def compute_expected_y_pred(y_train, fh):
# fitting
yt = y_train.copy()
t1 = Deseasonalizer(sp=12, model="multiplicative")
yt = t1.fit_transform(yt)
t2 = Detrender(PolynomialTrendForecaster(degree=1))
yt = t2.fit_transform(yt)
forecaster = NaiveForecaster()
forecaster.fit(yt, fh=fh)
# predicting
y_pred = forecaster.predict()
y_pred = t2.inverse_transform(y_pred)
y_pred = t1.inverse_transform(y_pred)
return y_pred
class ThetaForecaster(ExponentialSmoothing):
"""Theta method for forecasting.
The theta method as defined in [1]_ is equivalent to simple exponential
smoothing (SES) with drift (as demonstrated in [2]_).
The series is tested for seasonality using the test outlined in A&N. If
deemed seasonal, the series is seasonally adjusted using a classical
multiplicative decomposition before applying the theta method. The
resulting forecasts are then reseasonalised.
In cases where SES results in a constant forecast, the theta forecaster
will revert to predicting the SES constant plus a linear trend derived
from the training data.
Prediction intervals are computed using the underlying state space model.
Parameters
----------
initial_level : float, optional
The alpha value of the simple exponential smoothing, if the value is
set then
this will be used, otherwise it will be estimated from the data.
deseasonalize : bool, optional (default=True)
If True, data is seasonally adjusted.
sp : int, optional (default=1)
The number of observations that constitute a seasonal period for a
multiplicative deseasonaliser, which is used if seasonality is
detected in the
training data. Ignored if a deseasonaliser transformer is provided.
Default is
1 (no seasonality).
Attributes
----------
initial_level_ : float
The estimated alpha value of the SES fit.
drift_ : float
The estimated drift of the fitted model.
se_ : float
The standard error of the predictions. Used to calculate prediction
intervals.
References
----------
.. [1] Assimakopoulos, V. and Nikolopoulos, K. The theta model: a
decomposition approach to forecasting. International Journal of
Forecasting 16, 521-530, 2000.
https://www.sciencedirect.com/science/article/pii/S0169207000000662
.. [2] `Hyndman, Rob J., and Billah, Baki. Unmasking the Theta method.
International J. Forecasting, 19, 287-290, 2003.
https://www.sciencedirect.com/science/article/pii/S0169207001001431
Examples
--------
>>> from sktime.datasets import load_airline
>>> from sktime.forecasting.theta import ThetaForecaster
>>> y = load_airline()
>>> forecaster = ThetaForecaster(sp=12)
>>> forecaster.fit(y)
ThetaForecaster(...)
>>> y_pred = forecaster.predict(fh=[1,2,3])
"""
_fitted_param_names = ("initial_level", "smoothing_level")
_tags = {
"scitype:y": "univariate",
"ignores-exogeneous-X": True,
"capability:pred_int": True,
"requires-fh-in-fit": False,
"handles-missing-data": False,
}
def __init__(self, initial_level=None, deseasonalize=True, sp=1):
self.sp = sp
self.deseasonalize = deseasonalize
self.deseasonalizer_ = None
self.trend_ = None
self.initial_level_ = None
self.drift_ = None
self.se_ = None
super(ThetaForecaster, self).__init__(initial_level=initial_level, sp=sp)
def _fit(self, y, X=None, fh=None):
"""Fit to training data.
Parameters
----------
y : pd.Series
Target time series to which to fit the forecaster.
fh : int, list or np.array, optional (default=None)
The forecasters horizon with the steps ahead to to predict.
X : pd.DataFrame, optional (default=None)
Exogenous variables are ignored
Returns
-------
self : returns an instance of self.
"""
sp = check_sp(self.sp)
if sp > 1 and not self.deseasonalize:
warn("`sp` is ignored when `deseasonalise`=False")
if self.deseasonalize:
self.deseasonalizer_ = Deseasonalizer(sp=self.sp, model="multiplicative")
y = self.deseasonalizer_.fit_transform(y)
self.initialization_method = "known" if self.initial_level else "estimated"
# fit exponential smoothing forecaster
# find theta lines: Theta lines are just SES + drift
super(ThetaForecaster, self)._fit(y, fh=fh)
self.initial_level_ = self._fitted_forecaster.params["smoothing_level"]
# compute and store historical residual standard error
self.sigma_ = np.sqrt(self._fitted_forecaster.sse / (len(y) - 1))
# compute trend
self.trend_ = self._compute_trend(y)
return self
def _predict(self, fh, X=None):
"""Make forecasts.
Parameters
----------
fh : array-like
The forecasters horizon with the steps ahead to to predict.
Default is
one-step ahead forecast, i.e. np.array([1]).
X : pd.DataFrame, optional (default=None)
Exogenous time series
Returns
-------
y_pred : pandas.Series
Returns series of predicted values.
"""
y_pred = super(ThetaForecaster, self)._predict(fh, X)
# Add drift.
drift = self._compute_drift()
y_pred += drift
if self.deseasonalize:
y_pred = self.deseasonalizer_.inverse_transform(y_pred)
return y_pred
@staticmethod
def _compute_trend(y):
# Trend calculated through least squares regression.
coefs = _fit_trend(y.values.reshape(1, -1), order=1)
return coefs[0, 0] / 2
def _compute_drift(self):
fh = self.fh.to_relative(self.cutoff)
if np.isclose(self.initial_level_, 0.0):
# SES was constant, so revert to simple trend
drift = self.trend_ * fh
else:
# Calculate drift from SES parameters
n_timepoints = len(self._y)
drift = self.trend_ * (
fh
+ (1 - (1 - self.initial_level_) ** n_timepoints) / self.initial_level_
)
return drift
def _predict_quantiles(self, fh, X=None, alpha=None):
"""Compute/return prediction quantiles for a forecast.
private _predict_quantiles containing the core logic,
called from predict_quantiles and predict_interval
Parameters
----------
fh : int, list, np.array or ForecastingHorizon
Forecasting horizon
X : pd.DataFrame, optional (default=None)
Exogenous time series
alpha : list of float, optional (default=[0.5])
A list of probabilities at which quantile forecasts are computed.
Returns
-------
quantiles : pd.DataFrame
Column has multi-index: first level is variable name from y in fit,
second level being the values of alpha passed to the function.
Row index is fh. Entries are quantile forecasts, for var in col index,
at quantile probability in second col index, for the row index.
"""
# prepare return data frame
index = pd.MultiIndex.from_product([["Quantiles"], alpha])
pred_quantiles = pd.DataFrame(columns=index)
sem = self.sigma_ * np.sqrt(
self.fh.to_relative(self.cutoff) * self.initial_level_**2 + 1
)
y_pred = self._predict(fh, X)
# we assume normal additive noise with sem variance
for a in alpha:
pred_quantiles[("Quantiles", a)] = y_pred + norm.ppf(a) * sem
# todo: should this not increase with the horizon?
# i.e., sth like norm.ppf(a) * sem * fh.to_absolute(cutoff) ?
# I've just refactored this so will leave it for now
return pred_quantiles
def _update(self, y, X=None, update_params=True):
super(ThetaForecaster, self)._update(
y, X, update_params=False
) # use custom update_params routine
if update_params:
if self.deseasonalize:
y = self.deseasonalizer_.transform(self._y) # use updated y
self.initial_level_ = self._fitted_forecaster.params["smoothing_level"]
self.trend_ = self._compute_trend(y)
return self
class ThetaForecaster(ExponentialSmoothing):
"""
Theta method of forecasting.
The theta method as defined in [1]_ is equivalent to simple exponential
smoothing
(SES) with drift. This is demonstrated in [2]_.
The series is tested for seasonality using the test outlined in A&N. If
deemed
seasonal, the series is seasonally adjusted using a classical
multiplicative
decomposition before applying the theta method. The resulting forecasts
are then
reseasonalised.
In cases where SES results in a constant forecast, the theta forecaster
will revert
to predicting the SES constant plus a linear trend derived from the
training data.
Prediction intervals are computed using the underlying state space model.
Parameters
----------
initial_level : float, optional
The alpha value of the simple exponential smoothing, if the value is
set then
this will be used, otherwise it will be estimated from the data.
deseasonalize : bool, optional (default=True)
If True, data is seasonally adjusted.
sp : int, optional (default=1)
The number of observations that constitute a seasonal period for a
multiplicative deseasonaliser, which is used if seasonality is
detected in the
training data. Ignored if a deseasonaliser transformer is provided.
Default is
1 (no seasonality).
Attributes
----------
initial_level_ : float
The estimated alpha value of the SES fit.
drift_ : float
The estimated drift of the fitted model.
se_ : float
The standard error of the predictions. Used to calculate prediction
intervals.
References
----------
.. [1] `Assimakopoulos, V. and Nikolopoulos, K. The theta model: a
decomposition
approach to forecasting. International Journal of Forecasting 16,
521-530,
2000.
<https://www.sciencedirect.com/science/article/pii
/S0169207000000662>`_
.. [2] `Hyndman, Rob J., and Billah, Baki. Unmasking the Theta method.
International J. Forecasting, 19, 287-290, 2003.
<https://www.sciencedirect.com/science/article/pii
/S0169207001001431>`_
"""
_fitted_param_names = ("initial_level", "smoothing_level")
def __init__(self, initial_level=None, deseasonalize=True, sp=1):
self.sp = sp
self.deseasonalize = deseasonalize
self.deseasonalizer_ = None
self.trend_ = None
self.initial_level_ = None
self.drift_ = None
self.se_ = None
super(ThetaForecaster, self).__init__(initial_level=initial_level,
sp=sp)
def fit(self, y, X=None, fh=None):
"""Fit to training data.
Parameters
----------
y : pd.Series
Target time series to which to fit the forecaster.
fh : int, list or np.array, optional (default=None)
The forecasters horizon with the steps ahead to to predict.
X : pd.DataFrame, optional (default=None)
Exogenous variables are ignored
Returns
-------
self : returns an instance of self.
"""
y, _ = check_y_X(y, X)
sp = check_sp(self.sp)
if sp > 1 and not self.deseasonalize:
warn("`sp` is ignored when `deseasonalise`=False")
if self.deseasonalize:
self.deseasonalizer_ = Deseasonalizer(sp=self.sp,
model="multiplicative")
y = self.deseasonalizer_.fit_transform(y)
# fit exponential smoothing forecaster
# find theta lines: Theta lines are just SES + drift
super(ThetaForecaster, self).fit(y, fh=fh)
self.initial_level_ = self._fitted_forecaster.params["smoothing_level"]
# compute trend
self.trend_ = self._compute_trend(y)
self._is_fitted = True
return self
def _predict(self, fh, X=None, return_pred_int=False, alpha=DEFAULT_ALPHA):
"""
Make forecasts.
Parameters
----------
fh : array-like
The forecasters horizon with the steps ahead to to predict.
Default is
one-step ahead forecast, i.e. np.array([1]).
Returns
-------
y_pred : pandas.Series
Returns series of predicted values.
"""
y_pred = super(ThetaForecaster, self)._predict(fh,
X,
return_pred_int=False,
alpha=alpha)
# Add drift.
drift = self._compute_drift()
y_pred += drift
if self.deseasonalize:
y_pred = self.deseasonalizer_.inverse_transform(y_pred)
if return_pred_int:
pred_int = self.compute_pred_int(y_pred=y_pred, alpha=alpha)
return y_pred, pred_int
return y_pred
@staticmethod
def _compute_trend(y):
# Trend calculated through least squares regression.
coefs = _fit_trend(y.values.reshape(1, -1), order=1)
return coefs[0, 0] / 2
def _compute_drift(self):
fh = self.fh.to_relative(self.cutoff)
if np.isclose(self.initial_level_, 0.0):
# SES was constant, so revert to simple trend
drift = self.trend_ * fh
else:
# Calculate drift from SES parameters
n_timepoints = len(self._y)
drift = self.trend_ * (fh +
(1 -
(1 - self.initial_level_)**n_timepoints) /
self.initial_level_)
return drift
def _compute_pred_err(self, alphas):
"""
Get the prediction errors for the forecast.
"""
self.check_is_fitted()
n_timepoints = len(self._y)
self.sigma_ = np.sqrt(self._fitted_forecaster.sse / (n_timepoints - 1))
sem = self.sigma_ * np.sqrt(
self.fh.to_relative(self.cutoff) * self.initial_level_**2 + 1)
errors = []
for alpha in alphas:
z = _zscore(1 - alpha)
error = z * sem
errors.append(
pd.Series(error, index=self.fh.to_absolute(self.cutoff)))
return errors
def update(self, y, X=None, update_params=True):
super(ThetaForecaster, self).update(
y, X, update_params=False) # use custom update_params routine
if update_params:
if self.deseasonalize:
y = self.deseasonalizer_.transform(self._y) # use updated y
self.initial_level_ = self._fitted_forecaster.params[
"smoothing_level"]
self.trend_ = self._compute_trend(y)
return self
def test_transform_inverse_transform_equivalence(sp, model):
transformer = Deseasonalizer(sp=sp, model=model)
transformer.fit(y_train)
yit = transformer.inverse_transform(transformer.transform(y_train))
np.testing.assert_array_equal(y_train.index, yit.index)
np.testing.assert_array_almost_equal(y_train, yit)
def test_inverse_transform_time_index(sp, model):
transformer = Deseasonalizer(sp=sp, model=model)
transformer.fit(y_train)
yit = transformer.inverse_transform(y_test)
np.testing.assert_array_equal(yit.index, y_test.index)
class ThetaForecaster(ExponentialSmoothing):
"""Theta method for forecasting.
The theta method as defined in [1]_ is equivalent to simple exponential
smoothing (SES) with drift (as demonstrated in [2]_).
The series is tested for seasonality using the test outlined in A&N. If
deemed seasonal, the series is seasonally adjusted using a classical
multiplicative decomposition before applying the theta method. The
resulting forecasts are then reseasonalised.
In cases where SES results in a constant forecast, the theta forecaster
will revert to predicting the SES constant plus a linear trend derived
from the training data.
Prediction intervals are computed using the underlying state space model.
Parameters
----------
initial_level : float, optional
The alpha value of the simple exponential smoothing, if the value is
set then
this will be used, otherwise it will be estimated from the data.
deseasonalize : bool, optional (default=True)
If True, data is seasonally adjusted.
sp : int, optional (default=1)
The number of observations that constitute a seasonal period for a
multiplicative deseasonaliser, which is used if seasonality is
detected in the
training data. Ignored if a deseasonaliser transformer is provided.
Default is
1 (no seasonality).
Attributes
----------
initial_level_ : float
The estimated alpha value of the SES fit.
drift_ : float
The estimated drift of the fitted model.
se_ : float
The standard error of the predictions. Used to calculate prediction
intervals.
References
----------
.. [1] Assimakopoulos, V. and Nikolopoulos, K. The theta model: a
decomposition approach to forecasting. International Journal of
Forecasting 16, 521-530, 2000.
https://www.sciencedirect.com/science/article/pii/S0169207000000662
.. [2] `Hyndman, Rob J., and Billah, Baki. Unmasking the Theta method.
International J. Forecasting, 19, 287-290, 2003.
https://www.sciencedirect.com/science/article/pii/S0169207001001431
Examples
--------
>>> from sktime.datasets import load_airline
>>> from sktime.forecasting.theta import ThetaForecaster
>>> y = load_airline()
>>> forecaster = ThetaForecaster(sp=12)
>>> forecaster.fit(y)
ThetaForecaster(...)
>>> y_pred = forecaster.predict(fh=[1,2,3])
"""
_fitted_param_names = ("initial_level", "smoothing_level")
_tags = {
"ignores-exogeneous-X": True,
"capability:pred_int": True,
"requires-fh-in-fit": False,
"handles-missing-data": False,
}
def __init__(self, initial_level=None, deseasonalize=True, sp=1):
self.sp = sp
self.deseasonalize = deseasonalize
self.deseasonalizer_ = None
self.trend_ = None
self.initial_level_ = None
self.drift_ = None
self.se_ = None
super(ThetaForecaster, self).__init__(initial_level=initial_level,
sp=sp)
def _fit(self, y, X=None, fh=None):
"""Fit to training data.
Parameters
----------
y : pd.Series
Target time series to which to fit the forecaster.
fh : int, list or np.array, optional (default=None)
The forecasters horizon with the steps ahead to to predict.
X : pd.DataFrame, optional (default=None)
Exogenous variables are ignored
Returns
-------
self : returns an instance of self.
"""
sp = check_sp(self.sp)
if sp > 1 and not self.deseasonalize:
warn("`sp` is ignored when `deseasonalise`=False")
if self.deseasonalize:
self.deseasonalizer_ = Deseasonalizer(sp=self.sp,
model="multiplicative")
y = self.deseasonalizer_.fit_transform(y)
self.initialization_method = "known" if self.initial_level else "estimated"
# fit exponential smoothing forecaster
# find theta lines: Theta lines are just SES + drift
super(ThetaForecaster, self)._fit(y, fh=fh)
self.initial_level_ = self._fitted_forecaster.params["smoothing_level"]
# compute trend
self.trend_ = self._compute_trend(y)
return self
def _predict(self, fh, X=None, return_pred_int=False, alpha=DEFAULT_ALPHA):
"""Make forecasts.
Parameters
----------
fh : array-like
The forecasters horizon with the steps ahead to to predict.
Default is
one-step ahead forecast, i.e. np.array([1]).
Returns
-------
y_pred : pandas.Series
Returns series of predicted values.
"""
y_pred = super(ThetaForecaster, self)._predict(fh,
X,
return_pred_int=False,
alpha=alpha)
# Add drift.
drift = self._compute_drift()
y_pred += drift
if self.deseasonalize:
y_pred = self.deseasonalizer_.inverse_transform(y_pred)
if return_pred_int:
pred_int = self.compute_pred_int(y_pred=y_pred, alpha=alpha)
return y_pred, pred_int
return y_pred
@staticmethod
def _compute_trend(y):
# Trend calculated through least squares regression.
coefs = _fit_trend(y.values.reshape(1, -1), order=1)
return coefs[0, 0] / 2
def _compute_drift(self):
fh = self.fh.to_relative(self.cutoff)
if np.isclose(self.initial_level_, 0.0):
# SES was constant, so revert to simple trend
drift = self.trend_ * fh
else:
# Calculate drift from SES parameters
n_timepoints = len(self._y)
drift = self.trend_ * (fh +
(1 -
(1 - self.initial_level_)**n_timepoints) /
self.initial_level_)
return drift
def compute_pred_int(self, y_pred, alpha=DEFAULT_ALPHA):
"""
Compute/return prediction intervals for a forecast.
Must be run *after* the forecaster has been fitted.
If alpha is iterable, multiple intervals will be calculated.
public method including checks & utility
dispatches to core logic in _compute_pred_int
Parameters
----------
y_pred : pd.Series
Point predictions.
alpha : float or list, optional (default=0.95)
A significance level or list of significance levels.
Returns
-------
intervals : pd.DataFrame
A table of upper and lower bounds for each point prediction in
``y_pred``. If ``alpha`` was iterable, then ``intervals`` will be a
list of such tables.
"""
self.check_is_fitted()
alphas = check_alpha(alpha)
errors = self._compute_pred_err(alphas)
# compute prediction intervals
pred_int = [
pd.DataFrame({
"lower": y_pred - error,
"upper": y_pred + error
}) for error in errors
]
# for a single alpha, return single pd.DataFrame
if isinstance(alpha, float):
return pred_int[0]
# otherwise return list of pd.DataFrames
return pred_int
def _compute_pred_err(self, alphas):
"""Get the prediction errors for the forecast."""
self.check_is_fitted()
n_timepoints = len(self._y)
self.sigma_ = np.sqrt(self._fitted_forecaster.sse / (n_timepoints - 1))
sem = self.sigma_ * np.sqrt(
self.fh.to_relative(self.cutoff) * self.initial_level_**2 + 1)
errors = []
for alpha in alphas:
z = _zscore(1 - alpha)
error = z * sem
errors.append(
pd.Series(error, index=self.fh.to_absolute(self.cutoff)))
return errors
def _update(self, y, X=None, update_params=True):
super(ThetaForecaster, self)._update(
y, X, update_params=False) # use custom update_params routine
if update_params:
if self.deseasonalize:
y = self.deseasonalizer_.transform(self._y) # use updated y
self.initial_level_ = self._fitted_forecaster.params[
"smoothing_level"]
self.trend_ = self._compute_trend(y)
return self