# 六、numpy的常用函数

1. 读取文件

• 逻辑上可被解释为二维数组的文本文件：
  • 数据项1<分隔符>数据项2<分隔符>...<分隔符>数据项n

    numpy.loadtxt(
    文件路径,
    delimiter=分隔符(缺省一个空格),
    usecols=列序列(缺省所有列),
    unpack=是否展开列(缺省False), # 返回一个二维数组(False)或多个一维数组(True)

    dtype=元素类型(缺省float),      converters=转换器字典(缺省不做转换))

# k.pyimport numpy as npimport datetime as dtimport matplotlib.pyplot as mpimport matplotlib.dates as mddef dmy2ymd(dmy):    dmy = str(dmy, encoding='utf-8')    date = dt.datetime.strptime(dmy, '%d-%m-%Y').date()    ymd = date.strftime('%Y-%m-%d')    return ymddates, opening_prices, highest_prices, lowest_prices, closing_prices \= np.loadtxt('../data/aapl.csv', delimiter=',', usecols=(1, 3, 4, 5, 6), unpack=True, dtype='M8[D], f8, f8, f8, f8', \            converters={1: dmy2ymd})mp.figure('Candlestick', dpi=150, facecolor='lightgray')mp.title('Candlestick', fontsize=20)mp.xlabel('Date', fontsize=14)mp.ylabel('Price', fontsize=14)ax = mp.gca()# 设置水平坐标每个星期一为主刻度ax.xaxis.set_major_locator(md.WeekdayLocator(byweekday=md.MO))# 设置每一天为次刻度ax.xaxis.set_minor_locator(md.DayLocator())# 设置水平坐标主刻度标签格式ax.xaxis.set_major_formatter(md.DateFormatter('%d %b %Y'))mp.tick_params(labelsize=10)mp.grid(linestyle=':')dates = dates.astype(md.datetime.datetime)rise = closing_prices - opening_prices >= 0.01 # 阳线掩码，为bool数组fall = opening_prices - closing_prices >= 0.01 # 阴险掩码# 填充色fc = np.zeros(dates.size, dtype='3f4')fc[rise], fc[fall] = (1, 1, 1), (0, 0.5, 0)  # 设置红色和浅绿色# 边缘色ec = np.zeros(dates.size, dtype='3f4')ec[rise], ec[fall] = (1, 0, 0), (0, 0.5, 0)  # 设置红色和浅绿色mp.bar(dates, highest_prices - lowest_prices, 0.01, lowest_prices, color=fc, edgecolor=ec)mp.bar(dates, closing_prices - opening_prices, 0.6, opening_prices, color=fc, edgecolor=ec)# 自动调整水平坐标轴的日期标签mp.gcf().autofmt_xdate()mp.show()

2. 算数平均值

• 样本：S = [s1, s2, ..., sn]
• 算数平均值：m = (s1+s2+...+sn) / n
  测 误差
  s1 = s+d1
  s2 = s+d2
  ...
  sn = s+dn
  m =(s1+s2+...+sn) / n
  =(s+s+...+s) / n + (d1+d2+...+dn)/n
  =s + (d1+d2+...+dn)/n
  n->oo
• 算数平均值表示对真值的无偏估计。

numpy.mean(S)->m

# mean.pyimport numpy as npdef dmy2ymd(dmy):    dmy = str(dmy, encoding='utf-8')    date = dt.datetime.strptime(dmy, '%d-%m-%Y').date()    ymd = date.strftime('%Y-%m-%d')    return ymddates, opening_prices, highest_prices, lowest_prices, closing_prices \= np.loadtxt('../data/aapl.csv', delimiter=',', usecols=(1, 3, 4, 5, 6), unpack=True, dtype='M8[D], f8, f8, f8, f8', \            converters={1: dmy2ymd})mean = 0for cp in closing_prices:    mean += cpmean /= closing_prices.sizeprint(mean)mean = np.mean(closing_prices)print(mean)

351.03766666666667351.0376666666667

3. 加权平均值

• 样本：S = [s1, s2, ..., sn]
• 权重：W = [w1, w2, ..., wn]
• 加权平均值：
  a = (s1w1+s2w2+...+snwn)/(w1+w2+...+wn)
  numpy.average(S, weights=W)->a
• VWAP - 成交量加权平均价格
• TWAP - 时间加权平均价格

# VWAP.pyimport numpy as npclosing_prices, volumes = np.loadtxt('../data/aapl.csv', delimiter=',', usecols=(6, 7), unpack=True)# 原始计算方法vwap, wsum = 0, 0for closing_price, volume in zip(closing_prices, volumes):    vwap += closing_price * volume    wsum += volumevwap /= wsumprint(vwap)vwap = np.average(closing_prices, weights=volumes)print(vwap)

350.5895493532009350.5895493532009

# TWAP.pyimport numpy as npimport datetime as dtdef dmy2days(dmy):    dmy = str(dmy, encoding='utf-8')    date = dt.datetime.strptime(dmy, '%d-%m-%Y').date()    days = (date - dt.date.min).days    return daysdates, closing_prices = np.loadtxt('../data/aapl.csv', delimiter=',', usecols=(1, 6), unpack=True,                                    converters={1:dmy2days})twap = np.average(closing_prices, weights=dates)print(twap)

351.0377051146597

4. 最值

• max/min：返回一个数组中最大/最小元素
• argmax/argmin：返回一个数组中最大/最小元素的下标
• maximum/minimum：将两个同维数组中对应位置元素中最大/最小元素构成一个新的数组
• ptp：返回一个数组中最大值和最小值之差

# max.pyimport numpy as np# 产生9个[10, 100)区间的随机数a = np.random.randint(10, 100, 9).reshape(3, 3)print(a)print(np.max(a),np.min(a))print(np.argmax(a), np.argmin(a))b = np.random.randint(10, 100, 9).reshape(3, 3)print(b)print(np.maximum(a, b))print(np.minimum(a, b))print(np.ptp(a))

[[70 48 99] [85 78 99] [97 41 86]]99 412 7[[40 66 28] [34 11 90] [59 88 24]][[70 66 99] [85 78 99] [97 88 86]][[40 48 28] [34 11 90] [59 41 24]]58

# range.pyimport numpy as npdates, highest_prices, lowest_prices = np.loadtxt('../data/aapl.csv', delimiter=',', usecols=(1, 4, 5),                                                   unpack=True, dtype='U10, f8, f8')max_price = np.max(highest_prices)min_price = np.min(lowest_prices)print(min_price, 'to', max_price)max_index = np.argmax(highest_prices)min_index = np.argmin(lowest_prices)print(dates[max_index], 'and', dates[min_index])# 波动范围r = highest_prices - lowest_priceshighest_ptp = np.ptp(highest_prices)lowest_ptp = np.ptp(lowest_prices)print(highest_ptp, 'and', lowest_ptp)

333.53 to 364.916-02-2011 and 28-01-201124.859999999999957 and 26.970000000000027

5. 中位数

• 将多个样本按照大小排序，取中间位置的元素。

• np.median(无序样本)->中位数

  10 20 30 40 50
  ^ (a[(5-1)/2]+a[5/2])/2
  10 20 30 40 50 60
  ^ ^
  (a[(6-1)/2]+a[6/2])/2

• (a[(s-1)/2]+a[s/2])/2

# med.pyimport numpy as npclosing_prices = np.loadtxt('../data/aapl.csv', delimiter=',', usecols=(6))size = closing_prices.sizesorted_prices = np.msort(closing_prices)median = (sorted_prices[int((size - 1)/ 2)] + sorted_prices[int(size / 2)]) / 2print(median)median = np.median(closing_prices)print(median)

[336.1  339.32 345.03 344.32 343.44 346.5  351.88 355.2  358.16 354.54 356.85 359.18 359.9  363.13 358.3  350.56 338.61 342.62 342.88 348.16 353.21 349.31 352.12 359.56 360.   355.36 355.76 352.47 346.67 351.99]352.055352.055

6. 标准差

• 样本：S = [s1, s2, ..., sn]
• 平均值：m = (s1+s2+...+sn)/n
• 离差：D = [d1, d2, ..., dn], di = si-m
• 离差方：Q = [q1, q2, ..., qn], qi = di^2
• 总体方差：v = (q1+q2+...+qn)/n

• 总体标准差：s = sqrt(v)，方均根

• 样本方差：v' = (q1+q2+...+qn)/(n-1)
  • ${s^2} = \frac{1}{
    {n - 1}}(\sum\limits_{n - 1}^n {({x_i} - \bar x){)^2}}$
  • numpy.std(S)
• 样本标准差：s' = sqrt(v')，方均根
  • numpy.std(S, ddof=1)
  • ${s^2} = \frac{1}{
    {n - 1}}\sqrt {(\sum\limits_{n - 1}^n {({x_i} - \bar x){)^2}} }$

# var.pyimport numpy as npclosing_prices = np.loadtxt('../data/aapl.csv', delimiter=',', usecols=(6))mean = np.mean(closing_prices) # 算术平均值devs = closing_prices - mean # 离差dsqs = devs ** 2 # 离差方pvar = np.mean(dsqs) # 总体方差pstd = np.sqrt(pvar) # 总体标准差svar = np.sum(dsqs) / (dsqs.size - 1) # 样本方差sstd = np.sqrt(svar) # 样本标准差print(pstd, sstd)pstd = np.std(closing_prices)sstd = np.std(closing_prices, ddof=1)print(pstd, sstd)

7.080008325481608 7.2010428762608497.080008325481608 7.201042876260849

7. 时间数据处理

1. 按星期取平均值

# week.pyimport numpy as npimport datetime as dt# 转换器函数：日月年格式的字符串转换成星期def dmy2wday(dmy):    dmy = str(dmy, encoding='utf-8')    date = dt.datetime.strptime(dmy, '%d-%m-%Y').date()    wday = date.weekday() # 用0到6表示周一到周日    return wdaywdays, closing_prices = np.loadtxt('../data/aapl.csv', delimiter=',', usecols=(1, 6), unpack=True,                                    converters={1: dmy2wday})print(wdays)ave_closing_prices = np.zeros(5)for wday in range(ave_closing_prices.size):    '''    ave_closing_prices[wday] = np.take(        closing_prices, np.where(wdays == wday)).mean()    '''    '''    ave_closing_prices[wday] = \        closing_prices[np.where(wdays == wday)].mean()    '''    ave_closing_prices[wday] = \        closing_prices[wdays == wday].mean()for wday, ave_closing_price in zip(        ['MON', 'TUE', 'WED', 'THU', 'FRI'],        ave_closing_prices):    print(wday, np.round(ave_closing_price, 2))

[4. 0. 1. 2. 3. 4. 0. 1. 2. 3. 4. 0. 1. 2. 3. 4. 1. 2. 3. 4. 0. 1. 2. 3. 4. 0. 1. 2. 3. 4.]MON 351.79TUE 350.64WED 352.14THU 350.9FRI 350.02

1. 按星期汇总数据
   • 数组的轴向汇总
   • np.apply_along_axis(处理函数, 轴向, 数组)
   • 沿着数组中所指定的轴向，调用处理函数，并将每次调用的返回值重新组织成数组返回。

# axis.pyimport numpy as npdef foo(arg):    print('foo', arg)    return arg.sum()a = np.arange(1, 10).reshape(3, 3)print(a)b = np.apply_along_axis(foo, 0, a)print(b)c = np.apply_along_axis(foo, 1, a)print(c)

[[1 2 3] [4 5 6] [7 8 9]]foo [1 4 7]foo [2 5 8]foo [3 6 9][12 15 18]foo [1 2 3]foo [4 5 6]foo [7 8 9][ 6 15 24]

# sum.pyimport numpy as npimport datetime as dtdef dmy2wday(dmy):    dmy = str(dmy, encoding='utf-8')    date = dt.datetime.strptime(dmy, '%d-%m-%Y').date()    wday = date.weekday()    return wdaywdays, opening_prices, highest_prices, \    lowest_prices, closing_prices = np.loadtxt(        '../data/aapl.csv',        delimiter=',', usecols=(1, 3, 4, 5, 6),        unpack=True, converters={1: dmy2wday})wdays = wdays[:16]opening_prices = opening_prices[:16]highest_prices = highest_prices[:16]lowest_prices = lowest_prices[:16]closing_prices = closing_prices[:16]# 第一个星期一的索引first_monday = np.where(wdays == 0)[0][0]last_friday = np.where(wdays == 4)[0][-1]indices = np.arange(first_monday, last_friday + 1)indices = np.split(indices, 3)def week_summary(indices):    opening_price = opening_prices[indices[0]]    highest_price = highest_prices[indices].max()    lowest_price = lowest_prices[indices].min()    closing_price = closing_prices[indices[-1]]    return opening_price, highest_price, \        lowest_price, closing_pricesummaries = np.apply_along_axis(    week_summary, 1, indices)print(summaries)np.savetxt('../data/summary.csv', summaries, delimiter=',', fmt='%g')

[[335.8  346.7  334.3  346.5 ] [347.89 360.   347.64 356.85] [356.79 364.9  349.52 350.56]]

8. 卷积

• 卷积积分
• 激励函数：$g(t)$
• 单位激励下的响应函数：$f(t)$

• 响应函数：$\int{g(t)f(t)dt}$

  a = [1 2 3 4 5]  b = [6 7 8]  c = numpy.convolve(a, b, 卷积类型)                        40   61   82            - 有效卷积(valid)                   19   40   61   82   67       - 同维卷积(same)              6    19   40   61   82   67   40  - 完全卷积(full)     0    0    1    2    3    4    5    0    0    8    7    6         8    7    6              8    7    6                   8    7    6                        8    7    6                             8    7    6                                  8    7    6

# conv.pyimport numpy as npa = np.arange(1, 6) # 被卷积序列b = np.arange(6, 9) # 卷积核序列print(a, b)c = np.convolve(a, b, 'full')print(c)d = np.convolve(a, b, 'same')print(d)e = np.convolve(a, b, 'valid')print(e)

[1 2 3 4 5] [6 7 8][ 6 19 40 61 82 67 40][19 40 61 82 67][40 61 82]

9. 移动均线

简单平均a b c d e f g h i j [1/5 1/5 1/5 1/5 1/5] 卷积核(a+b+c+d+e)/5(b+c+d+e+f)/5(c+d+e+f+g)/5...(f+g+h+i+j)/5指数平均，加权平均[A, B, C, D, E]S=A+B+C+D+E(aA+bB+cC+dD+eE)/SaA/S+bB/S+cC/S+dD/S+eE/S[A/S, B/S, C/S, D/S, E/S] 卷积核

# smaema.pyimport numpy as npimport datetime as dtimport matplotlib.pyplot as mpimport matplotlib.dates as mddef dmy2ymd(dmy):    dmy = str(dmy, encoding='utf-8')    date = dt.datetime.strptime(dmy, '%d-%m-%Y').date()    ymd = date.strftime('%Y-%m-%d')    return ymddates, closing_prices = np.loadtxt('../data/aapl.csv', delimiter=',', usecols=(1, 6), unpack=True, dtype='M8[D], f8',             converters={1: dmy2ymd})# SMA1普通方法sma51 = np.zeros(closing_prices.size - 4)for i in range(sma51.size):    sma51[i] = closing_prices[i:i+5].mean()# SMA2卷积方法sma52 = np.convolve(closing_prices, np.ones(5)/5, 'valid')sma10 = np.convolve(closing_prices, np.ones(10)/10, 'valid')# EMA卷积方法weights = np.exp(np.linspace(-1, 0, 5))weights /= weights.sum()ema5 = np.convolve(closing_prices, weights[::-1], 'valid')weights = np.exp(np.linspace(-1, 0, 10))weights /= weights.sum()ema10 = np.convolve(closing_prices, weights[::-1], 'valid')mp.figure('Simple Moving Average', facecolor='lightgray')mp.title('Simple Moving Average', fontsize=20)mp.xlabel('Date', fontsize=14)mp.ylabel('Price', fontsize=14)ax = mp.gca()# 设置水平坐标每个星期一为主刻度ax.xaxis.set_major_locator(md.WeekdayLocator(byweekday=md.MO))# 设置每一天为次刻度ax.xaxis.set_minor_locator(md.DayLocator())# 设置水平坐标主刻度标签格式ax.xaxis.set_major_formatter(md.DateFormatter('%d %b %Y'))mp.tick_params(labelsize=10)mp.grid(linestyle=':')dates = dates.astype(md.datetime.datetime)mp.plot(dates, closing_prices, c='lightgray', label='Closing Price')mp.plot(dates[4:], sma51, c='orange', label='SMA-5(1)')mp.plot(dates[4:], sma52, c='green', label='SMA-5(2)', alpha=0.2, linewidth=4)mp.plot(dates[9:], sma10, c='dodgerblue', label='SMA-10')mp.plot(dates[4:], ema5, c='blue', label='EMA-5')mp.plot(dates[9:], ema10, c='red', label='EMA-10')# 自动调整水平坐标轴的日期标签mp.gcf().autofmt_xdate()mp.legend()mp.show()

10. 布林带

• 中轨：移动平均线
• 上轨：中轨+2x标准差
• 下轨：中轨-2x标准差

# ebb.pyimport numpy as npimport datetime as dtimport matplotlib.pyplot as mpimport matplotlib.dates as mddef dmy2ymd(dmy):    dmy = str(dmy, encoding='utf-8')    date = dt.datetime.strptime(dmy, '%d-%m-%Y').date()    ymd = date.strftime('%Y-%m-%d')    return ymddates, closing_prices = np.loadtxt('../data/aapl.csv', delimiter=',', usecols=(1, 6), unpack=True, dtype='M8[D], f8',             converters={1: dmy2ymd})weights = np.exp(np.linspace(-1, 0, 5))weights /= weights.sum()medios = np.convolve(closing_prices, weights[::-1], 'valid')stds = np.zeros(medios.size)for i in range(stds.size):    stds[i] = closing_prices[i:i+5].std()stds *= 2lowers = medios - stdsuppers = medios + stdsmp.figure('Simple Moving Average', facecolor='lightgray')mp.title('Simple Moving Average', fontsize=20)mp.xlabel('Date', fontsize=14)mp.ylabel('Price', fontsize=14)ax = mp.gca()# 设置水平坐标每个星期一为主刻度ax.xaxis.set_major_locator(md.WeekdayLocator(byweekday=md.MO))# 设置每一天为次刻度ax.xaxis.set_minor_locator(md.DayLocator())# 设置水平坐标主刻度标签格式ax.xaxis.set_major_formatter(md.DateFormatter('%d %b %Y'))mp.tick_params(labelsize=10)mp.grid(linestyle=':')dates = dates.astype(md.datetime.datetime)mp.plot(dates, closing_prices, c='lightgray', label='Closing Price')mp.plot(dates[4:], medios, c='Dodgerblue', label='Medio')mp.plot(dates[4:], lowers, c='limegreen', label='Lower')mp.plot(dates[4:], uppers, c='orange', label='upper')# 自动调整水平坐标轴的日期标签mp.gcf().autofmt_xdate()mp.legend()mp.show()

11. 线性模型

1    2   3    4      560   65  70   75    <80>

1. 线性预测
   a b c d e f
   aA+bB+cC=d bA+cB+dC=e > -> 求A B C
   cA+dB+eC=f /
   dA+eB+fC -> ?

$\left( {\begin{array}{

{20}{c}}

a&b&c\

b&c&d\

c&d&e

\end{array}} \right)\left( {\begin{array}{

{20}{c}}

A\

B\

C

\end{array}} \right) = \left( {\begin{array}{*{20}{c}}

d\

e\

f

\end{array}} \right)$

• np.linalg.lstsq(a, b)[0]

# ebb.pyimport numpy as npimport datetime as dtimport pandas as pdimport matplotlib.pyplot as mpimport matplotlib.dates as mddef dmy2ymd(dmy):    dmy = str(dmy, encoding='utf-8')    date = dt.datetime.strptime(dmy, '%d-%m-%Y').date()    ymd = date.strftime('%Y-%m-%d')    return ymddates, closing_prices = np.loadtxt('../data/aapl.csv', delimiter=',', usecols=(1, 6), unpack=True, dtype='M8[D], f8',             converters={1: dmy2ymd})N = 5pred_prices = np.zeros(    closing_prices.size - 2 * N + 1)for i in range(pred_prices.size):    a = np.zeros((N, N))    for j in range(N):        a[j, ] = closing_prices[i + j:i + j + N]    b = closing_prices[i + N:i + N * 2]    x = np.linalg.lstsq(a, b)[0]    pred_prices[i] = b.dot(x)mp.figure('Linear Prediction',          facecolor='lightgray')mp.title('Linear Prediction', fontsize=20)mp.xlabel('Date', fontsize=14)mp.ylabel('Price', fontsize=14)ax = mp.gca()# 设置水平坐标每个星期一为主刻度ax.xaxis.set_major_locator(md.WeekdayLocator(    byweekday=md.MO))# 设置水平坐标每一天为次刻度ax.xaxis.set_minor_locator(md.DayLocator())# 设置水平坐标主刻度标签格式ax.xaxis.set_major_formatter(md.DateFormatter(    '%d %b %Y'))mp.tick_params(labelsize=10)mp.grid(linestyle=':')dates = dates.astype(md.datetime.datetime)mp.plot(dates, closing_prices, 'o-', c='lightgray',        label='Closing Price')dates = np.append(dates,                  dates[-1] + pd.tseries.offsets.BDay())mp.plot(dates[2 * N:], pred_prices, 'o-',        c='orangered', linewidth=3,        label='Predicted Price')mp.legend()mp.gcf().autofmt_xdate()mp.show()

/Users/haoen110/miniconda3/lib/python3.7/site-packages/ipykernel_launcher.py:25: FutureWarning: `rcond` parameter will change to the default of machine precision times ``max(M, N)`` where M and N are the input matrix dimensions.To use the future default and silence this warning we advise to pass `rcond=None`, to keep using the old, explicitly pass `rcond=-1`./Users/haoen110/miniconda3/lib/python3.7/site-packages/pandas/plotting/_converter.py:129: FutureWarning: Using an implicitly registered datetime converter for a matplotlib plotting method. The converter was registered by pandas on import. Future versions of pandas will require you to explicitly register matplotlib converters.To register the converters:    >>> from pandas.plotting import register_matplotlib_converters    >>> register_matplotlib_converters()  warnings.warn(msg, FutureWarning)

1. 线性拟合
   kx + b = y
   kx1 + b = y1
   kx2 + b = y2
   ...
   kxn + b = yn
   / x1 1   / k   / y1 | x2 1 | X | b | = | y2 |
   | ... |   / | ...|
    xn 1/  yn /
   a x b
   = np.linalg.lstsq(a, b)[0]
   y = kx + b
   kx1 + b = y1' - y1
   kx2 + b = y2' - y2
   ...
   kxn + b = yn' - yn
   [y1 - (kx1 + b)]^2 +
   [y2 - (kx2 + b)]^2 + ... +
   [yn - (kxn + b)]^2 = loss = f(k, b)
   k, b? -> loss ->min

import datetime as dtimport numpy as npimport matplotlib.pyplot as mpimport matplotlib.dates as mddef dmy2ymd(dmy):    dmy = str(dmy, encoding='utf-8')    date = dt.datetime.strptime(        dmy, '%d-%m-%Y').date()    ymd = date.strftime('%Y-%m-%d')    return ymddates, opening_prices, highest_prices, \    lowest_prices, closing_prices = np.loadtxt(        '../data/aapl.csv', delimiter=',',        usecols=(1, 3, 4, 5, 6), unpack=True,        dtype='M8[D], f8, f8, f8, f8',        converters={1: dmy2ymd})trend_points = (highest_prices + lowest_prices +                closing_prices) / 3spreads = highest_prices - lowest_pricesresistance_points = trend_points + spreadssupport_points = trend_points - spreadsdays = dates.astype(int)a = np.column_stack((days, np.ones_like(days)))x = np.linalg.lstsq(a, trend_points)[0]trend_line = days * x[0] + x[1]x = np.linalg.lstsq(a, resistance_points)[0]resistance_line = days * x[0] + x[1]x = np.linalg.lstsq(a, support_points)[0]support_line = days * x[0] + x[1]mp.figure('Trend', facecolor='lightgray')mp.title('Trend', fontsize=20)mp.xlabel('Date', fontsize=14)mp.ylabel('Price', fontsize=14)ax = mp.gca()ax.xaxis.set_major_locator(md.WeekdayLocator(    byweekday=md.MO))ax.xaxis.set_minor_locator(md.DayLocator())ax.xaxis.set_major_formatter(md.DateFormatter(    '%d %b %Y'))mp.tick_params(labelsize=10)mp.grid(linestyle=':')dates = dates.astype(md.datetime.datetime)rise = closing_prices - opening_prices >= 0.01fall = opening_prices - closing_prices >= 0.01fc = np.zeros(dates.size, dtype='3f4')ec = np.zeros(dates.size, dtype='3f4')fc[rise], fc[fall] = (1, 1, 1), (0.85, 0.85, 0.85)ec[rise], ec[fall] = (0.85, 0.85, 0.85), (0.85, 0.85, 0.85)mp.bar(dates, highest_prices - lowest_prices, 0,       lowest_prices, color=fc, edgecolor=ec)mp.bar(dates, closing_prices - opening_prices, 0.8,       opening_prices, color=fc, edgecolor=ec)mp.scatter(dates, trend_points, c='dodgerblue',           alpha=0.5, s=60, zorder=2)mp.plot(dates, trend_line, c='dodgerblue',        linewidth=3, label='Trend')mp.scatter(dates, resistance_points, c='orangered',           alpha=0.5, s=60, zorder=2)mp.plot(dates, resistance_line, c='orangered',        linewidth=3, label='Resistance')mp.scatter(dates, support_points, c='limegreen',           alpha=0.5, s=60, zorder=2)mp.plot(dates, support_line, c='limegreen',        linewidth=3, label='Support')mp.legend()mp.gcf().autofmt_xdate()mp.show()

/Users/haoen110/miniconda3/lib/python3.7/site-packages/ipykernel_launcher.py:28: FutureWarning: `rcond` parameter will change to the default of machine precision times ``max(M, N)`` where M and N are the input matrix dimensions.To use the future default and silence this warning we advise to pass `rcond=None`, to keep using the old, explicitly pass `rcond=-1`./Users/haoen110/miniconda3/lib/python3.7/site-packages/ipykernel_launcher.py:30: FutureWarning: `rcond` parameter will change to the default of machine precision times ``max(M, N)`` where M and N are the input matrix dimensions.To use the future default and silence this warning we advise to pass `rcond=None`, to keep using the old, explicitly pass `rcond=-1`./Users/haoen110/miniconda3/lib/python3.7/site-packages/ipykernel_launcher.py:32: FutureWarning: `rcond` parameter will change to the default of machine precision times ``max(M, N)`` where M and N are the input matrix dimensions.To use the future default and silence this warning we advise to pass `rcond=None`, to keep using the old, explicitly pass `rcond=-1`.