如何利用python进行时间序列分析

# -*- coding:utf-8 -*-
import numpy as np
import pandas as pdfrom datetime import datetimeimport matplotlib.pylab as plt
# 读取数据，pd.read_csv默认生成DataFrame对象，需将其转换成Series对象df = pd.read_csv('AirPassengers.csv', encoding='utf-8', index_col='date')df.index = pd.to_datetime(df.index) # 将字符串索引转换成时间索引ts = df['x'] # 生成pd.Series对象# 查看数据格式ts.head()ts.head().index

ts['1949']

ts['1949-1' : '1949-6']

# -*- coding:utf-8 -*-
from statsmodels.tsa.stattools import adfuller
import pandas as pd
import matplotlib.pyplot as plt
import numpy as np
from statsmodels.graphics.tsaplots import plot_acf, plot_pacf
# 移动平均图
def draw_trend(timeSeries, size):
 f = plt.figure(facecolor='white')
 # 对size个数据进行移动平均
 rol_mean = timeSeries.rolling(window=size).mean()
 # 对size个数据进行加权移动平均
 rol_weighted_mean = pd.ewma(timeSeries, span=size)

 timeSeries.plot(color='blue', label='Original')
 rolmean.plot(color='red', label='Rolling Mean')
 rol_weighted_mean.plot(color='black', label='Weighted Rolling Mean')
 plt.legend(loc='best')
 plt.title('Rolling Mean')
 plt.show()

def draw_ts(timeSeries): f = plt.figure(facecolor='white')
 timeSeries.plot(color='blue')
 plt.show()

'''　　Unit Root Test
 The null hypothesis of the Augmented Dickey-Fuller is that there is a unit
 root, with the alternative that there is no unit root. That is to say the
 bigger the p-value the more reason we assert that there is a unit root
'''
def testStationarity(ts):
 dftest = adfuller(ts)
 # 对上述函数求得的值进行语义描述
 dfoutput = pd.Series(dftest[0:4], index=['Test Statistic','p-value','#Lags Used','Number of Observations Used'])
 for key,value in dftest[4].items():
  dfoutput['Critical Value (%s)'%key] = value
 return dfoutput

# 自相关和偏相关图，默认阶数为31阶
def draw_acf_pacf(ts, lags=31):
 f = plt.figure(facecolor='white')
 ax1 = f.add_subplot(211)
 plot_acf(ts, lags=31, ax=ax1)
 ax2 = f.add_subplot(212)
 plot_pacf(ts, lags=31, ax=ax2)
 plt.show()

ts_log = np.log(ts)
test_stationarity.draw_ts(ts_log)

test_stationarity.draw_trend(ts_log, 12)

diff_12 = ts_log.diff(12)
diff_12.dropna(inplace=True)
diff_12_1 = diff_12.diff(1)
diff_12_1.dropna(inplace=True)
test_stationarity.testStationarity(diff_12_1)

from statsmodels.tsa.seasonal import seasonal_decompose
decomposition = seasonal_decompose(ts_log, model="additive")

trend = decomposition.trend
seasonal = decomposition.seasonal
residual = decomposition.resid

rol_mean = ts_log.rolling(window=12).mean()
rol_mean.dropna(inplace=True)
ts_diff_1 = rol_mean.diff(1)
ts_diff_1.dropna(inplace=True)
test_stationarity.testStationarity(ts_diff_1)

ts_diff_2 = ts_diff_1.diff(1)
ts_diff_2.dropna(inplace=True)

from statsmodels.tsa.arima_model import ARMA
model = ARMA(ts_diff_2, order=(1, 1)) 
result_arma = model.fit( disp=-1, method='css')

predict_ts = result_arma.predict()
# 一阶差分还原diff_shift_ts = ts_diff_1.shift(1)diff_recover_1 = predict_ts.add(diff_shift_ts)# 再次一阶差分还原
rol_shift_ts = rol_mean.shift(1)
diff_recover = diff_recover_1.add(rol_shift_ts)
# 移动平均还原
rol_sum = ts_log.rolling(window=11).sum()
rol_recover = diff_recover*12 - rol_sum.shift(1)
# 对数还原
log_recover = np.exp(rol_recover)
log_recover.dropna(inplace=True)

ts = ts[log_recover.index] # 过滤没有预测的记录plt.figure(facecolor='white')
log_recover.plot(color='blue', label='Predict')
ts.plot(color='red', label='Original')
plt.legend(loc='best')
plt.title('RMSE: %.4f'% np.sqrt(sum((log_recover-ts)**2)/ts.size))
plt.show()

# 差分操作
def diff_ts(ts, d):
 global shift_ts_list
 # 动态预测第二日的值时所需要的差分序列
 global last_data_shift_list
 shift_ts_list = []
 last_data_shift_list = []
 tmp_ts = ts
 for i in d:
  last_data_shift_list.append(tmp_ts[-i])
  print last_data_shift_list
  shift_ts = tmp_ts.shift(i)
  shift_ts_list.append(shift_ts)
  tmp_ts = tmp_ts - shift_ts
 tmp_ts.dropna(inplace=True)
 return tmp_ts

# 还原操作
def predict_diff_recover(predict_value, d):
 if isinstance(predict_value, float):
  tmp_data = predict_value
  for i in range(len(d)):
   tmp_data = tmp_data + last_data_shift_list[-i-1]
 elif isinstance(predict_value, np.ndarray):
  tmp_data = predict_value[0]
  for i in range(len(d)):
   tmp_data = tmp_data + last_data_shift_list[-i-1]
 else:
  tmp_data = predict_value
  for i in range(len(d)):
   try:
    tmp_data = tmp_data.add(shift_ts_list[-i-1])
   except:
    raise ValueError('What you input is not pd.Series type!')
  tmp_data.dropna(inplace=True)
 return tmp_data

diffed_ts = diff_ts(ts_log, d=[12, 1])
model = arima_model(diffed_ts)
model.certain_model(1, 1)
predict_ts = model.properModel.predict()
diff_recover_ts = predict_diff_recover(predict_ts, d=[12, 1])
log_recover = np.exp(diff_recover_ts)

def proper_model(data_ts, maxLag):
 init_bic = sys.maxint
 init_p = 0
 init_q = 0
 init_properModel = None
 for p in np.arange(maxLag):
  for q in np.arange(maxLag):
   model = ARMA(data_ts, order=(p, q))
   try:
    results_ARMA = model.fit(disp=-1, method='css')
   except:
    continue
   bic = results_ARMA.bic
   if bic < init_bic:
    init_p = p
    init_q = q
    init_properModel = results_ARMA
    init_bic = bic
 return init_bic, init_p, init_q, init_properModel

from dateutil.relativedelta import relativedeltadef _add_new_data(ts, dat, type='day'):
if type == 'day':
  new_index = ts.index[-1] + relativedelta(days=1)
 elif type == 'month':
  new_index = ts.index[-1] + relativedelta(months=1)
 ts[new_index] = dat

def add_today_data(model, ts, data, d, type='day'):
 _add_new_data(ts, data, type) # 为原始序列添加数据
 # 为滞后序列添加新值
 d_ts = diff_ts(ts, d)
 model.add_today_data(d_ts[-1], type)

def forecast_next_day_data(model, type='day'):
 if model == None:
  raise ValueError('No model fit before')
 fc = model.forecast_next_day_value(type)
 return predict_diff_recover(fc, [12, 1])

ts_train = ts_log[:'1956-12']
ts_test = ts_log['1957-1':]

diffed_ts = diff_ts(ts_train, [12, 1])
forecast_list = []
for i, dta in enumerate(ts_test):
 if i%7 == 0:
  model = arima_model(diffed_ts)
  model.certain_model(1, 1)
 forecast_data = forecast_next_day_data(model, type='month')
 forecast_list.append(forecast_data)
 add_today_data(model, ts_train, dta, [12, 1], type='month')

predict_ts = pd.Series(data=forecast_list, index=ts['1957-1':].index)log_recover = np.exp(predict_ts)original_ts = ts['1957-1':]

# -*-coding:utf-8-*-
import pandas as pd
import numpy as np
from statsmodels.tsa.arima_model import ARMA
import sys
from dateutil.relativedelta import relativedelta
from copy import deepcopy
import matplotlib.pyplot as plt

class arima_model:

 def __init__(self, ts, maxLag=9):
  self.data_ts = ts
  self.resid_ts = None
  self.predict_ts = None
  self.maxLag = maxLag
  self.p = maxLag
  self.q = maxLag
  self.properModel = None
  self.bic = sys.maxint

 # 计算最优ARIMA模型，将相关结果赋给相应属性
 def get_proper_model(self):
  self._proper_model()
  self.predict_ts = deepcopy(self.properModel.predict())
  self.resid_ts = deepcopy(self.properModel.resid)

 # 对于给定范围内的p,q计算拟合得最好的arima模型，这里是对差分好的数据进行拟合，故差分恒为0
 def _proper_model(self):
  for p in np.arange(self.maxLag):
   for q in np.arange(self.maxLag):
    # print p,q,self.bic
    model = ARMA(self.data_ts, order=(p, q))
    try:
     results_ARMA = model.fit(disp=-1, method='css')
    except:
     continue
    bic = results_ARMA.bic
    # print 'bic:',bic,'self.bic:',self.bic
    if bic < self.bic:
     self.p = p
     self.q = q
     self.properModel = results_ARMA
     self.bic = bic
     self.resid_ts = deepcopy(self.properModel.resid)
     self.predict_ts = self.properModel.predict()

 # 参数确定模型
 def certain_model(self, p, q):
   model = ARMA(self.data_ts, order=(p, q))
   try:
    self.properModel = model.fit( disp=-1, method='css')
    self.p = p
    self.q = q
    self.bic = self.properModel.bic
    self.predict_ts = self.properModel.predict()
    self.resid_ts = deepcopy(self.properModel.resid)
   except:
    print 'You can not fit the model with this parameter p,q, '       'please use the get_proper_model method to get the best model'

 # 预测第二日的值
 def forecast_next_day_value(self, type='day'):
  # 我修改了statsmodels包中arima_model的源代码，添加了constant属性，需要先运行forecast方法，为constant赋值
  self.properModel.forecast()
  if self.data_ts.index[-1] != self.resid_ts.index[-1]:
   raise ValueError('''The index is different in data_ts and resid_ts, please add new data to data_ts.
   If you just want to forecast the next day data without add the real next day data to data_ts,
   please run the predict method which arima_model included itself''')
  if not self.properModel:
   raise ValueError('The arima model have not computed, please run the proper_model method before')
  para = self.properModel.params

  # print self.properModel.params
  if self.p == 0: # It will get all the value series with setting self.data_ts[-self.p:] when p is zero
   ma_value = self.resid_ts[-self.q:]
   values = ma_value.reindex(index=ma_value.index[::-1])
  elif self.q == 0:
   ar_value = self.data_ts[-self.p:]
   values = ar_value.reindex(index=ar_value.index[::-1])
  else:
   ar_value = self.data_ts[-self.p:]
   ar_value = ar_value.reindex(index=ar_value.index[::-1])
   ma_value = self.resid_ts[-self.q:]
   ma_value = ma_value.reindex(index=ma_value.index[::-1])
   values = ar_value.append(ma_value)

  predict_value = np.dot(para[1:], values) + self.properModel.constant[0]
  self._add_new_data(self.predict_ts, predict_value, type)
  return predict_value

 # 动态添加数据函数，针对索引是月份和日分别进行处理
 def _add_new_data(self, ts, dat, type='day'):
  if type == 'day':
   new_index = ts.index[-1] + relativedelta(days=1)
  elif type == 'month':
   new_index = ts.index[-1] + relativedelta(months=1)
  ts[new_index] = dat

 def add_today_data(self, dat, type='day'):
  self._add_new_data(self.data_ts, dat, type)
  if self.data_ts.index[-1] != self.predict_ts.index[-1]:
   raise ValueError('You must use the forecast_next_day_value method forecast the value of today before')
  self._add_new_data(self.resid_ts, self.data_ts[-1] - self.predict_ts[-1], type)

if __name__ == '__main__':
 df = pd.read_csv('AirPassengers.csv', encoding='utf-8', index_col='date')
 df.index = pd.to_datetime(df.index)
 ts = df['x']

 # 数据预处理
 ts_log = np.log(ts)
 rol_mean = ts_log.rolling(window=12).mean()
 rol_mean.dropna(inplace=True)
 ts_diff_1 = rol_mean.diff(1)
 ts_diff_1.dropna(inplace=True)
 ts_diff_2 = ts_diff_1.diff(1)
 ts_diff_2.dropna(inplace=True)

 # 模型拟合
 model = arima_model(ts_diff_2)
 # 这里使用模型参数自动识别
 model.get_proper_model()
 print 'bic:', model.bic, 'p:', model.p, 'q:', model.q
 print model.properModel.forecast()[0]
 print model.forecast_next_day_value(type='month')

 # 预测结果还原
 predict_ts = model.properModel.predict()
 diff_shift_ts = ts_diff_1.shift(1)
 diff_recover_1 = predict_ts.add(diff_shift_ts)
 rol_shift_ts = rol_mean.shift(1)
 diff_recover = diff_recover_1.add(rol_shift_ts)
 rol_sum = ts_log.rolling(window=11).sum()
 rol_recover = diff_recover*12 - rol_sum.shift(1)
 log_recover = np.exp(rol_recover)
 log_recover.dropna(inplace=True)

 # 预测结果作图
 ts = ts[log_recover.index]
 plt.figure(facecolor='white')
 log_recover.plot(color='blue', label='Predict')
 ts.plot(color='red', label='Original')
 plt.legend(loc='best')
 plt.title('RMSE: %.4f'% np.sqrt(sum((log_recover-ts)**2)/ts.size))
 plt.show()

# Note: The information criteria add 1 to the number of parameters
#  whenever the model has an AR or MA term since, in principle,
#  the variance could be treated as a free parameter and restricted
#  This code does not allow this, but it adds consistency with other
#  packages such as gretl and X12-ARIMA
 
from __future__ import absolute_import
from statsmodels.compat.python import string_types, range
# for 2to3 with extensions
 
from datetime import datetime
 
import numpy as np
from scipy import optimize
from scipy.stats import t, norm
from scipy.signal import lfilter
from numpy import dot, log, zeros, pi
from numpy.linalg import inv
 
from statsmodels.tools.decorators import (cache_readonly,
           resettable_cache)
import statsmodels.tsa.base.tsa_model as tsbase
import statsmodels.base.wrapper as wrap
from statsmodels.regression.linear_model import yule_walker, GLS
from statsmodels.tsa.tsatools import (lagmat, add_trend,
          _ar_transparams, _ar_invtransparams,
          _ma_transparams, _ma_invtransparams,
          unintegrate, unintegrate_levels)
from statsmodels.tsa.vector_ar import util
from statsmodels.tsa.ar_model import AR
from statsmodels.tsa.arima_process import arma2ma
from statsmodels.tools.numdiff import approx_hess_cs, approx_fprime_cs
from statsmodels.tsa.base.datetools import _index_date
from statsmodels.tsa.kalmanf import KalmanFilter
 
_armax_notes = """
 
  Notes
  -----
  If exogenous variables are given, then the model that is fit is
 
  .. math::
 
   \\phi(L)(y_t - X_t\\beta) = \\theta(L)\epsilon_t
 
  where :math:`\\phi` and :math:`\\theta` are polynomials in the lag
  operator, :math:`L`. This is the regression model with ARMA errors,
  or ARMAX model. This specification is used, whether or not the model
  is fit using conditional sum of square or maximum-likelihood, using
  the `method` argument in
  :meth:`statsmodels.tsa.arima_model.%(Model)s.fit`. Therefore, for
  now, `css` and `mle` refer to estimation methods only. This may
  change for the case of the `css` model in future versions.
"""
 
_arma_params = """ endog : array-like
  The endogenous variable.
 order : iterable
  The (p,q) order of the model for the number of AR parameters,
  differences, and MA parameters to use.
 exog : array-like, optional
  An optional arry of exogenous variables. This should *not* include a
  constant or trend. You can specify this in the `fit` method."""
 
_arma_model = "Autoregressive Moving Average ARMA(p,q) Model"
 
_arima_model = "Autoregressive Integrated Moving Average ARIMA(p,d,q) Model"
 
_arima_params = """ endog : array-like
  The endogenous variable.
 order : iterable
  The (p,d,q) order of the model for the number of AR parameters,
  differences, and MA parameters to use.
 exog : array-like, optional
  An optional arry of exogenous variables. This should *not* include a
  constant or trend. You can specify this in the `fit` method."""
 
_predict_notes = """
  Notes
  -----
  Use the results predict method instead.
"""
 
_results_notes = """
  Notes
  -----
  It is recommended to use dates with the time-series models, as the
  below will probably make clear. However, if ARIMA is used without
  dates and/or `start` and `end` are given as indices, then these
  indices are in terms of the *original*, undifferenced series. Ie.,
  given some undifferenced observations::
 
   1970Q1, 1
   1970Q2, 1.5
   1970Q3, 1.25
   1970Q4, 2.25
   1971Q1, 1.2
   1971Q2, 4.1
 
  1970Q1 is observation 0 in the original series. However, if we fit an
  ARIMA(p,1,q) model then we lose this first observation through
  differencing. Therefore, the first observation we can forecast (if
  using exact MLE) is index 1. In the differenced series this is index
  0, but we refer to it as 1 from the original series.
"""
 
_predict = """
  %(Model)s model in-sample and out-of-sample prediction
 
  Parameters
  ----------
  %(params)s
  start : int, str, or datetime
   Zero-indexed observation number at which to start forecasting, ie.,
   the first forecast is start. Can also be a date string to
   parse or a datetime type.
  end : int, str, or datetime
   Zero-indexed observation number at which to end forecasting, ie.,
   the first forecast is start. Can also be a date string to
   parse or a datetime type. However, if the dates index does not
   have a fixed frequency, end must be an integer index if you
   want out of sample prediction.
  exog : array-like, optional
   If the model is an ARMAX and out-of-sample forecasting is
   requested, exog must be given. Note that you'll need to pass
   `k_ar` additional lags for any exogenous variables. E.g., if you
   fit an ARMAX(2, q) model and want to predict 5 steps, you need 7
   observations to do this.
  dynamic : bool, optional
   The `dynamic` keyword affects in-sample prediction. If dynamic
   is False, then the in-sample lagged values are used for
   prediction. If `dynamic` is True, then in-sample forecasts are
   used in place of lagged dependent variables. The first forecasted
   value is `start`.
  %(extra_params)s
 
  Returns
  -------
  %(returns)s
  %(extra_section)s
"""
 
_predict_returns = """predict : array
   The predicted values.
 
"""
 
_arma_predict = _predict % {"Model" : "ARMA",
       "params" : """
   params : array-like
   The fitted parameters of the model.""",
       "extra_params" : "",
       "returns" : _predict_returns,
       "extra_section" : _predict_notes}
 
_arma_results_predict = _predict % {"Model" : "ARMA", "params" : "",
         "extra_params" : "",
         "returns" : _predict_returns,
         "extra_section" : _results_notes}
 
_arima_predict = _predict % {"Model" : "ARIMA",
        "params" : """params : array-like
   The fitted parameters of the model.""",
        "extra_params" : """typ : str {'linear', 'levels'}
 
   - 'linear' : Linear prediction in terms of the differenced
    endogenous variables.
   - 'levels' : Predict the levels of the original endogenous
    variables.\n""", "returns" : _predict_returns,
        "extra_section" : _predict_notes}
 
_arima_results_predict = _predict % {"Model" : "ARIMA",
          "params" : "",
          "extra_params" :
          """typ : str {'linear', 'levels'}
 
   - 'linear' : Linear prediction in terms of the differenced
    endogenous variables.
   - 'levels' : Predict the levels of the original endogenous
    variables.\n""",
          "returns" : _predict_returns,
          "extra_section" : _results_notes}
 
_arima_plot_predict_example = """  Examples
  --------
  > import statsmodels.api as sm
  > import matplotlib.pyplot as plt
  > import pandas as pd
  >
  > dta = sm.datasets.sunspots.load_pandas().data[['SUNACTIVITY']]
  > dta.index = pd.DatetimeIndex(start='1700', end='2009', freq='A')
  > res = sm.tsa.ARMA(dta, (3, 0)).fit()
  > fig, ax = plt.subplots()
  > ax = dta.ix['1950':].plot(ax=ax)
  > fig = res.plot_predict('1990', '2012', dynamic=True, ax=ax,
  ...      plot_insample=False)
  > plt.show()
 
  .. plot:: plots/arma_predict_plot.py
"""
 
_plot_predict = ("""
  Plot forecasts
      """ + '\n'.join(_predict.split('\n')[2:])) % {
      "params" : "",
       "extra_params" : """alpha : float, optional
   The confidence intervals for the forecasts are (1 - alpha)%
  plot_insample : bool, optional
   Whether to plot the in-sample series. Default is True.
  ax : matplotlib.Axes, optional
   Existing axes to plot with.""",
      "returns" : """fig : matplotlib.Figure
   The plotted Figure instance""",
      "extra_section" : ('\n' + _arima_plot_predict_example +
           '\n' + _results_notes)
      }
 
_arima_plot_predict = ("""
  Plot forecasts
      """ + '\n'.join(_predict.split('\n')[2:])) % {
      "params" : "",
       "extra_params" : """alpha : float, optional
   The confidence intervals for the forecasts are (1 - alpha)%
  plot_insample : bool, optional
   Whether to plot the in-sample series. Default is True.
  ax : matplotlib.Axes, optional
   Existing axes to plot with.""",
      "returns" : """fig : matplotlib.Figure
   The plotted Figure instance""",
    "extra_section" : ('\n' + _arima_plot_predict_example +
         '\n' +
         '\n'.join(_results_notes.split('\n')[:3]) +
        ("""
  This is hard-coded to only allow plotting of the forecasts in levels.
""") +
        '\n'.join(_results_notes.split('\n')[3:]))
      }
 
 
def cumsum_n(x, n):
 if n:
  n -= 1
  x = np.cumsum(x)
  return cumsum_n(x, n)
 else:
  return x
 
 
def _check_arima_start(start, k_ar, k_diff, method, dynamic):
 if start < 0:
  raise ValueError("The start index %d of the original series "
       "has been differenced away" % start)
 elif (dynamic or 'mle' not in method) and start < k_ar:
  raise ValueError("Start must be >= k_ar for conditional MLE "
       "or dynamic forecast. Got %d" % start)
 
 
def _get_predict_out_of_sample(endog, p, q, k_trend, k_exog, start, errors,
        trendparam, exparams, arparams, maparams, steps,
        method, exog=None):
 """
 Returns endog, resid, mu of appropriate length for out of sample
 prediction.
 """
 if q:
  resid = np.zeros(q)
  if start and 'mle' in method or (start == p and not start == 0):
   resid[:q] = errors[start-q:start]
  elif start:
   resid[:q] = errors[start-q-p:start-p]
  else:
   resid[:q] = errors[-q:]
 else:
  resid = None
 
 y = endog
 if k_trend == 1:
  # use expectation not constant
  if k_exog > 0:
   #TODO: technically should only hold for MLE not
   # conditional model. See #274.
   # ensure 2-d for conformability
   if np.ndim(exog) == 1 and k_exog == 1:
    # have a 1d series of observations -> 2d
    exog = exog[:, None]
   elif np.ndim(exog) == 1:
    # should have a 1d row of exog -> 2d
    if len(exog) != k_exog:
     raise ValueError("1d exog given and len(exog) != k_exog")
    exog = exog[None, :]
   X = lagmat(np.dot(exog, exparams), p, original='in', trim='both')
   mu = trendparam * (1 - arparams.sum())
   # arparams were reversed in unpack for ease later
   mu = mu + (np.r_[1, -arparams[::-1]] * X).sum(1)[:, None]
  else:
   mu = trendparam * (1 - arparams.sum())
   mu = np.array([mu]*steps)
 elif k_exog > 0:
  X = np.dot(exog, exparams)
  #NOTE: you shouldn't have to give in-sample exog!
  X = lagmat(X, p, original='in', trim='both')
  mu = (np.r_[1, -arparams[::-1]] * X).sum(1)[:, None]
 else:
  mu = np.zeros(steps)
 
 endog = np.zeros(p + steps - 1)
 
 if p and start:
  endog[:p] = y[start-p:start]
 elif p:
  endog[:p] = y[-p:]
 
 return endog, resid, mu
 
 
def _arma_predict_out_of_sample(params, steps, errors, p, q, k_trend, k_exog,
        endog, exog=None, start=0, method='mle'):
 (trendparam, exparams,
  arparams, maparams) = _unpack_params(params, (p, q), k_trend,
           k_exog, reverse=True)
 # print 'params:',params
 # print 'arparams:',arparams,'maparams:',maparams
 endog, resid, mu = _get_predict_out_of_sample(endog, p, q, k_trend, k_exog,
             start, errors, trendparam,
             exparams, arparams,
             maparams, steps, method,
             exog)
# print 'mu[-1]:',mu[-1], 'mu[0]:',mu[0]
 forecast = np.zeros(steps)
 if steps == 1:
  if q:
   return mu[0] + np.dot(arparams, endog[:p]) + np.dot(maparams,
                resid[:q]), mu[0]
  else:
   return mu[0] + np.dot(arparams, endog[:p]), mu[0]
 
 if q:
  i = 0 # if q == 1
 else:
  i = -1
 
 for i in range(min(q, steps - 1)):
  fcast = (mu[i] + np.dot(arparams, endog[i:i + p]) +
     np.dot(maparams[:q - i], resid[i:i + q]))
  forecast[i] = fcast
  endog[i+p] = fcast
 
 for i in range(i + 1, steps - 1):
  fcast = mu[i] + np.dot(arparams, endog[i:i+p])
  forecast[i] = fcast
  endog[i+p] = fcast
 
 #need to do one more without updating endog
 forecast[-1] = mu[-1] + np.dot(arparams, endog[steps - 1:])
 return forecast, mu[-1] #Modified by me, the former is return forecast
 
 
def _arma_predict_in_sample(start, end, endog, resid, k_ar, method):
 """
 Pre- and in-sample fitting for ARMA.
 """
 if 'mle' in method:
  fittedvalues = endog - resid # get them all then trim
 else:
  fittedvalues = endog[k_ar:] - resid
 
 fv_start = start
 if 'mle' not in method:
  fv_start -= k_ar # start is in terms of endog index
 fv_end = min(len(fittedvalues), end + 1)
 return fittedvalues[fv_start:fv_end]
 
 
def _validate(start, k_ar, k_diff, dates, method):
 if isinstance(start, (string_types, datetime)):
  start = _index_date(start, dates)
  start -= k_diff
 if 'mle' not in method and start < k_ar - k_diff:
  raise ValueError("Start must be >= k_ar for conditional "
       "MLE or dynamic forecast. Got %s" % start)
 
 return start
 
 
def _unpack_params(params, order, k_trend, k_exog, reverse=False):
 p, q = order
 k = k_trend + k_exog
 maparams = params[k+p:]
 arparams = params[k:k+p]
 trend = params[:k_trend]
 exparams = params[k_trend:k]
 if reverse:
  return trend, exparams, arparams[::-1], maparams[::-1]
 return trend, exparams, arparams, maparams
 
 
def _unpack_order(order):
 k_ar, k_ma, k = order
 k_lags = max(k_ar, k_ma+1)
 return k_ar, k_ma, order, k_lags
 
 
def _make_arma_names(data, k_trend, order, exog_names):
 k_ar, k_ma = order
 exog_names = exog_names or []
 ar_lag_names = util.make_lag_names([data.ynames], k_ar, 0)
 ar_lag_names = [''.join(('ar.', i)) for i in ar_lag_names]
 ma_lag_names = util.make_lag_names([data.ynames], k_ma, 0)
 ma_lag_names = [''.join(('ma.', i)) for i in ma_lag_names]
 trend_name = util.make_lag_names('', 0, k_trend)
 exog_names = trend_name + exog_names + ar_lag_names + ma_lag_names
 return exog_names
 
 
def _make_arma_exog(endog, exog, trend):
 k_trend = 1 # overwritten if no constant
 if exog is None and trend == 'c': # constant only
  exog = np.ones((len(endog), 1))
 elif exog is not None and trend == 'c': # constant plus exogenous
  exog = add_trend(exog, trend='c', prepend=True)
 elif exog is not None and trend == 'nc':
  # make sure it's not holding constant from last run
  if exog.var() == 0:
   exog = None
  k_trend = 0
 if trend == 'nc':
  k_trend = 0
 return k_trend, exog
 
 
def _check_estimable(nobs, n_params):
 if nobs <= n_params:
  raise ValueError("Insufficient degrees of freedom to estimate")
 
 
class ARMA(tsbase.TimeSeriesModel):
 
 __doc__ = tsbase._tsa_doc % {"model" : _arma_model,
         "params" : _arma_params, "extra_params" : "",
         "extra_sections" : _armax_notes %
         {"Model" : "ARMA"}}
 
 def __init__(self, endog, order, exog=None, dates=None, freq=None,
     missing='none'):
  super(ARMA, self).__init__(endog, exog, dates, freq, missing=missing)
  exog = self.data.exog # get it after it's gone through processing
  _check_estimable(len(self.endog), sum(order))
  self.k_ar = k_ar = order[0]
  self.k_ma = k_ma = order[1]
  self.k_lags = max(k_ar, k_ma+1)
  self.constant = 0 #Added by me
  if exog is not None:
   if exog.ndim == 1:
    exog = exog[:, None]
   k_exog = exog.shape[1] # number of exog. variables excl. const
  else:
   k_exog = 0
  self.k_exog = k_exog
 
 def _fit_start_params_hr(self, order):
  """
  Get starting parameters for fit.
 
  Parameters
  ----------
  order : iterable
   (p,q,k) - AR lags, MA lags, and number of exogenous variables
   including the constant.
 
  Returns
  -------
  start_params : array
   A first guess at the starting parameters.
 
  Notes
  -----
  If necessary, fits an AR process with the laglength selected according
  to best BIC. Obtain the residuals. Then fit an ARMA(p,q) model via
  OLS using these residuals for a first approximation. Uses a separate
  OLS regression to find the coefficients of exogenous variables.
 
  References
  ----------
  Hannan, E.J. and Rissanen, J. 1982. "Recursive estimation of mixed
   autoregressive-moving average order." `Biometrika`. 69.1.
  """
  p, q, k = order
  start_params = zeros((p+q+k))
  endog = self.endog.copy() # copy because overwritten
  exog = self.exog
  if k != 0:
   ols_params = GLS(endog, exog).fit().params
   start_params[:k] = ols_params
   endog -= np.dot(exog, ols_params).squeeze()
  if q != 0:
   if p != 0:
    # make sure we don't run into small data problems in AR fit
    nobs = len(endog)
    maxlag = int(round(12*(nobs/100.)**(1/4.)))
    if maxlag >= nobs:
     maxlag = nobs - 1
    armod = AR(endog).fit(ic='bic', trend='nc', maxlag=maxlag)
    arcoefs_tmp = armod.params
    p_tmp = armod.k_ar
    # it's possible in small samples that optimal lag-order
    # doesn't leave enough obs. No consistent way to fix.
    if p_tmp + q >= len(endog):
     raise ValueError("Proper starting parameters cannot"
          " be found for this order with this "
          "number of observations. Use the "
          "start_params argument.")
    resid = endog[p_tmp:] - np.dot(lagmat(endog, p_tmp,
              trim='both'),
            arcoefs_tmp)
    if p < p_tmp + q:
     endog_start = p_tmp + q - p
     resid_start = 0
    else:
     endog_start = 0
     resid_start = p - p_tmp - q
    lag_endog = lagmat(endog, p, 'both')[endog_start:]
    lag_resid = lagmat(resid, q, 'both')[resid_start:]
    # stack ar lags and resids
    X = np.column_stack((lag_endog, lag_resid))
    coefs = GLS(endog[max(p_tmp + q, p):], X).fit().params
    start_params[k:k+p+q] = coefs
   else:
    start_params[k+p:k+p+q] = yule_walker(endog, order=q)[0]
  if q == 0 and p != 0:
   arcoefs = yule_walker(endog, order=p)[0]
   start_params[k:k+p] = arcoefs
 
  # check AR coefficients
  if p and not np.all(np.abs(np.roots(np.r_[1, -start_params[k:k + p]]
           )) < 1):
   raise ValueError("The computed initial AR coefficients are not "
        "stationary\nYou should induce stationarity, "
        "choose a different model order, or you can\n"
        "pass your own start_params.")
  # check MA coefficients
  elif q and not np.all(np.abs(np.roots(np.r_[1, start_params[k + p:]]
            )) < 1):
   return np.zeros(len(start_params)) #modified by me
   raise ValueError("The computed initial MA coefficients are not "
        "invertible\nYou should induce invertibility, "
        "choose a different model order, or you can\n"
        "pass your own start_params.")
 
  # check MA coefficients
  # print start_params
  return start_params
 
 def _fit_start_params(self, order, method):
  if method != 'css-mle': # use Hannan-Rissanen to get start params
   start_params = self._fit_start_params_hr(order)
  else: # use CSS to get start params
   func = lambda params: -self.loglike_css(params)
   #start_params = [.1]*(k_ar+k_ma+k_exog) # different one for k"""
  Compute the score function at params.
 
  Notes
  -----
  This is a numerical approximation.
  """
  return approx_fprime_cs(params, self.loglike, args=(False,))
 
 def hessian(self, params):
  """
  Compute the Hessian at params,
 
  Notes
  -----
  This is a numerical approximation.
  """
  return approx_hess_cs(params, self.loglike, args=(False,))
 
 def _transparams(self, params):
  """
  Transforms params to induce stationarity/invertability.
 
  Reference
  ---------
  Jones(1980)
  """
  k_ar, k_ma = self.k_ar, self.k_ma
  k = self.k_exog + self.k_trend
  newparams = np.zeros_like(params)
 
  # just copy exogenous parameters
  if k != 0:
   newparams[:k] = params[:k]
 
  # AR Coeffs
  if k_ar != 0:
   newparams[k:k+k_ar] = _ar_transparams(params[k:k+k_ar].copy())
 
  # MA Coeffs
  if k_ma != 0:
   newparams[k+k_ar:] = _ma_transparams(params[k+k_ar:].copy())
  return newparams
 
 def _invtransparams(self, start_params):
  """
  Inverse of the Jones reparameterization
  """
  k_ar, k_ma = self.k_ar, self.k_ma
  k = self.k_exog + self.k_trend
  newparams = start_params.copy()
  arcoefs = newparams[k:k+k_ar]
  macoefs = newparams[k+k_ar:]
  # AR coeffs
  if k_ar != 0:
   newparams[k:k+k_ar] = _ar_invtransparams(arcoefs)
 
  # MA coeffs
  if k_ma != 0:
   newparams[k+k_ar:k+k_ar+k_ma] = _ma_invtransparams(macoefs)
  return newparams
 
 def _get_predict_start(self, start, dynamic):
  # do some defaults
  method = getattr(self, 'method', 'mle')
  k_ar = getattr(self, 'k_ar', 0)
  k_diff = getattr(self, 'k_diff', 0)
  if start is None:
   if 'mle' in method and not dynamic:
    start = 0
   else:
    start = k_ar
   self._set_predict_start_date(start) # else it's done in super
  elif isinstance(start, int):
   start = super(ARMA, self)._get_predict_start(start)
  else: # should be on a date
   #elif 'mle' not in method or dynamic: # should be on a date
   start = _validate(start, k_ar, k_diff, self.data.dates,
        method)
   start = super(ARMA, self)._get_predict_start(start)
  _check_arima_start(start, k_ar, k_diff, method, dynamic)
  return start
 
 def _get_predict_end(self, end, dynamic=False):
  # pass through so predict works for ARIMA and ARMA
  return super(ARMA, self)._get_predict_end(end)
 
 def geterrors(self, params):
  """
  Get the errors of the ARMA process.
 
  Parameters
  ----------
  params : array-like
   The fitted ARMA parameters
  order : array-like
   3 item iterable, with the number of AR, MA, and exogenous
   parameters, including the trend
  """
 
  #start = self._get_predict_start(start) # will be an index of a date
  #end, out_of_sample = self._get_predict_end(end)
  params = np.asarray(params)
  k_ar, k_ma = self.k_ar, self.k_ma
  k = self.k_exog + self.k_trend
 
  method = getattr(self, 'method', 'mle')
  if 'mle' in method: # use KalmanFilter to get errors
   (y, k, nobs, k_ar, k_ma, k_lags, newparams, Z_mat, m, R_mat,
    T_mat, paramsdtype) = KalmanFilter._init_kalman_state(params,
                 self)
 
   errors = KalmanFilter.geterrors(y, k, k_ar, k_ma, k_lags, nobs,
           Z_mat, m, R_mat, T_mat,
           paramsdtype)
   if isinstance(errors, tuple):
    errors = errors[0] # non-cython version returns a tuple
  else: # use scipy.signal.lfilter
   y = self.endog.copy()
   k = self.k_exog + self.k_trend
   if k > 0:
    y -= dot(self.exog, params[:k])
 
   k_ar = self.k_ar
   k_ma = self.k_ma
 
   (trendparams, exparams,
    arparams, maparams) = _unpack_params(params, (k_ar, k_ma),
             self.k_trend, self.k_exog,
             reverse=False)
   b, a = np.r_[1, -arparams], np.r_[1, maparams]
   zi = zeros((max(k_ar, k_ma)))
   for i in range(k_ar):
    zi[i] = sum(-b[:i+1][::-1]*y[:i+1])
   e = lfilter(b, a, y, zi=zi)
   errors = e[0][k_ar:]
  return errors.squeeze()
 
 def predict(self, params, start=None, end=None, exog=None, dynamic=False):
  method = getattr(self, 'method', 'mle') # don't assume fit
  #params = np.asarray(params)
 
  # will return an index of a date
  start = self._get_predict_start(start, dynamic)
  end, out_of_sample = self._get_predict_end(end, dynamic)
  if out_of_sample and (exog is None and self.k_exog > 0):
   raise ValueError("You must provide exog for ARMAX")
 
  endog = self.endog
  resid = self.geterrors(params)
  k_ar = self.k_ar
 
  if out_of_sample != 0 and self.k_exog > 0:
   if self.k_exog == 1 and exog.ndim == 1:
    exog = exog[:, None]
    # we need the last k_ar exog for the lag-polynomial
   if self.k_exog > 0 and k_ar > 0:
    # need the last k_ar exog for the lag-polynomial
    exog = np.vstack((self.exog[-k_ar:, self.k_trend:], exog))
 
  if dynamic:
   #TODO: now that predict does dynamic in-sample it should
   # also return error estimates and confidence intervals
   # but how"""
  Compute the log-likelihood for ARMA(p,q) model
 
  Notes
  -----
  Likelihood used depends on the method set in fit
  """
  method = self.method
  if method in ['mle', 'css-mle']:
   return self.loglike_kalman(params, set_sigma2)
  elif method == 'css':
   return self.loglike_css(params, set_sigma2)
  else:
   raise ValueError("Method %s not understood" % method)
 
 def loglike_kalman(self, params, set_sigma2=True):
  """
  Compute exact loglikelihood for ARMA(p,q) model by the Kalman Filter.
  """
  return KalmanFilter.loglike(params, self, set_sigma2)
 
 def loglike_css(self, params, set_sigma2=True):
  """
  Conditional Sum of Squares likelihood function.
  """
  k_ar = self.k_ar
  k_ma = self.k_ma
  k = self.k_exog + self.k_trend
  y = self.endog.copy().astype(params.dtype)
  nobs = self.nobs
  # how to handle if empty"css-mle",
   transparams=True, solver='lbfgs', maxiter=50, full_output=1,
   disp=5, callback=None, **kwargs):
  """
  Fits ARMA(p,q) model using exact maximum likelihood via Kalman filter.
 
  Parameters
  ----------
  start_params : array-like, optional
   Starting parameters for ARMA(p,q). If None, the default is given
   by ARMA._fit_start_params. See there for more information.
  transparams : bool, optional
   Whehter or not to transform the parameters to ensure stationarity.
   Uses the transformation suggested in Jones (1980). If False,
   no checking for stationarity or invertibility is done.
  method : str {'css-mle','mle','css'}
   This is the loglikelihood to maximize. If "css-mle", the
   conditional sum of squares likelihood is maximized and its values
   are used as starting values for the computation of the exact
   likelihood via the Kalman filter. If "mle", the exact likelihood
   is maximized via the Kalman Filter. If "css" the conditional sum
   of squares likelihood is maximized. All three methods use
   `start_params` as starting parameters. See above for more
   information.
  trend : str {'c','nc'}
   Whether to include a constant or not. 'c' includes constant,
   'nc' no constant.
  solver : str or None, optional
   Solver to be used. The default is 'lbfgs' (limited memory
   Broyden-Fletcher-Goldfarb-Shanno). Other choices are 'bfgs',
   'newton' (Newton-Raphson), 'nm' (Nelder-Mead), 'cg' -
   (conjugate gradient), 'ncg' (non-conjugate gradient), and
   'powell'. By default, the limited memory BFGS uses m=12 to
   approximate the Hessian, projected gradient tolerance of 1e-8 and
   factr = 1e2. You can change these by using kwargs.
  maxiter : int, optional
   The maximum number of function evaluations. Default is 50.
  tol : float
   The convergence tolerance. Default is 1e-08.
  full_output : bool, optional
   If True, all output from solver will be available in
   the Results object's mle_retvals attribute. Output is dependent
   on the solver. See Notes for more information.
  disp : bool, optional
   If True, convergence information is printed. For the default
   l_bfgs_b solver, disp controls the frequency of the output during
   the iterations. disp < 0 means no output in this case.
  callback : function, optional
   Called after each iteration as callback(xk) where xk is the current
   parameter vector.
  kwargs
   See Notes for keyword arguments that can be passed to fit.
 
  Returns
  -------
  statsmodels.tsa.arima_model.ARMAResults class
 
  See also
  --------
  statsmodels.base.model.LikelihoodModel.fit : for more information
   on using the solvers.
  ARMAResults : results class returned by fit
 
  Notes
  ------
  If fit by 'mle', it is assumed for the Kalman Filter that the initial
  unkown state is zero, and that the inital variance is
  P = dot(inv(identity(m**2)-kron(T,T)),dot(R,R.T).ravel('F')).reshape(r,
  r, order = 'F')
 
  """
  k_ar = self.k_ar
  k_ma = self.k_ma
 
  # enforce invertibility
  self.transparams = transparams
 
  endog, exog = self.endog, self.exog
  k_exog = self.k_exog
  self.nobs = len(endog) # this is overwritten if method is 'css'
 
  # (re)set trend and handle exogenous variables
  # always pass original exog
  k_trend, exog = _make_arma_exog(endog, self.exog, trend)
 
  # Check has something to estimate
  if k_ar == 0 and k_ma == 0 and k_trend == 0 and k_exog == 0:
   raise ValueError("Estimation requires the inclusion of least one "
       "AR term, MA term, a constant or an exogenous "
       "variable.")
 
  # check again now that we know the trend
  _check_estimable(len(endog), k_ar + k_ma + k_exog + k_trend)
 
  self.k_trend = k_trend
  self.exog = exog # overwrites original exog from __init__
 
  # (re)set names for this model
  self.exog_names = _make_arma_names(self.data, k_trend, (k_ar, k_ma),
           self.exog_names)
  k = k_trend + k_exog
 
  # choose objective function
  if k_ma == 0 and k_ar == 0:
   method = "css" # Always CSS when no AR or MA terms
 
  self.method = method = method.lower()
 
  # adjust nobs for css
  if method == 'css':
   self.nobs = len(self.endog) - k_ar
 
  if start_params is not None:
   start_params = np.asarray(start_params)
 
  else: # estimate starting parameters
   start_params = self._fit_start_params((k_ar, k_ma, k), method)
 
  if transparams: # transform initial parameters to ensure invertibility
   start_params = self._invtransparams(start_params)
 
  if solver == 'lbfgs':
   kwargs.setdefault('pgtol', 1e-8)
   kwargs.setdefault('factr', 1e2)
   kwargs.setdefault('m', 12)
   kwargs.setdefault('approx_grad', True)
  mlefit = super(ARMA, self).fit(start_params, method=solver,
          maxiter=maxiter,
          full_output=full_output, disp=disp,
          callback=callback, **kwargs)
  params = mlefit.params
 
  if transparams: # transform parameters back
   params = self._transparams(params)
 
  self.transparams = False # so methods don't expect transf.
 
  normalized_cov_params = None # TODO: fix this
  armafit = ARMAResults(self, params, normalized_cov_params)
  armafit.mle_retvals = mlefit.mle_retvals
  armafit.mle_settings = mlefit.mle_settings
  armafit.mlefit = mlefit
  return ARMAResultsWrapper(armafit)
 
 
#NOTE: the length of endog changes when we give a difference to fit
#so model methods are not the same on unfit models as fit ones
#starting to think that order of model should be put in instantiation...
class ARIMA(ARMA):
 
 __doc__ = tsbase._tsa_doc % {"model" : _arima_model,
         "params" : _arima_params, "extra_params" : "",
         "extra_sections" : _armax_notes %
         {"Model" : "ARIMA"}}
 
 def __new__(cls, endog, order, exog=None, dates=None, freq=None,
    missing='none'):
  p, d, q = order
  if d == 0: # then we just use an ARMA model
   return ARMA(endog, (p, q), exog, dates, freq, missing)
  else:
   mod = super(ARIMA, cls).__new__(cls)
   mod.__init__(endog, order, exog, dates, freq, missing)
   return mod
 
 def __init__(self, endog, order, exog=None, dates=None, freq=None,
     missing='none'):
  p, d, q = order
  if d > 2:
   #NOTE: to make more general, need to address the d == 2 stuff
   # in the predict method
   raise ValueError("d > 2 is not supported")
  super(ARIMA, self).__init__(endog, (p, q), exog, dates, freq, missing)
  self.k_diff = d
  self._first_unintegrate = unintegrate_levels(self.endog[:d], d)
  self.endog = np.diff(self.endog, n=d)
  #NOTE: will check in ARMA but check again since differenced now
  _check_estimable(len(self.endog), p+q)
  if exog is not None:
   self.exog = self.exog[d:]
  if d == 1:
   self.data.ynames = 'D.' + self.endog_names
  else:
   self.data.ynames = 'D{0:d}.'.format(d) + self.endog_names
  # what about exog, should we difference it automatically before
  # super call"""
  """
  #TODO: remove all these getattr and move order specification to
  # class constructor
  k_diff = getattr(self, 'k_diff', 0)
  method = getattr(self, 'method', 'mle')
  k_ar = getattr(self, 'k_ar', 0)
  if start is None:
   if 'mle' in method and not dynamic:
    start = 0
   else:
    start = k_ar
  elif isinstance(start, int):
    start -= k_diff
    try: # catch when given an integer outside of dates index
     start = super(ARIMA, self)._get_predict_start(start,
                 dynamic)
    except IndexError:
     raise ValueError("start must be in series. "
          "got %d" % (start + k_diff))
  else: # received a date
   start = _validate(start, k_ar, k_diff, self.data.dates,
        method)
   start = super(ARIMA, self)._get_predict_start(start, dynamic)
  # reset date for k_diff adjustment
  self._set_predict_start_date(start + k_diff)
  return start
 
 def _get_predict_end(self, end, dynamic=False):
  """
  Returns last index to be forecast of the differenced array.
  Handling of inclusiveness should be done in the predict function.
  """
  end, out_of_sample = super(ARIMA, self)._get_predict_end(end, dynamic)
  if 'mle' not in self.method and not dynamic:
   end -= self.k_ar
 
  return end - self.k_diff, out_of_sample
 
 def fit(self, start_params=None, trend='c', method="css-mle",
   transparams=True, solver='lbfgs', maxiter=50, full_output=1,
   disp=5, callback=None, **kwargs):
  """
  Fits ARIMA(p,d,q) model by exact maximum likelihood via Kalman filter.
 
  Parameters
  ----------
  start_params : array-like, optional
   Starting parameters for ARMA(p,q). If None, the default is given
   by ARMA._fit_start_params. See there for more information.
  transparams : bool, optional
   Whehter or not to transform the parameters to ensure stationarity.
   Uses the transformation suggested in Jones (1980). If False,
   no checking for stationarity or invertibility is done.
  method : str {'css-mle','mle','css'}
   This is the loglikelihood to maximize. If "css-mle", the
   conditional sum of squares likelihood is maximized and its values
   are used as starting values for the computation of the exact
   likelihood via the Kalman filter. If "mle", the exact likelihood
   is maximized via the Kalman Filter. If "css" the conditional sum
   of squares likelihood is maximized. All three methods use
   `start_params` as starting parameters. See above for more
   information.
  trend : str {'c','nc'}
   Whether to include a constant or not. 'c' includes constant,
   'nc' no constant.
  solver : str or None, optional
   Solver to be used. The default is 'lbfgs' (limited memory
   Broyden-Fletcher-Goldfarb-Shanno). Other choices are 'bfgs',
   'newton' (Newton-Raphson), 'nm' (Nelder-Mead), 'cg' -
   (conjugate gradient), 'ncg' (non-conjugate gradient), and
   'powell'. By default, the limited memory BFGS uses m=12 to
   approximate the Hessian, projected gradient tolerance of 1e-8 and
   factr = 1e2. You can change these by using kwargs.
  maxiter : int, optional
   The maximum number of function evaluations. Default is 50.
  tol : float
   The convergence tolerance. Default is 1e-08.
  full_output : bool, optional
   If True, all output from solver will be available in
   the Results object's mle_retvals attribute. Output is dependent
   on the solver. See Notes for more information.
  disp : bool, optional
   If True, convergence information is printed. For the default
   l_bfgs_b solver, disp controls the frequency of the output during
   the iterations. disp < 0 means no output in this case.
  callback : function, optional
   Called after each iteration as callback(xk) where xk is the current
   parameter vector.
  kwargs
   See Notes for keyword arguments that can be passed to fit.
 
  Returns
  -------
  `statsmodels.tsa.arima.ARIMAResults` class
 
  See also
  --------
  statsmodels.base.model.LikelihoodModel.fit : for more information
   on using the solvers.
  ARIMAResults : results class returned by fit
 
  Notes
  ------
  If fit by 'mle', it is assumed for the Kalman Filter that the initial
  unkown state is zero, and that the inital variance is
  P = dot(inv(identity(m**2)-kron(T,T)),dot(R,R.T).ravel('F')).reshape(r,
  r, order = 'F')
 
  """
  arima_fit = super(ARIMA, self).fit(start_params, trend,
           method, transparams, solver,
           maxiter, full_output, disp,
           callback, **kwargs)
  normalized_cov_params = None # TODO: fix this"typ %s not understood" % typ)
 
 predict.__doc__ = _arima_predict
 
 
class ARMAResults(tsbase.TimeSeriesModelResults):
 """
 Class to hold results from fitting an ARMA model.
 
 Parameters
 ----------
 model : ARMA instance
  The fitted model instance
 params : array
  Fitted parameters
 normalized_cov_params : array, optional
  The normalized variance covariance matrix
 scale : float, optional
  Optional argument to scale the variance covariance matrix.
 
 Returns
 --------
 **Attributes**
 
 aic : float
  Akaike Information Criterion
  :math:`-2*llf+2* df_model`
  where `df_model` includes all AR parameters, MA parameters, constant
  terms parameters on constant terms and the variance.
 arparams : array
  The parameters associated with the AR coefficients in the model.
 arroots : array
  The roots of the AR coefficients are the solution to
  (1 - arparams[0]*z - arparams[1]*z**2 -...- arparams[p-1]*z**k_ar) = 0
  Stability requires that the roots in modulus lie outside the unit
  circle.
 bic : float
  Bayes Information Criterion
  -2*llf + log(nobs)*df_model
  Where if the model is fit using conditional sum of squares, the
  number of observations `nobs` does not include the `p` pre-sample
  observations.
 bse : array
  The standard errors of the parameters. These are computed using the
  numerical Hessian.
 df_model : array
  The model degrees of freedom = `k_exog` + `k_trend` + `k_ar` + `k_ma`
 df_resid : array
  The residual degrees of freedom = `nobs` - `df_model`
 fittedvalues : array
  The predicted values of the model.
 hqic : float
  Hannan-Quinn Information Criterion
  -2*llf + 2*(`df_model`)*log(log(nobs))
  Like `bic` if the model is fit using conditional sum of squares then
  the `k_ar` pre-sample observations are not counted in `nobs`.
 k_ar : int
  The number of AR coefficients in the model.
 k_exog : int
  The number of exogenous variables included in the model. Does not
  include the constant.
 k_ma : int
  The number of MA coefficients.
 k_trend : int
  This is 0 for no constant or 1 if a constant is included.
 llf : float
  The value of the log-likelihood function evaluated at `params`.
 maparams : array
  The value of the moving average coefficients.
 maroots : array
  The roots of the MA coefficients are the solution to
  (1 + maparams[0]*z + maparams[1]*z**2 + ... + maparams[q-1]*z**q) = 0
  Stability requires that the roots in modules lie outside the unit
  circle.
 model : ARMA instance
  A reference to the model that was fit.
 nobs : float
  The number of observations used to fit the model. If the model is fit
  using exact maximum likelihood this is equal to the total number of
  observations, `n_totobs`. If the model is fit using conditional
  maximum likelihood this is equal to `n_totobs` - `k_ar`.
 n_totobs : float
  The total number of observations for `endog`. This includes all
  observations, even pre-sample values if the model is fit using `css`.
 params : array
  The parameters of the model. The order of variables is the trend
  coefficients and the `k_exog` exognous coefficients, then the
  `k_ar` AR coefficients, and finally the `k_ma` MA coefficients.
 pvalues : array
  The p-values associated with the t-values of the coefficients. Note
  that the coefficients are assumed to have a Student's T distribution.
 resid : array
  The model residuals. If the model is fit using 'mle' then the
  residuals are created via the Kalman Filter. If the model is fit
  using 'css' then the residuals are obtained via `scipy.signal.lfilter`
  adjusted such that the first `k_ma` residuals are zero. These zero
  residuals are not returned.
 scale : float
  This is currently set to 1.0 and not used by the model or its results.
 sigma2 : float
  The variance of the residuals. If the model is fit by 'css',
  sigma2 = ssr/nobs, where ssr is the sum of squared residuals. If
  the model is fit by 'mle', then sigma2 = 1/nobs * sum(v**2 / F)
  where v is the one-step forecast error and F is the forecast error
  variance. See `nobs` for the difference in definitions depending on the
  fit.
 """
 _cache = {}
 
 #TODO: use this for docstring when we fix nobs issue
 
 def __init__(self, model, params, normalized_cov_params=None, scale=1.):
  super(ARMAResults, self).__init__(model, params, normalized_cov_params,
           scale)
  self.sigma2 = model.sigma2
  nobs = model.nobs
  self.nobs = nobs
  k_exog = model.k_exog
  self.k_exog = k_exog
  k_trend = model.k_trend
  self.k_trend = k_trend
  k_ar = model.k_ar
  self.k_ar = k_ar
  self.n_totobs = len(model.endog)
  k_ma = model.k_ma
  self.k_ma = k_ma
  df_model = k_exog + k_trend + k_ar + k_ma
  self._ic_df_model = df_model + 1
  self.df_model = df_model
  self.df_resid = self.nobs - df_model
  self._cache = resettable_cache()
  self.constant = 0 #Added by me
 
 @cache_readonly
 def arroots(self):
  return np.roots(np.r_[1, -self.arparams])**-1
 
 @cache_readonly
 def maroots(self):
  return np.roots(np.r_[1, self.maparams])**-1
 
 @cache_readonly
 def arfreq(self):
  r"""
  Returns the frequency of the AR roots.
 
  This is the solution, x, to z = abs(z)*exp(2j*np.pi*x) where z are the
  roots.
  """
  z = self.arroots
  if not z.size:
   return
  return np.arctan2(z.imag, z.real) / (2*pi)
 
 @cache_readonly
 def mafreq(self):
  r"""
  Returns the frequency of the MA roots.
 
  This is the solution, x, to z = abs(z)*exp(2j*np.pi*x) where z are the
  roots.
  """
  z = self.maroots
  if not z.size:
   return
  return np.arctan2(z.imag, z.real) / (2*pi)
 
 @cache_readonly
 def arparams(self):
  k = self.k_exog + self.k_trend
  return self.params[k:k+self.k_ar]
 
 @cache_readonly
 def maparams(self):
  k = self.k_exog + self.k_trend
  k_ar = self.k_ar
  return self.params[k+k_ar:]
 
 @cache_readonly
 def llf(self):
  return self.model.loglike(self.params)
 
 @cache_readonly
 def bse(self):
  params = self.params
  hess = self.model.hessian(params)
  if len(params) == 1: # can't take an inverse, ensure 1d
   return np.sqrt(-1./hess[0])
  return np.sqrt(np.diag(-inv(hess)))
 
 def cov_params(self): # add scale argument"css" and k_ar > 0:
    exog = exog[k_ar:]
  if model.method == "css" and k_ar > 0:
   endog = endog[k_ar:]
  fv = endog - self.resid
  # add deterministic part back in
  #k = self.k_exog + self.k_trend
  #TODO: this needs to be commented out for MLE with constant
  #if k != 0:
  # fv += dot(exog, self.params[:k])
  return fv
 
 @cache_readonly
 def resid(self):
  return self.model.geterrors(self.params)
 
 @cache_readonly
 def pvalues(self):
 #TODO: same for conditional and unconditional"""
  Out-of-sample forecasts
 
  Parameters
  ----------
  steps : int
   The number of out of sample forecasts from the end of the
   sample.
  exog : array
   If the model is an ARMAX, you must provide out of sample
   values for the exogenous variables. This should not include
   the constant.
  alpha : float
   The confidence intervals for the forecasts are (1 - alpha) %
 
  Returns
  -------
  forecast : array
   Array of out of sample forecasts
  stderr : array
   Array of the standard error of the forecasts.
  conf_int : array
   2d array of the confidence interval for the forecast
  """
  if exog is not None:
   #TODO: make a convenience function for this. we're using the
   # pattern elsewhere in the codebase
   exog = np.asarray(exog)
   if self.k_exog == 1 and exog.ndim == 1:
    exog = exog[:, None]
   elif exog.ndim == 1:
    if len(exog) != self.k_exog:
     raise ValueError("1d exog given and len(exog) != k_exog")
    exog = exog[None, :]
   if exog.shape[0] != steps:
    raise ValueError("new exog needed for each step")
   # prepend in-sample exog observations
   exog = np.vstack((self.model.exog[-self.k_ar:, self.k_trend:],
        exog))
 
  forecast, ct = _arma_predict_out_of_sample(self.params,
            steps, self.resid, self.k_ar,
            self.k_ma, self.k_trend,
            self.k_exog, self.model.endog,
            exog, method=self.model.method)
  self.constant = ct
 
  # compute the standard errors
  fcasterr = self._forecast_error(steps)
  conf_int = self._forecast_conf_int(forecast, fcasterr, alpha)
 
  return forecast, fcasterr, conf_int
 
 def summary(self, alpha=.05):
  """Summarize the Model
 
  Parameters
  ----------
  alpha : float, optional
   Significance level for the confidence intervals.
 
  Returns
  -------
  smry : Summary instance
   This holds the summary table and text, which can be printed or
   converted to various output formats.
 
  See Also
  --------
  statsmodels.iolib.summary.Summary
  """
  from statsmodels.iolib.summary import Summary
  model = self.model
  title = model.__class__.__name__ + ' Model Results'
  method = model.method
  # get sample TODO: make better sample machinery for estimation
  k_diff = getattr(self, 'k_diff', 0)
  if 'mle' in method:
   start = k_diff
  else:
   start = k_diff + self.k_ar
  if self.data.dates is not None:
   dates = self.data.dates
   sample = [dates[start].strftime('%m-%d-%Y')]
   sample += ['- ' + dates[-1].strftime('%m-%d-%Y')]
  else:
   sample = str(start) + ' - ' + str(len(self.data.orig_endog))
 
  k_ar, k_ma = self.k_ar, self.k_ma
  if not k_diff:
   order = str((k_ar, k_ma))
  else:
   order = str((k_ar, k_diff, k_ma))
  top_left = [('Dep. Variable:', None),
     ('Model:', [model.__class__.__name__ + order]),
     ('Method:', [method]),
     ('Date:', None),
     ('Time:', None),
     ('Sample:', [sample[0]]),
     ('', [sample[1]])
     ]
 
  top_right = [
      ('No. Observations:', [str(len(self.model.endog))]),
      ('Log Likelihood', ["%#5.3f" % self.llf]),
      ('S.D. of innovations', ["%#5.3f" % self.sigma2**.5]),
      ('AIC', ["%#5.3f" % self.aic]),
      ('BIC', ["%#5.3f" % self.bic]),
      ('HQIC', ["%#5.3f" % self.hqic])]
 
  smry = Summary()
  smry.add_table_2cols(self, gleft=top_left, gright=top_right,
        title=title)
  smry.add_table_params(self, alpha=alpha, use_t=False)
 
  # Make the roots table
  from statsmodels.iolib.table import SimpleTable
 
  if k_ma and k_ar:
   arstubs = ["AR.%d" % i for i in range(1, k_ar + 1)]
   mastubs = ["MA.%d" % i for i in range(1, k_ma + 1)]
   stubs = arstubs + mastubs
   roots = np.r_[self.arroots, self.maroots]
   freq = np.r_[self.arfreq, self.mafreq]
  elif k_ma:
   mastubs = ["MA.%d" % i for i in range(1, k_ma + 1)]
   stubs = mastubs
   roots = self.maroots
   freq = self.mafreq
  elif k_ar:
   arstubs = ["AR.%d" % i for i in range(1, k_ar + 1)]
   stubs = arstubs
   roots = self.arroots
   freq = self.arfreq
  else: # 0,0 model
   stubs = []
  if len(stubs): # not 0, 0
   modulus = np.abs(roots)
   data = np.column_stack((roots.real, roots.imag, modulus, freq))
   roots_table = SimpleTable(data,
          headers=['   Real',
            '   Imaginary',
            '   Modulus',
            '  Frequency'],
          title="Roots",
          stubs=stubs,
          data_fmts=["%17.4f", "%+17.4fj",
             "%17.4f", "%17.4f"])
 
   smry.tables.append(roots_table)
  return smry
 
 def summary2(self, title=None, alpha=.05, float_format="%.4f"):
  """Experimental summary function for ARIMA Results
 
  Parameters
  -----------
  title : string, optional
   Title for the top table. If not None, then this replaces the
   default title
  alpha : float
   significance level for the confidence intervals
  float_format: string
   print format for floats in parameters summary
 
  Returns
  -------
  smry : Summary instance
   This holds the summary table and text, which can be printed or
   converted to various output formats.
 
  See Also
  --------
  statsmodels.iolib.summary2.Summary : class to hold summary
   results
 
  """
  from pandas import DataFrame
  # get sample TODO: make better sample machinery for estimation
  k_diff = getattr(self, 'k_diff', 0)
  if 'mle' in self.model.method:
   start = k_diff
  else:
   start = k_diff + self.k_ar
  if self.data.dates is not None:
   dates = self.data.dates
   sample = [dates[start].strftime('%m-%d-%Y')]
   sample += [dates[-1].strftime('%m-%d-%Y')]
  else:
   sample = str(start) + ' - ' + str(len(self.data.orig_endog))
 
  k_ar, k_ma = self.k_ar, self.k_ma
 
  # Roots table
  if k_ma and k_ar:
   arstubs = ["AR.%d" % i for i in range(1, k_ar + 1)]
   mastubs = ["MA.%d" % i for i in range(1, k_ma + 1)]
   stubs = arstubs + mastubs
   roots = np.r_[self.arroots, self.maroots]
   freq = np.r_[self.arfreq, self.mafreq]
  elif k_ma:
   mastubs = ["MA.%d" % i for i in range(1, k_ma + 1)]
   stubs = mastubs
   roots = self.maroots
   freq = self.mafreq
  elif k_ar:
   arstubs = ["AR.%d" % i for i in range(1, k_ar + 1)]
   stubs = arstubs
   roots = self.arroots
   freq = self.arfreq
  else: # 0, 0 order
   stubs = []
 
  if len(stubs):
   modulus = np.abs(roots)
   data = np.column_stack((roots.real, roots.imag, modulus, freq))
   data = DataFrame(data)
   data.columns = ['Real', 'Imaginary', 'Modulus', 'Frequency']
   data.index = stubs
 
  # Summary
  from statsmodels.iolib import summary2
  smry = summary2.Summary()
 
  # Model info
  model_info = summary2.summary_model(self)
  model_info['Method:'] = self.model.method
  model_info['Sample:'] = sample[0]
  model_info[' '] = sample[-1]
  model_info['S.D. of innovations:'] = "%#5.3f" % self.sigma2**.5
  model_info['HQIC:'] = "%#5.3f" % self.hqic
  model_info['No. Observations:'] = str(len(self.model.endog))
 
  # Parameters
  params = summary2.summary_params(self)
  smry.add_dict(model_info)
  smry.add_df(params, float_format=float_format)
  if len(stubs):
   smry.add_df(data, float_format="%17.4f")
  smry.add_title(results=self, title=title)
 
  return smry
 
 def plot_predict(self, start=None, end=None, exog=None, dynamic=False,
      alpha=.05, plot_insample=True, ax=None):
  from statsmodels.graphics.utils import _import_mpl, create_mpl_ax
  _ = _import_mpl()
  fig, ax = create_mpl_ax(ax)
 
 
  # use predict so you set dates
  forecast = self.predict(start, end, exog, dynamic)
  # doing this twice. just add a plot keyword to predict"predict_dates"):
   from pandas import TimeSeries
   forecast = TimeSeries(forecast, index=self.data.predict_dates)
   ax = forecast.plot(ax=ax, label='forecast')
  else:
   ax.plot(forecast)
 
  x = ax.get_lines()[-1].get_xdata()
  if out_of_sample:
   label = "{0:.0%} confidence interval".format(1 - alpha)
   ax.fill_between(x[-out_of_sample:], conf_int[:, 0], conf_int[:, 1],
       color='gray', alpha=.5, label=label)
 
  if plot_insample:
   ax.plot(x[:end + 1 - start], self.model.endog[start:end+1],
     label=self.model.endog_names)
 
  ax.legend(loc='best')
 
  return fig
 plot_predict.__doc__ = _plot_predict
 
 
class ARMAResultsWrapper(wrap.ResultsWrapper):
 _attrs = {}
 _wrap_attrs = wrap.union_dicts(tsbase.TimeSeriesResultsWrapper._wrap_attrs,
         _attrs)
 _methods = {}
 _wrap_methods = wrap.union_dicts(tsbase.TimeSeriesResultsWrapper._wrap_methods,
          _methods)
wrap.populate_wrapper(ARMAResultsWrapper, ARMAResults)
 
 
class ARIMAResults(ARMAResults):
 def predict(self, start=None, end=None, exog=None, typ='linear',
    dynamic=False):
  return self.model.predict(self.params, start, end, exog, typ, dynamic)
 predict.__doc__ = _arima_results_predict
 
 def _forecast_error(self, steps):
  sigma2 = self.sigma2
  ma_rep = arma2ma(np.r_[1, -self.arparams],
       np.r_[1, self.maparams], nobs=steps)
 
  fcerr = np.sqrt(np.cumsum(cumsum_n(ma_rep, self.k_diff)**2)*sigma2)
  return fcerr
 
 def _forecast_conf_int(self, forecast, fcerr, alpha):
  const = norm.ppf(1 - alpha/2.)
  conf_int = np.c_[forecast - const*fcerr, forecast + const*fcerr]
  return conf_int
 
 def forecast(self, steps=1, exog=None, alpha=.05):
  """
  Out-of-sample forecasts
 
  Parameters
  ----------
  steps : int
   The number of out of sample forecasts from the end of the
   sample.
  exog : array
   If the model is an ARIMAX, you must provide out of sample
   values for the exogenous variables. This should not include
   the constant.
  alpha : float
   The confidence intervals for the forecasts are (1 - alpha) %
 
  Returns
  -------
  forecast : array
   Array of out of sample forecasts
  stderr : array
   Array of the standard error of the forecasts.
  conf_int : array
   2d array of the confidence interval for the forecast
 
  Notes
  -----
  Prediction is done in the levels of the original endogenous variable.
  If you would like prediction of differences in levels use `predict`.
  """
  if exog is not None:
   if self.k_exog == 1 and exog.ndim == 1:
    exog = exog[:, None]
   if exog.shape[0] != steps:
    raise ValueError("new exog needed for each step")
   # prepend in-sample exog observations
   exog = np.vstack((self.model.exog[-self.k_ar:, self.k_trend:],
        exog))
  forecast, ct = _arma_predict_out_of_sample(self.params, steps, self.resid,
            self.k_ar, self.k_ma,
            self.k_trend, self.k_exog,
            self.model.endog,
            exog, method=self.model.method)
 
  #self.constant = ct
  d = self.k_diff
  endog = self.model.data.endog[-d:]
  forecast = unintegrate(forecast, unintegrate_levels(endog, d))[d:]
 
  # get forecast errors
  fcerr = self._forecast_error(steps)
  conf_int = self._forecast_conf_int(forecast, fcerr, alpha)
  return forecast, fcerr, conf_int
 
 def plot_predict(self, start=None, end=None, exog=None, dynamic=False,
      alpha=.05, plot_insample=True, ax=None):
  from statsmodels.graphics.utils import _import_mpl, create_mpl_ax
  _ = _import_mpl()
  fig, ax = create_mpl_ax(ax)
 
  # use predict so you set dates
  forecast = self.predict(start, end, exog, 'levels', dynamic)
  # doing this twice. just add a plot keyword to predict"predict_dates"):
   from pandas import TimeSeries
   forecast = TimeSeries(forecast, index=self.data.predict_dates)
   ax = forecast.plot(ax=ax, label='forecast')
  else:
   ax.plot(forecast)
 
  x = ax.get_lines()[-1].get_xdata()
  if out_of_sample:
   label = "{0:.0%} confidence interval".format(1 - alpha)
   ax.fill_between(x[-out_of_sample:], conf_int[:, 0], conf_int[:, 1],
       color='gray', alpha=.5, label=label)
 
  if plot_insample:
   import re
   k_diff = self.k_diff
   label = re.sub("D\d*\.", "", self.model.endog_names)
   levels = unintegrate(self.model.endog,
         self.model._first_unintegrate)
   ax.plot(x[:end + 1 - start],
     levels[start + k_diff:end + k_diff + 1], label=label)
 
  ax.legend(loc='best')
 
  return fig
 
 plot_predict.__doc__ = _arima_plot_predict
 
 
class ARIMAResultsWrapper(ARMAResultsWrapper):
 pass
wrap.populate_wrapper(ARIMAResultsWrapper, ARIMAResults)
 
 
if __name__ == "__main__":
 import statsmodels.api as sm
 
 # simulate arma process
 from statsmodels.tsa.arima_process import arma_generate_sample
 y = arma_generate_sample([1., -.75], [1., .25], nsample=1000)
 arma = ARMA(y)
 res = arma.fit(trend='nc', order=(1, 1))
 
 np.random.seed(12345)
 y_arma22 = arma_generate_sample([1., -.85, .35], [1, .25, -.9],
         nsample=1000)
 arma22 = ARMA(y_arma22)
 res22 = arma22.fit(trend='nc', order=(2, 2))
 
 # test CSS
 arma22_css = ARMA(y_arma22)
 res22css = arma22_css.fit(trend='nc', order=(2, 2), method='css')
 
 data = sm.datasets.sunspots.load()
 ar = ARMA(data.endog)
 resar = ar.fit(trend='nc', order=(9, 0))
 
 y_arma31 = arma_generate_sample([1, -.75, -.35, .25], [.1],
         nsample=1000)
 
 arma31css = ARMA(y_arma31)
 res31css = arma31css.fit(order=(3, 1), method="css", trend="nc",
        transparams=True)
 
 y_arma13 = arma_generate_sample([1., -.75], [1, .25, -.5, .8],
         nsample=1000)
 arma13css = ARMA(y_arma13)
 res13css = arma13css.fit(order=(1, 3), method='css', trend='nc')
 
# check css for p < q and q < p
 y_arma41 = arma_generate_sample([1., -.75, .35, .25, -.3], [1, -.35],
         nsample=1000)
 arma41css = ARMA(y_arma41)
 res41css = arma41css.fit(order=(4, 1), trend='nc', method='css')
 
 y_arma14 = arma_generate_sample([1, -.25], [1., -.75, .35, .25, -.3],
         nsample=1000)
 arma14css = ARMA(y_arma14)
 res14css = arma14css.fit(order=(4, 1), trend='nc', method='css')
 
 # ARIMA Model
 from statsmodels.datasets import webuse
 dta = webuse('wpi1')
 wpi = dta['wpi']
 
 mod = ARIMA(wpi, (1, 1, 1)).fit()