利用Python科学计算处理物理问题（和物理告个别）

import numpy as np
from matplotlib import pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
q1 = np.arange(0.01, 1, 0.01)
q2 = np.arange(0.01, 1 , 0.01) //生成一位基底
q1, q2 = np.meshgrid(q1, q2) // 混合成二维数组，形成二维基底pCDa = (1-q1)
pCDb = (np.sqrt((1-q1)**2+q1**2)-q1)
alpha = (pCDa + pCDb) / (pCDa + 4 * pCDb)
beta = pCDb / (pCDa + 4 * pCDb)
pCDp00 = ( q1* pCDa ** 2 ) / ( pCDa*pCDa +pCDb*pCDb)
pCDp10 = ( q1* pCDb ** 2 ) / ( pCDa*pCDa +pCDb*pCDb)
pCDp01 = ( (1-q1) / 2 ) * ( pCDa + pCDb ) ** 2 / ( pCDa*pCDa +pCDb*pCDb)
pCDp11 = ( (1-q1) / 2 ) * ( pCDa - pCDb ) ** 2 / ( pCDa*pCDa +pCDb*pCDb)
s_x_pCD= -pCDp00 * np.log2(pCDp00) - pCDp01 * np.log2(pCDp01) - pCDp10 * np.log2(pCDp10) - pCDp11 * np.log2(pCDp11)
s_pCD = -q1* np.log2(q1) - (1-q1) * np.log2(1-q1)
Q_MID1 = (s_x_pCD - s_pCD) *q2 /q2 #AB或CD的关联值fig = plt.figure()
ax = Axes3D(fig)
ax.plot_surface(q1,q2,Q_MID1)
ax.set_xlabel('value of q2')
ax.set_ylabel('value of q1')
ax.set_zlabel('the value of Q_MID1(pCD)')
plt.show()

我们希望求这个交点，算是求解非线性方程 f(x) = 0。

尝试用牛顿拉普森方法求解。（其实就是将试位法斜率用 f ' (x) 代替）

看起来还不错，我们试着放到项目上

割线法

plt.text(0,1,r'$max\ is\ 1.34$') //在0,1处标注最大值是1.34plt.ylabel('value of Q_MID ')
plt.xlabel('value of q1') //对x,y轴进行说明plt.title('Q_MID when q1=q2') //标题名称# 设置xtick和ytick的方向：in、out
plt.rcParams['xtick.direction'] = 'in'
plt.rcParams['ytick.direction'] = 'in'#设置横纵坐标的名称以及对应字体格式
font2 =
{'family' : 'Times New Roman',
'weight' : 'normal',
'size' : 14,
}
plt.ylabel('MID ', font2)

https://matplotlib.org/gallery/index.html //matplotlib官网文档，有着大量实例可以参考

https://mp.weixin.qq.com/s?__biz=MzU4NjIxODMyOQ==&mid=2247488568&idx=6&sn=f80d6f5c540058aa6c0d5de97e3a8f1d&chksm=fdfffa0eca887318bc426ef057707f7271624fb39a44595db8a042503a82ff638d707760ce02&mpshare=1&scene=23&srcid=&sharer_sharetime=1588477717826&sharer_shareid=aa22f4d6a3d78a43601dd5a37b40202c#rd

1. 散点图
Scatteplot是用于研究两个变量之间关系的经典和基本图。如果数据中有多个组，则可能需要以不同颜色可视化每个组。在Matplotlib，你可以方便地使用。# Import dataset
midwest = pd.read_csv("https://raw.githubusercontent.com/selva86/datasets/master/midwest_filter.csv")# Prepare Data
# Create as many colors as there are unique midwest['category']
categories = np.unique(midwest['category'])
colors = [plt.cm.tab10(i/float(len(categories)-1)) for i in range(len(categories))]# Draw Plot for Each Category
plt.figure(figsize=(16, 10), dpi= 80, facecolor='w', edgecolor='k')for i, category in enumerate(categories):
plt.scatter('area', 'poptotal',
data=midwest.loc[midwest.category==category, :],
s=20, c=colors[i], label=str(category))# Decorations
plt.gca().set(xlim=(0.0, 0.1), ylim=(0, 90000),
xlabel='Area', ylabel='Population')plt.xticks(fontsize=12); plt.yticks(fontsize=12)
plt.title("Scatterplot of Midwest Area vs Population", fontsize=22)
plt.legend(fontsize=12)
plt.show()

2. 带边界的气泡图
有时，您希望在边界内显示一组点以强调其重要性。在此示例中，您将从应该被环绕的数据帧中获取记录，并将其传递给下面的代码中描述的记录。encircle()from matplotlib import patches
from scipy.spatial import ConvexHull
import warnings; warnings.simplefilter('ignore')
sns.set_style("white")# Step 1: Prepare Data
midwest = pd.read_csv("https://raw.githubusercontent.com/selva86/datasets/master/midwest_filter.csv")# As many colors as there are unique midwest['category']
categories = np.unique(midwest['category'])
colors = [plt.cm.tab10(i/float(len(categories)-1)) for i in range(len(categories))]# Step 2: Draw Scatterplot with unique color for each category
fig = plt.figure(figsize=(16, 10), dpi= 80, facecolor='w', edgecolor='k')for i, category in enumerate(categories):
plt.scatter('area', 'poptotal', data=midwest.loc[midwest.category==category, :], s='dot_size', c=colors[i], label=str(category), edgecolors='black', linewidths=.5)# Step 3: Encircling
# https://stackoverflow.com/questions/44575681/how-do-i-encircle-different-data-sets-in-scatter-plot
def encircle(x,y, ax=None, **kw):
if not ax: ax=plt.gca()
p = np.c_[x,y]
hull = ConvexHull(p)
poly = plt.Polygon(p[hull.vertices,:], **kw)
ax.add_patch(poly)# Select data to be encircled
midwest_encircle_data = midwest.loc[midwest.state=='IN', :]# Draw polygon surrounding vertices
encircle(midwest_encircle_data.area, midwest_encircle_data.poptotal, ec="k", fc="gold", alpha=0.1)
encircle(midwest_encircle_data.area, midwest_encircle_data.poptotal, ec="firebrick", fc="none", linewidth=1.5)# Step 4: Decorations
plt.gca().set(xlim=(0.0, 0.1), ylim=(0, 90000),
xlabel='Area', ylabel='Population')plt.xticks(fontsize=12); plt.yticks(fontsize=12)
plt.title("Bubble Plot with Encircling", fontsize=22)
plt.legend(fontsize=12)
plt.show()

3. 带线性回归最佳拟合线的散点图
如果你想了解两个变量如何相互改变，那么最合适的线就是要走的路。下图显示了数据中各组之间最佳拟合线的差异。要禁用分组并仅为整个数据集绘制一条最佳拟合线，请从下面的调用中删除该参数。# Import Data
df = pd.read_csv("https://raw.githubusercontent.com/selva86/datasets/master/mpg_ggplot2.csv")
df_select = df.loc[df.cyl.isin([4,8]), :]# Plot
sns.set_style("white")
gridobj = sns.lmplot(x="displ", y="hwy", hue="cyl", data=df_select,
height=7, aspect=1.6, robust=True, palette='tab10',
scatter_kws=dict(s=60, linewidths=.7, edgecolors='black'))# Decorations
gridobj.set(xlim=(0.5, 7.5), ylim=(0, 50))
plt.title("Scatterplot with line of best fit grouped by number of cylinders",

每个回归线都在自己的列中
或者，您可以在其自己的列中显示每个组的最佳拟合线。你可以通过在里面设置参数来实现这一点。# Import Data
df = pd.read_csv("https://raw.githubusercontent.com/selva86/datasets/master/mpg_ggplot2.csv")
df_select = df.loc[df.cyl.isin([4,8]), :]# Each line in its own column
sns.set_style("white")
gridobj = sns.lmplot(x="displ", y="hwy",
data=df_select,
height=7,
robust=True,
palette='Set1',
col="cyl",
scatter_kws=dict(s=60, linewidths=.7, edgecolors='black'))# Decorations
gridobj.set(xlim=(0.5, 7.5), ylim=(0, 50))
plt.show()

4. 抖动图
通常，多个数据点具有完全相同的X和Y值。结果，多个点相互绘制并隐藏。为避免这种情况，请稍微抖动点，以便您可以直观地看到它们。这很方便使用# Import Data
df = pd.read_csv("https://raw.githubusercontent.com/selva86/datasets/master/mpg_ggplot2.csv")# Draw Stripplot
fig, ax = plt.subplots(figsize=(16,10), dpi= 80)
sns.stripplot(df.cty, df.hwy, jitter=0.25, size=8, ax=ax, linewidth=.5)# Decorations
plt.title('Use jittered plots to avoid overlapping of points', fontsize=22)
plt.show()

5. 计数图
避免点重叠问题的另一个选择是增加点的大小，这取决于该点中有多少点。因此，点的大小越大，周围的点的集中度就越大。# Import Data
df = pd.read_csv("https://raw.githubusercontent.com/selva86/datasets/master/mpg_ggplot2.csv")
df_counts = df.groupby(['hwy', 'cty']).size().reset_index(name='counts')# Draw Stripplot
fig, ax = plt.subplots(figsize=(16,10), dpi= 80)
sns.stripplot(df_counts.cty, df_counts.hwy, size=df_counts.counts*2, ax=ax)# Decorations
plt.title('Counts Plot - Size of circle is bigger as more points overlap', fontsize=22)
plt.show()

6. 边缘直方图
边缘直方图具有沿X和Y轴变量的直方图。这用于可视化X和Y之间的关系以及单独的X和Y的单变量分布。该图如果经常用于探索性数据分析（EDA）。# Import Data
df = pd.read_csv("https://raw.githubusercontent.com/selva86/datasets/master/mpg_ggplot2.csv")# Create Fig and gridspec
fig = plt.figure(figsize=(16, 10), dpi= 80)
grid = plt.GridSpec(4, 4, hspace=0.5, wspace=0.2)# Define the axes
ax_main = fig.add_subplot(grid[:-1, :-1])
ax_right = fig.add_subplot(grid[:-1, -1], xticklabels=[], yticklabels=[])
ax_bottom = fig.add_subplot(grid[-1, 0:-1], xticklabels=[], yticklabels=[])# Scatterplot on main ax
ax_main.scatter('displ', 'hwy', s=df.cty*4, c=df.manufacturer.astype('category').cat.codes, alpha=.9, data=df, cmap="tab10", edgecolors='gray', linewidths=.5)# histogram on the right
ax_bottom.hist(df.displ, 40, histtype='stepfilled', orientation='vertical', color='deeppink')
ax_bottom.invert_yaxis()# histogram in the bottom
ax_right.hist(df.hwy, 40, histtype='stepfilled', orientation='horizontal', color='deeppink')# Decorations
ax_main.set(title='Scatterplot with Histograms displ vs hwy', xlabel='displ', ylabel='hwy')
ax_main.title.set_fontsize(20)
for item in ([ax_main.xaxis.label, ax_main.yaxis.label] + ax_main.get_xticklabels() + ax_main.get_yticklabels()):
item.set_fontsize(14)xlabels = ax_main.get_xticks().tolist()
ax_main.set_xticklabels(xlabels)
plt.show()

相关图
Correlogram用于直观地查看给定数据帧（或2D数组）中所有可能的数值变量对之间的相关度量。# Import Dataset
df = pd.read_csv("https://github.com/selva86/datasets/raw/master/mtcars.csv")# Plot
plt.figure(figsize=(12,10), dpi= 80)
sns.heatmap(df.corr(), xticklabels=df.corr().columns, yticklabels=df.corr().columns, cmap='RdYlGn', center=0, annot=True)# Decorations
plt.title('Correlogram of mtcars', fontsize=22)
plt.xticks(fontsize=12)
plt.yticks(fontsize=12)
plt.show()

9. 矩阵图
成对图是探索性分析中的最爱，以理解所有可能的数字变量对之间的关系。它是双变量分析的必备工具。# Load Dataset
df = sns.load_dataset('iris')# Plot
plt.figure(figsize=(10,8), dpi= 80)
sns.pairplot(df, kind="scatter", hue="species", plot_kws=dict(s=80, edgecolor="white", linewidth=2.5))
plt.show()

# Prepare Data
df = pd.read_csv("https://github.com/selva86/datasets/raw/master/mtcars.csv")
x = df.loc[:, ['mpg']]
df['mpg_z'] = (x - x.mean())/x.std()
df['colors'] = ['red' if x < 0 else 'green' for x in df['mpg_z']]
df.sort_values('mpg_z', inplace=True)
df.reset_index(inplace=True)# Draw plot
plt.figure(figsize=(14,10), dpi= 80)
plt.hlines(y=df.index, xmin=0, xmax=df.mpg_z, color=df.colors, alpha=0.4, linewidth=5)# Decorations
plt.gca().set(ylabel='$Model$', xlabel='$Mileage$')
plt.yticks(df.index, df.cars, fontsize=12)
plt.title('Diverging Bars of Car Mileage', fontdict={'size':20})
plt.grid(linestyle='--', alpha=0.5)
plt.show()

发散型文本分散的文本类似于发散条，如果你想以一种漂亮和可呈现的方式显示图表中每个项目的价值，它更喜欢。# Prepare Data
df = pd.read_csv("https://github.com/selva86/datasets/raw/master/mtcars.csv")
x = df.loc[:, ['mpg']]
df['mpg_z'] = (x - x.mean())/x.std()
df['colors'] = ['red' if x < 0 else 'green' for x in df['mpg_z']]
df.sort_values('mpg_z', inplace=True)
df.reset_index(inplace=True)# Draw plot
plt.figure(figsize=(14,14), dpi= 80)
plt.hlines(y=df.index, xmin=0, xmax=df.mpg_z)
for x, y, tex in zip(df.mpg_z, df.index, df.mpg_z):
t = plt.text(x, y, round(tex, 2), horizontalalignment='right' if x < 0 else 'left',
verticalalignment='center', fontdict={'color':'red' if x < 0 else 'green', 'size':14})# Decorations
plt.yticks(df.index, df.cars, fontsize=12)
plt.title('Diverging Text Bars of Car Mileage', fontdict={'size':20})
plt.grid(linestyle='--', alpha=0.5)
plt.xlim(-2.5, 2.5)
plt.show()

面积图通过对轴和线之间的区域进行着色，区域图不仅强调峰值和低谷，而且还强调高点和低点的持续时间。高点持续时间越长，线下面积越大。import numpy as np
import pandas as pd# Prepare Data
df = pd.read_csv("https://github.com/selva86/datasets/raw/master/economics.csv", parse_dates=['date']).head(100)
x = np.arange(df.shape[0])
y_returns = (df.psavert.diff().fillna(0)/df.psavert.shift(1)).fillna(0) * 100# Plot
plt.figure(figsize=(16,10), dpi= 80)
plt.fill_between(x[1:], y_returns[1:], 0, where=y_returns[1:] >= 0, facecolor='green', interpolate=True, alpha=0.7)
plt.fill_between(x[1:], y_returns[1:], 0, where=y_returns[1:] <= 0, facecolor='red', interpolate=True, alpha=0.7)# Annotate
plt.annotate('Peak 1975', xy=(94.0, 21.0), xytext=(88.0, 28),
bbox=dict(boxstyle='square', fc='firebrick'),
arrowprops=dict(facecolor='steelblue', shrink=0.05), fontsize=15, color='white')# Decorations
xtickvals = [str(m)[:3].upper()+"-"+str(y) for y,m in zip(df.date.dt.year, df.date.dt.month_name())]
plt.gca().set_xticks(x[::6])
plt.gca().set_xticklabels(xtickvals[::6], rotation=90, fontdict={'horizontalalignment': 'center', 'verticalalignment': 'center_baseline'})
plt.ylim(-35,35)
plt.xlim(1,100)
plt.title("Month Economics Return %", fontsize=22)
plt.ylabel('Monthly returns %')
plt.grid(alpha=0.5)
plt.show()

有序条形图有序条形图有效地传达了项目的排名顺序。但是，在图表上方添加度量标准的值，用户可以从图表本身获取精确信息。# Prepare Data
df_raw = pd.read_csv("https://github.com/selva86/datasets/raw/master/mpg_ggplot2.csv")
df = df_raw[['cty', 'manufacturer']].groupby('manufacturer').apply(lambda x: x.mean())
df.sort_values('cty', inplace=True)
df.reset_index(inplace=True)# Draw plot
import matplotlib.patches as patchesfig, ax = plt.subplots(figsize=(16,10), facecolor='white', dpi= 80)
ax.vlines(x=df.index, ymin=0, ymax=df.cty, color='firebrick', alpha=0.7, linewidth=20)# Annotate Text
for i, cty in enumerate(df.cty):
ax.text(i, cty+0.5, round(cty, 1), horizontalalignment='center')# Title, Label, Ticks and Ylim
ax.set_title('Bar Chart for Highway Mileage', fontdict={'size':22})
ax.set(ylabel='Miles Per Gallon', ylim=(0, 30))
plt.xticks(df.index, df.manufacturer.str.upper(), rotation=60, horizontalalignment='right', fontsize=12)# Add patches to color the X axis labels
p1 = patches.Rectangle((.57, -0.005), width=.33, height=.13, alpha=.1, facecolor='green', transform=fig.transFigure)
p2 = patches.Rectangle((.124, -0.005), width=.446, height=.13, alpha=.1, facecolor='red', transform=fig.transFigure)
fig.add_artist(p1)
fig.add_artist(p2)
plt.show()

密度图密度图是一种常用工具，可视化连续变量的分布。通过“响应”变量对它们进行分组，您可以检查X和Y之间的关系。以下情况，如果出于代表性目的来描述城市里程的分布如何随着汽缸数的变化而变化。# Import Data
df = pd.read_csv("https://github.com/selva86/datasets/raw/master/mpg_ggplot2.csv")# Draw Plot
plt.figure(figsize=(16,10), dpi= 80)
sns.kdeplot(df.loc[df['cyl'] == 4, "cty"], shade=True, color="g", label="Cyl=4", alpha=.7)
sns.kdeplot(df.loc[df['cyl'] == 5, "cty"], shade=True, color="deeppink", label="Cyl=5", alpha=.7)
sns.kdeplot(df.loc[df['cyl'] == 6, "cty"], shade=True, color="dodgerblue", label="Cyl=6", alpha=.7)
sns.kdeplot(df.loc[df['cyl'] == 8, "cty"], shade=True, color="orange", label="Cyl=8", alpha=.7)# Decoration
plt.title('Density Plot of City Mileage by n_Cylinders', fontsize=22)
plt.legend()

直方密度线图带有直方图的密度曲线将两个图表传达的集体信息汇集在一起，这样您就可以将它们放在一个图形而不是两个图形中。# Import Data
df = pd.read_csv("https://github.com/selva86/datasets/raw/master/mpg_ggplot2.csv")# Draw Plot
plt.figure(figsize=(13,10), dpi= 80)
sns.distplot(df.loc[df['class'] == 'compact', "cty"], color="dodgerblue", label="Compact", hist_kws={'alpha':.7}, kde_kws={'linewidth':3})
sns.distplot(df.loc[df['class'] == 'suv', "cty"], color="orange", label="SUV", hist_kws={'alpha':.7}, kde_kws={'linewidth':3})
sns.distplot(df.loc[df['class'] == 'minivan', "cty"], color="g", label="minivan", hist_kws={'alpha':.7}, kde_kws={'linewidth':3})
plt.ylim(0, 0.35)# Decoration
plt.title('Density Plot of City Mileage by Vehicle Type', fontsize=22)
plt.legend()
plt.show()