Singular Value Decomposition (SVD) 1

Transcription

1 Singular Value Decomposition (SVD) 1 Let A m n be a matrix. Then σ is called one singular value associated with the singular vectors u R m 1 and v R n 1 for A provided that { Av = σu, A T u = σv. We further have { AV = UΣ, A T U = V Σ, where U and V are both unitary, and the diagonal terms in Σ are σ s, 0 s in off-diagonal terms. You may use the built-in function svd. 1 See Zheng-Liang Lu 398 / 433

2 Example: Low-rank Approximation for Image Compression This idea originates from Principal Component Analysis (PCA). 2 Use svd to calculate the principal components of the input image. Then we can have an image extremely similar to the origin one, but with a smaller image size by keeping the vectors associated with a few first largest of principal components. 2 See Zheng-Liang Lu 399 / 433

3 1 >> Lecture 6 2 >> 3 >> -- User-Controlled Input and Output 4 >> Zheng-Liang Lu 400 / 433

4 American Standard Code for Information Interchange (ASCII) 4 Everything in the computer is encoded in binary. ASCII codes represent text in computers, communications equipment, and other devices that use text. ASCII is a character-encoding scheme originally based on the English alphabet that encodes 128 specified characters into the 7-bit binary integers (see the next page). Unicode 3 became a standard for the modern systems from Unicode includes ASCII. 3 See Unicode 8.0 Character Code Charts. 4 The first version was in See ASCII. Zheng-Liang Lu 401 / 433

5 Zheng-Liang Lu 402 / 433

6 Import Tool One can load data from the specified file by importdata(filename), which can recognize the common file extensions. For example, txt, csv, jpg, bmp, wav, avi, and xls. 5 Note that the file name must be a string. If not, importdata interprets the file as a delimited ASCII file as default. importdata( -pastespecial ) loads data from the system clipboard rather than from a file. Try uiimport. 5 See supported-file-formats.html. Zheng-Liang Lu 403 / 433

7 Example 1 >> A = importdata('ngc6543a.jpg'); 2 >> image(a); % show image Zheng-Liang Lu 404 / 433

8 Example Use a text editor to create a space-delimited ASCII file with column headers like this: 1 Day1 Day2 Day3 Day4 Day5 Day6 Day Zheng-Liang Lu 405 / 433

9 1 clear; clc; 2 3 A = importdata('myfile01.txt', ' ', 1); 4 for k = [3, 5] 5 disp(a.colheaders{1, k}); % headers of columns 6 disp(a.data(:, k)); % numeric data 7 end Zheng-Liang Lu 406 / 433

10 Access to Delimited Text Files dlmread(filename, delimiter) reads ASCII-delimited file of numeric data. dlmwrite(filename, M, delimiter) writes the array M to the file using the specified delimiter to separate array elements. The default delimiter is the comma (,). dlmwrite(filename, M, -append ) appends the data to the end of the existing file. dlmread(filename, delimiter, R, C) reads data whose upper left corner is at row R and column C in the file. R and C start from 0. (R, C)=(0, 0) specifies the first element in the file. Zheng-Liang Lu 407 / 433

11 Example Write to the file. 1 >> M = gallery('integerdata', 100, [5 8], 0); 2 >> dlmwrite('myfile.txt', M, 'delimiter', '\t'); Read from the file. 1 >> dlmread('myfile.txt', '\t') 2 >> dlmread('myfile.txt', '\t', 2, 3) Zheng-Liang Lu 408 / 433

12 textread textread is useful for reading text files with known formats. [A, B, C,... ]=textread(filename, format, N) reads data from the file filename into the variables (A, B, C,... ) using the specified format, format, for N lines. format determines the number and types of return arguments. (See next page.) Without N, textread reads until the end of file. The common conversions are as follows: %d: signed integer values. %f: floating-point values. %s: strings. Zheng-Liang Lu 409 / 433

13 Example Create a file with the following lines: 1 Sally Level Yes 2 Arthur Level No 1 >> [names, types, x, y, answer] =... textread('mydata.dat', '%s %s %f %d %s') %... normal usage 1 >> [names, types, x, answer] =... textread('mydata.dat', '%s Level%d %f %*d %s',... 1) % check the difference! In %*f, * ignores the matching characters specified by *. Zheng-Liang Lu 410 / 433

14 Access to Excel Files xlsread(filename, sheet, xlrange) reads from the specified sheet and range. The variable sheet can be the sheet name 6 or the sheet number in the excel file. The variable xlrange is optional for the rectangular portion of the worksheet to read. For example, xlrange = B:B is used to import column B. To read a single value, use xlrange = B1:B1. 7 xlswrite(filename, A, sheet, xlrange) writes the array A to the specified range of the sheet. 6 The default sheet name is 工作表 1. 7 Contribution by Mr. Tsung-Yu Hsieh (MAT24409) on August 27, Zheng-Liang Lu 411 / 433

15 Example 1 >> values = {1, 2, 3; 4, 5, 'x'; 7, 8, 9}; 2 >> headers = {'First', 'Second', 'Third'}; 3 >> xlswrite('myexample.xlsx', [headers; values]);... % write 1 >> subseta = xlsread('myexample.xlsx', 1, 'B2 :... C3') % read 2 3 subseta = NaN Zheng-Liang Lu 412 / 433

16 Low-Level File I/O Low-level file I/O functions allow the most control over reading/writing data to a file. However, these functions require that you specify more detailed information about the files than the easier-to-use high-level functions, such as importdata. If the high-level functions cannot import your data, you may consider to use low-level file I/O. The normal procedure is as follow: 1. Open a file. 2. Read/write data into the file. 3. Close the file. Zheng-Liang Lu 413 / 433

17 Open Files fopen(filename, permission) create a file object in the memory: filename: the file name permission: file access type, specified as a string return an integer at least 3 as the file identifier 8 You can create a new file, make permission w. Be aware that you can repeatedly open a new file with the same file name. Note that fid is 1 if fopen fails to open the file with r. 8 Matlab reserves 1 and 2 for standard output and standard error, respectively. Zheng-Liang Lu 414 / 433

18 Low-Level File I/O Low-level file I/O functions allow the most control over reading/writing data to a file. However, these functions require that you specify more detailed information about the files than the easier-to-use high-level functions, such as importdata. If the high-level functions cannot import your data, you may consider to use low-level file I/O. The normal procedure is as follow: 1. Open a file. 2. Read/write data into the file. 3. Close the file. Zheng-Liang Lu 415 / 433

19 Open Files fopen(filename, permission) create a file object in the memory: filename: the file name permission: file access type, specified as a string return an integer at least 3 as the file identifier 9 You can create a new file, make permission w. Be aware that you can repeatedly open a new file with the same file name. Note that fid is 1 if fopen fails to open the file with r. 9 Matlab reserves 1 and 2 for standard output and standard error, respectively. Zheng-Liang Lu 416 / 433

20 Permission Codes See Zheng-Liang Lu 417 / 433

21 Example feof(fid), which refers to end-of-file, returns 1 if a previous operation set the end-of-file indicator for the specified file. fgetl(fid) returns the next line of the specified file, removing the newline characters. If the line contains only the end-of-file marker, then the return value is 1. 1 clear; clc; 2 3 f = fopen('fgetl.m', 'r'); 4 while ~feof(f) 5 disp(fgetl(f)); 6 end 7 fclose(f); Zheng-Liang Lu 418 / 433

22 Close Files fclose(fid) closes an opened file. fclose( all ) closes all opened files. fclose returns a status of 0 when the close operation is successful. Otherwise, it returns 1. Zheng-Liang Lu 419 / 433

23 fprintf Recall that fprintf(format, A,..., A n ) formats data and displays the results on the screen 11. fprintf(fid, format, A,..., A n ) applies format to all elements of arrays A 1,..., A n in column order, and writes the data to a text file. format: to specify a format for the output fields; it is a string. A 1,..., A n : arrays for the output fields. Matlab reserves the file identifier number 1 and 2 for standard output on the screen, and standard error, respectively. fprintf(1, This is standard output!\n ); fprintf(2, This is standard error!\n ); sprintf(format, A 1,..., A n ) returns the results as a string. 11 Recall disp. Zheng-Liang Lu 420 / 433

24 Example fprintf can print multiple numeric values and literal text to the screen. 1 >> A = [ ; ]; 2 >> format = 'X is %4.2f meters or %8.3f mm.\n'; 3 >> fprintf(format, A); % print on the screen 4 5 X is 9.90 meters or mm. 6 X is 8.80 meters or mm. 7 X is 7.70 meters or mm. %4.2f specifies that the first value in each line of output is a floating-point number with a field width of four digits, including two digits after the decimal point. Can you explain %8.3f? Zheng-Liang Lu 421 / 433

25 Escape Characters %%: percent sign \\: backslash \b: backspace \n: new line \t: horizontal tab Zheng-Liang Lu 422 / 433

26 Format Conversion Integer, signed: %d Integer, unsigned %u: base 10 %o: base 8 %x: base 16 Floating-point number %f: fixed-point notation %e: scientific notation, such as e+00 Characters %c: single character %s: string Note that the % can be followed by an optional field width to handle fixed width fields. Zheng-Liang Lu 423 / 433

27 Example: grep in UNIX/Linux findstr(s 1, S 2 ) returns the starting indices of any occurrences of the shorter of the two strings in the longer function grep(f, pattern) 2 f = fopen(f, 'r'); 3 cnt = 0; 4 while ~feof(f) 5 t = fgetl(f); 6 cnt = cnt + 1; 7 w = findstr(t, pattern); 8 if ~isempty(w) 9 fprintf('%d: %s\n', cnt, t); 10 end 11 end 12 fclose(f); 13 end 12 See Knuth Morris Pratt string searching algorithm (1974). Zheng-Liang Lu 424 / 433

28 Exercise Write a program which write a multiplication table into a text file Zheng-Liang Lu 425 / 433

29 1 clear; clc; 2 3 f = fopen('multiplicationtable.txt', 'w'); 4 for i = 1 : 9 5 for j = 1 : 9 6 fprintf(f, '%3d', i * j); 7 end 8 fprintf(f, '\n'); 9 end 10 fclose(f); Zheng-Liang Lu 426 / 433

30 fscanf fscanf(fid, format, i) reads data from the file specified by file identifier fid, converts it according to the string specified by format. fscanf can be used to skip specific characters in a sample file, and return only numeric data. Zheng-Liang Lu 427 / 433

31 Example 1 clear; clc; 2 str = { 3 ['78' char(176) 'C']; % char(176) is the... symbol of degree 4 ['72' char(176) 'C']; 5 ['64' char(176) 'C']; 6 ['66' char(176) 'C']; 7 ['49' char(176) 'C']}; 8 fid = fopen('data.txt', 'w'); 9 for i = 1 : length(str) 10 fprintf(fid, '%s\n', str{i}); 11 end 12 fclose(fid); 13 % main 14 fid = fopen('data.txt', 'r'); 15 A = fscanf(fid, ['%d' char(176) 'C']) 16 fclose(fid); Zheng-Liang Lu 428 / 433

32 Binary Files fread(fid, size, precision) interprets values in the file according to the form and size described by precision. fwrite(fid, A, precision) translates the values of A according to the form and size described by precision. Options for size are: N: read N elements into a column vector. inf : read to the end of the file. [M, N]: read elements to fill an M-by-N matrix, in column order. Zheng-Liang Lu 429 / 433

33 Options for precision are: uchar : unsigned integer, 8 bits. int64 : integer, 64 bits. uint64 : unsigned integer, 64 bits. float64 : floating point, 64 bits. Note that the number can be replaced by 8, 16, and 32. Zheng-Liang Lu 430 / 433

34 Example 1 clear; clc; 2 A = magic(3) 3 fid = fopen('magic3.txt', 'w'); 4 fwrite(fid, A, 'int32'); 5 fclose(fid); 6 fid = fopen('magic3.txt', 'r'); 7 fread(fid, [3 3], 'int32'); % try [3 1]? You may try different combinations, say, int64 in fwrite and int32 in fread. Zheng-Liang Lu 431 / 433

35 Access to Internet 13 urlread(url, name, value) returns the contents of a URL. 1 A =... urlread(' 2 f = fopen('matlab.html', 'w'); 3 fprintf(fid, '%s', A); 4 fclose(f); 5 dos('start matlab.html'); Try sendmail, ftp. 13 See Zheng-Liang Lu 432 / 433

36 Example: Yahoo Finance API Current market and historical data from the Yahoo! data server Blog: 研究雅虎股票 API (Yahoo finance stock API) Google: yahoo-finance-managed Historical Stock Data downloader by Josiah Renfree (2008) Zheng-Liang Lu 433 / 433