import numpy as np
import pandas as pd
from scipy.special import inv_boxcox
from scipy.stats import (
_distn_infrastructure,
boxcox,
rv_continuous,
rv_discrete,
)
from statsmodels.base.data import PandasData
from statsmodels.base.model import Results
from statsmodels.base.wrapper import (
ResultsWrapper,
populate_wrapper,
union_dicts,
)
[docs]class HoltWintersResults(Results):
"""
Results from fitting Exponential Smoothing models.
Parameters
----------
model : ExponentialSmoothing instance
The fitted model instance.
params : dict
All the parameters for the Exponential Smoothing model.
sse : float
The sum of squared errors.
aic : float
The Akaike information criterion.
aicc : float
AIC with a correction for finite sample sizes.
bic : float
The Bayesian information criterion.
optimized : bool
Flag indicating whether the model parameters were optimized to fit
the data.
level : ndarray
An array of the levels values that make up the fitted values.
trend : ndarray
An array of the trend values that make up the fitted values.
season : ndarray
An array of the seasonal values that make up the fitted values.
params_formatted : pd.DataFrame
DataFrame containing all parameters, their short names and a flag
indicating whether the parameter's value was optimized to fit the data.
resid : ndarray
An array of the residuals of the fittedvalues and actual values.
k : int
The k parameter used to remove the bias in AIC, BIC etc.
fittedvalues : ndarray
An array of the fitted values. Fitted by the Exponential Smoothing
model.
fittedfcast : ndarray
An array of both the fitted values and forecast values.
fcastvalues : ndarray
An array of the forecast values forecast by the Exponential Smoothing
model.
mle_retvals : {None, scipy.optimize.optimize.OptimizeResult}
Optimization results if the parameters were optimized to fit the data.
"""
def __init__(
self,
model,
params,
sse,
aic,
aicc,
bic,
optimized,
level,
trend,
season,
params_formatted,
resid,
k,
fittedvalues,
fittedfcast,
fcastvalues,
mle_retvals=None,
):
self.data = model.data
super(HoltWintersResults, self).__init__(model, params)
self._model = model
self._sse = sse
self._aic = aic
self._aicc = aicc
self._bic = bic
self._optimized = optimized
self._level = level
self._trend = trend
self._season = season
self._params_formatted = params_formatted
self._fittedvalues = fittedvalues
self._fittedfcast = fittedfcast
self._fcastvalues = fcastvalues
self._resid = resid
self._k = k
self._mle_retvals = mle_retvals
@property
def aic(self):
"""
The Akaike information criterion.
"""
return self._aic
@property
def aicc(self):
"""
AIC with a correction for finite sample sizes.
"""
return self._aicc
@property
def bic(self):
"""
The Bayesian information criterion.
"""
return self._bic
@property
def sse(self):
"""
The sum of squared errors between the data and the fittted value.
"""
return self._sse
@property
def model(self):
"""
The model used to produce the results instance.
"""
return self._model
@model.setter
def model(self, value):
self._model = value
@property
def level(self):
"""
An array of the levels values that make up the fitted values.
"""
return self._level
@property
def optimized(self):
"""
Flag indicating if model parameters were optimized to fit the data.
"""
return self._optimized
@property
def slope(self):
"""
An array of the slope values that make up the fitted values.
.. deprecated:: 0.12
Use the trend property instead.
"""
import warnings
warnings.warn(
"slope is deprecated and will be removed after 0.13", FutureWarning
)
return self._trend
@property
def trend(self):
"""
An array of the trend values that make up the fitted values.
"""
return self._trend
@property
def season(self):
"""
An array of the seasonal values that make up the fitted values.
"""
return self._season
@property
def params_formatted(self):
"""
DataFrame containing all parameters
Contains short names and a flag indicating whether the parameter's
value was optimized to fit the data.
"""
return self._params_formatted
@property
def fittedvalues(self):
"""
An array of the fitted values
"""
return self._fittedvalues
@property
def fittedfcast(self):
"""
An array of both the fitted values and forecast values.
"""
return self._fittedfcast
@property
def fcastvalues(self):
"""
An array of the forecast values
"""
return self._fcastvalues
@property
def resid(self):
"""
An array of the residuals of the fittedvalues and actual values.
"""
return self._resid
@property
def k(self):
"""
The k parameter used to remove the bias in AIC, BIC etc.
"""
return self._k
@property
def mle_retvals(self):
"""
Optimization results if the parameters were optimized to fit the data.
"""
return self._mle_retvals
@mle_retvals.setter
def mle_retvals(self, value):
self._mle_retvals = value
[docs] def predict(self, start=None, end=None):
"""
In-sample prediction and out-of-sample forecasting
Parameters
----------
start : int, str, or datetime, optional
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. Default is the the zeroth observation.
end : int, str, or datetime, optional
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. Default is the last observation in
the sample.
Returns
-------
forecast : ndarray
Array of out of sample forecasts.
"""
return self.model.predict(self.params, start, end)
[docs] def forecast(self, steps=1):
"""
Out-of-sample forecasts
Parameters
----------
steps : int
The number of out of sample forecasts from the end of the
sample.
Returns
-------
forecast : ndarray
Array of out of sample forecasts
"""
try:
freq = getattr(self.model._index, "freq", 1)
if isinstance(freq, int):
start = self.model._index.shape[0]
end = start + steps - 1
else:
start = self.model._index[-1] + freq
end = self.model._index[-1] + steps * freq
return self.model.predict(self.params, start=start, end=end)
except (AttributeError, ValueError):
# May occur when the index does not have a freq
return self.model._predict(h=steps, **self.params).fcastvalues
[docs] def summary(self):
"""
Summarize the fitted Model
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
from statsmodels.iolib.table import SimpleTable
model = self.model
title = model.__class__.__name__ + " Model Results"
dep_variable = "endog"
orig_endog = self.model.data.orig_endog
if isinstance(orig_endog, pd.DataFrame):
dep_variable = orig_endog.columns[0]
elif isinstance(orig_endog, pd.Series):
dep_variable = orig_endog.name
seasonal_periods = (
None
if self.model.seasonal is None
else self.model.seasonal_periods
)
lookup = {
"add": "Additive",
"additive": "Additive",
"mul": "Multiplicative",
"multiplicative": "Multiplicative",
None: "None",
}
transform = self.params["use_boxcox"]
box_cox_transform = True if transform else False
box_cox_coeff = (
transform if isinstance(transform, str) else self.params["lamda"]
)
if isinstance(box_cox_coeff, float):
box_cox_coeff = "{:>10.5f}".format(box_cox_coeff)
top_left = [
("Dep. Variable:", [dep_variable]),
("Model:", [model.__class__.__name__]),
("Optimized:", [str(np.any(self.optimized))]),
("Trend:", [lookup[self.model.trend]]),
("Seasonal:", [lookup[self.model.seasonal]]),
("Seasonal Periods:", [str(seasonal_periods)]),
("Box-Cox:", [str(box_cox_transform)]),
("Box-Cox Coeff.:", [str(box_cox_coeff)]),
]
top_right = [
("No. Observations:", [str(len(self.model.endog))]),
("SSE", ["{:5.3f}".format(self.sse)]),
("AIC", ["{:5.3f}".format(self.aic)]),
("BIC", ["{:5.3f}".format(self.bic)]),
("AICC", ["{:5.3f}".format(self.aicc)]),
("Date:", None),
("Time:", None),
]
smry = Summary()
smry.add_table_2cols(
self, gleft=top_left, gright=top_right, title=title
)
formatted = self.params_formatted # type: pd.DataFrame
def _fmt(x):
abs_x = np.abs(x)
scale = 1
if abs_x != 0:
scale = int(np.log10(abs_x))
if scale > 4 or scale < -3:
return "{:>20.5g}".format(x)
dec = min(7 - scale, 7)
fmt = "{{:>20.{0}f}}".format(dec)
return fmt.format(x)
tab = []
for _, vals in formatted.iterrows():
tab.append(
[
_fmt(vals.iloc[1]),
"{0:>20}".format(vals.iloc[0]),
"{0:>20}".format(str(bool(vals.iloc[2]))),
]
)
params_table = SimpleTable(
tab,
headers=["coeff", "code", "optimized"],
title="",
stubs=list(formatted.index),
)
smry.tables.append(params_table)
return smry
[docs] def simulate(
self,
nsimulations,
anchor=None,
repetitions=1,
error="add",
random_errors=None,
random_state=None,
):
r"""
Random simulations using the state space formulation.
Parameters
----------
nsimulations : int
The number of simulation steps.
anchor : int, str, or datetime, optional
First period for simulation. The simulation will be conditional on
all existing datapoints prior to the `anchor`. Type depends on the
index of the given `endog` in the model. Two special cases are the
strings 'start' and 'end'. `start` refers to beginning the
simulation at the first period of the sample, and `end` refers to
beginning the simulation at the first period after the sample.
Integer values can run from 0 to `nobs`, or can be negative to
apply negative indexing. Finally, if a date/time index was provided
to the model, then this argument can be a date string to parse or a
datetime type. Default is 'end'.
repetitions : int, optional
Number of simulated paths to generate. Default is 1 simulated path.
error : {"add", "mul", "additive", "multiplicative"}, optional
Error model for state space formulation. Default is ``"add"``.
random_errors : optional
Specifies how the random errors should be obtained. Can be one of
the following:
* ``None``: Random normally distributed values with variance
estimated from the fit errors drawn from numpy's standard
RNG (can be seeded with the `random_state` argument). This is the
default option.
* A distribution function from ``scipy.stats``, e.g.
``scipy.stats.norm``: Fits the distribution function to the fit
errors and draws from the fitted distribution.
Note the difference between ``scipy.stats.norm`` and
``scipy.stats.norm()``, the latter one is a frozen distribution
function.
* A frozen distribution function from ``scipy.stats``, e.g.
``scipy.stats.norm(scale=2)``: Draws from the frozen distribution
function.
* A ``np.ndarray`` with shape (`nsimulations`, `repetitions`): Uses
the given values as random errors.
* ``"bootstrap"``: Samples the random errors from the fit errors.
random_state : int or np.random.RandomState, optional
A seed for the random number generator or a
``np.random.RandomState`` object. Only used if `random_errors` is
``None``. Default is ``None``.
Returns
-------
sim : pd.Series, pd.DataFrame or np.ndarray
An ``np.ndarray``, ``pd.Series``, or ``pd.DataFrame`` of simulated
values.
If the original data was a ``pd.Series`` or ``pd.DataFrame``, `sim`
will be a ``pd.Series`` if `repetitions` is 1, and a
``pd.DataFrame`` of shape (`nsimulations`, `repetitions`) else.
Otherwise, if `repetitions` is 1, a ``np.ndarray`` of shape
(`nsimulations`,) is returned, and if `repetitions` is not 1 a
``np.ndarray`` of shape (`nsimulations`, `repetitions`) is
returned.
Notes
-----
The simulation is based on the state space model of the Holt-Winter's
methods. The state space model assumes that the true value at time
:math:`t` is randomly distributed around the prediction value.
If using the additive error model, this means:
.. math::
y_t &= \hat{y}_{t|t-1} + e_t\\
e_t &\sim \mathcal{N}(0, \sigma^2)
Using the multiplicative error model:
.. math::
y_t &= \hat{y}_{t|t-1} \cdot (1 + e_t)\\
e_t &\sim \mathcal{N}(0, \sigma^2)
Inserting these equations into the smoothing equation formulation leads
to the state space equations. The notation used here follows
[1]_.
Additionally,
.. math::
B_t &= b_{t-1} \circ_d \phi\\
L_t &= l_{t-1} \circ_b B_t\\
S_t &= s_{t-m}\\
Y_t &= L_t \circ_s S_t,
where :math:`\circ_d` is the operation linking trend and damping
parameter (multiplication if the trend is additive, power if the trend
is multiplicative), :math:`\circ_b` is the operation linking level and
trend (addition if the trend is additive, multiplication if the trend
is multiplicative), and :math:`\circ_s` is the operation linking
seasonality to the rest.
The state space equations can then be formulated as
.. math::
y_t &= Y_t + \eta \cdot e_t\\
l_t &= L_t + \alpha \cdot (M_e \cdot L_t + \kappa_l) \cdot e_t\\
b_t &= B_t + \beta \cdot (M_e \cdot B_t + \kappa_b) \cdot e_t\\
s_t &= S_t + \gamma \cdot (M_e \cdot S_t + \kappa_s) \cdot e_t\\
with
.. math::
\eta &= \begin{cases}
Y_t\quad\text{if error is multiplicative}\\
1\quad\text{else}
\end{cases}\\
M_e &= \begin{cases}
1\quad\text{if error is multiplicative}\\
0\quad\text{else}
\end{cases}\\
and, when using the additive error model,
.. math::
\kappa_l &= \begin{cases}
\frac{1}{S_t}\quad
\text{if seasonality is multiplicative}\\
1\quad\text{else}
\end{cases}\\
\kappa_b &= \begin{cases}
\frac{\kappa_l}{l_{t-1}}\quad
\text{if trend is multiplicative}\\
\kappa_l\quad\text{else}
\end{cases}\\
\kappa_s &= \begin{cases}
\frac{1}{L_t}\quad\text{if seasonality is
multiplicative}\\
1\quad\text{else}
\end{cases}
When using the multiplicative error model
.. math::
\kappa_l &= \begin{cases}
0\quad
\text{if seasonality is multiplicative}\\
S_t\quad\text{else}
\end{cases}\\
\kappa_b &= \begin{cases}
\frac{\kappa_l}{l_{t-1}}\quad
\text{if trend is multiplicative}\\
\kappa_l + l_{t-1}\quad\text{else}
\end{cases}\\
\kappa_s &= \begin{cases}
0\quad\text{if seasonality is multiplicative}\\
L_t\quad\text{else}
\end{cases}
References
----------
.. [1] Hyndman, R.J., & Athanasopoulos, G. (2018) *Forecasting:
principles and practice*, 2nd edition, OTexts: Melbourne,
Australia. OTexts.com/fpp2. Accessed on February 28th 2020.
"""
# check inputs
if error in ["additive", "multiplicative"]:
error = {"additive": "add", "multiplicative": "mul"}[error]
if error not in ["add", "mul"]:
raise ValueError("error must be 'add' or 'mul'!")
# Get the starting location
if anchor is None or anchor == "end":
start_idx = self.model.nobs
elif anchor == "start":
start_idx = 0
else:
start_idx, _, _ = self.model._get_index_loc(anchor)
if isinstance(start_idx, slice):
start_idx = start_idx.start
if start_idx < 0:
start_idx += self.model.nobs
if start_idx > self.model.nobs:
raise ValueError("Cannot anchor simulation outside of the sample.")
# get Holt-Winters settings and parameters
trend = self.model.trend
damped = self.model.damped_trend
seasonal = self.model.seasonal
use_boxcox = self.params["use_boxcox"]
lamda = self.params["lamda"]
alpha = self.params["smoothing_level"]
beta = self.params["smoothing_trend"]
gamma = self.params["smoothing_seasonal"]
phi = self.params["damping_trend"]
# if model has no seasonal component, use 1 as period length
m = max(self.model.seasonal_periods, 1)
n_params = (
2
+ 2 * self.model.has_trend
+ (m + 1) * self.model.has_seasonal
+ damped
)
mul_seasonal = seasonal == "mul"
mul_trend = trend == "mul"
mul_error = error == "mul"
# define trend, damping and seasonality operations
if mul_trend:
op_b = np.multiply
op_d = np.power
neutral_b = 1
else:
op_b = np.add
op_d = np.multiply
neutral_b = 0
if mul_seasonal:
op_s = np.multiply
neutral_s = 1
else:
op_s = np.add
neutral_s = 0
# set initial values
level = self.level
_trend = self.trend
season = self.season
# (notation as in https://otexts.com/fpp2/ets.html)
y = np.empty((nsimulations, repetitions))
# lvl instead of l because of E741
lvl = np.empty((nsimulations + 1, repetitions))
b = np.empty((nsimulations + 1, repetitions))
s = np.empty((nsimulations + m, repetitions))
# the following uses python's index wrapping
if start_idx == 0:
lvl[-1, :] = self.params["initial_level"]
b[-1, :] = self.params["initial_trend"]
else:
lvl[-1, :] = level[start_idx - 1]
b[-1, :] = _trend[start_idx - 1]
if 0 <= start_idx and start_idx <= m:
initial_seasons = self.params["initial_seasons"]
_s = np.concatenate(
(initial_seasons[start_idx:], season[:start_idx])
)
s[-m:, :] = np.tile(_s, (repetitions, 1)).T
else:
s[-m:, :] = np.tile(
season[start_idx - m : start_idx], (repetitions, 1)
).T
# set neutral values for unused features
if trend is None:
b[:, :] = neutral_b
phi = 1
beta = 0
if seasonal is None:
s[:, :] = neutral_s
gamma = 0
if not damped:
phi = 1
# calculate residuals for error covariance estimation
if use_boxcox:
fitted = boxcox(self.fittedvalues, lamda)
else:
fitted = self.fittedvalues
if error == "add":
resid = self.model._y - fitted
else:
resid = (self.model._y - fitted) / fitted
sigma = np.sqrt(np.sum(resid ** 2) / (len(resid) - n_params))
# get random error eps
if isinstance(random_errors, np.ndarray):
if random_errors.shape != (nsimulations, repetitions):
raise ValueError(
"If random_errors is an ndarray, it must have shape "
"(nsimulations, repetitions)"
)
eps = random_errors
elif random_errors == "bootstrap":
eps = np.random.choice(
resid, size=(nsimulations, repetitions), replace=True
)
elif random_errors is None:
if random_state is None:
eps = np.random.randn(nsimulations, repetitions) * sigma
elif isinstance(random_state, int):
rng = np.random.RandomState(random_state)
eps = rng.randn(nsimulations, repetitions) * sigma
elif isinstance(random_state, np.random.RandomState):
eps = random_state.randn(nsimulations, repetitions) * sigma
else:
raise ValueError(
"Argument random_state must be None, an integer, "
"or an instance of np.random.RandomState"
)
elif isinstance(random_errors, (rv_continuous, rv_discrete)):
params = random_errors.fit(resid)
eps = random_errors.rvs(*params, size=(nsimulations, repetitions))
elif isinstance(random_errors, _distn_infrastructure.rv_frozen):
eps = random_errors.rvs(size=(nsimulations, repetitions))
else:
raise ValueError("Argument random_errors has unexpected value!")
for t in range(nsimulations):
b0 = op_d(b[t - 1, :], phi)
l0 = op_b(lvl[t - 1, :], b0)
s0 = s[t - m, :]
y0 = op_s(l0, s0)
if error == "add":
eta = 1
kappa_l = 1 / s0 if mul_seasonal else 1
kappa_b = kappa_l / lvl[t - 1, :] if mul_trend else kappa_l
kappa_s = 1 / l0 if mul_seasonal else 1
else:
eta = y0
kappa_l = 0 if mul_seasonal else s0
kappa_b = (
kappa_l / lvl[t - 1, :]
if mul_trend
else kappa_l + lvl[t - 1, :]
)
kappa_s = 0 if mul_seasonal else l0
y[t, :] = y0 + eta * eps[t, :]
lvl[t, :] = l0 + alpha * (mul_error * l0 + kappa_l) * eps[t, :]
b[t, :] = b0 + beta * (mul_error * b0 + kappa_b) * eps[t, :]
s[t, :] = s0 + gamma * (mul_error * s0 + kappa_s) * eps[t, :]
if use_boxcox:
y = inv_boxcox(y, lamda)
sim = np.atleast_1d(np.squeeze(y))
if y.shape[0] == 1 and y.size > 1:
sim = sim[None, :]
# Wrap data / squeeze where appropriate
if not isinstance(self.model.data, PandasData):
return sim
_, _, _, index = self.model._get_prediction_index(
start_idx, start_idx + nsimulations - 1
)
if repetitions == 1:
sim = pd.Series(sim, index=index, name=self.model.endog_names)
else:
sim = pd.DataFrame(sim, index=index)
return sim
class HoltWintersResultsWrapper(ResultsWrapper):
_attrs = {
"fittedvalues": "rows",
"level": "rows",
"resid": "rows",
"season": "rows",
"trend": "rows",
"slope": "rows",
}
_wrap_attrs = union_dicts(ResultsWrapper._wrap_attrs, _attrs)
_methods = {"predict": "dates", "forecast": "dates"}
_wrap_methods = union_dicts(ResultsWrapper._wrap_methods, _methods)
populate_wrapper(HoltWintersResultsWrapper, HoltWintersResults)