第 5 章繪圖及模型建構 plotting & model building

Puff! The magic dragon,Live by the sea….

第 5章繪圖及模型建構plotting & model building


• 圖比表好• 表比文字好


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


• line ~ for theoretical calculation data

• data symbol ~ for experimental data


Requirements of correct plot

• P.243-244

• Read it NOW!


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


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')


-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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1

實部

虛部


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])


x=[1:.01:2];

y=cos(tan(x))-tan(sin(x));

plot(x,y)


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

1000

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x

p


Self testing

• pp.250-251

• Write down the program now!


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])


0 1 2 3 4 5-1

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x1

y1

-5 0 50

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x2

y2


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


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

3

3.5

4

4.5

5

5.5

6

plot(A) % n條曲線 (A的行向量對其索引的圖 )


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

4

5

6

7

8

•plot(x,A) % A矩陣對 x向量作圖


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 )


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


Data markers and line types


Data markers and line types

• plot(x,y,’o’)

• plot(x,y,x,y,’o’)


plot(x,y,’o’) plot(x,y,x,y,’o’)


plot(x,y,’ *’,x,y,’:’)


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)')


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

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3

3.5

4

x

Hyp

erbo

lic S

ine

and

Tang

ent

sinh(x)tanh(x)


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


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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1

1.2

1.4

1.6

1.8

2

x

Tang

ent

and

Sec

ant

tan(x)

sec(x)


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)


Graphical solution of equations圖解法求解方程式

• Example 5.2-1 load line analysis of a circuit


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


0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

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0.5

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0.9

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


The answer:i1=0.25 Ampere, v2=7.5V


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

0.22

0.23

0.24

0.25

0.26

0.27

0.28

0.29

v2(Volts)

i 1(Am

pere

)

load line

current


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


-1 0 1 2 3 4 5 6-1

0

1

2

3

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5

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7

y2 VS y1

Imag(z) VS Real(z)


Annotating plots 繪圖註解• Using TEX


Note for improving plots

• p.262-263


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


222

222

1.0)1(

02.0)01.01(100

xx

xxy


•Logarithmic plots

222

222

1.0)1(

02.0)01.01(100

xx

xxy


Note for using logarithmic scale

• P.264

• loglog(x,y)

• semilogx(x,y)

• semilogy(x,y)


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)')


100

101

102

10-1

100

frequency(rad/sec)

outp

ut/in

put

lowpass RC circuit frequency response(RC=0.1sec)


Controlling tick-mark spacing and labels

• help set

• help axes

• set(gca,'XTick',[xmin:dx:xmax],'YTick',[ymin:dy:ymax])


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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')


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')


11 22 33 44 55 660

5

10

15

月份

1997

qwyt

euw

drjh

ewfh

wek

jhfw

je23

4354

35


Stem, stairs, and bar plots

• stem(x,y)

• stairs(x,y)

• bar(x,y)

• See also table 5.3-1


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


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')


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')


Self test p.271


The plot editor


5.5 Function discovery

1. Linear function

2. Power function

3. Exponential function

• Experimental data

mxmx

m

bbexy

bxxy

bmxxy

)10(or ,)(

)(

)(


Step of function discovery

• P.277


Applications


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])


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

)


Heat transferpredict when will T(F) achieve 120F?

t(sec) T(F)

0 145

620 130

2266 103

3482 90

120t=?


model

• Linear model: T-68=mt+b

• Exponential model: T-68=b(10)mt


%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)')


%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)')


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

)


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.


Hydraulic engineering

• Fluid mechanics• Fluid engineering….

volumeV

pressurep

rate flow

f

Vrf

pcf


V( 杯 )

倒滿一杯需時 t(sec)

15 612 79 86 9


• % 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


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


5.6 Regression

• Correlation analysis

• Regression


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.


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.


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)


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


Self testing p.292


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


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


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


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


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


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


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 !


•Give some criticism on the following figure!


Scaling the data

• x’=x-xmin or x’=x-xmean

• x’=x/xmean or x’=x/xmax


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


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


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


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;


Plot the residual


Linear model (0,0)

m

iii

m

ii

m

iii

yxxa

aJ

yxaJ

xay

11

21

1

1

21

1

have we

0/let

)(


Quadratic model (0,0)

m

iii

m

ii

m

ii

m

iii

m

ii

m

ii

m

iiii

yxxaxa

yxxaxa

aJaJ

yxaxaJ

axaxay

11

22

1

31

1

2

1

32

1

41

21

1

22

21

322

1

have we

0/0/let

)(


Not pass (0,0)

001

0022

01

)(

)()(

yxxay

yxxaxxay


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

21

2313

2212

2111

2

1

0

y

Xa

yXa


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


Linear-in-the-parameters regression

• P.305 example

• Self testing


5.7 the basic fitting interface

• P.306


5.8 3-D Plots

• 3-d line plots

• Surface mesh plot

• Contour plots


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)

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x

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tz

tey

text

t

cos

sin05.0

05.0


[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


[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

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y

])[( 222 yyxxez

第 5 章 繪圖及模型建構 plotting & model building

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第 5 章繪圖及模型建構 plotting & model building