Glossary

Glossary

This page defines terms that look similar but have intentionally distinct meanings in the smooth package and its documentation. They are not interchangeable — each name encodes either a different surface (API parameter vs. dict key vs. math symbol), a different language (R vs. Python), or a different scope (component vs. model class).

When the wiki uses a particular spelling, the spelling carries information. If you are unsure which one you want, look it up here.

ARIMA: AR / MA / I family

Symbol	What it is	Where it appears
AR	Autoregressive model class / general term (uppercase).	Prose, page titles.
MA	Moving average model class / general term (uppercase).	Prose, page titles.
I	Integration order in ARIMA (uppercase).	Prose.
ar	Dict / list key carrying AR order values in the orders argument.	R orders=list(ar=...), Python orders={"ar": ...}.
ma	Dict / list key for MA order values.	Same as ar.
i	Dict / list key for integration order values.	Same as ar.
ar_order	Python-only scalar / list argument to ADAM / AutoADAM specifying the AR order(s) directly. Mutually exclusive with orders.	Python only.
i_order	Python-only scalar / list argument for integration order.	Python only.
ma_order	Python-only scalar / list argument for MA order.	Python only.
"ar"	Quoted string literal — same value as ar, but as a dict key in code.	Code examples.
AR component	The autoregressive part of a fitted ARIMA inside the state-space form. Has length = sum of the AR orders.	Internal documentation.
AR polynomial	The polynomial in the lag operator with AR coefficients (used in stability checks and forecasts).	Math sections.
ARMA	AR + MA, no integration.	Used in the arma= argument that fixes AR/MA coefficients.
ARIMA	AR + I + MA.	The full model class.
SARIMA	Seasonal ARIMA — ARIMA with one seasonal lag.	Conventional notation (p,d,q)(P,D,Q)[s].
SSARIMA	State Space ARIMA — ARIMA in SSOE form (separate function in R).	See SSARIMA.md.
MSARIMA	Multiple Seasonal ARIMA — optimised ARIMA implementation handling multiple seasonalities.	See MSARIMA.md.

Important: ar_order (Python scalar argument) and orders=dict(ar=...) (R-style dict) are different surfaces for the same concept. If both are passed in Python, orders wins and the scalar form is ignored with a UserWarning. See Orders-and-Lags.md.

ETS notation letters

The model parameter is a 3–4 character ETS code. The letters mean different things by position.

Position	Letter	Meaning
1 (Error)	A	Additive error
1 (Error)	M	Multiplicative error
2 (Trend)	N	No trend
2 (Trend)	A	Additive trend
2 (Trend)	Ad	Additive damped trend
2 (Trend)	M	Multiplicative trend
2 (Trend)	Md	Multiplicative damped trend
3 (Seasonal)	N	No seasonality
3 (Seasonal)	A	Additive seasonality
3 (Seasonal)	M	Multiplicative seasonality
Any	Z	Auto-select among A and M (or N where applicable).
Any	X	Auto-select restricted to additive options.
Any	Y	Auto-select restricted to multiplicative options.
Any	F	Full search across all 30 ETS variants.
Any	P	Pool of pure-additive vs. pure-multiplicative.
Any	S	"Safe" 19-model pool from the forecast package.
Any	C	Combination of selected models by IC weights.

The same letter (A, M, N) at different positions means different things. "AAA" is additive-error / additive-trend / additive-seasonality, and the three As do not share a definition — only a name.

State components vs. parameter names

Several short words label both a model component and an API parameter.

Word	As model component	As parameter
level	A latent state — the local level.	A value in initial=list(level=…) (R) / initial={"level": …} (Python).
trend	A latent state — the local trend.	A value in initial. The 2nd letter of the ETS code (different surface again).
seasonal	One or more latent seasonal states.	A value in initial. The 3rd letter of the ETS code.
arma	Fitted AR / MA coefficients.	The arma= argument that fixes them at user values.
phi	The damping coefficient.	The phi= argument that fixes or estimates it.
persistence	The vector of smoothing parameters in the state equation.	The persistence= argument that fixes them.
lags	The lag indices used by the state equation.	The lags= argument that supplies them.

Smoothing parameters

Symbol	Greek	Component it persists	Notes
alpha	α	Level	Always present in ETS.
beta	β	Trend	Only when trend ≠ N. Constrained by beta ≤ alpha under bounds="usual".
gamma	γ	Seasonal	Only when seasonal ≠ N. Constrained by gamma ≤ 1-alpha under bounds="usual". With multiple seasonalities, becomes a vector gamma1, gamma2, ….
delta	δ	Adaptive regressor coefficients	Only with regressors="adapt". One delta per regressor.
phi	φ	Damping	Only with damped trends (Ad, Md).

initial (parameter) vs. initial values

The initial argument controls how starting values are obtained; the initial values themselves are the numbers in the state vector at time t=0.

Surface	Possible values	Meaning
initial= (argument)	"backcasting"	Recursively initialise from data using backwards pass.
initial= (argument)	"optimal"	Estimate the initial state as part of the optimisation.
initial= (argument)	"two-stage"	First backcast, then re-estimate initials together with parameters.
initial= (argument)	"complete"	All states initialised together with parameters.
initial= (argument)	"provided" or list(...) (R) / dict(...) (Python)	Use the supplied values verbatim.
Initial values	Numeric vector	Actual starting values for level, trend, seasonal, regressors.

lags vs. lag

Term	Meaning
lags= (argument)	List of integer seasonal periods supplied at construction (e.g. [1, 12], [24, 168]).
lag	The integer offset in math notation, e.g. "the lag-12 seasonal effect".
lags_model_all	Internal flat list combining state lags and exogenous-regressor lags (Python only).
lags_model_seasonal	Internal list of seasonal-component lags only (Python only).

loss vs. lossValue

Term	Meaning
loss= (argument)	The user-chosen loss function ("likelihood", "MSE", "MAE", …). See Loss-Functions.md.
m$lossValue (R) / m.loss_value (Python)	The numeric value of the chosen loss at the converged optimum.

model (parameter) vs. model (object)

Term	Meaning
model= (argument)	The ETS code string or a vector / list of strings for selection.
model (in R) / fitted ADAM (Python)	The object returned by adam() / ADAM.fit().
Refit-model passed in	An R-only convention where a previously fitted object may be passed as model=… to reuse its structure. Not available in Python.

Status markers used in this wiki

Marker	Meaning
Yes	Implemented in this language.
R-only	Implemented in R only. Python implementation is on Roadmap.md.
TBA	"To be added" — historical marker. Going forward, TBA items live in Roadmap.md, not in reference pages.

See also

• Home
• Model-Specification
• Orders-and-Lags
• Persistence
• Initialisation
• Roadmap