puff! the magic dragon, live by the sea…. 第 5 章 繪圖及模型建構 plotting & model...
TRANSCRIPT
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Puff! The magic dragon,Live by the sea….
第 5章 繪圖及模型建構plotting & model building
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Puff! The magic dragon,Live by the sea….
• 圖比表好• 表比文字好
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Puff! The magic dragon,Live by the sea….
xy plotting functions
• y=f(x)• abscissa ~ x (independent variable)• ordinate ~ y (dependent variable)• Scale 刻度• tick mark 刻度標示• axis label 軸稱 or axial title• data symbol or point marker• Legend ~ 說明
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Puff! The magic dragon,Live by the sea….
• line ~ for theoretical calculation data
• data symbol ~ for experimental data
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Puff! The magic dragon,Live by the sea….
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Puff! The magic dragon,Live by the sea….
Requirements of correct plot
• P.243-244
• Read it NOW!
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Puff! The magic dragon,Live by the sea….
Plot, Label & Title commands
x=[0:.1:52];
y=.4*sqrt(1.8*x);
plot(x,y),xlabel(' 距離 ( 英里 )'),ylabel(' 距離 ( 英里 )'),...
title(' 火箭高度相對於飛行方向的距離函數 ')
grid on, axis([0 52 0 5])
Copy figure
Paste it in the word file
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Puff! The magic dragon,Live by the sea….
Plot of complex numbers
• plot(y)=plot(real(y),imag(y)) %if y is complex
z=.1+.9i;
n=[0:0.01:10];
plot(z.^n),xlabel('real part'),ylabel('imaginary part')
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Puff! The magic dragon,Live by the sea….
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.8
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實 部
虛部
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Puff! The magic dragon,Live by the sea….
The function plot command fplot
• fplot(‘string’,[xmin xmax])
• fplot(‘string’,[xmin xmax ymin ymax])
f='cos(tan(x))-tan(sin(x))';
fplot(f,[1 2])
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Puff! The magic dragon,Live by the sea….
x=[1:.01:2];
y=cos(tan(x))-tan(sin(x));
plot(x,y)
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Puff! The magic dragon,Live by the sea….
Plotting polynomials• x=[-6:0.01:6];• p=[3,2,-100,2,-7,90];• plot(x,polyval(p,x)),xlabel('x'),ylabel('p')
-6 -4 -2 0 2 4 6-3000
-2000
-1000
0
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x
p
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Puff! The magic dragon,Live by the sea….
Self testing
• pp.250-251
• Write down the program now!
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Puff! The magic dragon,Live by the sea….
Subplot 子圖subplot(m,n,p) % 將視窗分成 m 列 n 行的陣列平面 第 p 平面
x=[0:.01:5];y=exp(-1.2*x).*sin(10*x+5);subplot(1,2,1)plot(x,y),xlabel('x1'),ylabel('y1'),axis([0 5 -1 1])x=[-6:.01:6];y=abs(x.^3-100);subplot(1,2,2)plot(x,y),xlabel('x2'),ylabel('y2'),axis([-6 6 0 350])
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0 1 2 3 4 5-1
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x1
y1
-5 0 50
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x2
y2
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overlay plot 層疊圖• P.253
• A ~ m 列 *n 行• plot(A) % n 條曲線 (A 的行向量對其索引的圖 )
• plot(x,A) % A 矩陣對 x 向量作圖• plot(A,x) % x 向量對 A 矩陣作圖 (m 條曲線 )
• plot(A,B) % B 矩陣的行對 A 矩陣的行作圖
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a=[1 2 3;6 5 4;1 5 3]
a =
1 2 3 6 5 4 1 5 3
?plot(a)
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 31
1.5
2
2.5
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3.5
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plot(A) % n條曲線 (A的行向量對其索引的圖 )
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Puff! The magic dragon,Live by the sea….
x =
10 20 30 40
?a
a =
1 2 3 6 5 4 1 5 3 8 8 8
?plot(x,a)
10 15 20 25 30 35 401
2
3
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•plot(x,A) % A矩陣對 x向量作圖
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Puff! The magic dragon,Live by the sea….
x =
10 20 30 40
?a
a =
1 2 3 6 5 4 1 5 3 8 8 8
?plot(a,x)
1 2 3 4 5 6 7 810
15
20
25
30
35
40
plot(A,x) % x向量對 A矩陣作圖 (m條曲線 )
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Puff! The magic dragon,Live by the sea….
a =
1 2 3 6 5 4 1 5 3 8 8 8
?b
b =
1 2 3 2 4 6 3 6 9 4 8 12
?plot(a,b)1 2 3 4 5 6 7 8
0
2
4
6
8
10
12
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Data markers and line types
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Data markers and line types
• plot(x,y,’o’)
• plot(x,y,x,y,’o’)
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plot(x,y,’o’) plot(x,y,x,y,’o’)
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plot(x,y,’ *’,x,y,’:’)
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Puff! The magic dragon,Live by the sea….
Labeling curves and data
x=[0:.01:2];
y=sinh(x);
z=tanh(x);
plot(x,y,x,z,'--'),xlabel('x'), ...
ylabel('Hyperbolic Sine and Tangent'), ...
legend('sinh(x)','tanh(x)')
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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
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x
Hyp
erbo
lic S
ine
and
Tang
ent
sinh(x)tanh(x)
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Legend 的移動• plot 圖出來後 利用滑鼠左鍵移動之
• 格線 (grid line) 造成 legend 不明顯在圖畫好 列印前 加入axes(legend(‘string1’,‘string2’)) % 框與軸線設定一致
refresh % 強迫圖形重新繪出
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x=[0:.01:1];y=tan(x);z=sec(x);plot(x,y,x,z),xlabel('x'), ... ylabel('Tangent and Secant'),gtext('tan(x)'), ... text(.3,1.2,'sec(x)')
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
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ent
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tan(x)
sec(x)
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x=[-30:.01:30]; % 求解 x=tan(x)y=tan(x);z=6*x;
plot(x,y,x,z),xlabel('x'), ... ylabel('6x=tan(x) '), axis([-5 5 -100 100])[x,y]=ginput(5)
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Graphical solution of equations圖解法求解方程式
• Example 5.2-1 load line analysis of a circuit
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30
15
30
1
)2(0
)1()1(*16.0currentload
30load
15y powersuppl
2
1
12
11
2111
12.01
1
1
2
v
R
vv
Ri
vRiv
ei
ΩR
Vv
v
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0 2 4 6 8 10 12 14 16 18 200
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1
v2(Volts)
i 1(Am
pere
)
load line
current
v_2=[0:.01:20];i_11=.16*(exp(.12*v_2)-1);i_12=-1/30*v_2+.5;plot(v_2,i_11,v_2,i_12), grid,xlabel('v_2(Volts)'),... ylabel('i_1(Ampere)'),axis([0 20 0 1]),... gtext('load line'),gtext('current')
30
15
30
1
)2(0
)1()1(*16.0currentload
30load
15y powersuppl
2
1
12
11
2111
12.01
1
1
2
v
R
vv
Ri
vRiv
ei
ΩR
Vv
v
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The answer:i1=0.25 Ampere, v2=7.5V
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axis([7 8 .2 .3])% room-in view
7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 80.2
0.21
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v2(Volts)
i 1(Am
pere
)
load line
current
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The hold command
• 2 plot commandsp.258-259x=[-1:.01:1];y1=3+exp(-x).*sin(6*x);y2=4+exp(-x).*cos(6*x);plot((.1+.9i).^[0:.01:10]),hold,plot(y1,y2),... gtext('y2 VS y1'),gtext('Imag(z) VS Real(z)')
gtext 最好在都 plot 完之後再執行 , 以免亂掉
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-1 0 1 2 3 4 5 6-1
0
1
2
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7
y2 VS y1
Imag(z) VS Real(z)
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Annotating plots 繪圖註解• Using TEX
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Note for improving plots
• p.262-263
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Puff! The magic dragon,Live by the sea….
Special plot types
• Logarithmic plots
• Frequency response plots and filter circuits
• Controlling tick-mark spacing and labels
• Stem, stairs, and bar plots
• Separate Y axes
• Polar plots
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222
222
1.0)1(
02.0)01.01(100
xx
xxy
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•Logarithmic plots
222
222
1.0)1(
02.0)01.01(100
xx
xxy
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Note for using logarithmic scale
• P.264
• loglog(x,y)
• semilogx(x,y)
• semilogy(x,y)
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1
1
v.s./plot...
)sin(sin
RCsA
A
AAfrf
tAvtAv
i
o
io
ooii
RC=0.1;s=[1:100]*i;M=abs(1./(RC*s+1));loglog(imag(s),M),grid,xlabel('frequency(rad/sec)'),... ylabel('output/input'),... title('lowpass RC circuit frequency response(RC=0.1sec)')
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100
101
102
10-1
100
frequency(rad/sec)
outp
ut/in
put
lowpass RC circuit frequency response(RC=0.1sec)
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Controlling tick-mark spacing and labels
• help set
• help axes
• set(gca,'XTick',[xmin:dx:xmax],'YTick',[ymin:dy:ymax])
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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
y
x=[0:.01:2];y=.25*x.^2;plot(x,y),set(gca,'XTick',[0:.2:2],'YTick',[0:.1:1]),... xlabel('x'),ylabel('y')
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x=[1:6];y=[13,5,7,14,10,12];plot(x,y,'o',x,y),...set(gca,'XTicklabel',['11'; '22'; '33'; '44'; '55'; '66']),... set(gca,'XTick',[1:6]),axis([1 6 0 15]),xlabel(' 月份 '),...
ylabel('1997qwyteuwdrjhewfhwekjhfwje23435435')
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11 22 33 44 55 660
5
10
15
月 份
1997
qwyt
euw
drjh
ewfh
wek
jhfw
je23
4354
35
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Stem, stairs, and bar plots
• stem(x,y)
• stairs(x,y)
• bar(x,y)
• See also table 5.3-1
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Separate Y axes
• plotyy(x1,y1,x2,y2) % 產生左右兩條 Y 軸 左 y1 v.s. x1; 右 y2 v.s. x2
• plotyy(x1,y1,x2,y2,’type1’,’type2’)
• plotyy(x1,y1,x2,y2,’plot’,’stem’)%
左 y1 v.s. x1 plot
右 y2 v.s. x2 stem 樹狀圖
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1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60
10
20
1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60
10
20x=[1:6];y1=2*x;y2=3*x;plotyy(x,y1,x,y2,'plot','stem')
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Polar plot
orbit eccentricity = 0.5
1
2
3
4
30
210
60
240
90
270
120
300
150
330
180 0
theta=[0:pi/90:2*pi];r=2./(1-0.5*cos(theta));polar(theta,r),title('orbit eccentricity = 0.5')
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Self test p.271
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The plot editor
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5.5 Function discovery
1. Linear function
2. Power function
3. Exponential function
• Experimental data
mxmx
m
bbexy
bxxy
bmxxy
)10(or ,)(
)(
)(
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Step of function discovery
• P.277
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Applications
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deflection=[0,.09,.18,.28,.37,.46,.55,.65,.74];force=[0:100:800];%plot with linear scalesubplot(2,1,1)plot(force,deflection,'o'),... xlabel('force(lb)'),ylabel('deflection(in)'),... axis([0 800 0 .8])%curve fitting with a linep=polyfit(force,deflection,1);k=1/p(1)%plot the fitted linef=[0:2:800];x=f/k;subplot(2,1,2)plot(f,x,force,deflection,'o'),... xlabel('force(lb)'),ylabel('deflection(in)'),... axis([0 800 0 .8])
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0 100 200 300 400 500 600 700 8000
0.2
0.4
0.6
0.8
force(lb)
defle
ctio
n(in
)
0 100 200 300 400 500 600 700 8000
0.2
0.4
0.6
0.8
force(lb)
defle
ctio
n(in
)
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Heat transferpredict when will T(F) achieve 120F?
t(sec) T(F)
0 145
620 130
2266 103
3482 90
120t=?
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model
• Linear model: T-68=mt+b
• Exponential model: T-68=b(10)mt
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%input datatime=[0,620,2266,3482];temp=[145,130,103,90];%substract room temp.temp=temp-68;% use linear scalesubplot(2,2,1)plot(time,temp,time,temp,'o'),...xlabel('Time(sec)'),ylabel('temperature(F)')%use exponential plotsubplot(2,2,2)semilogy(time,temp,time,temp,'o'),...xlabel('Time(sec)'),ylabel('temperature(F)')
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%use linear fittingp=polyfit(time,log10(temp),1);m=p(1)b=10^p(2)%calculate the time needed to 120Ft_120=(log10(120-68)-log10(b))/m%estimation t=[0:10:4000];T=68+b*10.^(m*t);subplot(2,2,3)semilogy(t,T-68,time,temp,'o',t_120,120-68,'+'),... xlabel('time(sec)'),ylabel('temp(F)')%plot fitted linesubplot(2,2,4)plot(t,T,time,temp+68,'o',t_120,120,'+'),... xlabel('time(sec)'),ylabel('temp(F)')
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0 1000 2000 3000 400020
40
60
80
Time(sec)
tem
pera
ture
(F)
0 1000 2000 3000 400010
1
102
Time(sec)
tem
pera
ture
(F)
0 1000 2000 3000 400010
1
102
time(sec)
tem
p(F
)
0 1000 2000 3000 400080
100
120
140
160
time(sec)
tem
p(F
)
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Interpolation and extrapolation
• Are the interpolation or extrapolation correct?
• Are they still in the feasible region of the model?
• e.g. elastic ring, the linear elasticity is applicable in the regions of small deformation, not for large deformation.
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Hydraulic engineering
• Fluid mechanics• Fluid engineering….
volumeV
pressurep
rate flow
f
Vrf
pcf
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V( 杯 )
倒滿一杯需時 t(sec)
15 612 79 86 9
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• % problem description• cups=[6,9,12,15];• meas_times=[9,8,7,6];• meas_flow=1./meas_times;• %line fitting• p=polyfit(log10(cups),log10(meas_flow),1);• coeffs=[p(1),10^p(2)];• m=coeffs(1)• b=coeffs(2)• %log-log plot as well as linear plot• x=[6:.01:40];• y=b*x.^m;• subplot(2,1,1)• loglog(x,y,cups,meas_flow,'o'),grid,...• xlabel('V(cups)'),ylabel('1 cup full time(sec)'),...• axis([5 15 .1 .3])• % plot and predict 36 cups• subplot(2,1,2)• plot(x,1./y,cups,meas_times,'o'),grid,...• axis([5 36 0 10])• %calculate V=36 cups' fill time (sec)• V=36• f_36=b*V^m
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101
10-1
V(cups)
1 cu
p fu
ll tim
e(se
c)
5 10 15 20 25 30 350
2
4
6
8
10
4.2sec
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5.6 Regression
• Correlation analysis
• Regression
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The least squares method
• Find a line (y=mx+b), such that, the sum of the square errors between the point and line is min.
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n
ii
n
iii
n
ii
n
ii
n
ii
n
iii
ybyxaxabynbxa
ybaxJ
1111
22
1
22
1
2
222
)(
0,0
b
J
a
J
n
ii
n
ii
n
iii
n
ii
n
ii
ynbax
yxbxax
11
111
2
)(
)()(
Find a,b
Such that J is min.
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x=[1:9];y=[5,6,10,20,28,33,34,36,42];xp=[1:.01:9];for k=1:4 coeff=polyfit(x,y,k) yp(k,:)=polyval(coeff,xp); J(k)=sum((polyval(coeff,x)-y).^2);endsubplot(2,2,1)plot(xp,yp(1,:),x,y,'o'),axis([0 10 0 50])subplot(2,2,2)plot(xp,yp(2,:),x,y,'o'),axis([0 10 0 50])subplot(2,2,3)plot(xp,yp(3,:),x,y,'o'),axis([0 10 0 50])subplot(2,2,4)plot(xp,yp(4,:),x,y,'o'),axis([0 10 0 50])disp(J)
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0 5 100
10
20
30
40
50
0 5 100
10
20
30
40
50
0 5 100
10
20
30
40
50
0 5 100
10
20
30
40
50
J = 71.5389 56.6727 41.8838 4.6566S= 1562 1562 1562 1562r^2= .9542 .9637 .9732 .9970
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Self testing p.292
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Fitting other functions
• y=bxm
p=polyfit(log10(x),log10(y),1)
p(1)=m, p(2)=log10b
y=b(10)mx
p=polyfit(x,log10(y),1)
P(1)=m, p(2)= log10b
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Puff! The magic dragon,Live by the sea….
The quality of a curve fit
• Coefficient of determination r^2
fitting goodfor %99r
fitting good,1~,1
1
][
])([
2
2
2
1
2
1
2
rJ
S
Jr
yyS
yxfJ
m
ii
m
iii
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Regression and numerical accuracy
使用高階多項式的兩個危險處(1) 高階多項式在數據點間會有很大的繞行(2) 使用高階多項式其係數需要較大的精確度表現 (format long, or format long e)
e.g. fitting for the data in p.295
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0 .1
2 1.884
4 2.732
6 3.388
8 3.346
10 3
12 2.644
14 2.022
16 1.65
18 1.5838
20 1.35
22 1.0082
24 .718
26 .689
28 .4308
30 .203
32 .1652
34 -.073
36 -.002
38 -.1122
40 .106
x y
05773943.0
20369642.115178007.0
107777.7109209.1
101.20
2170577394277.0
07742036964239.171531517800652.0
1067777099161.710920896532.1
1009690135.21033551.6
2
3344
566
2
3344
5669
xx
xx
xxy
xx
xx
xxy
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x1=[0:2:40];y1=[.1,1.884,2.732,3.388,3.346,3,2.644,2.022,1.65,1.5838,1.35,1.0082,.718,.689,.4308,.203,.1652,-.073,-.002,-.1122,.106];[p, s]=polyfit(x1,y1,6);format longx=[0:.1:40];y = polyval(p,x);subplot(2,1,1)plot(x,y,x1,y1,'o')subplot(2,1,2)p2=round(p.*10^8)/10^8;y2 = polyval(p2,x);plot(x,y2,x1,y1,'o')pp2
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0 5 10 15 20 25 30 35 40-1
0
1
2
3
4
0 5 10 15 20 25 30 35 40-15
-10
-5
0
5
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p.297
• 不用高次式曲線擬合• 使用兩個三次式 ~ [0, 15]; (15, 40]
• 可以將小數為數縮減成 8 位 不失其精度 !
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•Give some criticism on the following figure!
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Scaling the data
• x’=x-xmin or x’=x-xmean
• x’=x/xmean or x’=x/xmax
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Year=[1990:1999];Veh_Flow=[2.1,3.4,4.5,5.3,6.2,6.6,6.8,7,7.4,7.8];p=polyfit(Year,Veh_Flow,3)
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 2.791805e-019.> In c:\matlab\toolbox\matlab\polyfun\polyfit.m at line 48p = 1.0e+007 * 0.0000 0.0000 0.0104 -6.9010
year 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999
車流量million 2.1 3.4 4.5 5.3 6.2 6.6 6.8 7 7.4 7.8
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Year=[1990:1999];Veh_Flow=[2.1,3.4,4.5,5.3,6.2,6.6,6.8,7,7.4,7.8];x=Year-1990;p=polyfit(x,Veh_Flow,3)J=sum((polyval(p,x)-Veh_Flow).^2);S=sum((Veh_Flow-mean(Veh_Flow)).^2);r2=1-J/Sp = 0.0087 -0.1851 1.5991 2.0362r2 = 0.9972
year 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999
車流量million 2.1 3.4 4.5 5.3 6.2 6.6 6.8 7 7.4 7.8
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2000 yield
polyval(p,10)ans = 8.1767 million cars
year 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999
車流量million 2.1 3.4 4.5 5.3 6.2 6.6 6.8 7 7.4 7.8
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Using residuals
• P.300 example!% time datax=[0:19];% contamination datay=[6,13,23,33,54,83,118,156,210,282,...350,440,557,685,815,990,1170,1350,1575
,1830];
%linear fittingp1=polyfit(x,y,1);% quadratic fittingp2=polyfit(x,y,2);% cubic fittingp3=polyfit(x,y,3);
% exponential fittingp4=polyfit(x,log10(y),1);% residualres1=polyval(p1,x)-y;res2=polyval(p2,x)-y;res3=polyval(p3,x)-y;res4=10.^polyval(p4,x)-y;
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Plot the residual
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Linear model (0,0)
m
iii
m
ii
m
iii
yxxa
aJ
yxaJ
xay
11
21
1
1
21
1
have we
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Quadratic model (0,0)
m
iii
m
ii
m
ii
m
iii
m
ii
m
ii
m
iiii
yxxaxa
yxxaxa
aJaJ
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1
2
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32
1
41
21
1
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322
1
have we
0/0/let
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Not pass (0,0)
001
0022
01
)(
)()(
yxxay
yxxaxxay
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Multiple linear regression
• y=a0+a1x1+a2x2
• Data (y, x1, x2)
n
nn
y
y
y
y
xx
xx
xx
xx
a
a
a
...
1
...
1
1
1
3
2
1
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2313
2212
2111
2
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0
y
Xa
yXa
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Example 5.6-3
x1=[0:3]';x2=[5,7,8,11]';y=[7.1,19.2,31,45]';X=[ones(size(x1)) x1 x2];a=X\yyp=X*a;Max_percent_error=100*max(abs((yp-y)./y))a = 0.8000 10.2429 1.2143Max_percent_error = 3.2193
y=0.8+10.2429x1+1.2143x2
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Linear-in-the-parameters regression
• P.305 example
• Self testing
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5.7 the basic fitting interface
• P.306
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5.8 3-D Plots
• 3-d line plots
• Surface mesh plot
• Contour plots
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t=[0:pi/50:10*pi];plot3(exp(-.05*t).*sin(t),exp(-.05*t).*cos(t),t),... xlabel('x'),ylabel('y',zlabel('z'),grid)
-1-0.5
00.5
1
-1
-0.5
0
0.5
10
10
20
30
40
x
z
tz
tey
text
t
cos
sin05.0
05.0
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[X,Y]=meshgrid(-2:.1:2);Z=X.*exp(-((X-Y.^2).^2+Y.^2));mesh(X,Y,Z),xlabel('x'),ylabel('y'),zlabel('z')
])[( 222 yyxxez
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[X,Y]=meshgrid(-2:.1:2);Z=X.*exp(-((X-Y.^2).^2+Y.^2));contour(X,Y,Z),xlabel('x'),ylabel('y'),zlabel('z')
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
x
y
])[( 222 yyxxez
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