Exponential Smoothing¶

durbyn provides four exponential smoothing models and a separate Croston method from the same module.

SES — Simple Exponential Smoothing¶

Best for series with no trend or seasonality.

from durbyn import SES

model = SES(alpha=0.3).fit(y, m=1)
fc = model.forecast(h=12, level=[80, 95])

Parameters¶

Parameter	Type	Default	Description
`initial`	`str`	`"optimal"`	Initialisation method: `"optimal"` or `"simple"`
`alpha`	`float \\| None`	`None`	Smoothing parameter (0-1). `None` for optimised
`lambda_`	`float \\| bool \\| None`	`None`	Box-Cox parameter. `True` for auto, `None` for none
`biasadj`	`bool`	`False`	Bias adjustment for Box-Cox back-transformation

Holt — Linear Trend Method¶

Extends SES with a linear trend component.

from durbyn import Holt

# Damped trend
model = Holt(damped=True).fit(y, m=1)
fc = model.forecast(h=12)

# Exponential (multiplicative) trend
model = Holt(exponential=True).fit(y, m=1)
fc = model.forecast(h=12)

Parameters¶

Parameter	Type	Default	Description
`damped`	`bool`	`False`	Use damped trend
`initial`	`str`	`"optimal"`	`"optimal"` or `"simple"`
`exponential`	`bool`	`False`	Multiplicative (exponential) trend
`alpha`	`float \\| None`	`None`	Level smoothing parameter
`beta`	`float \\| None`	`None`	Trend smoothing parameter
`phi`	`float \\| None`	`None`	Damping parameter
`lambda_`	`float \\| bool \\| None`	`None`	Box-Cox parameter
`biasadj`	`bool`	`False`	Bias adjustment for Box-Cox

HoltWinters — Triple Exponential Smoothing¶

Handles trend and seasonality.

from durbyn import HoltWinters

# Additive seasonality
model = HoltWinters(seasonal="additive").fit(y, m=12)
fc = model.forecast(h=12, level=[80, 95])

# Multiplicative seasonality with damped trend
model = HoltWinters(
    seasonal="multiplicative",
    damped=True,
).fit(y, m=12)
fc = model.forecast(h=24)

Parameters¶

Parameter	Type	Default	Description
`seasonal`	`str`	`"additive"`	`"additive"` or `"multiplicative"`
`damped`	`bool`	`False`	Damped trend
`initial`	`str`	`"optimal"`	`"optimal"` or `"simple"`
`exponential`	`bool`	`False`	Multiplicative trend
`alpha`	`float \\| None`	`None`	Level smoothing
`beta`	`float \\| None`	`None`	Trend smoothing
`gamma`	`float \\| None`	`None`	Seasonal smoothing
`phi`	`float \\| None`	`None`	Damping parameter
`lambda_`	`float \\| bool \\| None`	`None`	Box-Cox parameter
`biasadj`	`bool`	`False`	Bias adjustment

Note

m (seasonal period) is required at fit time and must be > 1 for seasonal models.

ETS — Error-Trend-Seasonal State Space¶

General framework encompassing all exponential smoothing models. Supports automatic model selection.

from durbyn import ETS

# Automatic model selection
model = ETS(model="ZZZ").fit(y, m=12)
fc = model.forecast(h=12)

# Specific model: Additive error, Additive trend, Multiplicative seasonal
model = ETS(model="AAM").fit(y, m=12)
fc = model.forecast(h=12, level=[80, 95])

# With damped trend
model = ETS(model="ZZZ", damped=True).fit(y, m=12)

Model String¶

The model parameter is a 3-character string "ETS" where:

E (Error): A = additive, M = multiplicative, Z = auto
T (Trend): N = none, A = additive, M = multiplicative, Z = auto
S (Seasonal): N = none, A = additive, M = multiplicative, Z = auto

Common models:

Model	Description
`"ZZZ"`	Fully automatic selection
`"ANN"`	Simple exponential smoothing (additive errors)
`"AAN"`	Holt's linear trend
`"AAA"`	Additive Holt-Winters
`"MAM"`	Multiplicative Holt-Winters

Parameters¶

Parameter	Type	Default	Description
`model`	`str`	`"ZZZ"`	3-char model specification
`damped`	`bool \\| None`	`None`	Damped trend. `None` for auto
`alpha`	`float \\| None`	`None`	Level smoothing
`beta`	`float \\| None`	`None`	Trend smoothing
`gamma`	`float \\| None`	`None`	Seasonal smoothing
`phi`	`float \\| None`	`None`	Damping parameter
`additive_only`	`bool`	`False`	Restrict to additive models
`lambda_`	`float \\| bool \\| None`	`None`	Box-Cox parameter
`biasadj`	`bool`	`False`	Bias adjustment
`opt_crit`	`str`	`"lik"`	Optimisation criterion: `"lik"`, `"amse"`, `"mse"`
`ic`	`str`	`"aicc"`	Information criterion: `"aicc"`, `"aic"`, `"bic"`
`bounds`	`str`	`"both"`	Parameter bounds: `"both"`, `"usual"`, `"admissible"`
`restrict`	`bool`	`True`	Apply admissibility restrictions
`allow_multiplicative_trend`	`bool`	`False`	Allow multiplicative trend models

Croston — Intermittent Demand (Simple)¶

Simple Croston wrapper from the ExponentialSmoothing module. For more advanced variants, see Intermittent Demand.

from durbyn import Croston

data = [0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 3, 0, 0, 0, 1, 0]

model = Croston().fit(data)
fc = model.forecast(h=12)

print(fc.mean)  # flat forecast (no prediction intervals)

Note

Croston's method produces flat forecasts without prediction intervals. The level parameter is ignored.