大PandasACF和statsmodel ACF有什么区别？

Question

我在计算股票收益的自相关函数。为此，我测试了两个函数：内置于Pandas中的autocorr函数和由statsmodels.tsa提供的acf函数。这在以下MWE中完成：

import pandas as pd
from pandas_datareader import data
import matplotlib.pyplot as plt
import datetime
from dateutil.relativedelta import relativedelta
from statsmodels.tsa.stattools import acf, pacf

ticker = 'AAPL'
time_ago = datetime.datetime.today().date() - relativedelta(months = 6)

ticker_data = data.get_data_yahoo(ticker, time_ago)['Adj Close'].pct_change().dropna()
ticker_data_len = len(ticker_data)

ticker_data_acf_1 =  acf(ticker_data)[1:32]
ticker_data_acf_2 = [ticker_data.autocorr(i) for i in range(1,32)]

test_df = pd.DataFrame([ticker_data_acf_1, ticker_data_acf_2]).T
test_df.columns = ['Pandas Autocorr', 'Statsmodels Autocorr']
test_df.index += 1
test_df.plot(kind='bar')

我注意到他们预测的数值不一样：

是什么导致了这种差异，应该使用哪些值？

Answer 1

Pandas和Statsmodels版本之间的区别在于均值减和归一化/方差除：

• autocorr只将原始序列的子序列传递给np.corrcoef。在该方法中，利用这些子序列的样本均值和样本方差来确定相关系数
• acf相反，使用总体序列样本均值和样本方差来确定相关系数。

长时间序列的差异可能会变小，但短时间序列的差异会很大。

与Matlab相比，Pandas autocorr函数可能对应于对（滞后）序列本身进行Matlab s xcorr（交叉校正），而不是Matlab的autocorr，后者计算样本自相关（从文档中猜测；我无法验证这一点，因为我没有访问Matlab的权限）。

请参阅本MWE以获得澄清：

import numpy as np
import pandas as pd
from statsmodels.tsa.stattools import acf
import matplotlib.pyplot as plt
plt.style.use("seaborn-colorblind")

def autocorr_by_hand(x, lag):
    # Slice the relevant subseries based on the lag
    y1 = x[:(len(x)-lag)]
    y2 = x[lag:]
    # Subtract the subseries means
    sum_product = np.sum((y1-np.mean(y1))*(y2-np.mean(y2)))
    # Normalize with the subseries stds
    return sum_product / ((len(x) - lag) * np.std(y1) * np.std(y2))

def acf_by_hand(x, lag):
    # Slice the relevant subseries based on the lag
    y1 = x[:(len(x)-lag)]
    y2 = x[lag:]
    # Subtract the mean of the whole series x to calculate Cov
    sum_product = np.sum((y1-np.mean(x))*(y2-np.mean(x)))
    # Normalize with var of whole series
    return sum_product / ((len(x) - lag) * np.var(x))

x = np.linspace(0,100,101)

results = {}
nlags=10
results["acf_by_hand"] = [acf_by_hand(x, lag) for lag in range(nlags)]
results["autocorr_by_hand"] = [autocorr_by_hand(x, lag) for lag in range(nlags)]
results["autocorr"] = [pd.Series(x).autocorr(lag) for lag in range(nlags)]
results["acf"] = acf(x, unbiased=True, nlags=nlags-1)

pd.DataFrame(results).plot(kind="bar", figsize=(10,5), grid=True)
plt.xlabel("lag")
plt.ylim([-1.2, 1.2])
plt.ylabel("value")
plt.show()

Statsmodels使用np.correlate来优化它，但这基本上就是它的工作原理。

Answer 2

如注释中所建议的，通过向statsmodels函数提供unbiased=True，可以减少但不能完全解决问题。使用随机输入：

import statistics

import numpy as np
import pandas as pd
from statsmodels.tsa.stattools import acf

DATA_LEN = 100
N_TESTS = 100
N_LAGS = 32

def test(unbiased):
  data = pd.Series(np.random.random(DATA_LEN))
  data_acf_1 = acf(data, unbiased=unbiased, nlags=N_LAGS)
  data_acf_2 = [data.autocorr(i) for i in range(N_LAGS+1)]
  # return difference between results
  return sum(abs(data_acf_1 - data_acf_2))

for value in (False, True):
  diffs = [test(value) for _ in range(N_TESTS)]
  print(value, statistics.mean(diffs))

输出：

False 0.464562410987
True 0.0820847168593