示例

|在Colab中打开|

# !pip install fracdiff
# !pip install matplotlib pandas pandas_datareader seaborn statsmodels

import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
import pandas_datareader
import seaborn
import statsmodels.tsa.stattools as stattools
from fracdiff import Fracdiff, FracdiffStat, fdiff

seaborn.set_style("white")

系数

from fracdiff.base import fdiff_coef

plt.figure(figsize=(24, 6))

plt.subplot(1, 2, 1)
plt.title("Coefficients of fractional differentiation for d=0.0-1.0")
for d in np.linspace(0.0, 1.0, 5):
    plt.plot(fdiff_coef(d, 6), label=f"d={d:.2f}")
plt.legend()

plt.subplot(1, 2, 2)
plt.title("Coefficients of fractional differentiation for d=1.0-2.0")
for d in np.linspace(1.0, 2.0, 5):
    plt.plot(fdiff_coef(d, 6), label=f"d={d:.2f}")
plt.legend()

plt.show()

S&P 500

def fetch_yahoo(ticker, begin="1998-01-01", end="2020-09-30"):
    """Return pandas.Series."""
    return pandas_datareader.data.DataReader(ticker, "yahoo", begin, end)["Adj Close"]


def fetch_fred(ticker, begin="1998-01-01", end="2020-09-30"):
    """Return pandas.Series."""
    return pandas_datareader.data.DataReader(ticker, "fred", begin, end).iloc[:, 0]

spx = fetch_yahoo("^GSPC")
spx.head()

Date
1997-12-31    970.429993
1998-01-02    975.039978
1998-01-05    977.070007
1998-01-06    966.580017
1998-01-07    964.000000
Name: Adj Close, dtype: float64

spx.shape

(5725,)

绘制分数差分

plt.figure(figsize=(24, 24))
plt.tight_layout()
plt.subplots_adjust(hspace=0.4)

for i, d in enumerate(np.linspace(0.1, 0.9, 9)):
    diff = fdiff(spx, d, mode="valid")
    diff = pd.Series(diff, index=spx.index[-diff.size :])
    plt.subplot(9, 1, i + 1)
    plt.title(f"S&P 500, {d:.1f}th differentiated")
    plt.plot(diff, linewidth=0.4)

plt.show()

fracdiff 的平稳性

def adfstat(d):  # noqa: D103
    diff = fdiff(spx, d, mode="valid")
    stat, *_ = stattools.adfuller(diff)
    return stat


def correlation(d):  # noqa: D103
    diff = fdiff(spx, d, mode="valid")
    corr = np.corrcoef(spx[-diff.size :], diff)[0, 1]
    return corr


ds = np.linspace(0.0, 1.0, 10)
stats = np.vectorize(adfstat)(ds)
corrs = np.vectorize(correlation)(ds)

# 5% critical value of stationarity
_, _, _, _, crit, _ = stattools.adfuller(spx)

# plot
fig, ax_stat = plt.subplots(figsize=(24, 8))
ax_corr = ax_stat.twinx()

ax_stat.plot(ds, stats, color="blue", label="ADF statistics (left)")
ax_corr.plot(ds, corrs, color="orange", label="correlation (right)")
ax_stat.axhline(y=crit["5%"], linestyle="--", color="k", label="5% critical value")

plt.title("Stationarity and memory of fractionally differentiated S&P 500")
fig.legend()
plt.show()

在保留内存的同时进行微分

X = spx.values.reshape(-1, 1)

fs = FracdiffStat(mode="valid")

Xdiff = fs.fit_transform(X)
_, pvalue, _, _, _, _ = stattools.adfuller(Xdiff.reshape(-1))
corr = np.corrcoef(X[-Xdiff.size :, 0], Xdiff.reshape(-1))[0][1]

print(f"* Order: {fs.d_[0]:.2f}")
print(f"* ADF p-value: {100 * pvalue:.2f} %")
print(f"* Correlation with the original time-series: {corr:.2f}")

* Order: 0.84
* ADF p-value: 3.41 %
* Correlation with the original time-series: 0.66

spx_diff = pd.Series(Xdiff.reshape(-1), index=spx.index[-Xdiff.size :])

fig, ax_s = plt.subplots(figsize=(24, 6))
plt.title("S&P 500 and its differentiation preserving memory")
ax_d = ax_s.twinx()

plot_s = ax_s.plot(spx, color="blue", linewidth=0.4, label="S&P 500 (left)")
plot_d = ax_d.plot(
    spx_diff,
    color="orange",
    linewidth=0.4,
    label=f"S&P 500, {fs.d_[0]:.2f} th diff (right)",
)
plots = plot_s + plot_d

ax_s.legend(plots, [p.get_label() for p in plots], loc=0)
plt.show()

其他财务数据

nt_yahoo = [
    ("S&P 500", "^GSPC"),
    ("Nikkei 225", "^N225"),
    ("Shanghai Comp", "^SSEC"),
    ("US 10y", "^TNX"),
    ("Apple", "AAPL"),
]
nt_fred = [
    ("USD/JPY", "DEXJPUS"),
    ("Gold", "GOLDPMGBD228NLBM"),
    ("Crude Oil", "DCOILWTICO"),
]

dfy = pd.DataFrame({name: fetch_yahoo(ticker) for name, ticker in nt_yahoo})
dff = pd.DataFrame({name: fetch_fred(ticker) for name, ticker in nt_fred})

prices = pd.concat([dfy, dff], axis=1).fillna(method="ffill").loc["1998-01-05":]

prices

	S&P 500	Nikkei 225	Shanghai Comp	US 10y	Apple	USD/JPY	Gold	Crude Oil
1998-01-05	977.070007	14956.839844	1220.473022	5.498	0.122484	133.99	284.40	16.95
1998-01-06	966.580017	14896.400391	1233.619995	5.473	0.146113	133.88	282.80	16.64
1998-01-07	964.000000	15028.169922	1244.069946	5.527	0.135022	131.70	281.60	16.91
1998-01-08	956.049988	15019.179688	1237.162964	5.465	0.140326	132.49	281.65	17.01
1998-01-09	927.690002	14995.099609	1239.901001	5.379	0.140326	131.52	278.70	16.65
...	...	...	...	...	...	...	...	...
2020-09-24	3246.590088	23087.820312	3331.521973	0.666	108.220001	105.42	1861.75	40.11
2020-09-25	3298.459961	23204.619141	3331.521973	0.659	112.279999	105.59	1859.70	40.06
2020-09-28	3351.600098	23511.619141	3331.521973	0.663	114.959999	105.50	1864.30	40.47
2020-09-29	3335.469971	23539.099609	3331.521973	0.645	114.089996	105.68	1883.95	39.03
2020-09-30	3363.000000	23185.119141	3331.521973	0.677	115.809998	105.58	1886.90	40.05

5933 rows × 8 columns

def stats(X):  # noqa: D103
    return [stattools.adfuller(X[:, i])[0] for i in range(X.shape[1])]


ds = np.linspace(0.0, 1.0, 11)

df_stats = pd.DataFrame(
    [stats(Fracdiff(d, mode="valid").fit_transform(prices.values)) for d in ds],
    index=ds,
    columns=prices.columns,
)

df_stats

	S&P 500	Nikkei 225	Shanghai Comp	US 10y	Apple	USD/JPY	Gold	Crude Oil
0.0	0.968324	-1.170435	-2.116867	-1.153880	4.951773	-2.178117	0.049131	-2.434859
0.1	0.921352	-1.159694	-2.157641	-1.191706	4.862630	-2.132538	0.028313	-2.412764
0.2	0.802339	-1.311269	-2.128508	-1.240243	4.745854	-2.145354	-0.028585	-2.369947
0.3	0.717212	-1.359395	-2.250318	-1.268093	4.592151	-2.120619	-0.055576	-2.324401
0.4	0.565597	-1.313894	-2.257933	-1.430662	4.374133	-2.204098	-0.117716	-2.297110
0.5	0.290000	-1.587270	-2.328407	-1.673761	4.006742	-2.392610	-0.172077	-2.330552
0.6	-0.022086	-1.551456	-2.549346	-1.959220	3.325249	-2.481928	-0.410898	-2.502265
0.7	-0.728352	-2.142965	-2.883885	-2.162136	2.125019	-3.043593	-0.847778	-2.952382
0.8	-1.865212	-3.419540	-4.061263	-3.273391	0.207860	-4.235373	-1.633992	-4.159692
0.9	-4.699825	-6.702866	-7.249331	-6.317477	-3.184654	-7.890231	-3.769678	-7.355233
1.0	-14.739458	-79.308045	-13.078267	-56.638622	-12.503627	-20.568982	-14.873720	-12.198547

_, _, _, _, crit, _ = stattools.adfuller(prices["S&P 500"].values)

df_stats.plot(figsize=(24, 8), ylim=(-30, 5))
plt.axhline(y=crit["5%"], linestyle="--", color="gray")
plt.title("ADF statistics of fractionally differentiated prices")
plt.show()

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使用 nbsphinx 生成。Jupyter 笔记本可以在这里找到。