Matlab learning notes

notes ：

• Although the documentation is comprehensive , But this article is mainly to speed up in one day , So please forgive me for my shortcomings .
• The code is based on vscode/py 3.6 Called matlab terminal

Basic mathematical operations and matrix operations

Basic grammar

Variable

• No declaration required
• use = Assign a value to a variable

Variable name

• Variable names are case sensitive （ I don't know if it's because windows Why
• Variable names can only be used by [0~9,a~z,A~z,_] form , And the variable name cannot start with a number .
• Reserved variables are not suitable for variable names
  • Use iskeyword Command can view program keywords , These keywords are not allowed to be used as variable names .

Variable	significance
ans	The result of the last sentence
i,j	Complex operator ,（ We need to pay attention here ）
Inf	infinite
eps	Floating point relative precision , namely 1.0 The distance to the next floating-point number ( The value is 2.2204e-16)
NaN	The digital
pi	PI

Variable names should not override built-in function names

stay MATLAB in , Call priority of variables (calling priority) Higher than function , Therefore, variable names should not override built-in functions .

for example ：

cos = 'changqingaas';
cos(8)
% disp(['ans is ', num2str(ans)])
clear; %  If a function is overwritten by a variable name , Call clear < Variable name > You can unbind the variable name on the function name 
cos(8)
% disp(['ans is ', num2str(ans)])

notes ：clear It's a dangerous order , Because if you do not add parameters after this command , It means to clear all variables in the current workspace .

 Output ：

ans =

    'n'


ans =

   -0.1455

Variable type

logical,char,numeric,cell,struct And the array or matrix composed of them .

Display format of numeric variables

Numeric variables , Default to double Stored in form .

Can pass format < Display format > Change the display format of numeric variables .

MATLAB Command line

• Use end of line ; Suppress output :
  • Use after a line of command ; Suppress output , Otherwise, the calculation result will be displayed on the terminal .
• | command | effect | Running results | | ———- | —————————————————————————————— | —————————————————————————————— | | clc | Clear the output of the terminal | | | clear | By default, all variables in the current workspace are cleared | | | who | Display all variables in the workspace in a simplified format | Your variable is : ans | | whos | Display all variables in the workspace in a complex format | Name Size Bytes Class Attributes ans 1x1 8 double | | which | Check the location of the built-in function source code file , And edit Command can be combined to view the source code of built-in functions . | |

Use MATLAB Do numerical calculations

Use MATLAB Calculate mathematical expressions

• After the mathematical expression is calculated , Its value is stored in the variable ans
• log Express ln
• exp(x) Express e^x

MATLAB Built in mathematical functions

• MATLAB Built in arithmetic operation function
  • Basic operation :
    • Add : +,sum,cumsum,movsum
    • reduce : -,diff
    • ride : .*,*,prod,cumprod
    • except : ./,.\,/,\
    • chengfang : .^,^
  • Modulus operation : mod,rem,idivide,ceil,fix,floor,round
• MATLAB Built in trigonometric function
  • sine : sin,sind,sinpi,asin,asind,sinh,asinh
  • cosine : cos,cosd,cospi,acos,acosd,cosh,acosh
  • tangent : tan,tand,atan,atand,atan2,atan2d,tanh,atanh
  • Cosecant : csc,cscd,acsc,acscd,csch,acsch
  • Secant : sec,secd,asec,asecd,sech,asech
  • Cotangent : cot,cotd,acot,acotd,coth,acoth
  • hypotenuse : hypot
  • transformation : deg2rad,rad2deg,cart2pol,cart2sph,pol2cart,sph2cart
• MATLAB Built in exponential logarithmic function :
  • exp,expm1,log,log10,log1p,log2,nextpow2,nthroot,pow2,reallog,realpow,realsqrt,sqr
• MATLAB Built in complex function :
  • abs,angle,complex,conj,cplxpair,i,imag,isreal,j,real,sign,unwrap

Use MATLAB Do matrix operations

Define the matrix

Input the matrix to the terminal

stay MATLAB in , Use [] Enclose the matrix contents to be input , Use spaces or commas , Separate inline variables , Use ; Separate each line .

Use the colon operator to create a vector

Use the colon operator : You can create a long vector , The syntax is as follows :

for example ：

Define a special matrix

command	The result
eye(n)	Get one n × n The identity matrix of
zeros(n1, n2)	Get one n1 × n2 Of all the 0 matrix
ones(n1, n2)	Get one n1 × n2 Of all the 1 matrix
diag(vector)	Get a vector vector The content in is the diagonal matrix of diagonal

The index of the matrix

• MATLAB The matrix in is represented by Column Sequentially stored . And the index subscript is from 1 Start .
• Matrices can be indexed in two ways : Index by one dimension and index by two dimensions . For a general matrix , The index order is as follows : \begin{bmatrix} 1 or (1,1) & 4 or (1,2) & 7 or (1,3) \\ 2 or (2,1) & 5 or (2,2) & 8 or (2,3) \\ 3 or (3,1) & 6 or (3,2) &9 or (3,3) \end{bmatrix}
• The index of matrix can use colon :, Means to select all rows or columns .
• The index of a matrix can be one or two vectors , Indicates that all rows or columns in the selected vector . | Primitive matrix | Indexes | The result || —————————————————————————————— | ———————- | ———————————————————————————- || \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6\\ 7 & 8 & 9 \end{bmatrix}\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6\\ 7 & 8 & 9 \end{bmatrix}\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6\\ 7 & 8 & 9 \end{bmatrix}\begin{bmatrix} 1 & 4 \\ 7 & 2\end{bmatrix}\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6\\ 7 & 8 & 9 \end{bmatrix}\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6\\ 7 & 8 & 9 \end{bmatrix}\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6\\ 7 & 8 & 9 \end{bmatrix}\begin{bmatrix} 1 & 2 \\ 7 & 8 \end{bmatrix}

Matrix operations

Operator of operation matrix

Operator	operation	form	Example
+	Add matrix and vector	A+b	[6 3] + 2 = [8 5]
-	Matrix subtracts from vector	A-b	[6 3] - 2 = [4 1]
+	Add the matrix to its corresponding position	A+B	[6 3] + [4 8] = [10 11]
-	The matrix is subtracted from the corresponding position of the matrix	A-B	[6 3] - [4 8] = [2 -5]
*	Matrix Matrix Multiplication	A*B	[6 3] * [4 8]' = 48
.*	Multiply the matrix by its corresponding position	A.*B	[6 3] * [4 8] = [24 24]
/	Matrix and matrix right division ( Equivalent to A*inv(B))	A/B	[6 3] / [4 8] = 0.6
\	Matrix and matrix left division ( Equivalent to inv(A)*B)	A\B	[6 3] / [4 8] = [0.06667 1.3333; 0 0]
./	The matrix and the corresponding position of the matrix are divided by	A./B	[6 3] ./ [4 8] = [1.5 0.375]
.\	The matrix and the corresponding position of the matrix are divided by left	A.\B	[6 3] .\ [4 8] = [0.6667 2.6667]
^	Matrix and vector power	A^b	[1 2; 3 4]^3 = [37 54; 81 118]
.^	Matrix and matrix corresponding position power	A.^B	[1 2; 3 4].^[1 2; 3 4] = [1 4; 27 256]

Function of operation matrix

For the following matrix

A=\begin{bmatrix} 1 & 2 & 3 \\ 0 & 5 & 6 \\ 7 & 0 & 9 \end{bmatrix}

Operate to demonstrate common functions of the operation matrix

For the above functions , Except for the first parameter , Other parameters are optional .

Structured programming and function definition

Structured programming

Process control statements and logical operators

Flow control statement	effect
if, elseif, else	if if The statement is true , Execute clause
switch, case, otherwise	according to switch The content of the statement determines which clause to execute
while	Repeat the clause until while The condition in is false
for	Execute the clause a fixed number of times
try, catch	Execute clauses and catch exceptions during execution
break	Out of the loop
continue	Go directly to the next cycle
end	End clause
pause	Suspend program
return	Return to the calling function

All the above loops and conditional statements should end with end closed .

MATLAB There are also the following logical operators :

Operator	significance
==	be equal to
~=	It's not equal to
&&	And ( Support logic short circuit )
`		`	or ( Support logic short circuit )

Example of process control statement

if sentence ：

if condition1
	statement1
elseif condition2
	statement2
else 
	statement3
end

switch sentence

switch expression
    case value1
        statement1
    case value2
        statement2
    otherwise 
        statement
end

while sentence

while expression
	statement
end

for sentence

for variable = start:increment:end
	commands
end

break sentence

x = 2; k = 0; error = inf;
error_threshold = 1e-32;

while error > error_threshold

    if k > 100
        break
    end

    x = x - sin(x) / cos(x);
    error = abs(x - pi);
    k = k + 1;
end

Try to pre allocate memory space when using circular statements

Problems that should be paid attention to when writing scripts

Add a statement at the beginning of the script to empty the workspace

At the beginning of each script , The following statement should be added , Clear the workspace cache and traces of previous program runs :

clear all	%  Clear variables in the workspace memory 
close all 	%  Close the image drawn by the program before 
clc			%  Clear the output of the program on the terminal before 

Semicolons should be added after operations and assignment statements ; Suppress output

Semicolons should be added to all operations and assignment statements ; Suppress output , If you need to output a variable to the terminal , It should be called disp Method .

Use ellipsis ... Splice multiple lines

stay MATLAB in , Ellipsis ... You can splice multiple lines of statements into one line , Using this statement flexibly can improve the readability of the code .

function

Similar to scripts , Functions can be stored Function name .m In file , It can also be defined in memory in the form of a function handle .

See built-in functions

We can use which Command to view the location of the built-in function source code file , And edit Command can be combined to view the source code of built-in functions .

Run the following statement to open MATLAB Built in mean Function source file :

edit(which('mean.m'))

With Function name .m File form definition function

stay MATLAB The format of the function defined in the file is as follows :

function [ Output variable name ] =  Function name ( Enter variable name ) 
%  Documentation for functions 

 function code 

• function It's a keyword , Declare that what is saved in this file is a function .
• The input variable and Output variables nonessential , Functions can have no input variables , You can also have no output variables .
• Function name Should and .m Same file name , And does not contain special characters ( It's better not to have Chinese ).

MATLAB Built in function parameters

Function parameter	significance
imputname	Enter a list of variable names
mfilename	Function source code file name
nargin	Enter the number of variables
nargout	Number of output variables
varargin	Variable length input parameter list
varargout	Variable length output parameter list

MATLAB It does not provide the syntax of specifying default parameter values and function overloading in other high-level languages , But use the built-in function parameters mentioned above flexibly , You can specify the default parameter value and method overloading to a certain extent :

MATLAB Function definition example 1

The surface program is used to calculate the displacement of a free falling body :x = x_{0} + v_{0}t + \frac{1}{2} g t^2

function x = freebody(x0,v0,t)
% calculation of free falling
% x0: initial displacement in m
% v0: initial velocity in m/sec
% t: the elapsed time in sec
% x: the depth of falling in m
x = x0 + v0.*t + 1/2*9.8*t.*t;

This function demonstrates a MATLAB Programming skills : Try to use .* Instead of *, Because the former is not only for parameters t Scalar case is available , Also for variables t In the case of vectors or matrices .

freebody(0, 0, 2)			%  obtain  19.6000
freebody(0, 0, [0 1 2 3])	%  obtain  [0 4.9000 19.6000 44.1000]
freebody(0, 0, [0 1; 2 3])	%  obtain  [0 4.9000; 19.6000 44.1000]

MATLAB Function definition example 2

The following function realizes the conversion from Fahrenheit temperature to Celsius temperature , This function can identify the number of input samples to be converted , When the number of samples to be converted is 0 when , Exit function .

function F2C()
while 1
    F_degree = input('tempreature in Fahrenheit: ', 's');
    F_degree = str2num(F_degree);
    if isempty(F_degree)
        return
    end
    C_degree = (F_degree-32)*5/9;
    disp(['tempreature in Celsius: ' num2str(C_degree)])
end

Define a function in the form of a function handle

We can also define functions in the form of function handles , This is closer to the mathematical definition of function , The syntax is as follows :

 Function handle  = @( The input variable )  Output variables  

You can call this method directly through the function handle .

f = @(x) exp(-2*x);
x = 0:0.1:2;
plot(x, f(x));

Data type and file reading and writing

data type

MATLAB The main data types in are as follows :

value type (numeric)

MATLAB The supported value types are shown in the following table :

value type	describe
double	Double precision floating point
single	Single-precision floating-point
int8	8 Bit signed integer
int16	16 Bit signed integer
int32	32 Bit signed integer
int64	64 Bit signed integer
uint8	8 Bit unsigned integer
uint16	16 Bit unsigned integer
uint32	32 Bit unsigned integer
uint64	64 Bit unsigned integer

stay MATLAB in , Variables of numeric type are defaulted to double Type of , You can use type conversion to convert it to other numeric types .

n = 3;
class(n)	 

n = int8(3);
class(n)	 

Output ：

ans =
    'double'
ans =
    'int8'

String type (char)

stay MATLAB in , The string type consists of a pair of single quotation marks ' Wrap a paragraph of text to define . standard ASCII Characters can be converted to corresponding ASCII code .

s1 = 'h';
uint16(s1)	%  obtain  104

Strings are stored in memory in the form of a character matrix , You can index the matrix and assign values to it :

str1 = 'hello';
str2 = 'world';

str3 = [str1 str2];
size(str3)		%  obtain  [1 10]

str4 = [str1; str2];
size(str4)		%  obtain  [2 5]

str = 'aardvark';
'a' == str				%  obtain  [1 1 0 0 0 1 0 0]
str(str == 'a') = 'Z'	%  obtain  'ZZrdvZrk'

Structure (structure)

stay MATLAB in , The structure is a key value pair

The basic use of structures

• Similar to most programming languages ,MATLAB Use . To access the fields in the structure :

student.name = 'John Doe';
student.id = '[email protected]';
student.number = 301073268;
student.grade = [100, 75, 73; ...
                 95, 91, 85.5; ...
                 100, 98, 72];
student

Using subscript expressions on the structure list can expand or reduce the structure list .

student.name = 'John Doe';
student.id = '[email protected]';
student.number = 301073268;
student.grade = [100, 75, 73; 95, 91, 85.5; 100, 98, 72];

student

student(2).name = 'Ann Lane';
student(2).id = '[email protected]';
student(2).number = 301078853;
student(2).grade = [95 100 90; 95 82 97; 100 85 100];

student(1) = [] %  Delete student The first item on the list 

Structures can be cascaded , That is, the value of the field in the structure can also be the structure :

A = struct('data', [3 4 7; 8 0 1], ...
'nest', struct('testnum', 'Test 1', ...
'xdata', [4 2 8], ...
'ydata', [7 1 6]));
A(2).data = [9 3 2; 7 6 5];
A(2).nest.testnum = 'Test 2';
A(2).nest.xdata = [3 4 2];
A(2).nest.ydata = [5 0 9];

A

Common functions of structures

function	effect
struct	Create structure
struct2cell	Convert the structure to a cell array
cell2struct	Convert the cell array into a structure
isstruct	Determine whether a variable is a structure
structfun	Apply a function to each field of the structure
fieldnames	Get all field names of the structure
isfield	Determine whether the structure contains a field
getfield	Get the value of a field in the structure
setfield	Assign a value to a field in the structure
rmfield	Delete a field in the structure
orderfields	Sort the structure fields

Cell array (cell)

stay MATLAB in , A cell array is a data structure that can hold different types of elements , Be similar to Python Lists in languages .

Basic use of cellular arrays

Use {} Define a cell array like a matrix :

A = { [1 4 3; 0 5 8; 7 2 9]		'Anne Smith' ;...
3+7i		-pi:pi:pi}

A(1,1)={[1 4 3; 0 5 8; 7 2 9]};
A(1,2)={'Anne Smith'};
A(2,1)={3+7i};
A(2,2)={-pi:pi:pi};

A{1,1}=[1 4 3; 0 5 8; 7 2 9];
A{1,2}='Anne Smith';
A{2,1}=3+7i;
A{2,2}=-pi:pi:pi;
A

The above three methods are equivalent , The second way is to use Cell index assignment , The third way is to use Content index assignment .

• There are two ways to access data in a cellular array , Namely : Cell index () and Content index {} Because the subset of cell array is still cell array , In the use of indexer content , We need to point out that we are going to visit a Sub cell array still The contents in the corresponding area of the cell array .
  • Use Cell index (), What we get is a Sub cell array .
    • Use Content index {}, What we get is The contents in the corresponding area of the cell array .

Common functions of cell array

Common functions of cell array

function	effect
cell	Create a cell array
iscell	Determine whether a variable is a cellular array
cell2mat	Turn the cell array into a matrix
cell2struct	Turn the cell array into a structure
mat2cell	Converts an array to an array of cells of a specified size
num2cell	Convert the array to a cell array of the same size
struct2cell	Convert the structure to a cell array
celldisp	Recursively display the contents of the cell array
cellplot	Draw the structure of cell array in image form
cellfun	Apply a function to each cell of the cell array

among mat2cell Function can specify the size of each cell in the cell array when converting .

a = magic(3)

b = num2cell(a)
%  obtain 
% [8] [1] [6]
% [3] [5] [7]
% [4] [9] [2]

c = mat2cell(a, [1 2], [2, 1])
%  obtain 
% [1x2 double] [6]
% [2x2 double] [2x1 double]

High dimensional cell array

A three-dimensional cell array can have That's ok (row), Column (column), layer (layer) Three dimensions . When indexing a cell array , The order of priority from high to low is : That's ok → Column → layer .

Use cat Function can splice a cell array on a specified dimension .

A function that determines the data type of a variable

The following functions can judge the type of variables :

function	effect
isinteger	Judge whether the input parameter is an integer array
islogical	Judge whether the input parameter is a logical quantity array
isnumeric	Judge whether the input parameter is a numeric array
isreal	Judge whether the input parameter is a real array
ischar	Judge whether the input parameter is a character array
iscell	Judge whether the input parameter is a cell array
isfloat	Determine whether the input array is a floating-point array
ishandle	Determine whether the input array is a valid graphic handle
isempty	Determine whether the input array is empty
isprime	Determine which array elements are prime numbers
isnan	Determine which array elements are NaN
isinf	Determine which array elements are Inf
isequal	Determine whether the array is equal

File read and write

The supported file types are as follows :

The contents of the document	Extension	Function to read the file	Write function to file
MATLAB data	*.mat	load	save
Excel form	*.xls,*.xlsx	xlsread	xlswrite
Space separated numbers	*.txt	load	save

Reading and writing MATLAB Formatted data

MATLAB The data in the workspace can be in *.mat The format is saved in a file . Use save Function stores data in a file , Use load Function to read data from a file .

• save The syntax of the function is as follows :
  • save(filename,variables) Put the variable variables Stored in a file in binary form .
  • save(filename,variables,'-ascii') Put the variable variables Save in text form .
• load The syntax of the function is as follows :
  • load(filename) Read data from binary form file .
  • load(filename,'-ascii') Read data from a text file .

The parameter filename and variables All in string format , If not specified variables Parameters , Then save all variables in the current workspace into a file .

Complex data formats , Such as struct and cell, Storage in binary format is not supported .

Reading and writing Excel form

Use xlsread and xlswrite Functions can be read and written Excel data , The grammar is as follows :

Read Excel The syntax of the document :[num,txt,raw] = xlsread(filename,sheet,xlRange)

Score = xlsread('04Score.xlsx')
Score = xlsread('04Score.xlsx', 'B2:D4')
[Score Header] = xlsread('04Score.xlsx')

write in Excel The grammar of :xlswrite(filename,A,sheet,xlRange)

M = mean(Score);
xlswrite('04Score.xlsx', M, 1, 'E2:E4');
xlswrite('04Score.xlsx', {'Mean'}, 1, 'E1');

Basic drawing

Drawing and decoration of drawing lines

Use plot() Function to draw lines

stay MATLAB in , Use plot() Function to draw lines , Its grammar is :

plot(x,y,LineSpec)

LineSpec: Line setting of drawing line , Three designations Linetype , The sign and Color Of Setter Make a string , The setting character does not distinguish the order . For details, please refer to Official documents .

Line symbols	Linetype setter	Mark	Tag setter	Color	Color setter
-	Solid line （ Default ）	o	circle	y	yellow
--	Dotted line		plus	m	magenta
:	Dotted line	*	asterisk	c	Blue
-.	Point line	.	spot	r	Red
		x	Cross	g	green
		s	square	b	Blue
		d	The diamond	w	white
		^	Upper triangle	k	black
		v	Lower triangle
		>	Right triangle
		<	Left triangle
		p	Pentagonal
		h	hexagon

notes ：Python Medium matplotlib The way of drawing is almost the same as here

Decorative lines

Use legend() Function to add a legend to the picture

Use legend(label1, ..., labelN) Function can add a legend to the image .

x=0:0.5:4*pi;
y=sin(x); h=cos(x); w=1./(1+exp(-x)); g=(1/(2*pi*2)^0.5).*exp((-1.*(x-2*pi).^2)./(2*2^2));
plot(x,y,'bd-' ,x,h,'gp:',x,w,'ro-' ,x,g,'c^-');		%  Draw multiple lines 
legend('sin(x)','cos(x)','Sigmoid','Gauss function');	%  Add legend 

Use title() and *label() Add titles and labels to the pictures

x = 0:0.1:2*pi; y1 = sin(x); y2 = exp(-x);
plot(x, y1, '--*', x, y2, ':o');
xlabel('t = 0 to 2\pi');
ylabel('values of sin(t) and e^{-x}')
title('Function Plots of sin(t) and e^{-x}');
legend('sin(t)','e^{-x}');

Use text() and annotation() Add notes to the picture

x = linspace(0,3); y = x.^2.*sin(x); plot(x,y);
line([2,2],[0,2^2*sin(2)]);
str = '$$ \int_{0}^{2} x^2\sin(x) dx $$';
text(0.25,2.5,str,'Interpreter','latex');
annotation('arrow','X',[0.32,0.5],'Y',[0.6,0.4]); 

Control the axis , Borders and grids

command	effect
grid on/off	Set mesh visibility
box on/off	Set border visibility
axis on/off	Set axis visibility
axis normal	Restore default behavior , Set the properties of frame aspect ratio mode and data aspect ratio mode to automatic
axis square	Use coordinate axes of the same length , Adjust the increment between data units accordingly
axis equal	Use the same data unit length along each axis
axis tight	Set the coordinate axis range equal to the data range , Make the axis frame closely around the data

The following example demonstrates axis Effect of command :

t = 0:0.1:2*pi; x = 3*cos(t); y = sin(t);
subplot(2, 2, 1); plot(x, y); axis normal
subplot(2, 2, 2); plot(x, y); axis square
subplot(2, 2, 3); plot(x, y); axis equal
subplot(2, 2, 4); plot(x, y); axis equal tight

Draw multiple lines

Draw multiple lines on an image

By default , Every time you execute plot() Function will clear the result of the last drawing , Multiple execution plot() Only the last drawing will be retained .

We can use hold on and hold off The command controls the refresh of the drawing area , Make multiple drawing results remain in the drawing area at the same time .

hold on		%  Bring up the brush , Start drawing a set of pictures 			
plot(cos(0:pi/20:2*pi));
plot(sin(0:pi/20:2*pi));
hold off	%  Put down the paintbrush , This group of pictures has been drawn 

Draw multiple images in one window

subplot

Operation of graphic objects

stay MATLAB in , Graphics are stored in memory in the form of objects , You can operate on it by getting its graphic handle .

Get graph handle

A graph handle is essentially a floating point number , You can uniquely identify a graphic object . The following functions are used to get the graphic handle .

Function	Purpose
gca()	Get the handle of the current coordinate axis
gcf()	Get the handle of the current image
allchild(handle_list)	Get the handle of all child objects of this object
ancestor(h,type)	Get the latest type Ancestor nodes of type
delete(h)	Delete an object
findall(handle_list)	Get the descendants of this object

All drawing functions also return handles to graphic objects .

Manipulate graphic attributes through graphic handles

Use get() and set() Function can access and modify the attributes of graphic objects . visit Official documents You can view the properties of all graphic objects .

• set(H,Name,Value)
• v = get(h,propertyName)

The following two examples demonstrate the use of graphic handles to manipulate graphic objects :

Change axis properties :

%  The first picture 
set(gca, 'FontSize', 25);

%  The second picture 
set(gca, 'XTick', 0:pi/2:2*pi);
set(gca, 'XTickLabel', 0:90:360);

%  The third picture 
set(gca, 'FontName', 'symbol');
set(gca, 'XTickLabel', {'0', 'p/2', 'p', '3p/2', '2p'});

Change Linetype

h = plot(x,y); 
set(h, 'LineStyle','-.', ...
	'LineWidth', 7.0, ...
	'Color', 'g');

Save the drawing to a file

Use saveas(fig,filename) The command saves drawing objects to a file , among fig For graphics handle ,filname For the file name .

saveas(gcf, 'myfigure.png')

Draw advanced charts

Two dimensional charts

Broken line diagram

function	Graphic description
loglog()	x Axis and y The axes take logarithmic coordinates
semilogx()	x The axis takes logarithmic coordinates ,y The axis takes linear coordinates
semilogy()	x The axis takes linear coordinates ,y The axis takes logarithmic coordinates
plotyy()	With two sets y Linear coordinate system of coordinate axis
ploar()	Polar system

Logarithmic coordinate system drawing line

The following example demonstrates the logarithmic coordinate system line :

x = logspace(-1,1,100); y = x.^2;

subplot(2,2,1);
plot(x,y);
title('Plot');

subplot(2,2,2);
semilogx(x,y);
title('Semilogx');

subplot(2,2,3);
semilogy(x,y);
title('Semilogy');

subplot(2,2,4);
loglog(x, y);
title('Loglog'); 

Logarithmic coordinate system can be added grid , To distinguish between linear coordinate system and logarithmic coordinate system .

set(gca, 'XGrid','on');

double y Axis line

plotyy() The return value of is array [ax,hlines1,hlines2], among :

• ax It's a vector , Save the handle of two coordinate system objects .
• hlines1 and hlines2 They are the handles of two graph lines .

x = 0:0.01:20;
y1 = 200*exp(-0.05*x).*sin(x);
y2 = 0.8*exp(-0.5*x).*sin(10*x);
[AX,H1,H2] = plotyy(x,y1,x,y2);
set(get(AX(1),'Ylabel'),'String','Left Y-axis')
set(get(AX(2),'Ylabel'),'String','Right Y-axis')
title('Labeling plotyy');
set(H1,'LineStyle','--'); set(H2,'LineStyle',':');

Statistical charts

function	Graphic description
hist()	Histogram
bar()	Two dimensional histogram
pie()	The pie chart
stairs()	Stairs
stem()	Needle pattern

Histogram

Histogram

Use bar() and bar3() Function to draw two-dimensional and three-dimensional histograms respectively

x = [1 2 5 4 8]; y = [x;1:5];
subplot(1,3,1); bar(x); title('A bargraph of vector x');
subplot(1,3,2); bar(y); title('A bargraph of vector y');
subplot(1,3,3); bar3(y); title('A 3D bargraph');

hist It is mainly used to view the frequency distribution of variables , and bar It is mainly used to view the statistical results of discrete quantities .

Use barh() Function can draw a histogram of vertical arrangement

x = [1 2 5 4 8];
y = [x;1:5];
barh(y);
title('Horizontal');

towards bar() Pass in 'stack' Parameter allows the histogram to be drawn in the form of a stack .

x = [1 2 5 4 8];
y = [x;1:5];
bar(y,'stacked');
title('Stacked');

The pie chart

Use pie() and pie3() You can draw two-dimensional and three-dimensional pie charts . Pass a bool The vector indicates whether each sector is offset .

a = [10 5 20 30];
subplot(1,3,1); pie(a);
subplot(1,3,2); pie(a, [0,0,0,1]);
subplot(1,3,3); pie3(a, [0,0,0,1]);

Ladder diagram and needle diagram : Draw a sequence of discrete numbers

stairs() and stem() Function is used to draw ladder graph and needle graph respectively , Used to represent a sequence of discrete numbers .

x = linspace(0, 4*pi, 40); y = sin(x);
subplot(1,2,1); stairs(y);
subplot(1,2,2); stem(y);

Three dimensional chart

Convert a two-dimensional map to a three-dimensional map

stay MATLAB in , All drawings are three-dimensional , A two-dimensional picture is just a projection of a three-dimensional picture . Click... In the graphics window Rotate 3D Button , You can drag and drop the mouse to view the 3D view of the drawing .

Convert a three-dimensional drawing to a two-dimensional drawing

Use imagesc() Function can convert a three-dimensional view into a two-dimensional top view , The height is indicated by the color of the dot .

[x, y] = meshgrid(-3:.2:3,-3:.2:3); z = x.^2 + x.*y + y.^2; 

subplot(1, 2, 1)
surf(x, y, z); 

subplot(1, 2, 2)
imagesc(z);

Use colorbar The command can add the legend of the corresponding relationship between color and height on the generated two-dimensional map , Use colormap Command can change the color scheme . For details, please refer to Official documents

Three dimensional drawing

Preparation before drawing a three-dimensional map

• Use meshgrid() Generate 2D mesh

meshgrid() Function expands the two input vectors by corresponding row expansion and column expansion to obtain two augmented matrices , The binary function can be applied to this matrix .

x = -2:1:2;
y = -2:1:2;
[X,Y] = meshgrid(x,y)
Z = X.^2 + Y.^2

Draw 3D lines

Use plot3() Function to draw a three-dimensional surface , Input should be three vectors .

x=0:0.1:3*pi; z1=sin(x); z2=sin(2.*x); z3=sin(3.*x);
y1=zeros(size(x)); y3=ones(size(x)); y2=y3./2;
plot3(x,y1,z1,'r',x,y2,z2,'b',x,y3,z3,'g'); grid on;
xlabel('x-axis'); ylabel('y-axis'); zlabel('z-axis');

Draw a 3D face

Use mesh() and surf() The command can draw 3D faces , The former does not fill the mesh while the latter .

x = -3.5:0.2:3.5; y = -3.5:0.2:3.5;
[X,Y] = meshgrid(x,y);
Z = X.*exp(-X.^2-Y.^2);
subplot(1,2,1); mesh(X,Y,Z);
subplot(1,2,2); surf(X,Y,Z);

Draw contour lines of 3D graphics

Use contour() and contourf() Function can draw contour lines of 3D graphics , The former does not fill the mesh while the latter .

towards contour() The function can change the details of the image by passing in parameters or manipulating the graphic handle :

x = -3.5:0.2:3.5; y = -3.5:0.2:3.5;
[X,Y] = meshgrid(x,y); Z = X.*exp(-X.^2-Y.^2);

subplot(1,3,1); contour(Z,[-.45:.05:.45]); axis square;
subplot(1,3,2); [C,h] = contour(Z); clabel(C,h); axis square;
subplot(1,3,3); contourf(Z); axis square;

Use meshc() and surfc() Function can draw its contour lines when drawing a three-dimensional figure .

x = -3.5:0.2:3.5; y = -3.5:0.2:3.5;
[X,Y] = meshgrid(x,y); Z = X.*exp(-X.^2-Y.^2);

subplot(1,2,1); meshc(X,Y,Z);
subplot(1,2,2); surfc(X,Y,Z);

Draw a three-dimensional volume

Use patch() Function can draw a three-dimensional body .

v = [0 0 0; 1 0 0 ; 1 1 0; 0 1 0; 0.25 0.25 1; 0.75 0.25 1; 0.75 0.75 1; 0.25 0.75 1];
f = [1 2 3 4; 5 6 7 8; 1 2 6 5; 2 3 7 6; 3 4 8 7; 4 1 5 8];

subplot(1,2,1); 
patch('Vertices', v, 'Faces', f, 'FaceVertexCData', hsv(6), 'FaceColor', 'flat');
view(3); axis square tight; grid on;

subplot(1,2,2); 
patch('Vertices', v, 'Faces', f, 'FaceVertexCData', hsv(8), 'FaceColor','interp');
view(3); axis square tight; grid on

Perspective and lighting of three-dimensional drawings

Adjust the perspective

Use view() Function can adjust the perspective ,view() Function accepts two floating-point arguments , Represent two azimuth angles respectively azimuth and elevation.

sphere(50); shading flat;
material shiny;
axis vis3d off;
view(-45,20);

Symbolic operations

Create symbolic variables

Create symbolic numbers

Use sym Function can create symbolic numbers . Using symbolic numbers can accurately save irrational numbers , There will be no error .

sym(1/3)	%  obtain  1/3
1/3			%  obtain  0.3333

Saving irrational numbers as symbolic numbers can avoid the error of converting them into floating-point numbers :

sin(sym(pi))	%  obtain  0
sin(pi)			%  obtain  1.2246e-16

Create symbolic variables

Use sym and syms You can create symbolic variables , The difference lies in :

sym Only one symbolic variable can be created at a time , and syms You can create multiple symbolic variables at a time .

syms a		% sym Command can only create one symbolic variable 
syms b c d	% syms Command can create multiple symbolic variables 

If the specified symbolic variable already exists ,sym It will not retain its original value , and syms Will clear its value .

syms x y
f = x+y;	%  Implicitly create symbolic variables f
sym f		%  Do not empty variables f Original value , namely f = x + y

syms x y
f = x+y;	%  Implicitly create symbolic variables f
syms f		%  Empty variables f Original value , namely f = f

Use sym You can create a symbolic variable matrix .

A = sym('a', [2 5])	%  Create a 2*5 Symbolic variable matrix of 
whos

The output is as follows :

A =
[ a1_1, a1_2, a1_3, a1_4, a1_5]
[ a2_1, a2_2, a2_3, a2_4, a2_5]
 
 
  Name      Size            Bytes  Class    Attributes
  A         2x5               112  sym      

A combination of sym and syms You can quickly create a series of variables with subscripts

clear all
syms(sym('a', [1 5]))
whos

The output is as follows :

Name      Size            Bytes  Class    Attributes
a1        1x1                 8  sym                
a2        1x1                 8  sym                
a3        1x1                 8  sym                
a4        1x1                 8  sym                
a5        1x1                 8  sym                

Symbolic operations

Simplification and substitution of symbolic expressions

Simplification of symbolic expressions

Use simplify() Functions can simplify symbolic expressions .

syms x a b c

simplify(sin(x)^2 + cos(x)^2);		%  obtain  1
simplify(exp(c*log(sqrt(a+b))));	%  obtain  (a + b)^(c/2)

The standard of expression simplification is uncertain , The following three functions simplify the expression according to different standards :

expand() Functions can expand expressions

syms x

f = (x ^2- 1)*(x^4 + x^3 + x^2 + x + 1)*(x^4 - x^3 + x^2 - x + 1);
expand(f);		%  obtain  x^10 - 1

factor() Functions can be factorized

syms x

g = x^3 + 6*x^2 + 11*x + 6;
factor(g);		%  obtain  (x + 3)*(x + 2)*(x + 1)

horner() Functions can change polynomials into nested forms

syms x

h = x^5 + x^4 + x^3 + x^2 + x;
horner(h);		%  obtain  x*(x*(x*(x*(x + 1) + 1) + 1) + 1)

Substitution of symbolic expressions

Use sub(expr, old, new) The function can express a symbolic expression expr Medium old Replace with new.

syms x y

f = x^2*y + 5*x*sqrt(y);
subs(f, x, 3);	%  obtain  9*y + 15*y^(1/2)

Find the analytical solution of the equation

Use solve(eqn,var) and solve(eqns,vars) The analytical solution of the equation can be obtained .

Solve the univariate equation

Use == Define an equation , And call solve Function solving .（ If not specified == The value to the right of the symbol , Then the default right side of the equation is 0.）

syms x

eqn = x^3 - 6*x^2 == 6 - 11*x;
solve(eqn);		%  obtain  [1 2 3]

Solve multivariable equations

For multivariable equations , We need to specify which variable to solve .

syms x y

eqn = [6*x^2 - 6*x^2*y + x*y^2 - x*y + y^3 - y^2 == 0];
solve(eqn, y);	%  obtain  [1, 2*x, -3*x]

Solve equations

towards solve() The function can be introduced into the equation system to solve the equation .

syms u v

eqns = [2*u + v == 0, u - v == 1];
S = solve(eqns,[u v]);	

The solution of the equation can be indexed by the variable name , And the solution can be substituted into other expressions .

S.u;				%  obtain  1/3
S.v;				%  obtain  -2/3
subs(3*v + u, S);	%  obtain  -3/5

Symbolic calculus

Find the limit

Use limit(expr, var, a) Function can evaluate the symbolic expression expr In variables var Tend to be a The limit of time , Add parameter 'left' or 'right' You can specify the left limit or the right limit .

syms x;
expr = 1/x;

limit(expr,x,0);			%  obtain NaN
limit(expr,x,0,'left');		%  obtain -Inf
limit(expr,x,0,'right');	%  obtain Inf

differential

Use diff(expr, var, n) Function can evaluate the symbolic expression expr The variable var Of n First order differential .

syms a b c x;
expr = a*x^2 + b*x + c;

diff(expr, a);		%  obtain  x^2
diff(expr, b);		%  obtain  x
diff(expr, x);		%  obtain  b + 2*a*x
diff(expr, x, 2);	%  obtain  2*a

integral

Use int(expr, var) Function can evaluate the symbolic expression expr The variable var The indefinite integral of . Use int(expr, var, [a, b]) Function can specify upper and lower bounds to calculate definite integral ,a and b It can be a symbolic expression .

syms x a b
expr = -2*x/(1+x^2)^2;

int(expr, x);			%  obtain  1/(x^2 + 1)
int(expr, x, [1, 2]);	%  obtain  -0.3
int(expr, x, [1, Inf]);	%  obtain  -0.5
int(expr, x, [a, b]);	%  obtain  1/(b^2 + 1) - 1/(a^2 + 1)

For some functions ,MATLAB Its integral cannot be obtained , At this time MATLAB Will return an unresolved (unsolved) Integral form of .

syms x
int(sin(sinh(x)));	%  An integral without solution ,MATLAB return  int(sin(sinh(x)),

Sum of series

Use symsum(expr, k, [a b]) Calculation series expr The index of k from a To b The sum of .

syms k x

symsum(k^2, k)			%  obtain  k^3/3 - k^2/2 + k/6
symsum(k^2, k, [0 10])	%  obtain  385
symsum(x^k/factorial(k),k,1,Inf)	%  obtain  exp(x) - 1

Taylor expansion

Use taylor(expr,var,a) Calculation expression expr stay var=a The Taylor series at .

syms x

taylor(exp(x))	%  obtain  x^5/120 + x^4/24 + x^3/6 + x^2/2 + x + 1
taylor(sin(x))	%  obtain  x^5/120 - x^3/6 + x
taylor(cos(x))	%  obtain  x^4/24 - x^2/2 + 1

The plot

You can draw images of symbolic expressions , Common drawing functions are as follows :

function	effect
fplot()	Draw a two-dimensional line image of a symbolic expression
fplot3()	Draw a 3D line image of a symbolic expression
ezpolar()	Draw the polar marking image of the symbolic expression
fmesh()	Draw a mesh image
fsurf()	Draw a colored face image
fcontour()	Draw a contour image
fimplicit()	Draw an image of implicit functional relationships

The following example shows the drawing of two-dimensional and three-dimensional line images

subplot(1, 2, 1)
syms x
f = x^3 - 6*x^2 + 11*x - 6;
fplot(f, x)

subplot(1, 2, 2)
syms t
fplot3(t^2*sin(10*t), t^2*cos(10*t), t)

The following example demonstrates the rendering of three-dimensional surfaces

syms x y
fsurf(x^2 + y^2)

The following example demonstrates the drawing of implicit function relation image

syms x y
eqn = (x^2 + y^2)^4 == (x^2 - y^2)^2;
fimplicit(eqn, [-1 1])

Numerical calculation

Numerical operation of polynomials

Use MATLAB Represents a polynomial

Use vectors to represent polynomials

stay MATLAB in , Polynomials can be represented by vectors , The elements in the vector are the coefficients of polynomials ( Descending order ): The first is the coefficient of the highest order of the polynomial , The last digit is the constant term .

For example, polynomials : f(x) = x^3 - 2x - 5f(x)=x3−2x−5

You can use vectors p = [1 0 -2 -5] Express .

Polynomial evaluation :polyval()

Use polyval(p, x) You can calculate polynomials p stay x The value at each point .

a = [9,-5,3,7]; x = -2:0.01:5;
f = polyval(a,x);
plot(x,f);

Multiplication of polynomials :conv()

Use conv(p1, p2) Function can be applied to two vectors p1 and p2 Convolution multiplication , Used to calculate the multiplication of polynomials .

For example, polynomials : f(x) = (x^2+1) (2x+7)f(x)=(x2+1)(2x+7)

have access to conv() Function to get the expanded polynomial :

p = conv([1 0 1], [2 7])

obtain p = [2 7 2 7].

Numerical operation of polynomials

Factorization of polynomials :roots()

Use roots(p) Functions can be polynomial p Do factorization , That is, the expression value is 0 The root of the .

p = roots([1 -3.5 2.75 2.125 -3.875 1.25])

obtain p = [2 -1, 1+0.5i, 1-0.5i, 0.5], Express x^5 -3.5 x^4 +2.75 x^3 + 2.125 x^2 + -3.875 x+1.25 = (x-2)(x+1)(x-1-0.5i)(x-1+0.5i)(x-0.5)x5−3.5x4+2.75x3+2.125x2+−3.875x+1.25=(x−2)(x+1)(x−1−0.5i)(x−1+0.5i)(x−0.5).

Differentiation of polynomials :polyder()

Use polyder(p) Functions can calculate the derivatives of polynomials .

For example, take the derivative of the following polynomial : f(x) = 5x^4 - 2x^2 + 1f(x)=5x4−2x2+1

p = polyder([5 0 -2 0 1]);

obtain p = [20 0 -4 0], Means the derivative obtained by calculation f’(x) = 20 x^3 - 4xf′(x)=20x3−4x.

The integral of a polynomial :polyint()

Use polyint(p, k) Functions can calculate polynomials p Integral , The constant term of the integration result is set as k.

For example, take the derivative of the following polynomial : f(x) = 5x^4 - 2x^2 + 1f(x)=5x4−2x2+1

p = polyint([5 0 -2 0 1], 3)

obtain p = [1 0 -0.6667 0 1 3], It means the integral obtained by calculation \int f(x) dx = x^5 -0.6667x^3 + x + 3∫f(x)dx=x5−0.6667x3+x+3.

Numerical operation of nonlinear expressions

equation ( Group ) Root seeking fsolve()

Use fsolve(fun, x0) Find the root of nonlinear equations ,fun Is the function handle of the equation to be solved ,x0 For the initial value .

Find the equation 1.2x+x\sin(x)+0.3=01.2x+xsin(x)+0.3=0 stay x=0x=0 Near solution .

f2 = @(x) (1.2*x+x*sin(x)+0.3);
fsolve(f2,0)	%  obtain  -0.3500

Solve equations \begin{aligned} \left\{ \begin{aligned} e^{-e^{-(x_1+x_2)}} - x_2(1+x_1^2) = 0 \\ x_1 \cos x_2 + x_2 \sin x_1 - \frac{1}{2} = 0 \end{aligned} \right. \end{aligned}⎩⎪⎨⎪⎧​e−e−(x1​+x2​)−x2​(1+x12​)=0x1​cosx2​+x2​sinx1​−21​=0​​

Set the initial value as [0, 0][0,0]

fun = @(x) [exp(-exp(-(x(1)+x(2))))-x(2)*(1+x(1)^2)...
    x(1)*cos(x(2)) + x(2)*sin(x(1)) - 0.5]
x0 = [0,0];
x = fsolve(fun,x0)	%  obtain [0.3532 0.6061]

Numerical differentiation

Find the difference :diff()

Use diff(X, n) Calculate the vector X Of n Order difference ,n The default is 1.

x = [1 2 5 2 1];
diff(x);	%  obtain  [1 3 -3 -1]
diff(x,1);	%  obtain  [1 3 -3 -1]
diff(x,2);	%  obtain  [2 -6 2]

Find the derivative :diff(y)./diff(x)

Use the definition of derivative

You can calculate the approximate derivative of a function at a certain point .

The following program calculates f(x) = x^3f(x)=x3 The value of the first and second derivatives of .

x = -2:0.005:2; y = x.^3;
m = diff(y)./diff(x);			%  Calculate the first derivative 
m2 = diff(m)./diff(x(1:end-1));	%  Calculate the second derivative 

plot(x,y,x(1:end-1),m,x(1:end-2),m2);
xlabel('x'); ylabel('y');
legend('f(x) =x^3','f''(x)','f''''(x)', 4);

Numerical integration

The principle of numerical integration

There are three common algorithms for calculating numerical integrals : Rectangular method , Trapezoidal method , Parabola method , They treat the graphs of differential intervals as rectangles , trapezoid , Parabola to calculate the area .

The following three methods are used to calculate f(x) = 4x^3f(x)=4x3 In the interval (0, 2)(0,2) Integral in .

h = 0.05; x = 0:h:2;


%  Use the rectangular method to calculate the approximate integral 
midpoint = (x(1:end-1)+x(2:end))./2;
y = 4*midpoint.^3;
s = sum(h*y)			%  obtain  15.9950

%  Use the trapezoidal method to calculate the approximate integral 
trapezoid = (y(1:end-1)+y(2:end))/2;
s = h*sum(trapezoid)	%  obtain  15.2246

%  Use parabola method to calculate numerical integration 
s = h/3*(y(1)+2*sum(y(3:2:end-2))+4*sum(y(2:2:end))+y(end))	%  obtain  15.8240

Numerical integral function :integral()

integral(),integral2(),integral3() For functions in xmin to xmax Carry out a heavy , double , Triple integral .

Their first argument should be a function handle , The following example demonstrates their usage :

Calculation \int_0^2 \frac{1}{x^3-2x-5}∫02x3−2x−51

f = @(x) 1./(x.^3-2*x-5);
integral(f,0,2)		%  obtain  -0.4605

Calculation \int_0^\pi \int_\pi^{2\pi} (y\sin(x) + x \cos(y)) dx dy∫0π∫π2π(ysin(x)+xcos(y))dxdy

f = @(x,y) y.*sin(x)+x.*cos(y);
integral2(f,pi,2*pi,0,pi)	%  obtain  -9.8696

Calculation \int_{-1}^1 \int_0^1 \int_0^\pi (y\sin(x) + z \cos(y)) dx dy dz∫−11∫01∫0π(ysin(x)+zcos(y))dxdydz

f = @(x,y,z) y.*sin(x)+z.*cos(y);
integral3(f,0,pi,0,1,-1,1)

Statistics and fitting

Statistics

Descriptive statistics (Descriptive Statistics)

Descriptive statistics mainly studies data The central trend (Central Tendency) and Variation

The central trend (Central Tendency)

function	effect
mean()	Calculate average
median()	Calculate the median
mode()	Calculate the mode
prctile(X,num)	data X Of num% What is the quantile
max()	Calculate the maximum
min()	Calculate the minimum

The following functions plot statistical charts :

function	effect
bar()	Draw a bar graph
stem()	Draw a needle chart
area()	Draw a fill chart
boxplot()	Draw box lines

x = 1:14;
freqy = [1 0 1 0 4 0 1 0 3 1 0 0 1 1];
subplot(1,3,1); bar(x,freqy); xlim([0 15]);
subplot(1,3,2); area(x,freqy); xlim([0 15]);
subplot(1,3,3); stem(x,freqy); xlim([0 15]);

Boxplot can highlight the quartile of data .

marks = [80 81 81 84 88 92 92 94 96 97];
boxplot(marks) 
% prctile(marks, [25 50 75])	%  obtain  [81 90 94]

Variation

The degree of dispersion

function	effect
std()	Calculate the standard deviation of the data
var()	Calculate the variance of the data

skewness (Skewness

function	effect
skewness()	Calculate the skewness of the data

Skewness reflects the degree of symmetry of data

• When the data deviates to the left , Its skewness is less than 0.
• When the data is completely symmetrical , Its skewness is equal to 0.
• When the data is biased to the right , Its skewness is greater than 0.

X = randn([10 3]);		%  structure 10*3 Matrix 
X(X(:,1)<0, 1) = 0; 	%  Shift the first column of data to the right 
X(X(:,3)>0, 3) = 0;		%  Shift the second column of data to the left 
boxplot(X, {'Right-skewed', 'Symmetric', 'Left-skewed'});
skewness(X);	%  obtain  [0.5162 -0.7539 -1.1234]

kurtosis (Kurtosis)

function	effect
kurtosis()	Calculate the kurtosis of the data

kurtosis (Kurtosis) It represents the peak value of probability density distribution curve at the mean value . Intuitive to see , The kurtosis reflects the sharpness of the peak .

Statistical inference (Inferential Statistics)

The core of inferential statistics is hypothesis testing . The following functions are used for hypothesis testing .

function	effect
ttest()	Conduct T test
ztest()	Conduct Z test
ranksum()	Perform rank sum test
signrank()	Sign rank test

load examgrades
x = grades(:,1);
y = grades(:,2);
[h,p] = ttest(x,y);

Carry out the above procedure , obtain [h p] = [0 0.9805], Indicates at the default significance level (5%) Next, we have no reason to refuse x And y Homodistribution .

fitting

Polynomial fitting

Univariate polynomial fitting :polyfit()

Use polyfit(x, y, n) Function on data x and y Conduct n Quadratic polynomial fitting .

x = [-1.2 -0.5 0.3 0.9 1.8 2.6 3.0 3.5];
y = [-15.6 -8.5 2.2 4.5 6.6 8.2 8.9 10.0];

for i = 1:3 %  Once each , secondary , Cubic fitting 
    p = polyfit(x, y, i);
    xfit = x(1):0.1:x(end); yfit = polyval(p, xfit);
    subplot(1, 3, i); plot(x, y, 'ro', xfit, yfit);
    legend('Data points', 'Fitted curve', 'Location', 'southeast');
end

Multivariate linear fitting :regress()

Use regress(y, X) Function on data X and y Do multiple linear regression .

load carsmall;
y = MPG; x1 = Weight; x2 = Horsepower;		%  Import dataset 
X = [ones(length(x1),1) x1 x2];				%  Build augmentation X matrix 
b = regress(y,X);							%  Do a linear regression 

%  The following is the drawing statement 
x1fit = min(x1):100:max(x1); x2fit = min(x2):10:max(x2);
[X1FIT,X2FIT] = meshgrid(x1fit,x2fit);
YFIT = b(1)+b(2)*X1FIT+b(3)*X2FIT;
scatter3(x1,x2,y,'filled'); hold on;
mesh(X1FIT,X2FIT,YFIT); hold off;
xlabel('Weight'); ylabel('Horsepower'); zlabel('MPG'); view(50,10);

Nonlinear fitting

For nonlinear fitting , You need to use the curve fitting toolbox . Enter... In the command window cftool() Open the curve fitting toolbox .

interpolation

One dimensional interpolation

function	effect
interp1(x,v) or interp1(x,v,xq)	linear interpolation
spline(x,v) or spline(x,v,xq)	Cubic spline interpolation
pchip(x,v) or pchip(x,v,xq)	Three times Hermite interpolation
mkpp(breaks,coefs)	Generate piecewise polynomials
ppval(pp,xq)	Calculate the interpolation result of piecewise polynomial

The following example demonstrates the use of interp1(x, v, xq) Perform linear interpolation and use spline(x, v, xq) Perform cubic spline interpolation . The meanings of the parameters are as follows :

• x,v: Sample points to be interpolated .
• xq: Query point , Function returns the interpolation result at these points .

%  Structural data 
x = linspace(0, 2*pi, 40); x_m = x; x_m([11:13, 28:30]) = NaN; 
y_m = sin(x_m);
plot(x_m, y_m, 'ro', 'MarkerFaceColor', 'r'); hold on; 	

%  Linear interpolation of data 
m_i = ~isnan(x_m);
y_i = interp1(x_m(m_i), y_m(m_i), x);
plot(x,y_i, '-b'); hold on;

%  Perform cubic spline interpolation on the data 
m_i = ~isnan(x_m);
y_i = spline(x_m(m_i), y_m(m_i), x);
plot(x,y_i, '-g'); 

legend('Original', 'Linear', 'Spline');

Two dimensional interpolation

Use interp2() Can carry out two-dimensional interpolation , To whom method Parameter input string can specify interpolation algorithm .

Method	explain	Continuity
'linear'	( Default ) The value inserted at the query point is based on the linear interpolation of the value at the grid point of the adjacent point in each dimension .	C0
'spline'	The value inserted at the query point is based on the cubic interpolation of the value at the grid point of the adjacent point in each dimension . Interpolation is based on cubic splines using non knot termination conditions .	C2
'nearest'	The value inserted at the query point is the value closest to the sample grid point .	Discontinuous
'cubic'	The value inserted at the query point is based on the cubic interpolation of the value at the grid point of the adjacent point in each dimension . Interpolation is based on cubic convolution .	C1
'makima'	The modified Akima Three times Hermite interpolation . The value inserted at the query point is based on the maximum number of times 3 Piecewise functions of polynomials , Use the values of adjacent grid points in each dimension for calculation . To prevent overshoot , Improved Akima The formula .	C1

%  Build sample points 
xx = -2:.5:2; yy = -2:.5:3; [x,y] = meshgrid(xx,yy); 
xx_i = -2:.1:2; yy_i = -2:.1:3; [x_i,y_i] = meshgrid(xx_i,yy_i);
z = x.*exp(-x.^2-y.^2);

%  linear interpolation 
subplot(1, 2, 1); 
z_i = interp2(xx,yy,z,x_i,y_i);
surf(x_i,y_i,z_i); hold on;
plot3(x,y,z+0.01,'ok','MarkerFaceColor','r'); hold on;

%  Cubic interpolation 
subplot(1, 2, 2); 
z_ic = interp2(xx,yy,z,x_i,y_i, 'spline');
surf(x_i,y_i,z_ic); hold on;
plot3(x,y,z+0.01,'ok','MarkerFaceColor','r'); hold on;