Matlab数据可视化(5)：二维数据可视化 I
2018-12-03
以下介绍数据的二维可视化。
一. 二维散点图
(源代码：scatter.m)
我们用著名的Iris数据集(Fisher, 1936)作为绘图实例。Iris数据集包含3种鸢尾花的150个样本数据，每个数据都有4个属性(花萼和花瓣的长度及宽度)。
1) 基本散点图
我们用其中两个属性值作为X和Y轴，另一个属性值表示点的大小。(图1)
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%% 加载数据集
[attrib className] = xlsread("iris.xlsx");
%% 绘制基本的散点图
figure("units","normalized","Position",[0.2359 0.3009 0.4094 0.6037]);
scatter(attrib(:,1),attrib(:,2),10*attrib(:,3),[0 1 0],"filled","Marker","^");
set(gca,"Fontsize",12);
title({"Iris数据集包含150个数据，每个数据含4个属性",...
"第32个属性值扩大后用来决定标志的大小",...
"标志使用自定义的大小和样式"});
xlabel("属性1"); ylabel("属性2");
box on;
set(gcf,"color",[1 1 1],"paperpositionmode","auto");
图 1
2) 散点矩阵图
plotmatrix可以将每两个属性组合的散点图以矩阵的形式绘制出来。通过观察如图2，我们知道，3个各类的花朵不能通过某两个属性就可以加以区分。
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%% 绘制散点矩阵图
% The output parameters have
% A matrix of handles to the objects created in H,
% A matrix of handles to the individual subaxes in AX,
% A handle to a big (invisible) axes that frames the subaxes in BigAx,
% A matrix of handles for the histogram plots in P.
% BigAx is left as the current axes so that a subsequent title, xlabel, or ylabel command is centered with respect to the matrix of axes.
figure("units","normalized","Position",[ 0.2359 0.3009 0.4094 0.6037]);
[H,AX,BigAx,P] = plotmatrix(attrib,"r.");
attribName = {[char(10) "花萼长度"],[char(10) "花萼宽度"],[char(10) "花瓣长度"],[char(10) "花瓣宽度"]};
% 添加标注
for i = 1:4
set(get(AX(i,1),"ylabel"),"string",["属性" num2str(i) attribName{i}]);
set(get(AX(4,i),"xlabel"),"string",["属性" num2str(i) attribName{i}]);
end
set(get(BigAx,"title"),"String",{"散点矩阵图 (数据含3类Iris花)", ...
"如图所示这3类不能通过某2个属性进行区分"},"Fontsize",14);
set(gcf,"color",[1 1 1],"paperpositionmode","auto");
图 2
3) 带直方图的散点图
我们可以在绘制两个属性的散点图的同时，绘制出各属性分布的直方图。(图3)
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%% 在轴线上画出分布图
figure("units","normalized","Position",[0.2589 0.3296 0.3859 0.5750]);
H = scatterhist(attrib(:,2),attrib(:,3),"nbins",50,"direction","out");
mainDataAxes = H(1);
xhistAxes = H(2);
yhistAxes = H(3);
set(get(mainDataAxes,"title"),"String","有一个类别基于如下2个属性是可分的","Fontsize",12);
set(get(mainDataAxes,"xlabel"),"String","x (属性2, 花萼宽度)","Fontsize",14);
set(get(mainDataAxes,"ylabel"),"String","y (属性3, 花瓣长度)","Fontsize",14);
set(gcf,"color",[1 1 1],"paperpositionmode","auto");
图 3
二. 用热图表示数据点分布的密度
(源代码：scattersmooth.m)
当数据量很大时，散点图之间会有显著的重叠，这时密度图往往能够更加清晰的反映出数据的分布情况。我们以用如下方法生成的1000个数据点为例进行说明。
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%% 生成数据
z = [repmat([1 2],1000,1) + randn(1000,2)*[1 .5; 0 1.32];...
repmat([9 1],1000,1) + randn(1000,2)*[1.4 .2; 0 0.98];...
repmat([4 8],1000,1) + randn(1000,2)*[1 .7; 0 0.71];];
如果直接绘制散点图(如图4)，我们会发现象非常严重的重叠问题。
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%% 绘制原始数据
figure("units","normalized","position",[ 0.4458 0.6296 0.1995 0.2759]);
plot(z(:,1),z(:,2),".");
title({"原始数据","散点视图有显著的重叠"});
set(gcf,"color",[1 1 1],"paperpositionmode","auto");
图 4
我们通过将数据进行抽象，绘制密度图来解决上述问题。之前，我们介绍过热图(heat map)的绘制，当时用到的是imagesc和pcolor，而这里我们利用绘制二维曲面的surf命令，接合适当的视角来完成。(图5)
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xx = z(:,1);
yy = z(:,2);
gridSize = 100;
% 选择colormap
colormap(summer);
% 建立网格
x=linspace(min(xx),max(xx),gridSize);
y=linspace(min(yy),max(yy),gridSize);
gridEval = zeros(length(x)-1,length(y)-1);
% 计算每个网格中点的频数
for cnt_x=1:length(x)-1
for cnt_y=1:length(y)-1
x_ind=intersect(find(xx>x(cnt_x)),find(xx<=x(cnt_x+1)));
xy_ind=intersect(find(yy(x_ind)>y(cnt_y)), find(yy(x_ind)<=y(cnt_y+1)));
gridEval(cnt_y, cnt_x)=length(xy_ind);
end
end
% surface函数绘制热图
surf((x(1:end-1)+ x(2:end))/2,(y(1:end-1)+y(2:end))/2,gridEval); view(2);
shading interp; hold on;
axis([min(xx),max(xx) min(yy),max(yy)]);
% 添加标注
title(["密度图, 网格大小: " num2str(gridSize) " x " num2str(gridSize) " 个网格"],"Fontsize",14);
xlabel("x","Fontsize",14);
ylabel("y","Fontsize",14);
axis tight;
h1 = gca; % 保存句柄，以便后面添加边框
% 添加颜色条
h=colorbar;
axes(h);ylabel("密度, 每个网格的点数","Fontsize",14);
set(gcf,"color",[1 1 1],"paperpositionmode","auto");
% 添加黑色的边框
axes(h1);
line(get(gca,"xlim"),repmat(min(get(gca,"ylim")),1,2),"color",[0 0 0],"linewidth",1);
line(get(gca,"xlim"),repmat(max(get(gca,"ylim")),1,2),"color",[0 0 0],"linewidth",2);
line(repmat(min(get(gca,"xlim")),1,2),get(gca,"ylim"),"color",[0 0 0],"linewidth",2);
line(repmat(max(get(gca,"xlim")),1,2),get(gca,"ylim"),"color",[0 0 0],"linewidth",1);
图 5
直接计算密度得到的密度可能会显得不太自然，我们可以第个网格点处对所有数据进行加权，这样便可以成到平滑的作用。(图6)
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xx = z(:,1);
yy = z(:,2);
sigma = 0.1;
gridSize = 100;
% 选择colormap
colormap(summer);
x=linspace(min(xx),max(xx),gridSize);
y=linspace(min(yy),max(yy),gridSize);
gridEval = zeros(length(x)-1,length(y)-1);
% 计算每个点处的高斯函数
for i = 1:length(x)-1
for j = 1:length(y)-1
%calculate a Gaussian function on the grid with each point in the center and add them up
gridEval(j,i) = gridEval(j,i) + sum(exp(-(((x(i)-xx).^2)./(2*sigma.^2) + ((y(j)-yy).^2)./(2*sigma.^2))));
end
end
% 绘制热图
surf((x(1:end-1)+ x(2:end))/2,(y(1:end-1)+y(2:end))/2,gridEval); view(2); shading interp;
axis([min(xx),max(xx) min(yy),max(yy)]);
h1 = gca; % 保存句柄，以便后面添加边框
% 添加标注
title(["平滑散点图, \sigma = " num2str(sigma) ", 网格数： " num2str(gridSize) " x" num2str(gridSize)],"Fontsize",14);
xlabel("x","Fontsize",14);
ylabel("y","Fontsize",14);
% 添加颜色条
h=colorbar;
axes(h);ylabel("Intensity","Fontsize",14);
% 添加黑色的边框
axes(h1);
line(get(gca,"xlim"),repmat(min(get(gca,"ylim")),1,2),"color",[0 0 0],"linewidth",1);
line(get(gca,"xlim"),repmat(max(get(gca,"ylim")),1,2),"color",[0 0 0],"linewidth",2);
line(repmat(min(get(gca,"xlim")),1,2),get(gca,"ylim"),"color",[0 0 0],"linewidth",2);
line(repmat(max(get(gca,"xlim")),1,2),get(gca,"ylim"),"color",[0 0 0],"linewidth",1);
set(gcf,"color",[1 1 1],"paperpositionmode","auto");
图 6
三. 双向误差图(bidirectional error bar)
(源代码：bidirerrorbars.m)
对某些数据而言，X轴和Y轴上的数据都有一定的波动(误差)范围，有时我们需要同时在图上反映出这些波动信息。这称为双向误差图。我们曾通过errorbar命令绘制过Y轴方向上的误差图，在此基础上，我们绘制双向误差图。
(图7)
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%% 加载数据
load flatPlateBoundaryLayerData
xx = laminarFlow(:,2);
yy = laminarFlow(:,3);
% 划定网格
xg=linspace(min(xx),max(xx),6);
yg=linspace(min(yy),max(yy),6);
% 计算各个网格中点的频数，以及它们的均值和方差
for cnt_x=1:length(xg)-1
x_ind=intersect(find(xx>xg(cnt_x)),find(xx<=xg(cnt_x+1)));
x(cnt_x)=mean(xx(x_ind));
e_x(cnt_x)=std(xx(x_ind));
end
for cnt_y=1:length(yg)-1
y_ind=intersect(find(yy>yg(cnt_y)),find(yy<=yg(cnt_y+1)));
y(cnt_y)=mean(yy(y_ind));
e_y(cnt_y)=std(yy(y_ind));
end
figure("units","normalized","Position",[0.0750 0.5157 0.5703 0.3889]);
axes("Position",[0.0676 0.1100 0.8803 0.8150]);
%% 绘制双向误差图
h = biDirErrBar(x,y,e_x,e_y);
%% 标注
set(get(h,"title"),"string","Laminar Flow的双向误差图","Fontsize",15);
set(get(h,"xlabel"),"string","位置测度","Fontsize",15);
set(get(h,"ylabel"),"string","速度测度","Fontsize",15);
set(gcf,"color",[1 1 1],"paperpositionmode","auto");
图 7
其中，自定义函数biDirErrBar在Matlab命令errorbar基础上，添加X轴方向上的误差。定义如下：
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function h = biDirErrBar(x,y,e_x,e_y)
% bidirErrBar(x,y,e_x,e_y) plots bi directional error bars and returns the
% handle to the axes for annotation
% x, y are the data vectors with n elements each. e_x and e_y are the error bars to be
% positioned at each point in xi,yi,
errorbar(x,y,e_y); hold on;
plot(x,y,"Color",[0 0 0]);
x_lower=x-e_x; x_upper=x+e_x;
y_lower=y-.0001*y; y_upper=y+.0001*y;
line([x_lower; x_upper],[y; y],"Color",[0 0 1]);
hold on; line([x_lower; x_lower], [y_lower; y_upper],"Color",[0 0 1]);
hold on; line([x_upper; x_upper], [y_lower; y_upper],"Color",[0 0 1]);
h = gca;
四. 二维关系图
我们知道，在关系矩阵中，1表示连接，0表示非去接。进一步，非零整数可以表示两个结点间的连接数。我们可以利用热图表示出这种扩展的关系。以维克多·雨果《悲剧世界》中人物同时登场的次数数量为实例，我们绘制二维的关系图。(图8)
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%% 加载数据
load characterCoOccurences
%% 设置图像和坐标轴
figure("units","normalized","Position",[ 0.1990 0.1324 0.4854 0.7731]);
mainAx = axes("position",[ 0.1361 0.0143 0.8042 0.8038]);
% 设置colormap。由于关系图是稀疏的，因此这里我们反转colormap的顺序
m = colormap(copper);
m = m(end:-1:1,:);
colormap(m);
% 用surf创建热图
r=surf(lesMiserables);view(2);
% 将edgecolor设置为半透明
set(r,"edgealpha",0.2);
set(gca,"clim",[min(lesMiserables(:)) max(lesMiserables(:))]);
% 在上方添加颜色条
h=colorbar("northoutside");
% 添加刻度标签
set(mainAx,"xAxisLocation","top","xtick",0:78,"xticklabel",{" " LABELS{:} " "},...
"ytick",0:78,"yticklabel",{" " LABELS{:} " "},"ticklength",[0 0],"fontsize",8);
axis tight;
rotateXLabels(gca,90);
% 调整颜色条和坐标轴的位置，因为他们会受琶刻度标签旋转的影响
set(h,"position",[0.1006 0.9409 0.8047 0.0128]);
set(get(h,"title"),"String","维克多·雨果《悲剧世界》中人物同时出场次数","Fontsize",14);
set(mainAx,"position",[ 0.1361 0.0143 0.8042 0.8038]);
set(gcf,"color",[1 1 1],"paperpositionmode","auto");
box on;
图 8
五. 绘制系统树图(dendrogram)
(源代码：dendrocluster.m)
图 9
我们首先利用pdist命令得到基因之间的两两距离。然后，linkage命令通过聚类的方法，得到层级树形层级结构。层级结构的可视化则由dendrogram命令完成。
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%% 加载数据
load 14cancer.mat
data = [Xtrain(find(ytrainLabels==9),genesSet); Xtest(find(ytestLabels==9),genesSet)];
%% 癌症样本及100个感兴趣的基因绘制系统树图
Z_genes = linkage(pdist(data"));
Z_samples = linkage(pdist(data));
%% 设定图像位置
figure("units","normalized","Position",[0.5641 0.2407 0.3807 0.6426]);
mainPanel = axes("Position",[.25 .08 .69 .69]);
leftPanel = axes("Position",[.08 .08 .17 .69]);
topPanel = axes("Position",[.25 .77 .69 .21]);
%% 较低的值颜色较浅
m = colormap(pink); m = m(end:-1:1,:);
colormap(m);
%% 绘制系统树图 - 展示所有基因的节点(如果没有参数“0”，则默认值为最多30个节点)
axes(leftPanel);h = dendrogram(Z_samples,"orient","left"); set(h,"color",[0.1179 0 0],"linewidth",2);
axes(topPanel); h = dendrogram(Z_genes,0);set(h,"color",[0.1179 0 0],"linewidth",2);
%% 获取系统树图和热图数据生成的次序
Z_samples_order = str2num(get(leftPanel,"yticklabel"));
Z_genes_order = str2num(get(topPanel,"xticklabel"));
axes(mainPanel);
surf(data(Z_samples_order,Z_genes_order),"edgecolor",[.8 .8 .8]);view(2);
set(mainPanel,"Xticklabel",[],"yticklabel",[]);
%% 对齐X轴和Y轴
set(leftPanel,"ylim",[1 size(data,1)],"Visible","Off");
set(topPanel,"xlim",[1 size(data,2)],"Visible","Off");
axes(mainPanel);axis([1 size(data,2) 1 size(data,1)]);
%% 添加标注
axes(mainPanel); xlabel("30个不同的基因","Fontsize",14);
colorbar("Location","northoutside","Position",[ 0.0584 0.8761 0.3082 0.0238]);
annotation("textbox",[.5 .87 .4 .1],"String",{"基因表达水平", "白血病"},"Linestyle","none","fontsize",14);
%% 在坐标轴不可见的情况下，显示标签
set(leftPanel,"yaxislocation","left");
set(get(leftPanel,"ylabel"),"string","样本","Fontsize",14);
set(findall(leftPanel, "type", "text"), "visible", "on");
set(gcf,"color",[1 1 1],"paperpositionmode","auto");
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