Cellular automata introduction
Cellular automata was originally proposed by Von Neumann in the 1950s to simulate the self-replication of biological cells. Cellular automata came to the attention of scientists in 1970 when John Holden Conway of Cambridge University designed a computer game called “The Game of Life”.
S. Wolram published a series of papers in 1983. In this paper, the models generated by 256 rules of elementary cellular machine are studied and their evolution is described by entropy. Cellular automata are classified into stationary, periodic, chaotic and complex types.
Cellular automata (CA) is a method used to simulate local rules and local connections. Cellular automata are typically defined on a grid, where each point represents a cell and a finite state. The change rule applies to each cell and simultaneously. The rules of change typically depend on the state of the cell and the state of its (4 or 8) neighbors.
Change rules for 3 cells & change rules for cell states typically depend on the state of a cell and the states of its (4 or 8) neighbors.
Application of cellular automata Cellular automata has been applied to physical simulation, biological simulation and other fields.
5 Cellular automata MATLAB programming combined with the above, we can understand cellular automata simulation needs to understand three points. One is cellular, which can be understood as a square block composed of one or more points in the matrix in MATLAB. Generally, we use a point in the matrix to represent a cellular. The second is the change rule. The change rule of cellular determines the state of cellular at the next moment. The third is the state of the cell. The state of the cell is self-defined and usually opposite, such as the living state or death state of the creature, the red light or the green light, the point with or without obstacles, and so on.
One dimensional cellular automata — traffic rulesDefinition: 6.1 Cells are distributed on a one-dimensional linear grid. 6.2 Cells have only two states: car and empty.
What is cellular automata discrete system: cellular is defined in finite time and space, and the state of cellular is finite. Dynamical system: The behavior of cellular automata is characterized by dynamics. Simplicity and complexity: Cellular automata simulate complex worlds by controlling interacting cells with simple rules.
(5) Rules (State transition function)Definition: The dynamic function of the cellular state at the next moment is determined according to the current state of the cell and its neighbors. Simply speaking, it is a state transfer function. Classification: Summation type: the state of a cell depends on and only depends on the current state of all its neighbors and its own current state. Legal: The sum rule belongs to the legal rule. However, if the rules of cellular automata are limited to summation type, cellular automata will be limited.6. Forest fires 
Macro continuous model:
1 Acceleration rule: do not exceed v max (2 grids /s) v_{Max} (2 grids /s) v Max (2 grids /s) 2 Collision prevention: do not exceed the vehicle distance
Theoretical analysis:
Result analysis: space-time trajectory
Two, some source code
%% NOTE
% Total runtime for this code was about 1- 3 minutes on the personal
% computer used by the original researcher
% This runtime is expected to vary from device to device, and is being used
% here to obtain a general comparison for runtime for the different
% models. It is not for a universal runtime value
%% Documentation
% n size of each dimension of our square cellular automata grid
% P_HIV fraction (probability) of cells initially infected by virus
% P_i probability of a healthy cell becoming infected if its
% neighborhood contains 1 I1 cell or X I2 cells
% P_v probability of a healthy cell becoming infected by coming
% in contact with a virus randomly (not from its
% neighborhood)
% P_rep Probability of a dead cell becoming replaced by a healthy
% cell
% P_repI Probability of a dead cell becoming replaced by an infected
% cell
% X Number of I2 cells in the neighborhood of an H cell that
% can cause it to become infected
% tau1 tau1 is the number of timesteps it takes for an acute
% infected cell to become latent.
% tau2 tau2 is the number of timesteps it takes for a latent
% infected cell to become dead.
% totalsteps totalsteps is the total number of steps of the CA (the % total number of weeks of simulations)
% grid our cellular automata (CA) grid
% tempgrid tempgrid is a temporary grid full of random numbers that is
% used to randomly add different states to our CA grid.
% taugrid taugrid is a grid the same size as our CA grid that stores
% the number of timesteps that a cell has been in state I_1.
% If the number reaches tau1, then the state changes to I_2.
% state state is a [5 x totalsteps] size matrix that stores
% the total number of cells in each state at each timestep
% and the last row stores sum of I_1 and I_2 at each timestep
% timestep each simulation step of the cellular automata
% 1 timestep = 1 week of time in the real world
% nextgrid nextgrid is a temporary grid. It is a copy of the CA grid
% from the previous simulation. It stores all the CA rule
% updates of the current timestep and stores it all back to
% the grid to display.
%% Clean-up
clc; % clears command window
clear all; % clears workspace and deletes all variables
close all; % closes all open figures
%% Parameters
n = 100; % meaning that our grid will have the dimensions n x n
P_HIV = 0.05; % initial grid will have P_hiv acute infected cells
P_i = 0.997; % probability of infection by neighbors
P_v = 0.00001; % probability of infection by random viral contact
P_rep = 0.99; % probability of dead cell being replaced by healthy
P_repI = 0.00001; % probability of dead cell being replaced by infected
X = 4; % there must be at least X I_2 neighbors to infect cell
tau1 = 4; % time delay for I_1 cell to become I_2 cell
tau2 = 1; % time delay for I_2 cell to become D cell
totalsteps = 600 ; % total number of weeks of simulation to be performed
savesteps = [3 7 11 15 20 25 50 100 150 200 250 300 350 400 450 500];
%timesteps for which we want to save simulation image
%% States
% State 1: H: Healthy (Color- Green)
% State 4: I_1: Active Infected (Color- Cyan)
% State 3: I_2: Latent Infected (Color- Blue)
% State 2: D: Dead (Color- Black)
%% Initial Grid
grid = ones(n); % creates our initial n x n matrix and fills all cells
% with value 1 (meaning H state - Healthy cell)
tempgrid = rand(n); % creates a grid of random values of the same size as
% our CA grid. Used to randomly add I_1 state to our grid
grid(tempgrid(:,:)<=P_HIV) = 4; % I_1 state added with probability P_HIV
% The following sets the edge values of the grid to H state
grid(:,1 n]) = 1; % to set every value in the first and last column to 1
grid([1 n],:) = 1; % to set every value in the first and last row to 1
% NOTE: Our CA only simulates from rows 2 to n- 1.and columns 2 to n1.
% This is to prevent the edge row and column cells from having an
% out-of-bounds error when checking the neighbors around them.
% The edge values are all set to H state so that it does not affect
% Rule 1 of the CA for cells next to them
% set(figure, 'OuterPosition'[100 30 740 740]) % sets figure window size
% % outerposition values are [left, bottom, width, height]
% print1(grid); % prints the initial grid
% saveas(gcf,strcat('Model1Timestep0.pdf'));
%% Simulation
nextgrid = grid;
for x=2:n- 1
for y=2:n- 1
% Rule 1
% If the cell is in H state and at least one of its neighbors
% is in I_1 state, or X of its neighbors is in I_2 state, then
% the cell becomes I_1 with a probability of P_i
% The cell may also becomes I_1 by randomly coming in
% contact with a virus from outside its neighborhood with a
% probability of P_v
if(grid(x,y)==1)
if((rand<=P_i && (grid(x- 1,y- 1) = =4 || grid(x- 1,y)==4 || ...
grid(x- 1,y+1) = =4 || grid(x,y- 1) = =4 || ...
grid(x,y+1) = =4 || grid(x+1,y- 1) = =4 || ...
grid(x+1,y)==4 || grid(x+1,y+1) = =4 || ...
((grid(x- 1,y- 1) = =3) + (grid(x- 1,y)==3) +... (grid(x- 1,y+1) = =3) + (grid(x,y- 1) = =3) +... (grid(x,y+1) = =3) + (grid(x+1,y- 1) = =3) +... (grid(x+1,y)==3) + (grid(x+1,y+1) = =3))>=X ))...
|| rand<=P_v)
end
end
% Rule 2
% If the cell is in state I_1, and has been in this state for
% tau1 timesteps, the cell becomes I_2 state (latent infected)
if((grid(x,y)==4))
taugrid(x,y) = taugrid(x,y)+1;
if(taugrid(x,y)==tau1)
nextgrid(x,y)=3;
taugrid(x,y)=0;
end
continue;
end
% Rule 3
% If the cell is in I_2 state, and has been in this state for
% tau2 timesteps, the cell becomes D state (dead)
% Since tau2=1.this rule is implemented every timestep
if((grid(x,y)==3))
nextgrid(x,y)=2;
continue;
end
% Rule 4 a and b
% If the cell is in D state, then the cell will become H state
% with probability P_rep (4.a) or I_1 state with probability
% P_repI (4.b)
if(grid(x,y)==2 && rand<=P_rep)
if(rand<=P_repI)
nextgrid(x,y)=4;
continue;
else
nextgrid(x,y)=1;
end
end
end
end
% to assign the updates of this timestep in nextgrid back to our grid
grid=nextgrid;
% to display the grid at the end of each timestep
print1(grid);
% to save the image of the simulation every 25 timesteps as a pdf
if(find(savesteps==timestep))
saveas(gcf,strcat('Model1Timestep',num2str(timestep),'.pdf'));
end
state(1,timestep) = sum(sum(grid(2:n- 1.2:n- 1) = =1));
state(2,timestep) = sum(sum(grid(2:n- 1.2:n- 1) = =2));
state(3,timestep) = sum(sum(grid(2:n- 1.2:n- 1) = =3));
state(4,timestep) = sum(sum(grid(2:n- 1.2:n- 1) = =4));
timestep=timestep+1; % to move to the next timestep
end
% The following lines of code are to display a graph of each state of
% cells during simulation
set(figure, 'OuterPosition'[200 100 700 500]) % sets figure window size
plot( 1:totalsteps , state(1, :).'g'.1:totalsteps , state(4, :).'c'.1:totalsteps , state(3, :).'b'.1:totalsteps , state(2, :).'k' , 'linewidth'.2 );
legend( 'Healthy'.'Acute Infected'.'Latent Infected'.'Dead'.'Location' ,'NorthEast' );
saveas(gcf,strcat('Model1GraphSingleRun.pdf'));
error(nargchk(1,Inf,nargin)) ;
% check the arguments
if ~isnumeric(x),
error('Numeric argument expected'); endif nargin==1,
y = [] ;
va = [] ;
else
va = varargin ;
if ischar(va{1}),
% optional arguments are
y = [];elseif isnumeric(va{1})
y = va{1}; va = va(2:end) ;
else
error('Invalid second argument');end
if mod(size(va),2) = =1,
error('Property-Value have to be pairs'); end end % get the axes to plot in hca=get(get(0.'currentfigure'),'currentaxes');
if isempty(hca).warning('No current axes found') ;
return ;
end
% get the current limits of the axis
% used for limit restoration later on
xlim = get(hca,'xlim'); ylim = get(hca,'ylim'); % setup datafor the vertical lines
xx1 = repmat(x(:). '.3.1); yy1 = repmat([ylim(:) ; nan],1,numel(x)) ;
Copy the code3. Operation results
Matlab version and references
1 matlab version 2014A
2 Reference [1] CAI Limei. MATLAB Image Processing — Theory, Algorithm and Case Analysis [M]. Tsinghua University Press, 2020. [2] Yang Dan, ZHAO Haibin, LONG Zhe. Examples of MATLAB Image Processing In detail [M]. Tsinghua University Press, 2013. [3] Zhou Pin. MATLAB Image Processing and Graphical User Interface Design [M]. Tsinghua University Press, 2013. [4] LIU Chenglong. Proficient in MATLAB Image processing [M]. Tsinghua University Press, 2015. [5] MENG Yifan, LIU Yijun. [6] Zhang Na, Liu Kun, Han Meilin, Chen Chen. A PCA-SVM based Face Recognition Method [J]. Science and Technology Vision, 2021,(07) A face recognition algorithm based on PCA and LDA fusion [J]. Electronic measurement technology, 2020,43(13) [7] Chen yan. [8] dai lirong, Chen wanmi, guo sheng. Analysis of face recognition method based on BP neural network [J]. Information and computer (theory edition), 2020,32(23). Research on face recognition based on skin color model and SURF algorithm [J]. Industrial control computer, 2014,27(02)