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《美国大学生数学建模竞赛中的MATLAB算法应用详解》 美国大学生数学建模竞赛（MCM/ICM）是一项极具挑战性的国际性赛事，旨在提升参赛者的数学、计算机科学和团队协作能力。在历年的比赛中，MATLAB因其强大的数值计算、图形绘制以及编程功能，成为了参赛者们解决复杂问题的首选工具。本压缩包资源"美赛各题常用算法程序与参考代码.rar"包含了A至F赛题的MATLAB算法代码，旨在帮助参赛者理解和应用各种算法，以提高解决问题的效率。 MATLAB是Matrix Laboratory的缩写，它提供了丰富的数学函数库，支持线性代数、统计分析、信号处理等众多领域，使得编写高效、简洁的算法代码成为可能。在数学建模中，MATLAB的应用主要包括以下几个方面： 1. 数据分析：MATLAB可以快速处理大量数据，进行统计分析、拟合曲线、数据可视化等操作，为模型构建提供依据。 2. 数值计算：MATLAB内置的数值求解器可以用于解决微分方程、优化问题、线性代数问题等，是处理复杂模型的核心工具。 3. 图形绘制：MATLAB的图形界面简单易用，可以生成高质量的2D和3D图形，有助于直观地展示模型结果和分析过程。 4. 算法实现：MATLAB的脚本语言简洁明了，适合实现各种算法，如搜索算法、优化算法、机器学习算法等。 5. 并行计算：MATLAB的并行计算工具箱可以充分利用多核处理器，加速大规模计算任务，对于时间敏感的竞赛项目尤为关键。 具体到每个赛题，例如： - A题可能涉及优化算法，如遗传算法、粒子群优化，用于寻找最佳解决方案。 - B题可能需要用到信号处理技术，比如滤波、频谱分析，处理现实世界中的信号数据。 - C题可能需要建立复杂的动态系统模型，利用MATLAB的仿真工具进行模拟。 - D题可能涉及到统计建模和预测，MATLAB的统计和机器学习工具箱提供了多种模型选择。 - E题可能需要进行图像处理或模式识别，MATLAB的图像处理工具箱功能强大。 - F题可能需要解决复杂网络问题，如最短路径算法、网络流问题，MATLAB可以提供相应的算法实现。 通过深入研究这些参考代码，参赛者不仅可以掌握MATLAB的基本用法，更能理解如何将数学理论与实际问题相结合，形成有效的建模策略。同时，这些代码也是学习和借鉴他人思路的好材料，有助于培养创新思维和问题解决能力。 "美赛各题常用算法程序与参考代码"这一资源为参赛者提供了宝贵的实践平台，通过学习和实践这些MATLAB程序，参与者可以在数学建模的道路上更进一步，提高自己的竞争力，同时也为未来的学术研究和职业生涯打下坚实的基础。
Matlab Toolbox for Dimensionality Reduction (v0.7.1b) ===================================================== Information ------------------------- Author: Laurens van der Maaten Affiliation: University of California, San Diego / Delft University of Technology Contact: lvdmaaten@gmail.com Release date: June 25, 2010 Version: 0.7.1b Installation ------------------------- Copy the drtoolbox/ folder into the $MATLAB_DIR/toolbox directory (where $MATLAB_DIR indicates your Matlab installation directory). Start Matlab and select 'Set path...' from the File menu. Click the 'Add with subfolders...' button, select the folder $MATLAB_DIR/toolbox/drtoolbox in the file dialog, and press Open. Subsequently, press the Save button in order to save your changes to the Matlab search path. The toolbox is now installed. Some of the functions in the toolbox use MEX-files. Precompiled versions of these MEX-files are distributed with this release, but the compiled version for your platform might be missing. In order to compile all MEX-files, type cd([matlabroot '/toolbox/drtoolbox']) in your Matlab prompt, and execute the function MEXALL. Features ------------------------- This Matlab toolbox implements 32 techniques for dimensionality reduction. These techniques are all available through the COMPUTE_MAPPING function or trhough the GUI. The following techniques are available: - Principal Component Analysis ('PCA') - Linear Discriminant Analysis ('LDA') - Multidimensional scaling ('MDS') - Probabilistic PCA ('ProbPCA') - Factor analysis ('FactorAnalysis') - Sammon mapping ('Sammon') - Isomap ('Isomap') - Landmark Isomap ('LandmarkIsomap') - Locally Linear Embedding ('LLE') - Laplacian Eigenmaps ('Laplacian') - Hessian LLE ('HessianLLE') - Local Tangent Space Alignment ('LTSA') - Diffusion maps ('DiffusionMaps') - Kernel PCA ('KernelPCA') - Generalized Discriminant Analysis ('KernelLDA') - Stochastic Neighbor Embedding ('SNE') - Symmetric Stochastic Neighbor Embedding ('SymSNE') - t-Distributed Stochastic Neighbor Embedding ('tSNE') - Neighborhood Preserving Embedding ('NPE') - Linearity Preserving Projection ('LPP') - Stochastic Proximity Embedding ('SPE') - Linear Local Tangent Space Alignment ('LLTSA') - Conformal Eigenmaps ('CCA', implemented as an extension of LLE) - Maximum Variance Unfolding ('MVU', implemented as an extension of LLE) - Landmark Maximum Variance Unfolding ('LandmarkMVU') - Fast Maximum Variance Unfolding ('FastMVU') - Locally Linear Coordination ('LLC') - Manifold charting ('ManifoldChart') - Coordinated Factor Analysis ('CFA') - Gaussian Process Latent Variable Model ('GPLVM') - Autoencoders using stack-of-RBMs pretraining ('AutoEncoderRBM') - Autoencoders using evolutionary optimization ('AutoEncoderEA') Furthermore, the toolbox contains 6 techniques for intrinsic dimensionality estimation. These techniques are available through the function INTRINSIC_DIM. The following techniques are available: - Eigenvalue-based estimation ('EigValue') - Maximum Likelihood Estimator ('MLE') - Estimator based on correlation dimension ('CorrDim') - Estimator based on nearest neighbor evaluation ('NearNb') - Estimator based on packing numbers ('PackingNumbers') - Estimator based on geodesic minimum spanning tree ('GMST') In addition to these techniques, the toolbox contains functions for prewhitening of data (the function PREWHITEN), exact and estimate out-of-sample extension (the functions OUT_OF_SAMPLE and OUT_OF_SAMPLE_EST), and a function that generates toy datasets (the function GENERATE_DATA). The graphical user interface of the toolbox is accessible through the DRGUI function. Usage ------------------------- Basically, you only need one function: mappedX = compute_mapping(X, technique, no_dims); Try executing the following code: [X, labels] = generate_data('helix', 2000); figure, scatter3(X(:,1), X(:,2), X(:,3), 5, labels); title('Original dataset'), drawnow no_dims = round(intrinsic_dim(X, 'MLE')); disp(['MLE estimate of intrinsic dimensionality: ' num2str(no_dims)]); mappedX = compute_mapping(X, 'Laplacian', no_dims, 7); figure, scatter(mappedX(:,1), mappedX(:,2), 5, labels); title('Result of dimensionality reduction'), drawnow It will create a helix dataset, estimate the intrinsic dimensionality of the dataset, run Laplacian Eigenmaps on the dataset, and plot the results. All functions in the toolbox can work both on data matrices as on PRTools datasets (http://prtools.org). For more information on the options for dimensionality reduction, type HELP COMPUTE_MAPPING in your Matlab prompt. Information on the intrinsic dimensionality estimators can be obtained by typing the HELP INTRINSIC_DIM. Other functions that are useful are the GENERATE_DATA function and the OUT_OF_SAMPLE and OUT_OF_SAMPLE_EST functions. The GENERATE_DATA function provides you with a number of artificial datasets to test the techniques. The OUT_OF_SAMPLE function allows for out-of-sample extension for the techniques PCA, LDA, LPP, NPE, LLTSA, Kernel PCA, and autoencoders. The OUT_OF_SAMPLE_EST function allows you to perform an out-of-sample extension using an estimation technique, that is generally applicable. Many of the available functions are also available through the GUI, which can be executed by running the function DRGUI. Pitfalls ------------------------- When you run certain code, you might receive an error that a certain file is missing. This is because in some parts of the code, MEX-functions are used. I provide a number of precompiled versions of these MEX-functions in the toolbox. However, the MEX-file for your platform might be missing. To fix this, type in your Matlab: mexall Now you have compiled versions of the MEX-files as well. This fix also solves slow execution of the shortest path computations in Isomap. If you encounter an error considering CSDP while running the FastMVU-algorithm, the binary of CSDP for your platform is missing. If so, please obtain a binary distribution of CSDP from https://projects.coin-or.org/Csdp/ and place it in the drtoolbox/techniques directory. Make sure it has the right name for your platform (csdp.exe for Windows, csdpmac for Mac OS X (PowerPC), csdpmaci for Mac OS X (Intel), and csdplinux for Linux). Many methods for dimensionality reduction perform spectral analyses of sparse matrices. You might think that eigenanalysis is a well-studied problem that can easily be solved. However, eigenanalysis of large matrices turns out to be tedious. The toolbox allows you to use two different methods for eigenanalysis: - The original Matlab functions (based on Arnoldi methods) - The JDQR functions (based on Jacobi-Davidson methods) For problems up to 10,000 datapoints, we recommend using the 'Matlab' setting. For larger problems, switching to 'JDQR' is often worth trying. Papers ------------------------- For more information on the implemented techniques and for a theoretical and empirical comparison, please have a look at the following papers: - L.J.P. van der Maaten, E.O. Postma, and H.J. van den Herik. Dimensionality Reduction: A Comparative Review. Tilburg University Technical Report, TiCC-TR 2009-005, 2009. Version history ------------------------- Version 0.7.1b: - Small bugfixes. Version 0.7b: - Many small bugfixes and speed improvements. - Added out-of-sample extension for manifold charting. - Added first version of graphical user interface for the toolbox. The GUI was developed by Maxim Vedenev with the help of Susanth Vemulapalli and Maarten Huybrecht. I made some changes in the initial version of the GUI code. - Added implementation of Gaussian Process Latent Variable Model (GPLVM). - Removed Simple PCA as probabilistic PCA is more appropriate. Version 0.6b: - Resolved bug in LLE that was introduced with v0.6b. - Added implementation of t-SNE. - Resolved small bug in data generation function. - Improved RBM implementation in au
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