Naive Forecasting Methods

Naive forecasting methods serve as simple benchmark models for more complex forecasting approaches. Despite their simplicity, they often perform surprisingly well, especially for highly volatile or unpredictable data. Durbyn implements four naive methods:

Formula Interface (primary usage)

Example 1: Naive forecast

using Durbyn, Durbyn.Grammar

data = (sales = [120, 135, 148, 152, 141, 158, 170, 165, 180, 195],)

# Naive: forecast = last observation
spec = NaiveSpec(@formula(sales = naive_term()))
fitted = fit(spec, data)
fc = forecast(fitted, h = 12)

Example 2: Seasonal naive forecast

using Durbyn, Durbyn.Grammar

# Monthly sales data with yearly seasonality
data = (sales = [100, 110, 125, 140, 155, 170,
                 160, 150, 135, 120, 105, 95,
                 105, 115, 130, 145, 160, 175,
                 165, 155, 140, 125, 110, 100],)

# Seasonal naive: forecast = value from same month last year
spec = SnaiveSpec(@formula(sales = snaive_term()), m = 12)
fitted = fit(spec, data)
fc = forecast(fitted, h = 12)

Example 3: Random walk with drift

using Durbyn, Durbyn.Grammar

# Trending data
data = (value = cumsum(randn(50) .+ 0.5),)

# RW with drift: forecast = last value + h * average change
spec = RwSpec(@formula(value = rw_term(drift = true)))
fitted = fit(spec, data)
fc = forecast(fitted, h = 10)

# Access drift coefficient
fitted.fit.drift      # Drift value
fitted.fit.drift_se   # Drift standard error

Example 4: Mean forecast

using Durbyn, Durbyn.Grammar

# Stationary data around a mean
data = (temp = 20.0 .+ randn(100),)

# Mean: forecast = sample mean for all horizons
spec = MeanfSpec(@formula(temp = meanf_term()))
fitted = fit(spec, data, m = 12)
fc = forecast(fitted, h = 12)

# Access mean
fitted.fit.mu_original  # Mean on original scale

Example 5: Model comparison

using Durbyn, Durbyn.Grammar

data = (y = [100, 105, 102, 108, 115, 120, 118, 125, 130, 128],)

# Compare naive methods
naive_spec = NaiveSpec(@formula(y = naive_term()))
rw_spec = RwSpec(@formula(y = rw_term(drift = true)))
mean_spec = MeanfSpec(@formula(y = meanf_term()))

naive_fit = fit(naive_spec, data)
rw_fit = fit(rw_spec, data)
mean_fit = fit(mean_spec, data, m = 1)

# Compare residual variance
naive_fit.fit.sigma2  # Naive residual variance
rw_fit.fit.sigma2     # RW with drift residual variance

Example 6: Panel data / grouped fitting

using Durbyn, Durbyn.TableOps, Durbyn.ModelSpecs, Durbyn.Grammar

# Stacked table with :product column
panel = PanelData(tbl; groupby = :product, date = :date, m = 12)

# Fit seasonal naive to each group
spec = SnaiveSpec(@formula(sales = snaive_term()))
fitted = fit(spec, panel)
fc = forecast(fitted, h = 12)

Example 7: Box-Cox transformation

using Durbyn, Durbyn.Grammar

# Positive data with increasing variance
data = (sales = exp.(cumsum(randn(50) .* 0.1 .+ 0.05)),)

# Apply log transformation (lambda = 0) with bias adjustment
spec = NaiveSpec(@formula(sales = naive_term()), lambda = 0.0, biasadj = true)
fitted = fit(spec, data)
fc = forecast(fitted, h = 12)

Base API (array interface)

using Durbyn

y = randn(100)

# Naive forecast
fit_naive = naive(y)
fc = forecast(fit_naive; h = 12)

# Seasonal naive (monthly data)
y_seasonal = repeat([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], 3)
fit_snaive = snaive(y_seasonal, 12)
fc = forecast(fit_snaive; h = 24)

# Random walk with drift
y_trend = cumsum(randn(100))
fit_rw = rw(y_trend; drift = true)
fc = forecast(fit_rw; h = 10)

# Mean forecast
fit_mean = meanf(y, 1)
fc = forecast(fit_mean, 12)

NaiveFit exposes fitted, residuals, sigma2, lag, drift, drift_se, and optional Box-Cox parameters (lambda, biasadj).

Methodology

Naive Method

The naive forecast uses the last observed value as the forecast for all future horizons:

\[\hat{y}_{T+h|T} = y_T \quad \text{for all } h = 1, 2, \ldots\]

This is equivalent to a random walk model without drift.

Seasonal Naive Method

The seasonal naive method uses the observation from the same season in the previous cycle:

\[\hat{y}_{T+h|T} = y_{T+h-m \cdot k}\]

where $m$ is the seasonal period and $k = \lceil h/m \rceil$ is the number of complete seasonal cycles back.

For example, with monthly data ($m = 12$), the forecast for January next year uses the value from January this year.

Random Walk with Drift

The random walk with drift includes a linear trend based on the average historical change:

\[\hat{y}_{T+h|T} = y_T + h \cdot b\]

where the drift term $b$ is estimated as the mean of first differences:

\[b = \frac{1}{n}\sum_{t=2}^{T} (y_t - y_{t-1})\]

where $n$ is the number of valid consecutive pairs. This approach is more robust than the endpoint formula $(y_T - y_1)/(T-1)$ because it uses all available consecutive pairs rather than just endpoints, making it more reliable when data has missing values or gaps.

Mean Method

The mean method uses the historical average as the forecast:

\[\hat{y}_{T+h|T} = \bar{y} = \frac{1}{T}\sum_{t=1}^{T} y_t\]

This assumes the data fluctuates around a constant mean with no trend or seasonality.

Prediction Intervals

Naive methods produce prediction intervals that widen with the forecast horizon, reflecting increasing uncertainty over time.

Residual variance ($σ²$): The residual variance used for prediction intervals is computed as the mean squared error (MSE) of residuals without centering: $σ² = \frac{1}{n}\sum_{t} e_t^2$. This matches R's forecast package behavior.

Naive / Random Walk (without drift)

Forecast variance grows linearly with horizon:

\[\text{Var}(\hat{y}_{T+h|T}) = h \cdot \sigma^2\]

The standard error is:

\[\text{SE}(h) = \sqrt{h} \cdot \sigma\]

Seasonal Naive

Variance increases with the number of complete seasonal cycles:

\[\text{Var}(\hat{y}_{T+h|T}) = \lceil h/m \rceil \cdot \sigma^2\]

where $m$ is the seasonal period. The variance increases in steps each time a new seasonal cycle is entered.

Random Walk with Drift

Includes additional uncertainty from the drift estimate:

\[\text{Var}(\hat{y}_{T+h|T}) = h \cdot \sigma^2 + h^2 \cdot \text{SE}(b)^2\]

where $\text{SE}(b) = s_d / \sqrt{n}$ is the standard error of the drift coefficient, $s_d$ is the standard deviation of the first differences, and $n$ is the number of valid consecutive pairs.

Mean Method

Uses the t-distribution for intervals:

\[\hat{y}_{T+h|T} \pm t_{1-\alpha/2, T-1} \cdot s \cdot \sqrt{1 + 1/T}\]

where $s$ is the sample standard deviation and $T$ is the number of observations.

When to Use Each Method

Missing Value Handling

All naive methods handle missing values gracefully:

• Leading missings: Skipped when finding the starting point
• Trailing missings: The last valid observation is used
• Scattered missings: Converted to NaN internally; fitted values use forward-fill from the most recent valid lagged value

y = [1.0, missing, 3.0, 4.0, missing, 6.0]
fit = naive(y)  # Uses 6.0 (last valid) for forecasts
# fit.fitted[3] = 1.0 (lagged from y[1], since y[2] is missing)

Forward-fill behavior: When computing fitted values, if the lagged value is missing, the method forward-fills from the most recent valid fitted value. This matches the R forecast package's lagwalk() behavior and ensures fitted values are available wherever actual values exist, even when there are gaps in the series.

For seasonal naive, if a particular seasonal position has missing values, the method searches backwards through prior seasonal cycles to find the most recent valid observation at that position.

Box-Cox Transformation

All naive methods support Box-Cox transformation for variance stabilization:

# Log transformation (lambda = 0)
fit = naive(y; lambda = 0.0)

# Square root transformation (lambda = 0.5)
fit = naive(y; lambda = 0.5)

# Automatic lambda selection
fit = naive(y; lambda = :auto)

# With bias adjustment for back-transformation
fit = naive(y; lambda = 0.0, biasadj = true)

Lambda requirements:

• $λ ≤ 0$: Requires strictly positive values (non-positive values are treated as missing with a warning)
• $λ > 0$: Allows negative values and zeros via signed transformation
  • For x=0: the transform produces $-1/λ$ (finite value)

The biasadj option applies a bias correction when transforming forecasts back to the original scale, which can improve accuracy for skewed distributions. When enabled, bias adjustment is applied to both fitted values and point forecasts.

References

• Hyndman, R. J., & Athanasopoulos, G. (2021). Forecasting: Principles and Practice (3rd ed.). OTexts. https://otexts.com/fpp3/
• Makridakis, S., Wheelwright, S. C., & Hyndman, R. J. (1998). Forecasting: Methods and Applications (3rd ed.). John Wiley & Sons.