Time-Series Forecasting: ARIMA, Prophet, and Beyond

Forecasting begins with a specific question: what information available today can help us estimate an outcome tomorrow, next week, or next quarter?

A time series contains more than a collection of values. Their order matters. Yesterday's demand may help explain today's demand. An unexpected disruption may leave an effect that persists. The same weekday may behave similarly across successive weeks.

ARIMA gives us a disciplined language for describing some of these dependencies. Its value comes from making the assumptions explicit enough to inspect, challenge, and test.

This guide starts with ARIMA and its extensions, then compares related forecasting families: Prophet, exponential smoothing, complex-seasonality models, multivariate systems, boosted trees, and neural networks. It connects their assumptions to practical forecasting decisions. The examples are illustrative; no dataset has been fitted and no forecasting performance is claimed.

Understand the three building blocks

ARIMA stands for AutoRegressive Integrated Moving Average. Its familiar notation, ARIMA(p, d, q), describes three modeling choices:

Component	Parameter	Meaning
Autoregressive	p	Number of nonseasonal autoregressive lags in a standard contiguous specification
Integrated	d	Number of ordinary differences applied
Moving average	q	Number of nonseasonal lagged innovations in a standard contiguous specification

An autoregressive term uses past values. Differencing changes the series being modeled. A moving-average term uses past innovations: the unpredictable shocks relative to the model's information set.¹²

These terms describe structure, not a ranking of sophistication. ARIMA(1, 1, 1) is not inherently better than ARIMA(1, 0, 0). The appropriate specification depends on the series and the forecasting task.

AR: when past values carry information

An AR(1) model can be written as:

$$y_t = c + \phi y_{t-1} + \varepsilon_t$$

Today's value depends on yesterday's value, an intercept, and a new innovation.

Imagine a stable operational metric fluctuating around 100. A hypothetical model might predict $\hat y_t = 20 + 0.8 y_{t-1}$. If the last observed value is 120, the next forecast is 116. Continuing recursively produces forecasts that move toward 100—the model's implied long-run mean, because $20 / (1 - 0.8) = 100$.

Use AR when: dependence on recent values is useful and the modeled series satisfies the required stationarity conditions.

Watch for: confusing a persistent upward trend with stable autoregressive dependence. A series that keeps changing its level may need a different treatment. In an AR(1), stationarity requires $|\phi| < 1$; higher-order AR models have more general root conditions.

Operationally, ask whether “high yesterday” tends to mean “high today,” and whether that relationship appears stable across the period relevant to deployment.

MA: when unexpected shocks have short memory

An MA(1) model has the form:

$$y_t = \mu + \varepsilon_t + \theta \varepsilon_{t-1}$$

The current outcome contains a new shock and a contribution from the previous shock. Innovations are unobserved; fitting and filtering the model provide estimates of the relevant past shocks.

For an illustrative operational process, an unexpected disruption today may also affect tomorrow's recorded workload. An MA component can represent a short-lived pattern of this kind. It does not establish why the disruption happened.

Use MA when: the series is stationary and a finite history of innovations provides a useful description of its dependence.

Watch for: the name. This is not a rolling arithmetic average of recent observations. A seven-day moving-average smoothing chart and an MA(7) stochastic model are different objects.

An MA(q) shock contributes directly for q subsequent steps. The way an AR process propagates shocks is different and can extend indefinitely, with effects decaying under stationarity.

ARMA: combine value dependence and shock dependence

ARMA(p, q) combines autoregressive and moving-average components without differencing. It corresponds to ARIMA(p, 0, q).

Use it when: a stationary series is described economically by both components. For example, an operational measurement may show persistent movement together with a short-lived response to surprises.

Watch for: unnecessary parameters. A small ARMA model can sometimes describe a dependence pattern more compactly than a high-order AR model, but adding both components automatically can also create poorly identified or unstable fits.

In practice, the question is whether the extra component improves reliable forecasting enough to justify its complexity.

The I in ARIMA: model changes when levels are unstable

First differencing transforms the observations into changes:

$$\Delta y_t = y_t - y_{t-1}$$

Suppose a series is 100, 104, 107, 113. Its first differences are 4, 3, and 6. Instead of modeling the levels directly, an integrated model can describe the behavior of these changes and reconstruct forecasts on the original scale.

ARIMA(0, 1, 0) without drift is a random walk: its point forecast at every future horizon is the latest observed value. With drift, its expected path changes by an estimated amount per time step. This simple model is an important benchmark.²

Use ARIMA with d > 0 when: differencing is justified by the structure of nonstationarity and the differenced series can be modeled adequately.

Watch for: overdifferencing. More differencing does not mean more learning. It can introduce unnecessary dependence, increase noise, and damage forecasts. A deterministic trend, a one-off level shift, and a stochastic trend should not automatically receive identical treatment.

If your customer base doubles after an acquisition, a differencing operation does not explain the acquisition or guarantee that the old relationships remain useful.

Stationarity: a condition to investigate, not a checkbox

Weak stationarity means a constant mean, finite constant variance, and autocovariance that depends on lag rather than calendar time. This does not mean all observations look alike. A stationary process can still fluctuate substantially.

Plots, domain knowledge, and tests contribute different evidence. Two commonly used tests start from different null hypotheses:

Test	Null hypothesis	What a small p-value suggests
Augmented Dickey–Fuller (ADF)	A unit root is present	Evidence against that unit-root null under the chosen specification
KPSS	Level or trend stationarity, depending on specification	Evidence against that form of stationarity

The inclusion of an intercept or deterministic trend matters, as do lag choices. A failure to reject is not proof that the null is true. Short samples and structural breaks can complicate interpretation.³⁴

For example, inspect whether a test result changes after a known business reorganization. A statistical signal that spans two different operating regimes may be answering a less useful question than a model fitted to the relevant regime.

Variance changes also need attention. Differencing is not a universal variance-stabilizing transformation. A log transformation may be useful for suitable positive series, but it requires care with zeros, negatives, and transformation back to the original scale.

SARIMA: repeated patterns at a seasonal lag

SARIMA adds seasonal autoregressive, differencing, and moving-average terms:

$$\operatorname{SARIMA}(p,d,q)(P,D,Q)_m$$

Symbol	Interpretation
P	Seasonal autoregressive order
D	Seasonal differencing order
Q	Seasonal moving-average order
m	Observations per seasonal cycle

Seasonal differencing compares an observation with the value one seasonal cycle earlier: $y_t - y_{t-m}$. For monthly data with an annual cycle, $m = 12$; for daily data with a weekly cycle, $m = 7$. The cycle must match the observation frequency and the actual pattern.⁵

Consider monthly subscription renewals with a recurring annual pattern. SARIMA can represent dependence both between adjacent months and between corresponding months across years.

Use it when: a meaningful, reasonably stable seasonal cycle appears in an adequately long history.

Watch for: calendar effects and multiple seasonalities. Hourly demand can exhibit daily and weekly patterns simultaneously. A standard single-seasonal SARIMA specification does not directly represent multiple separate seasonal periods. Calendar regressors, Fourier terms with ARIMA errors, or other forecasting families may be better candidates.

Do not choose $m = 12$ simply because it appears in many examples. Twelve hourly observations describe half a day, not a year.

ARIMAX and SARIMAX: when history alone is insufficient

Demand may depend on holidays, planned campaigns, weather, prices, and other external information. A useful formulation is regression with ARIMA errors:

$$y_t = \beta_0 + \beta^T x_t + n_t$$

where $n_t$ follows an ARIMA process. With seasonal dynamics, the error model can be seasonal as well.

This combines an explanatory component with time dependence remaining after that component. Terminology varies: “ARIMAX” and “SARIMAX” can refer to differing parameterizations, so inspect the equation and implementation rather than relying on the label alone.⁶⁷

Use external predictors when: they provide information available at the moment the forecast is made and improve performance in realistic backtests.

Future values are crucial. Holiday dates are known. A planned campaign schedule may be known. Future weather needs a forecast or scenario. Using the actual future temperature during evaluation can overstate the forecasting system's real capability.

Intervals conditional on supplied future predictors generally do not automatically include uncertainty about those predictors. If the weather forecast is uncertain, a forecast of demand conditional on one weather path is only part of the uncertainty story.⁶

Also avoid a causal leap: adding price to a forecasting equation does not automatically identify the effect of changing price. “External predictor” is not a guarantee of statistical exogeneity.

ACF and PACF: clues for candidate models

The autocorrelation function, ACF, summarizes correlation across lags. The partial autocorrelation function, PACF, isolates a lag association after accounting for intervening lags in the relevant linear projection.

Classic population patterns offer useful clues: an AR(p) process has a PACF cutoff after p, while an MA(q) process has an ACF cutoff after q. Mixed models generally do not provide such a simple cutoff in either function.²

Treat sample plots as noisy evidence. A single spike can be chance. Several spikes can result from omitted seasonality or an inappropriate transformation. Looking at many lags also increases the opportunity to find apparently interesting patterns.

A practical workflow is to examine the original series, decide what transformation or differencing is defensible, and then use ACF and PACF to propose a small set of candidates. Avoid reading p and q from a chart as though they were printed answers.

A practical selection table

Observed structure or task	Starting candidate	Critical check
Stable series with useful lagged values	AR	Stability and remaining residual dependence
Stable series with short shock memory	MA	Whether a finite innovation history fits
Stable series needing both components	ARMA	Parsimony and parameter stability
Levels unstable, changes adequately stable	ARIMA	Appropriate differencing
Meaningful repeated seasonal dependence	SARIMA	Correct period and sufficient seasonal history
Useful external inputs without seasonal dynamics	Regression with ARIMA errors / ARIMAX	Future predictor availability
External inputs and seasonal dynamics	SARIMAX	Calendar structure and backtesting
Several interacting endogenous series	VAR or VECM	Multivariate assumptions and data needs
Volatility itself is the target	GARCH family	Mean forecast ≠ conditional variance
Sparse intermittent demand	Intermittent-demand methods	Zeros and irregular arrivals

The last three rows are neighboring approaches, not simply larger ARIMA specifications. Choosing another family can be the correct outcome of understanding ARIMA well.

Validate the forecast you will actually use

Start by defining the forecast origin, horizon, update frequency, and available information. A one-hour-ahead forecasting task differs from a four-week staffing decision, even when both use the same series.

For a concrete example, suppose a service desk plans staffing seven days ahead. At each historical forecast date, train using only information then available, forecast the next seven days, and compare with what happened. Repeat across origins. Keep an untouched final period for final assessment after model selection.⁸

Decide whether deployment will refit parameters daily, update the model state with new observations, or leave the model unchanged for a longer interval. Reproduce that behavior in evaluation. Those are different procedures and can yield different results.

Compare against credible baselines. If requests repeat by weekday, “same weekday last week” is a serious baseline. If levels behave like a random walk, “latest value” can be hard to beat.

For this example, assess average absolute error, large misses, and understaffing consequences across the seven-day horizon. A model that improves tomorrow's prediction but worsens days five through seven may be unsuitable for the decision.

AIC or AICc can help screen comparable candidate models. They do not replace out-of-sample evaluation. Be particularly careful comparing information criteria across transformations or differencing choices that change the modeled likelihood or effective data.

Every data-dependent preparation step belongs inside the appropriate training window. Interpolation using future observations, scaling on the full dataset, and selecting variables with the final test period can all compromise the exercise.

Diagnostics: what remains unexplained?

After fitting, inspect the residual series, its ACF, and signs of changing variance or systematic bias. The Ljung–Box test assesses residual autocorrelation over selected lags; appropriate degrees-of-freedom adjustments matter when parameters have been estimated.⁹

A non-significant result does not certify independence, normality, model correctness, or forecasting superiority. It means the test did not find sufficient evidence against its null at the selected settings.

For the service desk, a residual pattern every Monday suggests an operational feature or seasonal effect remains unmodeled. A sudden run of positive errors after a product launch suggests the historical relationship may have shifted. Adding more AR terms indiscriminately can hide the underlying problem rather than solve it.

Prediction intervals deserve their own evaluation. Measure coverage as well as width at the horizons that matter. Wide intervals can cover most outcomes while being operationally unhelpful; narrow intervals can look attractive while missing too often. Model-based uncertainty does not necessarily account fully for parameter selection, future structural breaks, or uncertain external inputs.

Why a flat forecast is not automatically a failed forecast

Suppose a model predicts a nearly constant future level. It is tempting to conclude that the algorithm learned nothing.

But a stationary process can have a long-horizon conditional mean approaching its long-run mean. A random walk without drift has a point forecast equal to its last observation. Neither prediction requires the future realized path to be flat.

Imagine repeated coin-flip outcomes coded as −1 and +1. A future expected value of zero does not imply the next observed outcome will be zero. Likewise, a time-series point forecast summarizes a conditional expectation or other target; it is not a drawing of the eventual noisy trajectory.

Judge the forecast against its target and baseline. A visually exciting curve is not evidence of useful predictive information.

A stronger forecasting workflow

For a monthly sales series, I would define the decision before choosing orders: are we budgeting the next month, stocking the next quarter, or exploring a campaign scenario?

Next, I would inspect the observation calendar, missing values, reporting changes, outliers, and seasonal history. A missing month is not automatically a month with zero sales.

I would establish naive and seasonal-naive benchmarks, then compare a modest set of justified ARIMA-family candidates. If campaign plans are available in advance, I would test an external-regressor model under the same information constraints.

After that come residual checks, rolling-origin performance, interval evaluation, and failure analysis around unusual periods. Convergence warnings, implausible parameters, or sensitivity to small data changes deserve investigation before deployment.

Finally, I would define monitoring and fallback behavior. If an input feed fails or the process changes, the forecasting service needs a defensible operational response, not merely a previously fitted model.

This is where the subject becomes especially interesting from an engineering perspective: choosing ARIMA orders is one step in a system that also manages data quality, time alignment, evidence, uncertainty, and decisions.

Beyond ARIMA: related families, different assumptions

Prophet, ETS, TBATS, and neural forecasting models belong in the same comparison because they can support similar decisions. They are not all descendants of ARIMA. Each makes different choices about how to represent the signal.

There are genuine connections: some exponential-smoothing models have ARIMA equivalents, dynamic harmonic regression can use ARIMA errors, and TBATS can include ARMA errors. Those connections do not make every forecasting algorithm an ARIMA variant.

A useful distinction is also local versus global modeling. A local model is fitted separately to each series. A global model learns shared parameters from multiple series. Standard ARIMA and Prophet workflows commonly fit local models; boosted trees and neural models can be trained globally. Global learning is not the same thing as explicitly modeling interactions between series, as a VAR does.

Prophet: calendar structure and changes in trend

Prophet is a decomposable forecasting model. A simplified additive formulation separates trend, seasonality, holiday effects, and an error term:

$$y(t) = g(t) + s(t) + h(t) + \varepsilon_t$$

This is a conceptual starting point; multiplicative effects and additional regressors change the formulation. The original work emphasizes interpretable components and analyst involvement in business forecasting.¹⁰

The trend can change slope at changepoints. Their flexibility is controlled through regularization: excessive flexibility can follow temporary fluctuations, while insufficient flexibility can miss meaningful changes. A fitted historical changepoint is not evidence that the model can foresee a future acquisition, policy change, or product launch.¹¹

Seasonal effects use Fourier terms. Holiday effects use supplied event information, and additional regressors require historical and future values. Conditional seasonalities can represent patterns that differ between operating conditions.¹²

Example: a retailer has several years of daily sales, a weekly shopping rhythm, an annual cycle, holiday closures, and campaigns planned in advance. Prophet is a candidate because these components are meaningful and can be examined individually.

A useful review would ask whether the apparent annual pattern is supported by enough history, whether holiday windows match actual shopping behavior, and whether campaign effects persist across different years. Plotting a component makes it inspectable; it does not prove that the component is correctly estimated.

Use it when: calendar effects, interpretable trend structure, and analyst-guided adjustments are important to the task.

Watch for: residual autocorrelation, unrealistic trend extrapolation, and unsupported seasonal behavior. Standard Prophet does not include the ARMA error structure used by ARIMA-family models. If residuals retain useful short-term dependence, the fitted calendar components may be insufficient.

Prophet generally models the observed outcome directly rather than requiring an ARIMA-style differencing step. That does not remove the need to examine stability, structural breaks, or changing variance.

Future regressor availability is as important here as in SARIMAX. A campaign plan known at the forecast origin is legitimate input. The campaign's eventual realized sales uplift is not.

Exponential smoothing: an evolving level, trend, and seasonality

Exponential-smoothing methods update components as observations arrive. Simple exponential smoothing addresses a changing level; Holt adds a trend; Holt–Winters includes seasonality. Damped-trend variants moderate long-range trend continuation.

The ETS framework specifies Error, Trend, and Seasonal components. Additive and multiplicative forms express different relationships between the level and the size of variation. Some forms require positive data.¹³

Example: a distributor needs next month's aggregate demand for a product with a stable annual pattern. An ETS candidate may describe the useful structure with relatively few components.

Use it when: evolving level, trend, and seasonality provide an adequate representation, especially as a serious baseline alongside ARIMA.

Watch for: unusual events and external drivers that the standard component model does not explicitly explain. A simple annual pattern may not represent multiple overlapping cycles.

ETS should not be dismissed as “just smoothing.” The practical question is whether its forecasts serve the decision reliably. If it performs comparably to a larger model with less operational complexity, that is useful evidence in its favor.

TBATS and dynamic harmonic regression: complex seasonality

Hourly electricity demand may have daily and weekly cycles; daily activity may exhibit weekly and annual patterns. Representing these with one seasonal lag can be restrictive.

TBATS combines trigonometric seasonal representations with possible Box–Cox transformation, ARMA errors, trend, and seasonal components. Its framework was developed for complex patterns, including multiple or non-integer seasonal periods.¹⁴

Use it when: complex seasonal structure is central and a local statistical model is appropriate.

Watch for: fitting cost and implementation limitations. The standard R forecast implementation ignores external regressors passed through its extra arguments; do not assume TBATS supports campaign or weather inputs simply because another model does.¹⁵

Dynamic harmonic regression uses Fourier terms as regressors, potentially for several seasonal periods, and can model the remaining errors with ARIMA. It also allows other regressors where the formulation supports them.⁶

Example: for hourly demand, compare a Fourier-plus-ARIMA model with known calendar information against TBATS. If temperature forecasts are important, explicitly assess how each candidate will incorporate them.

These approaches differ in how seasonal structure evolves and how the remaining dependence is represented. Choose seasonal complexity through training and validation rather than adding every conceivable cycle.

VAR and VECM: several series modeled jointly

A vector autoregression, or VAR, models each variable using lags of itself and the other variables in the system. A vector error correction model, or VECM, represents short-term changes while retaining cointegrating relationships between appropriate nonstationary variables.¹⁶

Example: an analyst wants to forecast several related operational measures jointly, such as incoming orders and dispatched orders. Their temporal relationships may matter in both directions.

Use VAR when: joint lagged dynamics are useful and the chosen representation satisfies the relevant stability assumptions. Consider VECM when integration and cointegration justify its error-correction structure.

Watch for: parameter growth. Adding variables and lags can quickly exhaust the information in a short history. Cointegration requires evidence and a defensible specification; two rising series are not automatically cointegrated.

This differs from supplying a future campaign schedule to SARIMAX. A multivariate system forecasts jointly modeled variables rather than treating all other inputs as externally supplied future paths.

Predictive relationships, including Granger-predictive relationships, do not by themselves establish intervention effects.

Gradient boosting: turn forecasting into supervised learning

XGBoost, LightGBM, and CatBoost can serve as numerical predictors in a forecasting pipeline. The pipeline creates inputs such as lagged outcomes, past-only rolling summaries, calendar features, and external variables. Frameworks such as MLForecast support this approach across many series.¹⁷

Example: a retailer forecasts thousands of store–product combinations using sales history, product attributes, store characteristics, and planned promotions. A shared model may learn useful patterns across related series.

Use it when: nonlinear feature interactions and cross-series learning are plausible sources of improvement.

Watch for: the timestamp of every feature. A trailing seven-day average used to forecast day t must not include the actual outcome on day t.

For multi-step forecasting, define the strategy. A recursive approach feeds earlier predictions into later steps. A direct approach fits horizon-specific predictions. Their training, feature construction, and error behavior differ.

Suppose a 14-day forecast requires yesterday's sales for every forecasted date. After the first step, some of those values are not yet observed. A training table that silently supplies the eventual actual values will produce an unrealistic evaluation.

Ordinary boosted-tree predictions also do not automatically provide calibrated prediction intervals or reliable extrapolation beyond the observed feature ranges. Those requirements need explicit treatment.

Neural forecasting: different architectures for different structures

“Deep learning” is not one forecasting model. Several architectures can be relevant:

Model	Main idea	When to consider it	Important qualification
LSTM	A recurrent network with gated state updates	Sequential structure and relevant covariates	Learning long-range dependencies is not guaranteed by the architecture
N-BEATS	Fully connected residual blocks with backcast and forecast outputs	Forecasting from target history, often across many related series	Original N-BEATS differs from exogenous-input extensions
N-HiTS	Multirate sampling and hierarchical interpolation	Multiscale structure and longer forecast horizons	Evaluate the actual horizon and operating conditions
TFT	Recurrent processing, attention, gating, and variable selection	Multiple horizons with static, past-observed, and known-future inputs	Interpretation aids do not establish causal relationships

N-BEATS offers generic and interpretable configurations; the interpretable design should not be assumed for every configuration. N-HiTS emphasizes different frequencies and scales. TFT explicitly distinguishes categories of input availability.¹⁸¹⁹²⁰

Example: for a large retail panel, product category can be static, historical sales are past-observed, and next week's planned discount can be known in advance. That distinction is useful whether the chosen model is TFT or something simpler.

Use neural models when: the amount and diversity of relevant training information, relationships across series, and required outputs justify the additional complexity.

Watch for: small-data overfitting, unstable tuning, training cost, and weak validation. A network can sometimes learn across many short series, so “each series must be very long” is too simplistic. But a sophisticated architecture cannot create information absent from the training population.

Published benchmark results are evidence about those experiments. They are not a guarantee that the same architecture will win on your workload.

ARIMA, Prophet, and ETS: a direct practical comparison

Question	ARIMA / SARIMAX	Prophet	ETS
Main representation	Autoregressive and innovation dynamics, with optional integration and seasonality	Trend, seasonal functions, holiday effects, and optional regressors	Evolving level, trend, and seasonal states
How is nonstationarity approached?	Differencing and/or specified deterministic or regression terms	Explicit trend representation and changepoints	Updating component states
Calendar events	Supplied regressors can represent them	Explicit holiday/event interface	Not explicit in standard ETS
Additional predictors	Supported in regression-with-ARIMA-errors formulations	Supported, with future inputs required	Not standard ETS functionality
Remaining short-term dependence	ARMA structure is explicit	Check residuals; no standard ARMA error component	Represented through the specified state-space dynamics
A reasonable starting scenario	Useful temporal dependence in a series or regression residuals	Calendar-driven business series with meaningful trend structure	A series adequately described by evolving components

This table is a starting guide, not a performance ranking. A model's ability to represent a feature does not establish that the feature improves forecasts.

The same weekly sales series can justify comparing all three approaches. Their different assumptions make the comparison informative.

One forecasting task, several defensible candidates

Suppose a retailer needs a 28-day sales forecast for inventory planning across 200 stores. Daily data show weekly patterns, annual effects, planned promotions, and several store openings.

First define the target: observed sales or unconstrained demand? If stockouts suppress sales, neither Prophet nor ARIMA automatically recovers lost demand. The training target may already reflect an operational constraint.

Then build a candidate set:

Candidate	Reason to test it	A failure to investigate
Seasonal naive	Establish the value of repeating the weekly pattern	Missed campaign or trend effects
ETS	Test whether components are sufficient	Missed external drivers
SARIMAX or harmonic regression with ARIMA errors	Combine lag structure with calendar and campaign inputs	Unstable estimates or incorrectly supplied future predictors
Prophet	Test decomposable trend and calendar structure	Overextended trends or remaining serial dependence
Global boosted trees	Share patterns across stores and model interactions	Leakage, poor handling of new stores, or unreliable recursive forecasts
A justified neural candidate	Test shared learning and multi-horizon representations	Complexity without a repeatable forecasting benefit

This is an illustrative study design, not a recommendation to train every algorithm on every project. Start with the candidates the data and decision justify.

Use identical forecast origins, targets, horizons, and information availability. Allow each model reasonable tuning using training and validation periods. Record whether it refits or only updates between origins.

Evaluate both store-level and aggregate errors. A model can look good in total while making unacceptable errors for small stores. If forecasts are produced at several hierarchy levels, consider whether reconciliation is needed to keep the totals coherent.

Track bias, large misses, interval coverage where intervals are produced, fitting failures, and computation cost. Also inspect promotion periods and recently opened stores separately. Decide how stockouts and closures affect scoring before seeing the results.

Choose the simplest operationally acceptable candidate with credible evidence for the required decision. A more complex model earns its place by delivering a benefit that survives the comparison.

A map of related forecasting families

mindmap
  root((Forecasting))
    Classical dynamics
      ARIMA and SARIMA
      SARIMAX
      VAR and VECM
    Components and calendars
      ETS and Holt-Winters
      Prophet
    Complex seasonality
      TBATS
      Harmonic regression with ARIMA errors
    Learned nonlinear functions
      XGBoost and LightGBM and CatBoost
      LSTM
      N-BEATS and N-HiTS
      TFT

Whatever the family, the common requirements remain: time-correct inputs, meaningful baselines, realistic backtests, uncertainty assessment, and monitoring after deployment.

LinkedIn introduction

ARIMA remains one of my favourite subjects to study. Understanding it more deeply has also led me to explore the alternatives—and the assumptions behind them.

When should I consider Prophet? When might ETS be enough? What changes when the data have multiple seasonal cycles, external drivers, or hundreds of related series?

I have expanded my article into a practical guide to time-series forecasting, starting with ARIMA and moving through Prophet, exponential smoothing, TBATS, multivariate models, gradient boosting, and neural approaches such as N-BEATS, N-HiTS, and TFT.

For each family, I examine what it represents, when it is worth considering, and what can go wrong. The article also includes a retail forecasting example showing how to compare candidates under the same real-world constraints.

One distinction matters: Prophet is a related forecasting approach, not an ARIMA derivative. These methods belong in the same conversation because they can support similar decisions, while making different assumptions.

Author's note

This is my own learning project and reflects my ongoing study of time-series forecasting. I use an LLM to help polish the language and presentation. I take responsibility for the published content. If you spot inaccuracies, missing nuances, or assumptions that deserve closer examination, I would appreciate an explanation or reference.

Question guiding this work: What evidence would make me trust this forecast for the decision it is supposed to support? That means credible baselines, information available at forecast time, realistic backtesting, and checking errors where they matter operationally.

Related notes

• Shorter focus on ARIMA ideas: ARIMA
• Model building and validation habits: Building a Reliable ML Model

Technical references

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Originally published on LinkedIn, 8 November 2026.

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1. statsmodels: ARIMA specification ↩

2. Hyndman and Athanasopoulos: ARIMA models ↩↩↩

3. statsmodels: Augmented Dickey–Fuller test ↩

4. statsmodels: KPSS test ↩

5. statsmodels: SARIMAX specification ↩

6. Hyndman and Athanasopoulos: Dynamic regression models ↩↩↩

7. statsmodels: SARIMAX and ARIMA FAQ ↩

8. Hyndman and Athanasopoulos: The forecaster's toolbox ↩

9. statsmodels: Ljung–Box test ↩

10. Taylor and Letham: Forecasting at Scale ↩

11. Prophet documentation: Trend changepoints ↩

12. Prophet documentation: Seasonality, holidays, and regressors ↩

13. Hyndman and Athanasopoulos: Exponential smoothing ↩

14. De Livera, Hyndman, and Snyder: Forecasting time series with complex seasonal patterns ↩

15. forecast package: TBATS ↩

16. statsmodels: VAR and VECM ↩

17. Nixtla: MLForecast ↩

18. Oreshkin et al.: N-BEATS ↩

19. Challu et al.: N-HiTS ↩

20. Lim et al.: Temporal Fusion Transformers ↩