Solutions for Geog 210C Labchris/Medrano_GEO 210C/GEO 210C...Solutions for Geog 210C Lab Assignment #1 - 68/70 STUDENT NAME: F. Antonio Medrano DATE COMPLETED: Contents Instructions

Solutions for Geog 210C Lab Assignment #1 - 68/70

STUDENT NAME: F. Antonio MedranoDATE COMPLETED:

Contents

Instructions

Preliminaries

1/1: Using kernel0 for 1-D Kernels

1/2: Using kernel0 for 2-D Kernels

1/3: Kernel as PDF Verification

1/4: Scaling Kernels, 1-D Gaussian PDF Verification

1/5: Scaling 2-D Kernels

1/6: Recentering 1-D Kernel Locations

1/7: Recentering 2-D Kernel Locations

Preliminaries #2

2/1: Plotting Kernels

2/2: Plotting a Summed Kernel

2/3: Plotting the 1-D Gaussian Kernel Density Estimate

2/4: Experimenting with more Kernels

Preliminaries #3

2/1: 2-D Kernel Density Estimation

2/2: Other Point Patterns

3/3: Play with many Kernel Types and properties

Instructions

You can insert a variety of text formats in your M-file, using:

Cell -> Insert Text Markup

the differences between the above text formats are only visible in the published html file.

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For each Lab assignment, send an e-mail to the READER [email protected] with:

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Good luck, and remember to put your name at the top of this file...

Phaedon C. Kyriakidis, April 2009.

Solutions for Geog 210C Lab Assignment #1 file:///C:/Documents%20and%20Settings/Micah/My%20Documents/Doc...

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Preliminaries

clear; clc;

1/1: Using kernel0 for 1-D Kernels 5/5, a better interpretation is

expected next time

% Create kernels over a fixed region and bandwidth and various typesdx0= -4:0.05:4;bandW1 = 1;kernType = [0 1 3 5]; % rectangular, triangular, Gaussian, and Epanechnikov

% display the four kernels of different types in one figurefor i = 1:4 subplot(2,2,i); output1(:,i) = kernel0({dx0}, kernType(i), bandW1, 1);end% The graphs show kernels of different shapes. Cool!

1/2: Using kernel0 for 2-D Kernels 4/5, interpretation???

% Create kernels over a fixed region and bandwidth and various typesdx0= -4:0.05:4;dy0= -4:0.05:4;bandW2 = [1 1];kernType = [0 1 3 5];


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% display the four kernels of different types in one figurefor i = 1:4 subplot(2,2,i); output2(:,:,i) = kernel0({dx0; dy0}, kernType(i), bandW2, 1);end

close all;

1/3: Kernel as PDF Verification

% To be a valid Probability Density Function, the kernel must have all% positive values and the area (or volume) under the kernel must add up to% 1. NOTE: In all calculations, the results calculate the values in this% order of kernel type --> [rectangular, triangular, Gaussian, Epanechnikov]

1-Dimensional

% First the easy part, do any of the four kernels have any negative values?anyNeg1 = any(any(output1 < 0) < 0)% nope, none of the four kernels have negative values.

% now, calculate the area under each kernel using zero order interpolationarea1 = sum(output1*0.05)

% just for kicks we see that we get the same answer with 1st order% interpolationarea1 = trapz(output1*0.05)

% Let's find the mean value of each kernelmean1 = sum([dx0; dx0; dx0; dx0]' .* output1 * 0.05)


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% They are all tiny numbers, quite close to zero in fact. This is a good% thing.

% Let's find the variance for each kernelvar1 = sum([dx0; dx0; dx0; dx0]'.^2 .* output1 * 0.05)% the Gaussian variance comes out to 1, which was the bandwidth that we% gave it. The others are less than one, but we can't interpret these since% they're not Gaussian.

anyNeg1 =

0

area1 =

1.0250 1.0000 0.9999 0.9994

area1 =

1.0250 1.0000 0.9999 0.9994

mean1 =

1.0e-16 *

0.4163 -0.0997 0.0553 0.0217

var1 =

0.3587 0.1662 0.9990 0.1994

2-Dimensional

% check for any negative valuesanyNeg2 = any(any(any(output2 < 0) < 0) < 0)%

% calculate the volumesarea2 = sum(sum(output2 * 0.05.^2))% all of the areas are very close to 1, wonderful!

% create a variable grids which is a 4-layer position grid for the four% kernelsgrid2 = dx0'*dy0;grids = cat(3, grid2, grid2, grid2, grid2);

%calculate the means and variancesmean2 = sum(sum(grids .* output2 * 0.05^2))var2 = sum(sum(grids.^2 .* output2 * 0.05^2))% the mean for all of them is zero! The variance for the Gaussian is 1, all% other variances are smaller.

anyNeg2 =

0


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area2(:,:,1) =

1.0506

area2(:,:,2) =

1.0000

area2(:,:,3) =

0.9999

area2(:,:,4) =

0.9988

mean2(:,:,1) =

0

mean2(:,:,2) =

0

mean2(:,:,3) =

0

mean2(:,:,4) =

0

var2(:,:,1) =

0.1287

var2(:,:,2) =

0.0276

var2(:,:,3) =

0.9979

var2(:,:,4) =

0.0398

1/4: Scaling Kernels, 1-D Gaussian PDF Verification 5/5

% Rescale the kerlnels to have different bandwidths and recalculate


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% everything we've done alreadybandW1_4 = [1.5 0.5];

% display the two kernels of different bandwidths in separate windowsfigure(1)output1_4(:,1) = kernel0({dx0}, 3, bandW1_4(1), 1);figure(2)output1_4(:,2) = kernel0({dx0}, 3, bandW1_4(2), 1);

% First the easy part, do either of the kernels have any negative values?anyNeg1_4 = any(any(output1_4 < 0) < 0)% nope, none of the four kernels have negative values.

% now, calculate the area under each kernel using zero order interpolationarea1_4 = sum(output1_4*0.05)% The area is equal to 1 for both.

% Let's find the mean value of each kernelmean1_4 = sum([dx0; dx0]' .* output1_4 * 0.05)% They are both tiny numbers, quite close to zero in fact. This is a good% thing.

% Let's find the variance for each kernelvar1_4 = sum([dx0; dx0]'.^2 .* output1_4 * 0.05)% The first Gaussian variance comes out to 2.1020, the square root of that% is 1.45, which is pretty close to the bandwidth of 1.5 that we gave the% original function. The others are less than one. The second variance is% 0.25, the square root of which is 0.5, which was the bandwidth we gave% it. Bandwidth equals standard deviation.

anyNeg1_4 =

0

area1_4 =

0.9927 1.0000

mean1_4 =

1.0e-16 *

0.4749 -0.1587

var1_4 =

2.1020 0.2500


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1/5: Scaling 2-D Kernels 5/5

% Create kernels with alternate bandwidth and various typesdx0= -4:0.05:4;


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dy0= -4:0.05:4;bandW2_5 = [0.5 2];kernType = [0 1 3 5];

% display the four kernels of different types in one figurefor i = 1:4 subplot(2,2,i) output2_5(:,:,i) = kernel0({dx0; dy0}, kernType(i), bandW2_5, 1);end

close all;

1/6: Recentering 1-D Kernel Locations 4/5... comment on-why the shift

may be useful

% Make a Gaussian kernel that is offset to the right by 2kGaus1new = kernel0({dx0-2}, 3, bandW1, 0);plot(dx0,kGaus1new,'.');


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1/7: Recentering 2-D Kernel Locations 5/5

% Plot a Gaussian kernel centered on the location xy = [2 2]. Note the odd% scale direction for the y-axis.figure(1)xy = [2 2];kGaus2new = kernel0({dx0 - xy(1); dy0 - xy(2)}, 3, bandW2, 0);imagesc(dx0, dy0, flipud(kGaus2new));axis square;colorbar;title('2D Gaussian Kernel (bandwidths: 1,1)');

% Plot a triangular kernel centered on the location xy = [-1 3]. Not again% the odd scale direction for the y-axisfigure(2)xy = [-1 3];kGaus2new = kernel0({dx0 - xy(1); dy0 - xy(2)}, 2, bandW2, 0);imagesc(dx0, dy0, flipud(kGaus2new));axis square;colorbar;title('2D triangular Kernel (bandwidths: 1,1)');


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close all;

Preliminaries #2

clear; clc;


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m1 = 30; s1 = 10; m2 = 60; s2 = 10;randn('state', 345);xEv = [normrnd(m1, s1, 50, 1); normrnd(m2, s2, 50, 1)];nBins = 25;univstats(xEv,1,[0 2 nBins]);hold on;plot(xEv,zeros(100,1),'*');

# of data used 100Mean 46.5903Std deviation 17.6666Minimum 7.9586Lower quartile 34.7585Median 45.2884Upper quartile 60.8633Maximum 81.2099 Finished UNIVSTATS: Version #1

2/1: Plotting Kernels 5/5

kernType = 3; % Gaussian kernelx30 = 30;x60 = 60;x = 0 : 0.5 : 100;dx30 = abs(x-x30);dx60 = abs(x-x60);bandW = 10;fx30 = kernel0({dx30},kernType,bandW,0);fx60 = kernel0({dx60},kernType,bandW,0);plot(x,fx30,'.',x,fx60,'+');% This shows the two original PDFs that were used to generate the dataset.


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% Each of these PDFs have an area of 1, so if we combine them they won't be% a proper PDF unless we scale them as well.

2/2: Plotting a Summed Kernel 5/5

f = fx30 + fx60;% f isn't a proper PDF though, it's too big. Let's scale it.fcorr = f / 2; % much better!plot(x,fcorr,'r^');legend('data histogram', 'data points', 'kernel30', 'kernel60', ... 'combined scaled kernel')legend('Location', 'BestOutside')hold off;

% Check if fcorr is a proper PDFarea = sum(fcorr*0.5)% This density is a proper PDF, because the area is 1

area =

0.9994


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2/3: Plotting the 1-D Gaussian Kernel Density Estimate 5/5

bandW = 2;kernType = 3;f = kerneldensity(xEv, 1, [], {x}, kernType, bandW, 1, 0);

univstats(xEv,1,[0 2 nBins]);hold on;plot(x,f,'ro')hold off;

# of rows in DataIn = 100# of data actually used = 100Computing kernel for datum #10Computing kernel for datum #20Computing kernel for datum #30Computing kernel for datum #40Computing kernel for datum #50Computing kernel for datum #60Computing kernel for datum #70Computing kernel for datum #80Computing kernel for datum #90Computing kernel for datum #100 Computing integral of estimated density... Integral of estimated discrete density = 1Output density normalized to a unit integral Finished KERNELDENSITY: Version #1# of data used 100Mean 46.5903Std deviation 17.6666


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Minimum 7.9586Lower quartile 34.7585Median 45.2884Upper quartile 60.8633Maximum 81.2099 Finished UNIVSTATS: Version #1

2/4: Experimenting with more Kernels 5/5

kernType = [0 1 3 5]; % do four kinds of kernelsbandW = [2 3 5 4]; %make them various bandwidths

for i = 1:4 figure(i); output1(:,i) = kerneldensity(xEv, 1, [], {x}, kernType(i), bandW(i), 1, 3);end% The bandwidth and kernel type both have a major effect on the smoothness% of the plot. A Gaussian kernel, being non-compact, in general gives the% smoothest graph at the cost of losing detail. The rectangular kernel% gives the most pointy results. The others are more in the middle, with% the triangular a little less pointy than the rectangular, and the% Epanechnikov is even less pointy, closer to the Gaussian. The smoother% the kernel, the smoother the final PDF, but you might lose out on some% detail. If you make your kernel narrow and pointy, you might miss out on% some of the macro-structure of your data. It's best to have some physics% justification for your kernel type and bandwidth.

# of rows in DataIn = 100# of data actually used = 100Computing kernel for datum #10Computing kernel for datum #20


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Computing kernel for datum #30Computing kernel for datum #40Computing kernel for datum #50Computing kernel for datum #60Computing kernel for datum #70Computing kernel for datum #80Computing kernel for datum #90Computing kernel for datum #100 Computing integral of estimated density... Integral of estimated discrete density = 1Output density normalized to a unit integral Finished KERNELDENSITY: Version #1# of rows in DataIn = 100# of data actually used = 100Computing kernel for datum #10Computing kernel for datum #20Computing kernel for datum #30Computing kernel for datum #40Computing kernel for datum #50Computing kernel for datum #60Computing kernel for datum #70Computing kernel for datum #80Computing kernel for datum #90Computing kernel for datum #100 Computing integral of estimated density... Integral of estimated discrete density = 1Output density normalized to a unit integral Finished KERNELDENSITY: Version #1# of rows in DataIn = 100# of data actually used = 100Computing kernel for datum #10Computing kernel for datum #20Computing kernel for datum #30Computing kernel for datum #40Computing kernel for datum #50Computing kernel for datum #60Computing kernel for datum #70Computing kernel for datum #80Computing kernel for datum #90Computing kernel for datum #100 Computing integral of estimated density... Integral of estimated discrete density = 0.99929Output density normalized to a unit integral Finished KERNELDENSITY: Version #1# of rows in DataIn = 100# of data actually used = 100Computing kernel for datum #10Computing kernel for datum #20Computing kernel for datum #30Computing kernel for datum #40Computing kernel for datum #50Computing kernel for datum #60Computing kernel for datum #70Computing kernel for datum #80Computing kernel for datum #90Computing kernel for datum #100


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Computing integral of estimated density... Integral of estimated discrete density = 0.99998Output density normalized to a unit integral Finished KERNELDENSITY: Version #1


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close all;

Preliminaries #3

clear; clc;


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load pointData.mat

% the area is 100x100, and there are 100 points in each one, so the lambda% is equal to 0.01 (assuming a 1x1 unit area) for all three data sets. The% lambda does not contain any information on the spatial distribution of% the three point patterns though, since all the patterns have the same% lambda but very different distributions.

2/1: 2-D Kernel Density Estimation 5/5

% plot the data setplot(pointsRand(:,1),pointsRand(:,2),'*'); axis([0 100 0 100]);axis square;

now make a 2-D Kernel Density estimation

x = 0:100;y = 0:100;kernType = 3;bandW = [3 3];figure(1)f = kerneldensity(pointsRand,[1 2],[],{x,y},kernType,bandW,1,2);colorbar;% The pattern looks fairly random, with a few clusters and a few sparse% areas. This is reflected by the colors of the density estimation too,% with some regions of dark blue and some regions of bright red. The% intensity ranges from 0 to 0.0005.

# of rows in DataIn = 100


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# of data actually used = 100Computing kernel for datum #10Computing kernel for datum #20Computing kernel for datum #30Computing kernel for datum #40Computing kernel for datum #50Computing kernel for datum #60Computing kernel for datum #70Computing kernel for datum #80Computing kernel for datum #90Computing kernel for datum #100 Computing integral of estimated density... Integral of estimated discrete density = 0.96629Output density normalized to a unit integral Irregular rasterAxes limits-0.5 100.5 -0.5 100.5 Finished RASTERMAP: Version #1 Finished KERNELDENSITY: Version #1

2/2: Other Point Patterns 5/5

figure(2)f = kerneldensity(pointsClus,[1 2],[],{x,y},kernType,bandW,1,2);colorbar;% This data has clusters of points, where most of the area is dark blue%representing low density, with small regions of extremeley dense bright% red clusters. These dense clusters reach higher density estimation


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% values, up to 0.0025.

figure(3)f = kerneldensity(pointsStrat,[1 2],[],{x,y},kernType,bandW,1,2);colorbar;% This data is mostly regularly distributed, so most of the density is in% the middle range of light blue. There is only one fairly minor cluster,% so the peak density only reaches a value of 0.00035

# of rows in DataIn = 100# of data actually used = 100Computing kernel for datum #10Computing kernel for datum #20Computing kernel for datum #30Computing kernel for datum #40Computing kernel for datum #50Computing kernel for datum #60Computing kernel for datum #70Computing kernel for datum #80Computing kernel for datum #90Computing kernel for datum #100 Computing integral of estimated density... Integral of estimated discrete density = 0.97524Output density normalized to a unit integral Irregular rasterAxes limits-0.5 100.5 -0.5 100.5 Finished RASTERMAP: Version #1 Finished KERNELDENSITY: Version #1# of rows in DataIn = 100# of data actually used = 100Computing kernel for datum #10Computing kernel for datum #20Computing kernel for datum #30Computing kernel for datum #40Computing kernel for datum #50Computing kernel for datum #60Computing kernel for datum #70Computing kernel for datum #80Computing kernel for datum #90Computing kernel for datum #100 Computing integral of estimated density... Integral of estimated discrete density = 0.97419Output density normalized to a unit integral Irregular rasterAxes limits-0.5 100.5 -0.5 100.5 Finished RASTERMAP: Version #1 Finished KERNELDENSITY: Version #1


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close all;

3/3: Play with many Kernel Types and properties 5/5


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% different discretization for the non-rectangular kernelsxd = 0:0.5:100;yd = 0:0.5:100;

% combine all the data into a single data structuredata = cat(3,pointsRand, pointsClus, pointsStrat);

for i = 1:3 figure(4*i-3)

kernType = 0; % rectangular kernel bandW = [3 3]; f = kerneldensity(data(:,:,i),[1 2],[],{x,y},kernType,bandW,1,2); colorbar;

figure(4*i-2) kernType = 1; % triangular kernel bandW = [10 10]; f = kerneldensity(data(:,:,i),[1 2],[],{xd,yd},kernType,bandW,1,2); colorbar;

figure(4*i-1) kernType = 3; % gaussian kernel bandW = [1 8]; f = kerneldensity(data(:,:,i),[1 2],[],{xd,yd},kernType,bandW,1,2); colorbar;

figure(4*i) kernType = 5; % Epanechnikov bandW = [10 5]; f = kerneldensity(data(:,:,i),[1 2],[],{xd,yd},kernType,bandW,1,2); colorbar;end

% So many choices!!! It takes a lot of artistic "feel" as well as cold% science to choose the right kernel type, bandwidth size, and bandwidth% dimensions ratio to choose something that makes physical sense as well as% containing the right amount of detail (not too smooth or too rough). It's% something that you definitely would improve on with the more experience% you have. And there may be times when you want employ assymetrical% kernels, such as if you're modeling a process where there is wind blowing% pollen or ash, or water currents pushing particulate matter through% rivers or oceans.

# of rows in DataIn = 100# of data actually used = 100Computing kernel for datum #10Computing kernel for datum #20Computing kernel for datum #30Computing kernel for datum #40Computing kernel for datum #50Computing kernel for datum #60Computing kernel for datum #70Computing kernel for datum #80Computing kernel for datum #90Computing kernel for datum #100 Computing integral of estimated density... Integral of estimated discrete density = 1.3319Output density normalized to a unit integral Irregular rasterAxes limits


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-0.5 100.5 -0.5 100.5 Finished RASTERMAP: Version #1 Finished KERNELDENSITY: Version #1# of rows in DataIn = 100# of data actually used = 100Computing kernel for datum #10Computing kernel for datum #20Computing kernel for datum #30Computing kernel for datum #40Computing kernel for datum #50Computing kernel for datum #60Computing kernel for datum #70Computing kernel for datum #80Computing kernel for datum #90Computing kernel for datum #100 Computing integral of estimated density... Integral of estimated discrete density = 0.94987Output density normalized to a unit integral Irregular rasterAxes limits-0.25 100.25 -0.25 100.25 Finished RASTERMAP: Version #1 Finished KERNELDENSITY: Version #1# of rows in DataIn = 100# of data actually used = 100Computing kernel for datum #10Computing kernel for datum #20Computing kernel for datum #30Computing kernel for datum #40Computing kernel for datum #50Computing kernel for datum #60Computing kernel for datum #70Computing kernel for datum #80Computing kernel for datum #90Computing kernel for datum #100 Computing integral of estimated density... Integral of estimated discrete density = 0.94236Output density normalized to a unit integral Irregular rasterAxes limits-0.25 100.25 -0.25 100.25 Finished RASTERMAP: Version #1 Finished KERNELDENSITY: Version #1# of rows in DataIn = 100# of data actually used = 100Computing kernel for datum #10Computing kernel for datum #20Computing kernel for datum #30Computing kernel for datum #40Computing kernel for datum #50Computing kernel for datum #60Computing kernel for datum #70Computing kernel for datum #80Computing kernel for datum #90


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Computing kernel for datum #100 Computing integral of estimated density... Integral of estimated discrete density = 0.95888Output density normalized to a unit integral Irregular rasterAxes limits-0.25 100.25 -0.25 100.25 Finished RASTERMAP: Version #1 Finished KERNELDENSITY: Version #1# of rows in DataIn = 100# of data actually used = 100Computing kernel for datum #10Computing kernel for datum #20Computing kernel for datum #30Computing kernel for datum #40Computing kernel for datum #50Computing kernel for datum #60Computing kernel for datum #70Computing kernel for datum #80Computing kernel for datum #90Computing kernel for datum #100 Computing integral of estimated density... Integral of estimated discrete density = 0.99Output density normalized to a unit integral Irregular rasterAxes limits-0.5 100.5 -0.5 100.5 Finished RASTERMAP: Version #1 Finished KERNELDENSITY: Version #1# of rows in DataIn = 100# of data actually used = 100Computing kernel for datum #10Computing kernel for datum #20Computing kernel for datum #30Computing kernel for datum #40Computing kernel for datum #50Computing kernel for datum #60Computing kernel for datum #70Computing kernel for datum #80Computing kernel for datum #90Computing kernel for datum #100 Computing integral of estimated density... Integral of estimated discrete density = 0.95786Output density normalized to a unit integral Irregular rasterAxes limits-0.25 100.25 -0.25 100.25 Finished RASTERMAP: Version #1 Finished KERNELDENSITY: Version #1# of rows in DataIn = 100# of data actually used = 100


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Computing kernel for datum #10Computing kernel for datum #20Computing kernel for datum #30Computing kernel for datum #40Computing kernel for datum #50Computing kernel for datum #60Computing kernel for datum #70Computing kernel for datum #80Computing kernel for datum #90Computing kernel for datum #100 Computing integral of estimated density... Integral of estimated discrete density = 0.93689Output density normalized to a unit integral Irregular rasterAxes limits-0.25 100.25 -0.25 100.25 Finished RASTERMAP: Version #1 Finished KERNELDENSITY: Version #1# of rows in DataIn = 100# of data actually used = 100Computing kernel for datum #10Computing kernel for datum #20Computing kernel for datum #30Computing kernel for datum #40Computing kernel for datum #50Computing kernel for datum #60Computing kernel for datum #70Computing kernel for datum #80Computing kernel for datum #90Computing kernel for datum #100 Computing integral of estimated density... Integral of estimated discrete density = 0.96927Output density normalized to a unit integral Irregular rasterAxes limits-0.25 100.25 -0.25 100.25 Finished RASTERMAP: Version #1 Finished KERNELDENSITY: Version #1# of rows in DataIn = 100# of data actually used = 100Computing kernel for datum #10Computing kernel for datum #20Computing kernel for datum #30Computing kernel for datum #40Computing kernel for datum #50Computing kernel for datum #60Computing kernel for datum #70Computing kernel for datum #80Computing kernel for datum #90Computing kernel for datum #100 Computing integral of estimated density... Integral of estimated discrete density = 1.3504Output density normalized to a unit integral


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Irregular rasterAxes limits-0.5 100.5 -0.5 100.5 Finished RASTERMAP: Version #1 Finished KERNELDENSITY: Version #1# of rows in DataIn = 100# of data actually used = 100Computing kernel for datum #10Computing kernel for datum #20Computing kernel for datum #30Computing kernel for datum #40Computing kernel for datum #50Computing kernel for datum #60Computing kernel for datum #70Computing kernel for datum #80Computing kernel for datum #90Computing kernel for datum #100 Computing integral of estimated density... Integral of estimated discrete density = 0.95043Output density normalized to a unit integral Irregular rasterAxes limits-0.25 100.25 -0.25 100.25 Finished RASTERMAP: Version #1 Finished KERNELDENSITY: Version #1# of rows in DataIn = 100# of data actually used = 100Computing kernel for datum #10Computing kernel for datum #20Computing kernel for datum #30Computing kernel for datum #40Computing kernel for datum #50Computing kernel for datum #60Computing kernel for datum #70Computing kernel for datum #80Computing kernel for datum #90Computing kernel for datum #100 Computing integral of estimated density... Integral of estimated discrete density = 0.94056Output density normalized to a unit integral Irregular rasterAxes limits-0.25 100.25 -0.25 100.25 Finished RASTERMAP: Version #1 Finished KERNELDENSITY: Version #1# of rows in DataIn = 100# of data actually used = 100Computing kernel for datum #10Computing kernel for datum #20Computing kernel for datum #30Computing kernel for datum #40Computing kernel for datum #50Computing kernel for datum #60Computing kernel for datum #70


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Computing kernel for datum #80Computing kernel for datum #90Computing kernel for datum #100 Computing integral of estimated density... Integral of estimated discrete density = 0.96181Output density normalized to a unit integral Irregular rasterAxes limits-0.25 100.25 -0.25 100.25 Finished RASTERMAP: Version #1 Finished KERNELDENSITY: Version #1


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close all

Published with MATLAB® 7.4


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