我们的大脑通常最多能感知三维空间，超过三维就很难想象了。尽管是三维，理解起来也很费劲，所以大多数情况下都使用二维平面。

<tt-image data-tteditor-tag="tteditorTag" contenteditable="false" class="syl1561879714051 ql-align-center" data-render-status="finished" data-syl-blot="image" style="box-sizing: border-box; cursor: text; text-align: left; color: rgb(34, 34, 34); font-family: "PingFang SC", "Hiragino Sans GB", "Microsoft YaHei", "WenQuanYi Micro Hei", "Helvetica Neue", Arial, sans-serif; font-size: 16px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-indent: 0px; text-transform: none; white-space: pre-wrap; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255); text-decoration-style: initial; text-decoration-color: initial; display: block;"> image

<input class="pgc-img-caption-ipt" placeholder="图片描述(最多50字)" value="" style="box-sizing: border-box; outline: 0px; color: rgb(102, 102, 102); position: absolute; left: 187.5px; transform: translateX(-50%); padding: 6px 7px; max-width: 100%; width: 375px; text-align: center; cursor: text; font-size: 12px; line-height: 1.5; background-color: rgb(255, 255, 255); background-image: none; border: 0px solid rgb(217, 217, 217); border-radius: 4px; transition: all 0.2s cubic-bezier(0.645, 0.045, 0.355, 1) 0s;"></tt-image>

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不过，我们仍然可以绘制出多维空间，今天就来用 Python 的 plotly 库绘制下三维到六维的图，看看长什么样。

数据我们使用一份来自 UCI 的真实汽车数据集，该数据集包括 205 个样本和 26 个特征，从中选择 6 个特征来绘制图形：

<tt-image data-tteditor-tag="tteditorTag" contenteditable="false" class="syl1561879714059 ql-align-center" data-render-status="finished" data-syl-blot="image" style="box-sizing: border-box; cursor: text; text-align: left; color: rgb(34, 34, 34); font-family: "PingFang SC", "Hiragino Sans GB", "Microsoft YaHei", "WenQuanYi Micro Hei", "Helvetica Neue", Arial, sans-serif; font-size: 16px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-indent: 0px; text-transform: none; white-space: pre-wrap; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255); text-decoration-style: initial; text-decoration-color: initial; display: block;"> image

<input class="pgc-img-caption-ipt" placeholder="图片描述(最多50字)" value="" style="box-sizing: border-box; outline: 0px; color: rgb(102, 102, 102); position: absolute; left: 187.5px; transform: translateX(-50%); padding: 6px 7px; max-width: 100%; width: 375px; text-align: center; cursor: text; font-size: 12px; line-height: 1.5; background-color: rgb(255, 255, 255); background-image: none; border: 0px solid rgb(217, 217, 217); border-radius: 4px; transition: all 0.2s cubic-bezier(0.645, 0.045, 0.355, 1) 0s;"></tt-image>

基础工作

安装好 plotly 包：

<pre spellcheck="false" style="box-sizing: border-box; margin: 5px 0px; padding: 5px 10px; border: 0px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-variant-numeric: inherit; font-variant-east-asian: inherit; font-weight: 400; font-stretch: inherit; font-size: 16px; line-height: inherit; font-family: inherit; vertical-align: baseline; cursor: text; counter-reset: list-1 0 list-2 0 list-3 0 list-4 0 list-5 0 list-6 0 list-7 0 list-8 0 list-9 0; background-color: rgb(240, 240, 240); border-radius: 3px; white-space: pre-wrap; color: rgb(34, 34, 34); letter-spacing: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-style: initial; text-decoration-color: initial;">pip install plotly
</pre>

加载数据集（文末会提供）：

<pre spellcheck="false" style="box-sizing: border-box; margin: 5px 0px; padding: 5px 10px; border: 0px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-variant-numeric: inherit; font-variant-east-asian: inherit; font-weight: 400; font-stretch: inherit; font-size: 16px; line-height: inherit; font-family: inherit; vertical-align: baseline; cursor: text; counter-reset: list-1 0 list-2 0 list-3 0 list-4 0 list-5 0 list-6 0 list-7 0 list-8 0 list-9 0; background-color: rgb(240, 240, 240); border-radius: 3px; white-space: pre-wrap; color: rgb(34, 34, 34); letter-spacing: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-style: initial; text-decoration-color: initial;">import pandas as pd
data = pd.read_csv("cars.csv")
</pre>

下面我们先绘制基础的二维图表，使用两个 RPM 和 Speed 两个特征即可：

绘制 2-D 图

<tt-image data-tteditor-tag="tteditorTag" contenteditable="false" class="syl1561879714067" data-render-status="finished" data-syl-blot="image" style="box-sizing: border-box; cursor: text; color: rgb(34, 34, 34); font-family: "PingFang SC", "Hiragino Sans GB", "Microsoft YaHei", "WenQuanYi Micro Hei", "Helvetica Neue", Arial, sans-serif; font-size: 16px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: pre-wrap; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255); text-decoration-style: initial; text-decoration-color: initial; display: block;"> image

<input class="pgc-img-caption-ipt" placeholder="图片描述(最多50字)" value="" style="box-sizing: border-box; outline: 0px; color: rgb(102, 102, 102); position: absolute; left: 187.5px; transform: translateX(-50%); padding: 6px 7px; max-width: 100%; width: 375px; text-align: center; cursor: text; font-size: 12px; line-height: 1.5; background-color: rgb(255, 255, 255); background-image: none; border: 0px solid rgb(217, 217, 217); border-radius: 4px; transition: all 0.2s cubic-bezier(0.645, 0.045, 0.355, 1) 0s;"></tt-image>

代码实现如下：

<pre spellcheck="false" style="box-sizing: border-box; margin: 5px 0px; padding: 5px 10px; border: 0px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-variant-numeric: inherit; font-variant-east-asian: inherit; font-weight: 400; font-stretch: inherit; font-size: 16px; line-height: inherit; font-family: inherit; vertical-align: baseline; cursor: text; counter-reset: list-1 0 list-2 0 list-3 0 list-4 0 list-5 0 list-6 0 list-7 0 list-8 0 list-9 0; background-color: rgb(240, 240, 240); border-radius: 3px; white-space: pre-wrap; color: rgb(34, 34, 34); letter-spacing: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-style: initial; text-decoration-color: initial;">import plotly
import plotly.graph_objs as go

绘制散点图

fig1 = go.Scatter(x=data['curb-weight'],
y=data['price'],
mode='markers')

绘制布局

mylayout = go.Layout(xaxis=dict(title="curb-weight"),
yaxis=dict( title="price"))

绘图 html

plotly.offline.plot({"data": [fig1],
"layout": mylayout},
auto_open=True)
</pre>

保存为 html 文件打开可以生成交互界面，也可以保存为 png 图片。

下面增加特征来绘制三维图。

绘制 3-D 图

可以使用 plotly 的 plot.Scatter3D 方法绘制三维图：

<tt-image data-tteditor-tag="tteditorTag" contenteditable="false" class="syl1561879714081" data-render-status="finished" data-syl-blot="image" style="box-sizing: border-box; cursor: text; color: rgb(34, 34, 34); font-family: "PingFang SC", "Hiragino Sans GB", "Microsoft YaHei", "WenQuanYi Micro Hei", "Helvetica Neue", Arial, sans-serif; font-size: 16px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: pre-wrap; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255); text-decoration-style: initial; text-decoration-color: initial; display: block;"> image

<input class="pgc-img-caption-ipt" placeholder="图片描述(最多50字)" value="" style="box-sizing: border-box; outline: 0px; color: rgb(102, 102, 102); position: absolute; left: 187.5px; transform: translateX(-50%); padding: 6px 7px; max-width: 100%; width: 375px; text-align: center; cursor: text; font-size: 12px; line-height: 1.5; background-color: rgb(255, 255, 255); background-image: none; border: 0px solid rgb(217, 217, 217); border-radius: 4px; transition: all 0.2s cubic-bezier(0.645, 0.045, 0.355, 1) 0s;"></tt-image>

代码实现如下：

<pre spellcheck="false" style="box-sizing: border-box; margin: 5px 0px; padding: 5px 10px; border: 0px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-variant-numeric: inherit; font-variant-east-asian: inherit; font-weight: 400; font-stretch: inherit; font-size: 16px; line-height: inherit; font-family: inherit; vertical-align: baseline; cursor: text; counter-reset: list-1 0 list-2 0 list-3 0 list-4 0 list-5 0 list-6 0 list-7 0 list-8 0 list-9 0; background-color: rgb(240, 240, 240); border-radius: 3px; white-space: pre-wrap; color: rgb(34, 34, 34); letter-spacing: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-style: initial; text-decoration-color: initial;">fig1 = go.Scatter3d(x=data['curb-weight'],
y=data['horsepower'],
z=data['price'],
marker=dict(opacity=0.9,
reversescale=True,
colorscale='Blues',
size=5),
line=dict (width=0.02),
mode='markers')
mylayout = go.Layout(scene=dict(xaxis=dict( title="curb-weight"),
yaxis=dict( title="horsepower"),
zaxis=dict(title="price")),)
plotly.offline.plot({"data": [fig1],
"layout": mylayout},
auto_open=True,
filename=("3DPlot.html"))
</pre>

如何绘制更高维度的图呢？显然无法通过扩展坐标轴的形式，不过有个小技巧就是制造一个虚拟维度，可以用不同颜色、形状大小、形状类别来入手。这样就可以显示第四个维度了。

绘制 4-D 图

下面我们将第四个变量——车辆油耗（city-mpg）添加到原先的三维图中，用颜色深浅表示，这样就绘制出了四维图。可以看到当其他三个指标（马力、车身重量、车价格）越高时：车辆油耗是越少的。

<tt-image data-tteditor-tag="tteditorTag" contenteditable="false" class="syl1561879714102" data-render-status="finished" data-syl-blot="image" style="box-sizing: border-box; cursor: text; color: rgb(34, 34, 34); font-family: "PingFang SC", "Hiragino Sans GB", "Microsoft YaHei", "WenQuanYi Micro Hei", "Helvetica Neue", Arial, sans-serif; font-size: 16px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: pre-wrap; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255); text-decoration-style: initial; text-decoration-color: initial; display: block;"> image

<input class="pgc-img-caption-ipt" placeholder="图片描述(最多50字)" value="" style="box-sizing: border-box; outline: 0px; color: rgb(102, 102, 102); position: absolute; left: 187.5px; transform: translateX(-50%); padding: 6px 7px; max-width: 100%; width: 375px; text-align: center; cursor: text; font-size: 12px; line-height: 1.5; background-color: rgb(255, 255, 255); background-image: none; border: 0px solid rgb(217, 217, 217); border-radius: 4px; transition: all 0.2s cubic-bezier(0.645, 0.045, 0.355, 1) 0s;"></tt-image>

绘制 5-D 图

基于这样的思想，我们还可以通过修改圆形大小再增加一个维度——发动机尺寸（engine-size）变成五维图：

<tt-image data-tteditor-tag="tteditorTag" contenteditable="false" class="syl1561879714111" data-render-status="finished" data-syl-blot="image" style="box-sizing: border-box; cursor: text; color: rgb(34, 34, 34); font-family: "PingFang SC", "Hiragino Sans GB", "Microsoft YaHei", "WenQuanYi Micro Hei", "Helvetica Neue", Arial, sans-serif; font-size: 16px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: pre-wrap; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255); text-decoration-style: initial; text-decoration-color: initial; display: block;"> image

<input class="pgc-img-caption-ipt" placeholder="图片描述(最多50字)" value="" style="box-sizing: border-box; outline: 0px; color: rgb(102, 102, 102); position: absolute; left: 187.5px; transform: translateX(-50%); padding: 6px 7px; max-width: 100%; width: 375px; text-align: center; cursor: text; font-size: 12px; line-height: 1.5; background-color: rgb(255, 255, 255); background-image: none; border: 0px solid rgb(217, 217, 217); border-radius: 4px; transition: all 0.2s cubic-bezier(0.645, 0.045, 0.355, 1) 0s;"></tt-image>

我们仍然可以比较容易地地发现：车越贵，发动机尺寸越大这样的规律。

绘制 6-D 图

接着还可以通过更改形状的方式增加第六个维度——车门数，圆形表示四车门，方形表示两车门。通常两个车门的都是昂贵的豪华跑车，在图中也可以看出方形主要集中在价格比较高的区域。

<tt-image data-tteditor-tag="tteditorTag" contenteditable="false" class="syl1561879714117" data-render-status="finished" data-syl-blot="image" style="box-sizing: border-box; cursor: text; color: rgb(34, 34, 34); font-family: "PingFang SC", "Hiragino Sans GB", "Microsoft YaHei", "WenQuanYi Micro Hei", "Helvetica Neue", Arial, sans-serif; font-size: 16px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: pre-wrap; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255); text-decoration-style: initial; text-decoration-color: initial; display: block;"> image

<input class="pgc-img-caption-ipt" placeholder="图片描述(最多50字)" value="" style="box-sizing: border-box; outline: 0px; color: rgb(102, 102, 102); position: absolute; left: 187.5px; transform: translateX(-50%); padding: 6px 7px; max-width: 100%; width: 375px; text-align: center; cursor: text; font-size: 12px; line-height: 1.5; background-color: rgb(255, 255, 255); background-image: none; border: 0px solid rgb(217, 217, 217); border-radius: 4px; transition: all 0.2s cubic-bezier(0.645, 0.045, 0.355, 1) 0s;"></tt-image>

这样我们就从普通的二维图扩展到了高维图，当然还可以继续拓展，不过分辨起来会越来越困难。