# Source code for statsmodels.tsa.filters.bk_filter

```
import numpy as np
from scipy.signal import fftconvolve

from statsmodels.tools.validation import array_like, PandasWrapper

[docs]def bkfilter(x, low=6, high=32, K=12):
"""
Filter a time series using the Baxter-King bandpass filter.

Parameters
----------
x : array_like
A 1 or 2d ndarray. If 2d, variables are assumed to be in columns.
low : float
Minimum period for oscillations, ie., Baxter and King suggest that
the Burns-Mitchell U.S. business cycle has 6 for quarterly data and
1.5 for annual data.
high : float
Maximum period for oscillations BK suggest that the U.S.
business cycle has 32 for quarterly data and 8 for annual data.
K : int
Lead-lag length of the filter. Baxter and King propose a truncation
length of 12 for quarterly data and 3 for annual data.

Returns
-------
ndarray
The cyclical component of x.

--------
statsmodels.tsa.filters.cf_filter.cffilter
The Christiano Fitzgerald asymmetric, random walk filter.
statsmodels.tsa.filters.bk_filter.hpfilter
Hodrick-Prescott filter.
statsmodels.tsa.seasonal.seasonal_decompose
Decompose a time series using moving averages.
statsmodels.tsa.seasonal.STL
Season-Trend decomposition using LOESS.

Notes
-----
Returns a centered weighted moving average of the original series. Where
the weights a[j] are computed ::

a[j] = b[j] + theta, for j = 0, +/-1, +/-2, ... +/- K
b = (omega_2 - omega_1)/pi
b[j] = 1/(pi*j)(sin(omega_2*j)-sin(omega_1*j), for j = +/-1, +/-2,...

and theta is a normalizing constant ::

theta = -sum(b)/(2K+1)

References
----------
Baxter, M. and R. G. King. "Measuring Business Cycles: Approximate
Band-Pass Filters for Economic Time Series." *Review of Economics and
Statistics*, 1999, 81(4), 575-593.

Examples
--------
>>> import statsmodels.api as sm
>>> import pandas as pd
>>> index = pd.DatetimeIndex(start='1959Q1', end='2009Q4', freq='Q')
>>> dta.set_index(index, inplace=True)

>>> cycles = sm.tsa.filters.bkfilter(dta[['realinv']], 6, 24, 12)

>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> cycles.plot(ax=ax, style=['r--', 'b-'])
>>> plt.show()

.. plot:: plots/bkf_plot.py
"""
# TODO: change the docstring to ..math::?
# TODO: allow windowing functions to correct for Gibb's Phenomenon?
# adjust bweights (symmetrically) by below before demeaning
# Lancosz Sigma Factors np.sinc(2*j/(2.*K+1))
pw = PandasWrapper(x)
x = array_like(x, 'x', maxdim=2)
omega_1 = 2. * np.pi / high  # convert from freq. to periodicity
omega_2 = 2. * np.pi / low
bweights = np.zeros(2 * K + 1)
bweights[K] = (omega_2 - omega_1) / np.pi  # weight at zero freq.
j = np.arange(1, int(K) + 1)
weights = 1 / (np.pi * j) * (np.sin(omega_2 * j) - np.sin(omega_1 * j))
bweights[K + j] = weights  # j is an idx
bweights[:K] = weights[::-1]  # make symmetric weights
bweights -= bweights.mean()  # make sure weights sum to zero
if x.ndim == 2:
bweights = bweights[:, None]
x = fftconvolve(x, bweights, mode='valid')
# get a centered moving avg/convolution

return pw.wrap(x, append='cycle', trim_start=K, trim_end=K)
```