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数学建模简介及数学建模常用方法数学建模，简单来说，就是用数学的语言和方法来描述和解决实际问题的过程。
它就像是一座桥梁，将现实世界中的复杂问题与数学的抽象世界连接起来，让我们能够借助数学的强大工具找到解决问题的有效途径。
在我们的日常生活中，数学建模无处不在。
比如，当我们规划一次旅行，考虑路线、时间和费用的最优组合时；当企业要决定生产多少产品才能实现利润最大化时；当交通部门设计道路规划以减少拥堵时，这些背后都有着数学建模的身影。
那么，数学建模具体是怎么一回事呢？数学建模首先要对实际问题进行观察和分析，明确问题的关键所在，确定需要考虑的因素和变量。
然后，根据这些因素和变量，运用数学知识建立相应的数学模型。
这个模型可以是一个方程、一个函数、一个图表，或者是一组数学关系。
接下来，通过对模型进行求解和分析，得到理论上的结果。
最后，将这些结果与实际情况进行对比和验证，如果结果不符合实际，就需要对模型进行修正和改进，直到得到满意的结果。
数学建模的过程并不是一帆风顺的，往往需要不断地尝试和调整。
但正是这种挑战，让数学建模充满了魅力和乐趣。
接下来，让我们了解一下数学建模中常用的一些方法。
第一种常用方法是线性规划。
线性规划是研究在一组线性约束条件下，如何使一个线性目标函数达到最优的数学方法。
比如说，一个工厂要生产两种产品，每种产品需要不同的资源和时间，而工厂的资源和时间是有限的，那么如何安排生产才能使利润最大呢？这时候就可以用线性规划来解决。
第二种方法是微分方程模型。
微分方程可以用来描述一些随时间变化的过程，比如人口的增长、传染病的传播、物体的运动等。
通过建立微分方程，并求解方程，我们可以预测未来的发展趋势，从而为决策提供依据。
第三种是概率统计方法。
在很多情况下，我们面临的问题具有不确定性，比如市场需求的波动、天气的变化等。
概率统计方法可以帮助我们处理这些不确定性，通过收集和分析数据，估计概率分布，进行假设检验等，为决策提供风险评估和可靠性分析。
数学建模方法数学建模，简单来说，就是用数学的语言和方法来描述和解决现实世界中的问题。
它就像是一座桥梁，连接着抽象的数学理论和复杂的实际情况。
那数学建模到底是怎么一回事呢？想象一下，你要规划一个城市的交通系统，让车辆能够高效通行，减少拥堵；或者要预测某种疾病的传播趋势，以便采取有效的防控措施；又或者要设计一个最优的生产流程，降低成本提高效率。
这些实际问题都可以通过数学建模来解决。
数学建模的第一步，是要对问题进行清晰的理解和定义。
这可不是一件简单的事情，需要我们仔细观察问题的背景、条件和目标。
比如说，如果要解决交通拥堵的问题，我们就得先了解这个城市的道路布局、车辆流量的规律、人们的出行习惯等等。
只有把这些都搞清楚了，才能准确地把实际问题转化为数学语言。
接下来，就是要做出合理的假设。
现实问题往往非常复杂，包含了太多的因素。
为了能够用数学方法来处理，我们必须对一些次要的因素进行简化和假设。
但要注意，这些假设不能太过于偏离实际情况，否则建立的模型就没有实用价值了。
有了假设之后，就可以选择合适的数学工具和方法来建立模型。
这就像是选择合适的工具来完成一项工作。
如果问题涉及到变量之间的线性关系，可能会用到线性规划；如果是要研究随机现象，概率统计就派上用场了；要是问题与变化的过程有关，微分方程可能就是一个好的选择。
建立好模型之后，就需要对模型进行求解和分析。
这可能需要运用数学运算、计算机编程等手段。
通过求解，我们可以得到一些结果，但这些结果并不是最终的答案，还需要对它们进行分析和解释。
看看这些结果是否合理，是否符合我们的预期。
比如说，通过一个数学模型计算出某个交通路口的最优信号灯时间设置，但如果这个时间设置在实际中根本无法实现，那就说明模型可能存在问题，需要重新调整和改进。
在模型求解和分析的过程中，还需要对模型进行检验。
可以用实际的数据来验证模型的准确性，如果模型的预测结果与实际情况相差较大，那就得重新审视模型的假设、参数和求解方法，对模型进行修正和完善。
数学建模的主要建模方法数学建模是指运用数学方法和技巧对复杂的实际问题进行抽象、建模、分析和求解的过程。
它是解决实际问题的一个重要工具，在科学研究、工程技术和决策管理等领域都有广泛的应用。
数学建模的主要建模方法包括数理统计法、最优化方法、方程模型法、概率论方法、图论方法等。
下面将分别介绍这些主要建模方法。
1.数理统计法：数理统计法是基于现有的数据进行概率分布的估计和参数的推断，以及对未知数据的预测。
它适用于对大量数据进行分析和归纳，提取有用的信息。
数理统计法可以通过描述统计和推断统计两种方式实现。
描述统计主要是对数据进行可视化和总结，如通过绘制直方图、散点图等图形来展示数据的分布特征；推断统计则采用统计模型对数据进行拟合，进行参数估计和假设检验等。
2.最优化方法：最优化方法是研究如何在给定的约束条件下找到一个最优解或近似最优解的方法。
它可以用来寻找最大值、最小值、使一些目标函数最优等问题。
最优化方法包括线性规划、非线性规划、整数规划、动态规划等方法。
这些方法可以通过建立数学模型来描述问题，并通过优化算法进行求解。
3.方程模型法：方程模型法是通过建立数学方程或函数来描述问题，并利用方程求解的方法进行求解。
这种方法适用于可以用一些基本的方程来描述的问题。
方程模型法可以采用微分方程、代数方程、差分方程等不同类型的方程进行建模。
通过求解这些方程，可以得到问题的解析解或数值解。
4.概率论方法：概率论方法是通过概率模型来描述和分析不确定性问题。
它可以用来处理随机变量、随机过程和随机事件等问题。
概率论方法主要包括概率分布、随机变量、概率计算、条件概率和贝叶斯推理等内容。
利用概率论的方法，可以对问题进行建模和分析，从而得到相应的结论和决策。
5.图论方法：图论方法是研究图结构的数学理论和应用方法。
它通过把问题抽象成图，利用图的性质和算法来分析和求解问题。
图论方法主要包括图的遍历、最短路径、最小生成树、网络流等内容。
什么是数学建模数学建模是指运用数学的理论、方法和技术，以模型为基础，通过对实际问题进行抽象、建模、求解和验证，为实际问题的研究和决策提供可靠依据的过程。
数学建模可以帮助我们更好地理解、分析、解决实际问题。
它是一种综合运用数学、物理、计算机科学和其他相关学科知识的跨学科研究领域，可以应用于各个领域的问题，包括自然科学、工程技术、社会科学、医学、金融等。
数学建模的过程一般包括以下几个步骤：1. 定义问题和目标。
在这个阶段，我们需要对实际问题进行全面的了解，明确研究的目标和需要解决的问题是什么，确定问题的限制和条件。
2. 建立模型。
在这个阶段，我们需要根据实际问题的特点和需要解决的问题，选择适当的模型类型，建立数学模型。
模型应该尽可能简明明了，能够比较好地描述实际问题，并且便于求解。
3. 求解模型。
在这个阶段，我们需要根据所建立的模型，采用数学和计算机科学等相关方法，对模型进行求解，得到具体的结果和解决方案。
4. 验证模型。
在这个阶段，我们需要根据模型的求解结果，进行模型的验证。
验证模型的正确性和可靠性，以及对模型的结果进行误差分析和敏感性分析，以保证模型的可行性和实用性。
5. 应用模型。
在这个阶段，我们需要将模型的结果应用于实际问题的解决中。
根据模型的结果，提出相应的决策和措施，实现问题的解决和优化。
数学建模具有广泛的应用领域和重要性。
在物理、化学、生物学和工程技术等领域，数学建模可以帮助我们解决复杂的系统问题，如气候模型、流体力学模型、生物进化模型等。
在社会科学领域，数学建模可以应用于经济学、管理学、社会学等领域，对社会现象进行建模和预测，如人口增长模型、市场模型、网络模型等。
在医学领域，数学建模可以帮助我们研究疾病的发展和治疗方法，如病毒传播模型、治疗模型等。
在金融领域，数学建模可以帮助我们分析风险和投资策略，如股票价格模型、期权评估模型等。
总之，数学建模是一种重要的跨学科研究领域，以模型为基础，运用数学和相关学科知识，对实际问题进行抽象、建模、求解和验证，为实际问题的研究和决策提供可靠依据，具有广泛的应用领域和重要性。
数学专业的数学建模数学建模是数学专业中重要的一门课程，它通过数学的方法和技巧解决实际问题。
本文将介绍数学建模的定义、应用领域、建模过程以及数学专业学生在数学建模中的作用。
一、数学建模的定义数学建模是将实际问题转化为数学问题，并应用数学方法和工具解决这些问题的过程。
它是数学与现实世界之间的桥梁，通过数学的抽象和建模能力，解决现实问题，提高生产效益和科学研究水平。
二、数学建模的应用领域数学建模广泛应用于各个领域，包括经济、生态、环境、物理、工程等。
在经济领域，数学建模可以帮助企业分析市场需求，制定最优营销策略；在生态领域，数学建模可以评估生物多样性，分析环境问题；在物理领域，数学建模可以解释物质运动规律；在工程领域，数学建模可以优化工艺流程，提高工程效率。
三、数学建模的过程数学建模的过程一般包括问题的分析、建立数学模型、求解模型和对结果的验证。
首先，需要对实际问题进行充分的分析，明确问题的要求和限制条件；其次，根据问题的特点，运用数学知识建立数学模型，将实际问题抽象为数学符号和方程；然后，对建立的数学模型进行求解，可以使用数值计算、优化算法等方法得到解析结果；最后，对结果进行验证，比较实际情况和模型预测，评估模型的准确性和可行性。
四、数学专业学生在数学建模中的作用数学专业学生在数学建模中发挥着重要的作用。
首先，他们具备扎实的数学基础和数学思维能力，能够快速理解和应用数学方法解决问题；其次，数学专业学生熟练掌握常用的数学工具和软件，能够高效地进行数学计算和模型求解；此外，他们对数学理论有深入的研究，能够通过对数学模型的优化和改进提升模型的准确性和可靠性。
总结：数学建模作为数学专业中重要的课程，对于培养学生的数学思维和解决实际问题的能力具有重要意义。
通过数学建模，学生能够将所学的数学知识应用到实际中，提升自己的综合素质。
希望广大学生能够重视数学建模的学习，不断提高自己的数学建模能力，为社会的发展做出贡献。
数学建模基础知识引言：数学建模是一门以数学为工具、以实际问题为研究对象、以模型为核心的学科。
它通过将实际问题抽象为数学模型，并利用数学方法对模型进行分析和求解，从而得到问题的解决方案。
在数学建模中，有一些基础知识是必不可少的，本文将介绍数学建模的基础知识，包括概率与统计、线性代数、微积分和优化算法。
一、概率与统计概率与统计是数学建模的基础。
概率论用于描述随机现象的规律性，统计学则用于从观测数据中推断总体的特征。
在数学建模中，需要根据实际问题的特点选择合适的概率模型，并利用统计方法对模型进行参数估计。
1.1 概率模型概率模型是概率论的基础，在数学建模中常用的概率模型包括离散型随机变量模型和连续型随机变量模型。
离散型随机变量模型适用于描述离散型随机事件，如投硬币的结果、掷骰子的点数等；连续型随机变量模型适用于描述连续型随机事件，如身高、体重等。
在选择概率模型时，需要根据实际问题的特点进行合理选择。
1.2 统计方法统计方法用于从观测数据中推断总体的特征。
在数学建模中，经常需要根据样本数据对总体参数进行估计。
常用的统计方法包括点估计和区间估计。
点估计用于估计总体参数的具体值，如均值、方差等；区间估计则用于给出总体参数的估计范围。
另外，假设检验和方差分析也是数学建模中常用的统计方法。
二、线性代数线性代数是数学建模的重要工具之一。
它研究线性方程组的解法、向量空间与线性变换等概念。
在线性方程组的求解过程中，常用的方法包括高斯消元法、矩阵的逆和特征值分解等。
线性代数还广泛应用于图论、网络分析等领域。
2.1 线性方程组线性方程组是线性代数的基础，它可以用矩阵和向量的形式来表示。
求解线性方程组的常用方法有高斯消元法、矩阵的逆矩阵和克拉默法则等。
高斯消元法通过矩阵的初等行变换将线性方程组转化为简化行阶梯形式，从而求得方程组的解。
2.2 向量空间与线性变换向量空间是线性代数的核心概念，它由若干个向量组成，并满足一定的运算规则。
数学建模是什么
数学建模是指利用数学工具和方法分析和解决实际问题的过程，是一种跨学科的综合性应用科学研究方法。
数学建模的基本步骤包括：问题建模、假设、模型的构建、模型求解和模型评价。
在这个过程中，数学建模的核心是模型的构建和求解，其中模型的构建需要理解实际问题的基本特征和数学方法的应用，而模型求解则需要掌握数学分析、数值计算等技能和方法。
数学建模的应用范围非常广泛，包括但不限于自然科学、社会科学、经济学、工程学等领域的问题。
数学建模在现实生活中的应用包括：企业生产、物流配送、城市交通规划、自然资源评估、环境保护、金融、医学等各个领域。
数学建模的方法多种多样，常见的数学方法包括：微积分、线性代数、概率论、统计学、优化理论等。
通过对实际问题的建模、数学方法的应用和模型求解的计算和分析，数学建模可进一步为决策提供科学依据和参考。
数学建模的主要特点是模型化思维、跨学科交叉和创新性思维。
在这个过程中，数学建模要求研究者对问题进行深入的分析和研究，要对数学方法的应用有较大的理解和掌握，并且要结合实际考虑模型的可行性。
数学建模的创新性思维则要求研究者在模型的构建和求解中体现出一定的创新性和思维深度。
无论是学术界还是实际应用领域，数学建模的应用都已经深入到各个角落。
在数学建模中，数学是一种工具性语言，
而模型则是实际问题的一种映射。
数学建模不仅促进了数学研究和应用之间的相互促进和发展，还连接了传统学科和新兴学科之间的桥梁，推动了知识的跨领域传播和交流。
数学建模有哪些方法
数学建模是指将实际问题用数学的方法进行描述和分析的过程。
常见的数学建模方法有以下几种：
1. 形式化建模：将实际问题抽象成数学模型，通过符号和公式的形式进行描述和求解。
2. 统计建模：利用统计学的方法对数据进行收集、整理和分析，从中提取规律和模式，对未知的情况进行预测和决策。
3. 数值模拟：利用计算机和数值方法对问题进行模拟和求解，通过近似计算得到结果。
4. 最优化建模：通过建立优化模型，寻找使目标函数达到最大或最小值的最优解。
5. 离散建模：将连续的问题离散化，转化为离散的数学模型进行分析和求解。
6. 动态建模：对问题进行时间序列的分析和建模，预测未来的变化和趋势。
7. 图论建模：将问题抽象成图的形式，利用图的相关理论和算法进行分析和求解。
8. 概率建模：利用概率论的方法对问题进行建模和分析，从中推断出一些未知的情况。
以上是一些常见的数学建模方法，具体的方法选择要根据实际问题的特点和要求进行判断和决策。
数学建模的原理
数学建模是一种以数学方法和工具为基础，对现实问题进行抽象和表达的过程。
其原理可以简单概括为以下几个步骤。
1. 问题抽象：将现实问题转化为数学模型。
在这一步骤中，需要明确问题的目标、限制条件和相关因素，并对它们进行数学化的描述。
2. 假设建立：基于对问题的理解和分析，提出相关的假设并建立相应的数学关系。
这些数学关系可以是方程、函数、概率模型等，用来表达问题中的变量间的关系。
3. 模型求解：利用数学方法，对所建立的数学模型进行求解。
这包括求解方程组、优化问题、概率分布等。
通常需要运用数学分析、优化方法、概率统计等工具以及计算机编程进行模型求解。
4. 模型评价：对得到的解进行评价，检验模型的有效性和可行性。
这可以通过与现实数据对比、敏感性分析、误差分析等方式来进行。
5. 结果分析：根据模型的求解结果，对问题的解释和分析。
分析模型的局限性、推断模型的适用范围，探究问题的深层次原因等。
6. 结论表达：将建模过程和结果进行总结和表达。
可以通过报告、论文、演示等形式对建模过程和结果进行系统化的呈现。
在数学建模过程中，需要深入理解问题本质和实际应用背景，结合数学理论和方法，进行抽象和简化，以符合现实问题的特点和需求。
同时，建模者需要具备良好的数学基础、逻辑思维能力、计算机编程技能等，并注重模型的可靠性、有效性和实用性。
We construct four models to study leaf classification, relationships between leaf shape and leaf distribution, correlations between leaf shape and tree profile, and total leaf mass of a tree. Model 1 deals with the classification of leaves. We focus primarily on the most conspicuous characteristic of leaves, namely, shape. We create seven geometric parameters to quantify the shape. Then we select six common types of leaves to construct a database. By calculating the deviation index of the parameters of a sample leaf from those of typical leaves, we can classify the leaf. To illustrate this classification process, we use a maple leaf as a test case. Model 2 studies the relationship between leaf shape and leaf distribution. First, we simplify a tree into an idealized model and then introduce the concept of solar altitude. By analyzing the overlapping individual shadows through considering the relafionship between leaf length and internode length under different solar alfitudes, we find that the leaf shape and distribution are optimized to maximize sunlight exposure according to the solar altitude. We apply the model to three test types of trees. Model 3 discusses the possible association between tree profile and leaf shape. Based on the similarity between the leaf veins and branch structure of trees, we propose that leaf shape is a two-dimensional mimic of the tree profile. Employing the method of Model 1, we set several parameters reflecting the general shape of each tree and compare them with those of its leaves. With the help of statistical tools, we demonstrate a rough association between tree profile and leaf shape. Model 4 estimates the total leaf mass of a tree given size characterisfics. Carbon dioxide (CO2) sequestrafion rate and tree age are introduced to establish the link between leaf mass and tree size. Since a unit mass of a leaf sequesters CO2 at a constant rate, the CO2 sequestration rate has a quadraticrelafionship with the age of the tree, and the size the tree experiences logisfic growth. We tac kle four main subproblems: • classification of leaves, • the relationship between leaf distribution and leaf shape, • the relationship between the tree profile and the leaf shape, and • calculation of the total leaf mass of a tree. To tackle the first prob lem, we select a set of parameters to quantify the characters of the leaf shape and use the leaf shape as the main standard for our classification process. For the second question, we use the overlapping area that one leaf's shadow casts on the leaf directly under it as the link between the leaf distribution and the leaf shape, since the leaf shape affects the overlapping. We assume that the leaf distribution tries to minimize the overlapping area. As for the third question, we set parameters for the tree profile and compare those with the parameters for the tree's leaf shape to judge whether there is a relation between tree profile and leaf shape. We use age to link the size of tree and the total weight of its leaves, because the tree size has an obvious relationship with its age and the age affects a tree's sequestration of carbon dioxide, which affects the total weight of a tree's leaves. Assumptions•The trees are all individual ("open grown") trees, such as are typically planted along streets, in yards, and in parks. Our calculation does not apply to densely raised trees, as in typical reforestation projects where large numbers of trees are planted close together. • The shape of the leaves does not reflect special uses for the trees, such as to resist ex tremely windy, cold, parched, wet, or dry conditions, or to produce food. • The type of the leaf distribution (leaf length and internode distance relation) reflects the tree's natural tendency to sunlight. • The tree profile that we consider is the part ab ove ground, includingthe trurvk, the branches, and leaves. • All parts of a leaf can lie flat, and the thickness or protrusion of veins can be neglected. • Leaves are the only part of the tree that reacts in photosynthesis and respiration, so that the carbon dioxide sequestration of a tree is the sum of ¿he sequestration of the leaves.A Close Look at Leaves 207•The sequestration of a tree or a leaf is the net amount of CO2 fixed in a tree, which is the difference between the CO2 released in respiration and the CO2 absorbed in photosynthesis.•The trees are in healthy, mature, and stable condition. Trees of the same species have same characteristics.Model 1: Leaf ClassificationDecisive ParametersTo classify the shape of a leaf, we set seven parameters and establish a database for comparison.Rectangularity We define the ratio of the area of the leaf to the area of its minimum bounding rectangle as the leaf's rectangularity (Figure 1).Figure 1. Figure 2. Figure 3. Figure 4.The photographs of leaves in Figures 1-4 are reproduced (with overlays by the authors of this paper) from Knight et al. [2010], by kind permission of that paper's authors.Aspect Ratio The aspect ratio is the ratio of the height of the minimum bounding rectangle to its width. (Figure 2).Circularity To evaluate how round a leaf is, we consider that ratio if the ex-circle to thein-circle. (Figure 3).Form Factor Form factor, a famous shape description parameter, is calculated as208 The UMAP Journal 33.3(2012)where A is the area of the leaf and P is its perimeter.Edge Regularity Area Index Although the aspect ratio and the rectangularity of two leaves may be similar, the contour or the exact shape of two leaves may vary greatly. To take the different contour of the leaf into consideration, we join every convex dot along the contour and develop what we call the bounding polygon area. The ratio between the leaf area and this bounding polygon area is the edge regularity area index. The closer this ratio is to 1, the less jagged and smoother the leaf's contour is (Figure 3).Edge Regularity Perimeter Index Similarly, we develop another parameter, the bounding polygon perimeter, the perimeter of the polygon when we join the convex dots of a leaf. We define the ratio of the bounding polygon perimeter to the perimeter of the leaf to be the edge regularity perimeter index. The smaller this ratio, the more jagged and irregular the contour of the leaf is (Figure 3).Proportional Index Since it is also highly critical to capture the spatial distribution of different portions of a leaf along its vertical axis, we divide the minimum bounding rectangle into four horizontal blocks of equal height, and then calculate the proportion of the leaf area in a particular region to the total leaf, which we refer to as the proportional index ¡PI) for that region (Figure 4). Hence, the PI is a vector of length four.Common Types of LeavesWe develop a database of the six most common leaf types in North America (Figure 6),using the seven parameters discussed above. Table 2 gives the values of the parameters for each leaf type, as measured from scans of photos of leaves in Knight et al. [2010]. ComparisonGiven a specific leaf, we calculate the seven characteristics of it and compare them with our database by calculating the squared deviation of each parameter of the given leaf from the corresponding standard parameter of each category. We realize that some of the parameters are somehow more important than others. So in an effort to make our model more accurateA Close Look at Leaves 209Figure 5. The six most common seen leaf types in North America. (The photos, from Knight et al. [2010], are reproduced by kind permission of that paper's authors.)Table 1. Parameter values for the six leaf types.TypeRectangularity Aspect Ratio Circularity Form Factor ER Area Index ER Perimeter Index PIi PI2 PI3 PI410.6627 0.8615 0.8140 0.9139 0.9322 0.8727 0.0649 0.2958 0.3439 0.295420.5902 0.6600 0.5432 0.6206 0.8780 0.8889 0.0769 0.3555 0.4243 0.143330.6250 0.1800 0.4564 0.2823 0.9091 0.9384 0.1179 0.2208 0.4139 0.247440.4772 0.6383 0.3454 0.2470 0.8500 0.8602 0.1909 0.3892 0.3047 0.115250.4876 0.4792 0.3123 0.3662 0.7880 0.8231 0.1299 0.3606 0.4123 0.097060.6576 0.3111 0.3311 0.4956 0.8895 0.9903 0.2920 0.4187 0.2677 0.0220and reliable, we introduce a weighted index of deviation ID, with7ID = 'where each /j is the squared deviation, except that. 1 'We determine the weights via the Analytical Hierarchy Process (AHP) [Saatyl982]. We build a 7 x 7 matrix reciprocal matrix by pair comparison:210 The UMAP Journal 33.3(2012)R AR C FF ERAI ERPI PIR AR C FF ERAI ERPI PI/ 1 3 1 4 2 2 \ 71/3 1 1/3 1 1/2 1/2 31 3 1 42 2 71/4 1 1/4 1 1/3 1/3 21/2 2 1/2 3 1 1 41/2 2 1/2 3 1 1 41/7 1/3 1/7 1/2 1/4 1/4 1 /The meaning of the number in each cell is explained in Table 2. The numbers themselves are based on our own subjective decisions.Table 2. The multiplication table of Djo.Intensity of Value1 3 5 7 9 2,4, 6, 8 ReciprocalsInterpretationRequirements i and j have equal value. Requirement i has a slightly higher value than j. Requirement i has a strongly higher value than j. Requirement i has a very strongly higher value than j. Requirement i has an absolutely higher value than j. Intermediate scales between two adjacent judgments. Requirement i has a lower value than j.We then input the matrix into a Matlab program that calculates the weight Wi of each factor, as given in Table 3.FactorWeightR0.0480Table 3. AHP-derived weights.AR C FF0.1583 0.0480 0.2048ERAI0.0855ERPI0.0855PI0.3701We test the consistency of the preferences for this instance of the AHP. For good consistency [Alonso and Lama ta 2006, 446-447]:•The principal eigenvalue A^ax of the matrix should be close to the number n of alternatives, here 7; we get A^ax = 7.05.•The consistency index CI = (Amax ~ ''^)/ip- ~ 1) should be close to 0; we get CI = 0.009. •The consistency ratio CR = CI/RI (where RI is the average value of CI for random matrices) should be less than 0.01; we get CR = 0.006.Hence, our decision method displays perfectly acceptable consistency and the weights are reasonable.A Close Look at Leaves 111Model TestingWe use a maple leaf of Figure 6 to test our classification model. Visually, it resembles Category 4 most.Figure 6. Test maple leaf.Now we test this hypothesis with our model. First, we process the image of the leaf, calculating rectangularity, aspect ratio, circularity, form factor, edge regularity area index, edge regularity perimeter index, and the proportional index, with values as in Table 2. The values of the seven parameters are shown in Table 4.FactorMeasured valueR0.355Table 4. Parameter values for the sample maple leaf.AR C FF ERAI ERPI0.908 0.269 0.157 0.625 0.379PIi0.097PI20.463PI30.431PI40.009Finally, we use our weights to calculate the index of deviation ID of the maple leaf from each of the six categories of leaves considered earlier. We show the results in Table 5. Table 5. Index of deviation of maple leaf from six common leaf categories.CategoryIndex of deviation In10.2720.1230.2340.0850.2460.18Since the index of deviation between the given maple leaf and Category 4 is smallest, the model predicts that the maple leaf falls into Category which conclusion is consistent with our initial hypothesis.ConclusionOur model is robust under reasonable conditions, as can be seen from the testing above. However, since our database contains only the six commonly-seen leaf types in North America, the variety in the database has room for improvement.212 The UMAP Journal 33.3(2012)Model 2: Leaf Distribution and Leaf ShapeIntroductionGenetic and environmental factors contribute to the pattern of leaf veins and tissue, thereby determining leaf shape. In this model, we investigate how leaf distribution influences leaf shape.Idealized Leaf Distribution Model We construct an idealized model that immensely simplifies the complex situation: The tree is made up of a branch perpendicular to the ground surface, and two identical leaves grown on the branch ipsilaterally (on the same side) and horizontally. The leaves face upward and point toward the sun in the sky. We suppose that the tree is at latitude L (Northern Hemisphere). Let the greatest average solar altitude in a year, which is attained at noon on the vernal equinox, be a. Figure 7 illustrates our primitive model of a tree at noon on the vernal equinox.Figure 7. Primitive model of a tree, at noon on the vernal equinox.Analysis of Overlapping Areas Our focus is the partly shaded leaf in Figure 7. The output of the model is what proportion of the leaf (PL) is shaded. We divide the situation into three scenarios, depending on the influence of the angle a on PL.A Close Look at Leaves 213Solar Altitude Near 90° This situatio n usually takes place in tropical regions, where leaf shapes are typically broad and wide and the tree crown usually contains only one layer of leaves. This can be explained in terms of Figure 7: With a near 90°, the shaded part of the lower leaf would be too big to supply enough solar energy for photosynthesis, and the greatest absorption of energy can be achieved by a broad leaf shape.Solar Altitude Near 0° This situation usually takes place in frigid zones, where leaves are typically acicular (needle-shaped) and the tree crown contains dense layers of closely-grown leaves. In terms of Figure 7: With a near 0°, the shaded part of the lower leaf would approach zero, allowing a much more concentrated distribution of leaves than in other situations. In addition, the maximum absorption of energy can be best achievedby needle-like leaves.Solar Altitude within Normal Range This scenario is typical in the temperate zone on earth, where sunlight irradiates the leaves in a tilted way. It is also the case in which our idealized model is the most suitable. Another crucial factor that we control in this case is the distance h between the two points connecting the leaves and the branch. We assume that a tree's leaf distribution tries to minimize the overlapping area between leaves, so our model investigates the quantitative relationship between the overlapping area and the shape of the leaf. To simplify the model, we model the leaf as a rhombus, whose major axis has length Lmajor arid whose minor axis has length Lminor- Also, we fix the area of the leaf as A, to ensure constant exposure area to the sun. With area fixed, now we only need to change the length of the major axis to change the shape of the leaf (see Figure 8). "majorFigure 8. Two leaves of the same area but different lengths of major axis.214 The UMAP Journal 33.3(2012)Also, since we have fixed the area of the leaf and just adjust its shape, the minimum proportion of the lower leaf shaded isE =^overlappingwhere ^overlapping IS the smallest overlapping area. The most efficient situation is for both leaves to be totally exposed to sunlight, as in Figure 9a: For some value h = hg, we achieve E = 0.Figure 9a. Upper leaf does not overlap lower one.Figure 9b. Upper leaf overlaps lower one.What ii h < ho, as in Figure 9b? We can easily give the relationship among h, Lmajor/ and E for a given fixed solar altitude a:E =-¿'major tan a — h= 1h-i-majorFor fixed h and a, the overlap area increases as the length of the leaf increase. The closer Lmajor is to h/ tan a, the smaller the overlap. From our discussion, the best leaf distribution occurs when h — ho, which means h = Lmajor tan a.Model TestingWe need to test whether this relation between leaf distribution and leaf shape is right. We offer data on leaf length ¿major and internode distance h of several kinds of trees and use our formula to calculate the respective solar altitudes of the trees. By converting the solar altitude into latitude, we can predict the origin of a tree! We choose species native to China:•LigustrumquihouiCarr. (waxy-leaf privet or Quihou privet, a semievergreen to evergreen shrub);A Glose Look at Leaves 215•Osmanthusfragrans (sweet olive, tea olive, or fragrant olive, an evergreen shrub or small tree that is the city flower of Hangzhou, China); and•Gamelliajapónica (Japanese camellia)as our test trees. Table 6 shows the results.Table 6. Test of model for leaf shape asTree kindLigustrumquihouiCarr. Osmanthusfragrans Camellia japónicaima,or2 10 6h2.5 18.5 9a function of latitude.Calculated tana1.25 1.85 1.50Latitude Predicted True38.7° 28.4° 33.7°35-35° 23-29° 32-36°The predicted latitudes of origin are close to the true latitudes, confirming our hypothesis of a relationship between leaf distribution and leaf shape.Model 3: Tree Profile and Leaf ShapeHypothesisSince•the vein structure determines the leaf shape;•the branch structure deternünes the tree profile; and•to some degree, the leaf veins resemble branches,we have a wild hypothesis that the leaf shape is two-dimensional mimic of the tree profile. Comparison of Leaf Shape and Tree ContourThe leaf shape is two-dimensional, so it is relatively easy to study its parameters. However, the tree profile is three-dimensional, so it is important to find a two-dimensional characteristic of a tree to use for comparison. Since the longitudinal section of a particular tree reflects its general size characteristics, we focus on that.Tree Profile ClassificationIn the leaf classification model, there are 6 general classes of leaves. Since we are comparing only the general resemblance between leaf and tree, we216 The UMAP Journal 33.3 {2012)incorporate Class 5 (elliptic leaf with serrated margin) into Class 2 (elliptic leaf, smooth margin). As a result, we get 5 classes of leaves and 5 respective types of trees:•Class 1: Cordate (Texas redbud)•Class 2 and Class 5: Elliptic (camphor tree)•Class 3: Subulate (pine)•Class 4: Palmate (oak)•Class 6: Obovate (mockernut hickory)Parameters of the Tree We appoint three parameters for the longitudinal section that can be compared with those of the leaf shape, namely, rectangularity, aspect ratio, and circularity. Table 7 shows the measurements for both trees and leaves.Table 7. Comparison of leaf parameters and tree parameters.ClassRectangularity (R) Leaf Tree Aspect Ratio (AR) Leaf Tree Circularity (C) Leaf Tree10.6627 0.62810.8615 0.79140.6396 0.58002 and 50.5902 0.68460.6600 0.72430.5698 0.592830.6250 0.51800.1800 0.66010.1834 0.289540.4772 0.52920.6383 0.79800.3069 0.407060.6576 0.62380.3111 0.67500.2889 0.3866For each of the parameter types, we drew a scatterplot, calculated the correlation, and investigated the statistical significance of the resulting line of best fit. Aspect ratio (AR) and circularity (C) were each statistically significant, pointing to linear relationships; rectangularity (R) was not.ConclusionThe tests of aspect ratio and circularity support the theory that leaf shape is a two-dimensional mimic of the tree contour. Thus, the shape of leaf resembles the shape of tree to some extent.A Close Look at Leaves 217Model 4: Leaf MassIntroductionA simple way to calculate the total leaf mass is to multiply the number of leaves by the mass of a single leaf. Our method is more accurate and less demanding, in that our model is (surprisingly!) independent of these two factors but dependent on a more reliable factor of a grown tree: photosynthesis. Our methodology of estimating the leaf mass of a tree is based on three variables:•tree age; • growth rate, which is determined by tree species; and • general type (hardwood or conifer). In other words, given the age and type of a tree, we can estimate the total mass of leaves. In this model, CO2 is used as a calculating medium.Leaf Mass and Tree AgeLeaf IVIass and CO2 Sequestration Trees sequester CO2 from the atmosphere in their leaves but mostly elsewhere in the tree. A tree's ability to sequester CO2 is measured in terms of mass As of CO2 (in pounds) per gram of leaf. Hardwood trees sequester more CO2 per gram of leaf than conifers. A tree's ability to sequester CO2 is different from its ability to absorb it, since the tree also releases CO2 into the atmosphere as part of its respiration. In other words, CO2 sequestration = CO2 absorption — CO2 release. Now we need only to estimate the weight of CO2 sequestered by the tree and then calculate the total mass of the leaves as the ratio of the niass of CO2 sequestered to the mass of CO2 sequestered per leaf:CO2 Sequestration and Tree Age The relationship between the amount of CO2 sequestered, the age of a tree, and the type of tree is given in a table by the Energy Information Administration [1998, Table 2, 8-9], which also divides trees based on their growth rate: fast, moderate, or slow. For each growth rate, we graphed the annual sequestration rate vs. age of the tree and fitted a quadratic model (see Figure 10 for conifer example).218 The UMAP Journal 33.3 {2012)Annual Carbon Sequestration Rates for Conifer—-je —Ouadratic Regression= Q.0038x'2+0.2793x+0.3986 = 0.9999YnrAgsFigure 10. Annual CO2 sequestration rates, in pounds of carbon per tree per year, for three rates of growth of conifer trees of increasing age.We were surprised to find that the curves fit the data perfectly! (This fact strongly suggests that the original table values were not measured but calculated from such a model.) From the equations of the fitted curves, we can easily estimate the CO2 sequestered for a tree of a given age and growth rate and consequently calculate the mass of the leaves.Tree Age and Tree SizeAbove, we used the age of a tree as a link between the two leaf mass and the size characteristics of the tree. Since we now know the relationship between the age of a tree (of a particular growth rate) and its total leaf mass, now we only need to work out the relationship between the age of the tree and the size characteristics of it. Tree size is the accumulation of growth, which is a biological phenomenon of increase with time. In its life cycle, a tree experiences logistic growth, leading to a model for its "size," or profile, P (height, mass, diameter) asP = yti (1 - e''^^)"', hence yl = ^^4 In (l - hP'^'^) ,where A is the age of the tree and the ki are constants that depend on the species of tree. Leaf Mass and Tree SizeFinally, we get to answer the question of whether there is a relationship between leaf mass and tree size characteristics. Putting together our earlier models, we have the relationships in Figure 11.A Close Look at Leaves 219CO2Se!niesirationR3leL Direct LJJfe. I — 1 by SpeedsTree Age • * ^ -^ ^reeStzeFigure 11.According to our earlier results, leaf mass and tree age are related to each other through CO2 sequestration, and we have just determined a function between tree age and tree size.Strengths and WeaknessesModel 1Strengths: Our model is based on quantitative analysis, so the classification process is both objective and efficient. Our model is based on categories of leaf types that are the most typical and common.Weakness: We divide leaves into only six categories, which may not cover all leaf types. Model 2Strengths: We have taken into consideration three climate conditions (tropical zone, temperate zone, and frigid zone) in discussing the relationship between the leaf distribution and the leaf shape. The results of our model conform to the data that we found.220 The UMAP Journal 33.3(2012)Weakness: We consider the leaf distribution on a single branch but have not considered the inner-influence between different leaves of different branches.Model 3 Strength: The whole process uses data and quantitative analysis as foundations, so the output is objective and reasonable.Weakness: We have limited categories of tree profiles.Model 4Strength: We use the carbon sequestration rate and age as the media to calculate thetotal mass of leaves, which is better than trying to estimate the number of leaves and the average weight of each.Weakness: The data are from a source that does not refer to the method of arriving at the data.Letter to a Science Journal EditorDear Editor:We present to you our key findings. We firstfocus on the possible influence on leaf shape of the leaf distribution on the tree. For survival reasons, a tree should develop an optimal leaf distribution and shape pattern that adjust to the specific region of its origin, thereby gaining the most nutrients for photosynthesis by maximizing the exposure area to stinshine. We demonstrate a mathematical relationship among solar altitude, leaf shape, and leaf distribution. Based on this finding, we may be able to determine the best location for replanting or assisted-migration of a tree species by observing its leaf distribution. Our second key finding is a rough relationship between the tree's profile and its leaves. In fact, we hypothesize that a leaf is a two-dimensional mimic of the tree. For several trees, we compared the shape of the leaf and the contour of the tree, finding similarities between certain characteristics.A Close Look at Leaves 221This finding is another instance of the natural world containing examples of self-similarity, a mathematical concept that means that an object is approximately similar to a part of itself, as is the case for the mathematical objects of the Koch snowflake and the Mandelbrot set. The third part of our study deals with the relationship between tree sizecharacteristics and the total mass of the leaves. The two are linked by the CO2 sequestration rate and the age of the tree. Hence, we can estimate the total mass of the leaves given some profile parameters of a tree, such as its height, diameter, volume, age, and type. This findingmight have potential for agricultural and environmental uses, such as a new method to estimate tea production or wood production, or estimation of the CO2 sequestration effect of a forest as an alleviator of global warming. In hope of publishing our research in your journal, we enclose our research paper for you to examine and judge. We are convinced that our research on leaves promises to contribute to a variety of areas. Sincerely yours.Team 14990AcknowledgmentThe authors thank David Knight, James Painter, and Matthew Potter of the Dept. of Electrical Engineering at Stanford University for permission to reproduce photos of leaves from their paper iCnight et al. [2010].ReferencesAlonso, José Antonio, and M^ [Maria] Teresa Lamata. 2006. Consistency in the analytic hierarchy process: A new approach. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 14 (4): 445^59. http://hera.ugr.es/doi/16515833.pdf. Du, Ji-Xiang, Xiao-Feng Wang, and Guo-Jun Zhang. 2007. Leaf shape based plant species recognition. Applied Mathematics and Computation 185 (2) (February 2007): 883-893. Energy Information Administration, U.S. Department of Energy. 1998. Method for calculating carbon sequestration by trees in urban and suburban settings, ftp : //ftp. eia.doe. gov/pub/oiaf/1605/cdroin/pdf / sequester.pdf .Im, C, H. Nishida, and T.L. Kunil. 1998. Recognizing plant species by leaf shapes—a case study of the Acer family. In Proceedings of 1998 IEEE222 The UMAP Journal 33.3(2012)International Conference on Pattern Recognition, Brisbane, August 1998, vol. 2,1171-1173. Knight, David, James Painter, and Matthew Potter. 2010. Automatic plant leaf classification for a mobile field guide: An android application. /~jpainter/docuinents/Plantyo20Leafyo 20Classification.pdf and http :///class/ee368/Proj ect_10/Reports/ Knight_Painter_Potter_PlantLeafClassification.pdf.Saaty, Thomas L. 1982. Strategy and Organization, The Analytical Hierarchy Process for Decisions in a Complex World. Belmont, CA: Lifetime Learning Pub.Tsukaya, Hirokazu 2006. Mechanism of leaf-shape determination. Annual Review of Plant Biology 57 (1): 477-^96.Wang, Z., Z. Chi, and D. Feng. 2003. Shape based leaf image retrieval. IEEE Proceedings: Vision, Image, and Signal Processing 150 (1) (February 2003): 34-^3.。
