时间序列分解算法:STL

1. 详解

STL (Seasonal-Trend decomposition procedure based on Loess) [1] 为时序分解中一种常见的算法,基于LOESS将某时刻的数据\(Y_v\)分解为趋势分量(trend component)、周期分量(seasonal component)和余项(remainder component):

\[Y_v = T _v + S_v + R_v \quad v= 1, \cdots, N \]

STL分为内循环(inner loop)与外循环(outer loop),其中内循环主要做了趋势拟合与周期分量的计算。假定\(T_v^{(k)}\)\(S_v{(k)}\)为内循环中第k-1次pass结束时的趋势分量、周期分量,初始时\(T_v^{(k)} = 0\);并有以下参数:

  • \(n_{(i)}\)内层循环数,
  • \(n_{(o)}\)外层循环数,
  • \(n_{(p)}\)为一个周期的样本数,
  • \(n_{(s)}\)为Step 2中LOESS平滑参数,
  • \(n_{(l)}\)为Step 3中LOESS平滑参数,
  • \(n_{(t)}\)为Step 6中LOESS平滑参数。

每个周期相同位置的样本点组成一个子序列(subseries),容易知道这样的子序列共有共有\(n_(p)\)个,我们称其为cycle-subseries。内循环主要分为以下6个步骤:

  • Step 1: 去趋势(Detrending),减去上一轮结果的趋势分量,\(Y_v - T_v^{(k)}\)
  • Step 2: 周期子序列平滑(Cycle-subseries smoothing),用LOESS (\(q=n_{n(s)}\), \(d=1\))对每个子序列做回归,并向前向后各延展一个周期;平滑结果组成temporary seasonal series,记为$C_v^{(k+1)}, \quad v = -n_{(p)} + 1, \cdots, -N + n_{(p)} $;
  • Step 3: 周期子序列的低通量过滤(Low-Pass Filtering),对上一个步骤的结果序列\(C_v^{(k+1)}\)依次做长度为\(n_(p)\)\(n_(p)\)\(3\)的滑动平均(moving average),然后做LOESS (\(q=n_{n(l)}\), \(d=1\))回归,得到结果序列\(L_v^{(k+1)}, \quad v = 1, \cdots, N\);相当于提取周期子序列的低通量;
  • Step 4: 去除平滑周期子序列趋势(Detrending of Smoothed Cycle-subseries),\(S_v^{(k+1)} = C_v^{(k+1)} - L_v^{(k+1)}\)
  • Step 5: 去周期(Deseasonalizing),减去周期分量,\(Y_v - S_v^{(k+1)}\)
  • Step 6: 趋势平滑(Trend Smoothing),对于去除周期之后的序列做LOESS (\(q=n_{n(t)}\), \(d=1\))回归,得到趋势分量\(T_v^{(k+1)}\)

外层循环主要用于调节robustness weight。如果数据序列中有outlier,则余项会较大。定义

\[h = 6 * median(|R_v|) \]

对于位置为\(v\)的数据点,其robustness weight为

\[\rho_v = B(|R_v|/h) \]

其中\(B\)函数为bisquare函数:

\[B(u) = \left \{ { \matrix { {(1-u^2)^2 } & {for \quad 0 \le u < 1} \cr { 0} & {for \quad u \ge 1} \cr } } \right. \]

然后每一次迭代的内循环中,在Step 2与Step 6中做LOESS回归时,邻域权重(neighborhood weight)需要乘以\(\rho_v\),以减少outlier对回归的影响。STL的具体流程如下:

outer loop:
	计算robustness weight;
	inner loop:
		Step 1 去趋势;
		Step 2 周期子序列平滑;
		Step 3 周期子序列的低通量过滤;
		Step 4 去除平滑周期子序列趋势;
		Step 5 去周期;
		Step 6 趋势平滑;

为了使得算法具有足够的robustness,所以设计了内循环与外循环。特别地,当\(n_{(i)}\)足够大时,内循环结束时趋势分量与周期分量已收敛;若时序数据中没有明显的outlier,可以将\(n_{(o)}\)设为0。

R提供STL函数,底层为作者Cleveland的Fortran实现。Python的statsmodels实现了一个简单版的时序分解,通过加权滑动平均提取趋势分量,然后对cycle-subseries每个时间点数据求平均组成周期分量:

def seasonal_decompose(x, model="additive", filt=None, freq=None, two_sided=True):
	_pandas_wrapper, pfreq = _maybe_get_pandas_wrapper_freq(x)
    x = np.asanyarray(x).squeeze()
    nobs = len(x)
    ...
    if filt is None:
      if freq % 2 == 0:  # split weights at ends
        filt = np.array([.5] + [1] * (freq - 1) + [.5]) / freq
        else:
          filt = np.repeat(1./freq, freq)

    nsides = int(two_sided) + 1
    # Linear filtering via convolution. Centered and backward displaced moving weighted average.
    trend = convolution_filter(x, filt, nsides)
    if model.startswith('m'):
        detrended = x / trend
    else:
        detrended = x - trend

    period_averages = seasonal_mean(detrended, freq)

    if model.startswith('m'):
        period_averages /= np.mean(period_averages)
    else:
        period_averages -= np.mean(period_averages)

    seasonal = np.tile(period_averages, nobs // freq + 1)[:nobs]

    if model.startswith('m'):
        resid = x / seasonal / trend
    else:
        resid = detrended - seasonal

    results = lmap(_pandas_wrapper, [seasonal, trend, resid, x])
    return DecomposeResult(seasonal=results[0], trend=results[1],
                           resid=results[2], observed=results[3])

R版STL分解带噪音点数据的结果如下图:

data = read.csv("artificialWithAnomaly/art_daily_flatmiddle.csv")
View(data)
data_decomp <- stl(ts(data[[2]], frequency = 1440/5), s.window = "periodic", robust = TRUE)
plot(data_decomp)

statsmodels模块的时序分解的结果如下图:

import statsmodels.api as sm
import matplotlib.pyplot as plt
import pandas as pd
from date_utils import get_gran, format_timestamp


dta = pd.read_csv('artificialWithAnomaly/art_daily_flatmiddle.csv',
                                   usecols=['timestamp', 'value'])
dta = format_timestamp(dta)
dta = dta.set_index('timestamp')
dta['value'] = dta['value'].apply(pd.to_numeric, errors='ignore')
dta.value.interpolate(inplace=True)
res = sm.tsa.seasonal_decompose(dta.value, freq=288)
res.plot()
plt.show()

2. 参考资料

[1] Cleveland, Robert B., William S. Cleveland, and Irma Terpenning. "STL: A seasonal-trend decomposition procedure based on loess." Journal of Official Statistics 6.1 (1990): 3.

posted @ 2017-08-18 16:15  Treant  阅读(20979)  评论(6编辑  收藏