【作者】 李瑛；

【导师】 吕良福；

【作者基本信息】 天津大学 ， 数学， 2021， 硕士

【摘要】 实际应用中的许多不确定性的问题都归结于通过相似性度量找到共同性以建立模型。衡量向量相似性的度量函数分为余弦距离、皮尔森距离等角度度量和欧氏距离、曼哈顿距离、切比雪夫距离等长度度量,衡量矩阵相似性是利用特征向量构成的可逆矩阵将对称矩阵分解为一个对角元素为特征值的对角矩阵。本文利用处理高阶数组的工具张量,将多维度大容量的数据构成张量,并提出了张量距离模型和张量分解聚类模型及其应用,主要工作为:1.张量距离模型,将同阶张量纤维化后,利用欧氏距离遍历纤维,确定欧氏距离的最小值及最小值所对应的索引。在空调运维中,根据同一地点不同建筑或同一建筑不同位置与空调运行阶段有关的各参数时空特性与多维特性,按规律性的活动时确定日类型、划分时间段后构造历史温度特征张量,爬虫室外温度平均插值后构造同阶预测温度特征张量,将历史温度特征张量与预测温度特征张量纤维化后利用欧氏距离遍历纤维相似性,确定最小值索引及其对应的纤维,逐小时输出预测时间段空调系统的室内温度及其运行策略。通过挖掘不断增长的历史数据中有效信息预测空调供暖问题中的室内温度舒适度和供水温度。2.张量分解聚类模型,先将张量分解为核张量,既可以保存张量的大部分信息,又压缩了存储空间,再利用K-Means聚类算法对核张量进行聚类,将相似度大的聚为一类,将差异性大的分成不同类,达到将无标签的数据集分为不同的类。将RGB彩色图像每个颜色通道的像素矩阵作为切片构成三阶张量,可以更好地保持了像素点间的空间关系而无需进行灰度处理。对单张图像张量分解后进行聚类分析,对比单张图像直接聚类分析,实验结果表明张量分解的秩越大,图像保存的信息越多;对多张图像张量分解后利用人脸LFW数据集输出与样本中心最相似的5张图像与最不相似的5张图象。实验证明了张量分解在处理数据集聚类的可行性,可将不同的人脸分离开。更多还原

【Abstract】 Many uncertain problems in practical applications are due to find commonality through similarity measures to build models.Measure the similarity of vectors with angle measures such as cosine distance or Pearson distance,and length metrics such as Euclidean distance,Manhattan distance,or Chebyshev distance.Measure the matrix similarity is to decompose into a diagonal matrix whose diagonal elements are eigenvalues by using an invertible matrix composed of eigenvectors.This thesis uses the tool tensor for processing high-order arrays to form tensors with multi-dimensional and large-capacity data,and proposes tensor distance algorithm and tensor decomposition clustering algorithm and their applications,the main work includes:1.Tensor distance algorithm,after fiberizing the tensor of the same order,the tensor distance model use the Euclidean distance to traverse the fiber to determine the minimum value and the corresponding index.Considering the space-time and multi-dimensional characteristics of air conditioning maintenance data in different buildings at the same location or in different locations of the same building,after dividing the day type,the time interval and expanding the outdoor temperature,the tensor of historical temperature was constructed.Interpolating outdoor temperature data through web crawler,the tensor of forecast temperature was constructed.Measuring distance of each input parameter among these tensors,the index of the smallest value and the corresponding fiber were found,hourly output the indoor temperature and its water supply temperature during the forecast period.Predict indoor temperature comfort and water supply temperature in air conditioning and heating problems by mining effective information in the ever-growing historical data.2.Tensor decomposition clustering algorithm,a tensor can decompose into a kernel tensor,which can not only save most of the information of the tensor,but also compress the storage space,and then use the K-Means clustering algorithm to cluster the kernel tensors,the core tensors with large similarity are clustered into one category,and the core tensors with large differences are divided into different categories,so as to divide the unlabeled data sets into different categories.Taking the pixel matrix of each color channel of the RGB color image as a slice to form a third-order tensor,the spatial relationship between the pixels can be better maintained without grayscale processing.Contrast clustering result after decomposing an image tensor with directly cluster result,tensor decomposition can save the storage structure and the effective information of the image.Clustering after decomposing multiple image tensors,group the images with large similarity into one category,and divide the large differences into different categories,so as to divide the unlabeled data set into different categories.Use the face dataset LFW to output five images with the most similar to the center of the sample and five most dissimilar images.Experiments have proved the feasibility of tensor decomposition in processing data set clustering,which can separate different faces.更多还原