所在位置与所属功能区，画出该城区取样点的分布图，同时可以的确定该城区五个功能区的规划情况，以便于(in order to facilitate)我后面进行相关分析。We use the Excel tool for 319 sampling points according to its location and the functional area, draw the distribution map of the city of sampling points, and can determine the planning of the city five functional areas, the correlation analysis for me.同时使用反距离加权插值法，并用Mathematica软件画出城区海拔等高线（白色线圈160,渐变20，如图2所示）及地势图（图3），规定Y轴正方向为正北，如图所示：At the same time, using the inverse distance weighted interpolation method, and use Mathematica software to draw the city elevation contours (white coil 160, gradient 20, as shown in Figure 2) and relief map (Figure 3), provides positive direction of Y axis was due north, as shown in figure.:结合图1、图2和图3可以看出该城区的东北部分多为山区，地势较高，海拔在60米至160米之间，中部海拔较低，在20至60米范围内，这里多为生活区及交通区，人类对环境的影响较大，南部海拔最低，大部分为20米以下，地势低洼，这里地区为化工区，工厂聚集，各种重金属在此累积，浓度明显偏高。并且此城区是东北高，西南低的地势，易形成空气对流和雨水冲刷，也对重金属的污染传播产生影响，这也是为什么我们在下面给出的空间分布图可以很明显的看出除Cu外各个重金属元素浓度大都由西南向东北方向逐渐稀释的原因之一。According to figure 1, figure 2 and figure 3,we can see the northeast part of the city is a mountainous area, higher ground, at an elevation of 60 meters to 160 meters, the middle elevation is low, at 20 to 60 meters range, here for the living area and the traffic area, the human impact on the environment is larger, the south elevation is the lowest, mostly below 20 meters, low-lying areas, here for chemical industrial zone, factory assembled, various heavy metals accumulated significantly higher, concentration obviously on the high side. And this city is the Northeast high, southwest is low, easy to form the air convection and rain erosion, impact of pollution of heavy metals, which is why we are given in the following spatial distribution map can clearly see that except for one reason Cu different heavy metals concentration is from southwest to northeast direction gradually diluted.B.计算重金属元素的统计特征值及分析B. calculation the value of statistical characteristics and analysis of heavy metal element 我们使用MiniTab15软件对8种重金属元素的统计特征值进行计算得出下表1：We use the MiniTab15 software to calculate the statistical characteristics of 8 heavy metals and obtain the table 1:从表1可以看出，各种重金属的平均含量均比其相应的土壤背景值含量高，Hg的最大值达299.7ng/g ,比是其相应土壤背景值含量的8.6倍，而且Hg的标准差最大，说明其浓度波动很明显，区域分布不均匀，可能是集中累积在某个污染源位置。而As和Ni平均值稍高于其平均背景值含量，且标准差最小，说明重金属分布较为均匀。该城区已经普遍受到重金属污染。Can be seen from table 1, the average contents of various heavy metals are higher than the corresponding content of the soil background values , a maximum value of Hg up to 299.7 ng/g is 8.6 times that of its corresponding content of the soil background values and a maximum standard deviations of Hg, explain the concentration fluctuation obviously, uneven regional distribution, is likely to be concentrated accumulated in a pollution source location. While the As and Ni average slightly higher than the average content of background value, and the minimum standard deviation, distribution of heavy metals is relatively uniform. The city has generally pollution by heavy metal.C.计算二元曲面的空间插值函数Calculate the surface space of binary interpolation function根据已有数据可知，重金属的浓度具有空间相关性，对空间一点，存在一个三元反距离加权插值函数为，因为海拔值对重金属浓度的影响不大，所以在这里先研究二元反距离加权插值函数，三元插值函数将在模型二中给出。According to the existing data, spatial correlation with the concentrations of heavy metals, the point in space (x, y, H), there is a three yuan inverse distance weighted interpolation function Z=f (x, y, H), because the altitude value has little effect on the concentration of heavy metals, so here to study binary inverse distance weighted interpolation function Z=f (x, y), three variables interpolation function Z=f (x, y, H) will be given in model 2.模型一：重金属元素浓度数据的二元插值函数Model: bivariate interpolation function of heavy metal concentration data设有个点，平面坐标为，重金属元素的浓度值为，由其确定的反距离加权插值函数为With points, plane coordinates , heavy metals concentration values for , , be determined by inverse distance weighted interpolation function is 其中到点的水平距离是。 是一个大于0的常数，称为加权幂指数，可以调节插值函数曲面的形状，越大，在节点处函数曲面越平坦；越小，在节点处函数曲面越尖锐。容易看出， 是 的权重。The horizontal distance of to is . is a constant greater than 0, called the weighted exponential interpolation function, can adjust the surface shape, larger , at a node function surface is flat; the smaller the , the node function surface more sharp. Easy to see, is the weight of .模型二：重金属元素浓度数据的三元插值函数Model two: the three element function of heavy metal concentration data 为了综合考虑海拔对各重金属浓度在空间分布中的影响，我们对二元插值函数进行如下改进，得到一个三元插值函数。In order to consider the effect of altitude on the concentrations of heavy metals in spatial distribution, we were improved as follows on the two element interpolation function, get a three element interpolation function . 设有个点，平面坐标为，重金属浓度值为，反距离加权插值函数为With , plane coordinates, heavy metal concentration values for , inverse distance weighted interpolation function其中点到点的距离为，。 是一个大于0的常数，称为加权幂指数，可以调节插值函数曲面的形状，越大，在节点处函数曲面越平坦；越小，在节点处函数曲面越尖锐。可以看出， 是 的权重。The distance of point to is , . is a constant greater than 0, called the weighted exponential interpolation function, can adjust the surface shape, the larger the , a node function surface is flat; the smaller the , the node function surface more sharp. As you can see, is the weight of.重金属元素浓度数据的三元插值函数，可以利用Mathematica程序计算得到，(详见附录（2）)。ternary interpolation function of heavy metals concentration data , can be calculated using Mathematica program, (see the appendix (2).模型求解（针对模型一）Solving the model (direct at the model one) 对重金属元素浓度数据作反距离加权平均法(Shepard方法)插值，以为权数，加权幂指数为2，数据以形式，利用Mathematica程序（见附录（1）6，可以得到8种重金属浓度的空间分布等高图，同时将数据形式改为也可以得到该城区的等海拔高线图（图2）及该城区地势图（图3）(计算过程见附录（4）)。并且程序中倒数第四行和倒数第七行中的整数值可以在区间1,319中进行调节，得到Shepard的近邻法，即取附近个近邻节点作曲面插值，我们经过反复试验调节值，并比较所得等高线图，发现取得所有节点时等高线更为明显，但是计算量较大。Inverse distance weighted average method of heavy metal concentration data (the Shepard mathod) interpolation, with di as the weights, the weighted exponent P is 2, the data using the form of data=x value, y value, concentration , using Mathematica program (see Appendix (1) 6, we can get 8 kinds of heavy metal concentration in the spatial distribution of high figure, while the data form revised to data=x value, y value, the elevation can also get the city such as elevation contour map (Figure 2) and the urban landform map (Figure 3) (see calculation process in Appendix (4). And the integer k in fourth line from the bottom and the bottom seventh lines in the program value can be adjusted in the interval 1319, get Shepard neighbour method, namely take k neighbor node near (x,y) as a curved surface interpolation, we adjust the K value by repeating the experiment, and comparing the resulte of contour map, found when get all nodes , the contour is more apparent, but the large amount of calculation. 需要注意的是以下图形中浓度由高到低，颜色由浅变深，白色部分为浓度最高，深紫色为最低。Need to be aware of is from high to low concentration in the following graphic, color change from shallow to deep, the white part is the highest concentration, purple is the lowest. 对于结果的分析Analysis of the results 利用反距离加权插值的方法得到8种主要重金属在该城区含量的空间分布图(图4至图10) ，由图可见重金属含量的空间分布具有较好的结构性和空间相关性，空间分布的方向性明显，从各图中可以很明显的看出除Cu外各个重金属元素浓度大都由西南逐渐向东北方向稀释，并且由于采样点的均匀分布，提高了反距离加权插值法的精确度，空间各元素分布的浓度值的估算也较为准确。Inverse distance weighted interpolation method are used to get the eight kinds of space of the main content of heavy metals in the urban area map (figure 4 and figure 10), the figure shows the spatial distribution of heavy metal content has good structural and spatial correlation and clear direction, from the graph can clearly see each heavy metal element concentration except Cu mostly from southwest to northeast gradually diluted, and because the sample point of uniform distribution, improve the precision of the inverse distance weighted interpolation method, the spatial distribution of each element density estimation is more accurate. 5.1.2 关于环境的评价模型evaluation model about the environment 5.1.2.1数据准备data preparation由于题目所给出的数据众多，我们首先提取各个区域的各种重金属数据，并且求出各种重金属的一些统计特征值，在本题中，我们主要应用其平均值，我们用平均值建立所需要的向量，我们用分别代表生活区、工业区、山区、交通区、公园绿地区的数据向量，分别对应，本题中有以下数据，见表2：Because of the problems are many data, we first extract all kinds of heavy metal data of each region, and find out some statistic characteristic values in a variety of heavy metals, in this case, we mainly use the average value, we need set up the needed vector by using the average vector, we use respectively represent life area, industrial area, mountainous area, traffic area, park green area data vector, As, Cd, Cr, Cu, Hg, Ni, Pb, Zn respectively corresponding to 1,2,3,4,5,6,7,8 in J, , there are several according to the title, see table 2: 对于环境的评级标准，我们不妨以其自然背景值的平均值作为基准，用下述公式来确定各级的评判准值，具体公式为：For environmental rating standards, we take the average value of the natural background values as reference, use the following formula to determine the levels of evaluation criterion, the specific formula: 其中时为一级，用表示。其中时为二级，用表示。其中时为三级，用表示。其中时为四级,用表示。其中时为五级，用表示。Regional average of the heavy metal content is cTheir average natural background is cn The 0<pi<1 is a class, denoted by S1. The 1<pi<2 is two, S2.The 2<pi<3 is three, S3. The 3<pi<4 is four, S4.The pi>5 is five, S5.其具体数据表如下表3所示：The specific data table as shown in table 3:5.1.2.2模型的建立与求解Establishing and solving of the model模型1.2的建立The establishment of models 1.2 A建立隶属度模型The establishment of membership degree model设有个监测点, 以 表示, 每个测点中包含种污染因子, 各测点污染因子实测值用向量的形式表示为: With I monitoring points to said, each measuring point contains pollution factor, each measuring point pollution factor to the measured values in the form of a vector as follows:将各污染因子按其污染强度划分为p 个评价等级, 各等级标准用向量形式表示为:Each factor according to its pollution intensity is divided into p a rating, the rating criteria in the form of a vector as follows: 取作为评价基准，按隶属函数分别计算各区污染因子指标对一级标准的隶属度。(Take as the evaluation criterion, according to membership function respectively to calculate the membership degree of each factor of pollution index level of standards.) 有单因素隶属函数确定出各测点对评价基准的分向隶属度记为(A single factor determining membership function of each measuring point on the benchmark points to note for membership)： 取为评价标准，按如下隶属函数关系确定个等级标准限值对一级标准的隶属度(Take S1 as the evaluation standard, determine the grade standard limit of Sj on a standard membership according to the following membership function)：记为(Remember to)：2.2 计算模糊贴近度(calculate fuzzy closeness ) 各区相对于各评价等级标准的模糊贴近度可用下列公式计算The fuzzy closeness of each area cj relative to the scale of the grade standard can be caclulated using the following formula: 定量的考察出各区与评价基准的综合贴近度，此距离越大，说明该店环境质量越好(Quantitative study of the and benchmark comprehensive similarity degree, the farther the distance, the better the quality of store environment)。3.0模型的求解(The solution of the model )先计算隶属度把上面的数据带入上述式子，算出、对的隶属向量的值,如下表4所示：To calculate membership first put the data into the formula above, calculate,ci sj to si subordinate vector of values, are shown in table 4 below: :在上述计算中，由于有三个超标量，因此取。In the above calculation, since there are three superscalar, so e=0.68.再计算模糊贴近度，经过计算与的相对距离分别为：0.68、0.53、0.09、0.41、0.35, 与的相对距离分别为：0、0.08、0.17、0.24、0.32.Then calculate the fuzzy degree of nearness, after calculation ,the relative distance of and were: 0.68, 0.53, 0.09, 0.41, 0.35, the relative distance between and were: 0, 0.08, 0.17, 0.24, 0.32.然后计算各区相对于各评价等级标准的模糊贴近度，其计算结果如下表5所示.Then calculate the fuzzy with respect to each evaluation grade standard approach, the results are as shown in Table 5.根据上述计算结果我们可以得出如下结论：（1）即生活区的环境为二级，即为尚清洁；（2）即工业区的环境为四级，即为重度污染；（3）即山区的环境为一级，即为清洁；（4）即交通区的环境为三级，即为轻度污染；（5）即公园绿地区的环境为二级，即为尚清洁。 通过对比各重金属在各个功能区的空间分布可知我们的结论是符合现实情况的，具有很高的精确度和可信度。Based on the above results we can draw the following conclusions:(1) C1 is the living environment for the level two, which is still clean;(2) the C2 industrial zone environment for level four, namely to severe pollution;(3) C3 is the mountainous environment for level one, is clean;(4) C4 traffic environment into level three, namely light pollution;(5) C5 is the park environment for level two, which is still clean.By comparing the spatial distribution of heavy metals in various functional areas that our conclusion is in conformity with the reality, has the very high precision and credibility.5.2 问题二的模型的建立与求解(Establishing and solving the model problem two)5.2.1模型原理(The principle of model)设样本为,其中表示样本评价指标的个数，表示样本对评价指标进行监测的次数。Set samples for, where represents a number of sample evaluation index, said the number of samples to monitor the evaluation index 主成分分析法具体步骤为：The principal component analysis method for concrete steps: （1）计算相关系数矩阵(Calculate the correlation coefficient matrix R) ， 为原变量与的相关系数( as the correlation coefficient of original variables and )，其计算公式为：(The calculation formula is as follows:)（2） 计算特征值与特征向量(Calculation of characteristic value and characteristic vector) 使特征值按由大到小的顺序排列：，且对应的特征向量为，要求的模为1。Make characteristic value according to the order from large to small: , and feature vector corresponding to, requirements of mould is 1.计算主成分贡献率和累计贡献率(Calculation of principal component contribution rate and accumulated contribution rate)贡献率： ， 累计贡献率：，主成分个数为满足的的最小值。Contribution rate: , the accumulative contribution rate: , principal component number is the minimum meet .（3） 计算主成分载荷：(The main component of load calculation)5.2.2模型建立model building以生活区为例，调查中共取了44个点采集数据，即为44，得出8种重金属的浓