Seasonal Average

Time Series

Medium

Problem

Theory

A seasonal average is the mean value for each season (period within a cycle) across multiple cycles. It captures recurring patterns that repeat at regular intervals.

\bar{y}_s = \frac{1}{n} \sum_{i=1}^{n} y_{s + (i-1) \cdot m}

where s is the season index (1 to $$), m is the number of seasons per cycle, and n is the number of cycles.

The Formula

For a time series with seasonal period m, the seasonal average for season s is:

\bar{y}_s = \frac{1}{n} \sum_{j=1}^{n} y_{s,j}

where y_{s,j} is the value for season s in cycle j, and n is the total number of cycles.

Example: Monthly data (12 seasons).

Average for January: Mean of all January values across years.

\bar{y}_{\text{Jan}} = \frac{y_{\text{Jan},2020} + y_{\text{Jan},2021} + y_{\text{Jan},2022}}{3}

Worked Example

Monthly sales data over 3 years:

Year 1: [120, 110, 130, 140, 150, 160, 170, 165, 155, 145, 135, 200]

Year 2: [125, 115, 135, 145, 155, 165, 175, 170, 160, 150, 140, 210]

Year 3: [130, 120, 140, 150, 160, 170, 180, 175, 165, 155, 145, 220]

Seasonal averages:

January (month 1): \frac{120 + 125 + 130}{3} = 125

February (month 2): \frac{110 + 115 + 120}{3} = 115

March (month 3): \frac{130 + 135 + 140}{3} = 135

...

December (month 12): \frac{200 + 210 + 220}{3} = 210

Result: [125, 115, 135, 145, 155, 165, 175, 170, 160, 150, 140, 210]

Interpretation

High seasonal average: Season typically has high values.

Low seasonal average: Season typically has low values.

Pattern: Identifies which seasons are strong vs weak.

Example: Retail sales high in December (holiday shopping), low in January (post-holiday).

Use case: Budgeting, inventory planning, staffing.

Seasonal Indices

Normalize seasonal averages to overall mean:

\text{SI}_s = \frac{\bar{y}_s}{\bar{y}_{\text{overall}}}

Interpretation:

• \text{SI}_s = 1.2: Season 20% above average
• \text{SI}_s = 0.8: Season 20% below average
• \text{SI}_s = 1.0: Season at average

Example:

Overall mean: 150

December average: 210

December index: \frac{210}{150} = 1.4 (40% above average)

Additive vs Multiplicative Seasonality

Additive model:

y_t = \text{Trend}_t + \text{Seasonal}_s + \text{Error}_t

Seasonal component: \text{Seasonal}_s = \bar{y}_s - \bar{y}_{\text{overall}}

Multiplicative model:

y_t = \text{Trend}_t \times \text{Seasonal}_s \times \text{Error}_t

Seasonal component: \text{Seasonal}_s = \frac{\bar{y}_s}{\bar{y}_{\text{overall}}}

Choose additive when: Seasonal variation constant in magnitude.

Choose multiplicative when: Seasonal variation proportional to level (common in business data).

Seasonal Decomposition

Classical decomposition:

1. Compute trend (moving average with period m)
2. Detrend: y_t - \text{Trend}_t (additive) or y_t / \text{Trend}_t (multiplicative)
3. Compute seasonal averages from detrended data
4. Center seasonal components (sum to zero or product to m)
5. Remainder: \text{Remainder}_t = y_t - \text{Trend}_t - \text{Seasonal}_s

Result: Separates trend, seasonality, and noise.

Worked Decomposition Example

Data: [10, 15, 12, 16, 20, 18, 22, 30, 25, 28, 35, 32] (quarterly over 3 years)

Step 1: Compute trend (4-quarter moving average)

Centered MA: [NaN, NaN, 13.25, 15.75, 17.00, 19.00, 22.50, 23.75, 26.25, 28.75, 30.00, NaN]

Step 2: Detrend (multiplicative)

Ratios: [NaN, NaN, 0.906, 1.016, 1.176, 0.947, 0.978, 1.263, 0.952, 0.974, 1.167, NaN]

Step 3: Seasonal averages

Q1: 0.906, 0.978 → Average 0.942

Q2: 1.016, 1.263 → Average 1.140

Q3: 1.176, 0.952 → Average 1.064

Q4: 0.947, 0.974, 1.167 → Average 1.029

Step 4: Normalize (ensure product = 4)

Product = 1.173, adjust by dividing each by (1.173/4)^{1/4} = 1.040

Final seasonal indices: [0.906, 1.096, 1.023, 0.989]

Forecasting with Seasonal Averages

Naive seasonal forecast:

\hat{y}_{t+h} = \bar{y}_s

where s = ((t+h-1) \mod m) + 1

Example: Forecast next January using average of past Januaries.

Improved forecast (trend + seasonal):

\hat{y}_{t+h} = \text{Trend}_{t+h} + \bar{y}_s

or

\hat{y}_{t+h} = \text{Trend}_{t+h} \times \text{SI}_s

Trend extrapolation: Linear, exponential, or model-based.

Seasonal Adjustment

Remove seasonality to reveal underlying trend:

Additive:

y_{t,\text{adj}} = y_t - \text{Seasonal}_s

Multiplicative:

y_{t,\text{adj}} = \frac{y_t}{\text{SI}_s}

Use case: Compare values across seasons fairly, analyze trend without seasonal noise.

Example: Unemployment rate seasonally adjusted to remove predictable variations.

X-12-ARIMA and X-13-ARIMA-SEATS

Official methods used by government agencies:

Advanced seasonal adjustment procedures.

Features:

• Handles trading day effects
• Holiday adjustments
• Outlier detection
• ARIMA modeling of irregular component

Software: U.S. Census Bureau provides implementation.

Application: Official economic statistics (GDP, CPI, unemployment).

Centered Moving Average for Detrending

Symmetric moving average:

\text{CMA}_t = \frac{1}{m} \sum_{i=-k}^{k} y_{t+i}

where m = 2k+1 for odd m.

For even m (e.g., m=12):

\text{CMA}_t = \frac{1}{2m} \left(y_{t-m/2} + 2\sum_{i=-m/2+1}^{m/2-1} y_{t+i} + y_{t+m/2}\right)

Purpose: Removes seasonality while preserving trend.

Note: Requires data on both sides (not causal).

Seasonality Tests

Visual inspection:

Plot data and look for repeating patterns.

Autocorrelation function (ACF):

Significant peaks at lags m, 2m, 3m, ... indicate seasonality.

Seasonal subseries plot:

Plot all values for each season. If means differ substantially, seasonality exists.

Formal tests:

• QS test (Maravall)
• OCSB test (Osborn-Chui-Smith-Birchenhall)
• Kruskal-Wallis test

Dealing with Changing Seasonality

Problem: Seasonal pattern evolves over time.

Example: Air conditioning sales seasonality changes with climate trends.

Solutions:

1. Moving seasonal averages:

Compute seasonal averages using only recent cycles.

\bar{y}_{s,t} = \frac{1}{k} \sum_{j=t-km+1}^{t} y_{s,j}

Use last k cycles instead of all history.

2. STL decomposition:

Seasonal-Trend decomposition using Loess.

Allows seasonal component to change gradually.

3. State space models:

Time-varying seasonal parameters.

Seasonal Dummies in Regression

Indicator variables for each season:

y_t = \beta_0 + \sum_{s=1}^{m-1} \beta_s D_{s,t} + \epsilon_t

where D_{s,t} = 1 if observation t is in season s, 0 otherwise.

Interpretation: \beta_s is difference between season s and reference season.

Advantage: Incorporate seasonality in regression framework.

Example: Quarterly data with Q4 as reference.

Q1, Q2, Q3 dummy variables capture seasonal effects.

Fourier Seasonality

Represent seasonality with sine and cosine terms:

y_t = \beta_0 + \sum_{k=1}^{K} \left[\alpha_k \sin\left(\frac{2\pi k t}{m}\right) + \gamma_k \cos\left(\frac{2\pi k t}{m}\right)\right] + \epsilon_t

Advantage: Smooth seasonal pattern, parsimonious (fewer parameters than dummies).

Disadvantage: Assumes smooth periodic pattern.

Use when: Long seasonal period (e.g., m=52 for weekly data).

Calendar Adjustments

Trading day effects:

Number of weekdays in month varies.

Solution: Adjust for trading days.

y_{t,\text{adj}} = y_t \times \frac{\text{Avg days}}{\text{Actual days}}

Holiday effects:

Easter, Thanksgiving, Chinese New Year dates vary.

Approach: Include holiday dummy variables or use specialized methods (X-13).

Length-of-month adjustment:

February has fewer days.

Per-day rate: \frac{y_t}{\text{Days in month}_t}

Seasonal Subseries Plot

Method:

Plot all observations for each season in separate panels.

Example: Monthly data.

12 panels, each showing all January values, all February values, etc.

Horizontal line: Seasonal average for that season.

Interpretation:

• Variation within season shows volatility
• Trend within season shows evolution
• Difference across panels shows seasonal pattern

Holt-Winters Seasonal Smoothing

Triple exponential smoothing:

Level: \ell_t = \alpha(y_t - s_{t-m}) + (1-\alpha)(\ell_{t-1} + b_{t-1})

Trend: b_t = \beta(\ell_t - \ell_{t-1}) + (1-\beta)b_{t-1}

Seasonal: s_t = \gamma(y_t - \ell_t) + (1-\gamma)s_{t-m}

Forecast:

\hat{y}_{t+h} = \ell_t + hb_t + s_{t+h-m\lfloor (h-1)/m \rfloor}

Advantage: Adaptive to changing patterns, all components updated each period.

Seasonal ARIMA Models

SARIMA(p,d,q)(P,D,Q)_m:

Combines non-seasonal and seasonal components.

Non-seasonal: AR(p), Differences(d), MA(q)

Seasonal: AR(P) at lag m, Differences(D) at lag m, MA(Q) at lag m

Example: SARIMA(1,1,1)(1,1,1)_{12} for monthly data.

Seasonal differencing:

\nabla_m y_t = y_t - y_{t-m}

Removes seasonal pattern.

Periodogram and Spectral Analysis

Periodogram:

I(\omega) = \frac{1}{T} \left|\sum_{t=1}^{T} y_t e^{-i\omega t}\right|^2

Peaks at frequencies: Indicate periodicity.

Seasonal frequency: \omega = \frac{2\pi}{m}

Example: Monthly data (m=12).

Peak at \omega = \frac{\pi}{6} indicates yearly seasonality.

Use: Detect multiple seasonal patterns (weekly and yearly).

Multiple Seasonality

Complex patterns: Multiple seasonal cycles.

Example: Hourly electricity demand.

• Daily pattern (24 hours)
• Weekly pattern (7 days)
• Yearly pattern (365 days)

TBATS model:

Trigonometric, Box-Cox, ARMA, Trend, Seasonal.

Handles multiple seasonalities with different periods.

STR (Seasonal-Trend decomposition with Regression):

Decomposes series with multiple seasonal patterns.

Seasonal Breaks

Structural changes in seasonal pattern:

Pattern shifts at specific point.

Example: Retail sales seasonality changes after major policy change.

Detection: CUSUM tests, Chow tests on seasonal dummies.

Modeling: Allow seasonal parameters to differ before/after breakpoint.

Dummy variables:

D_{s,t} \times \mathbb{1}_{t > \tau}

where \tau is breakpoint.

Interpolation of Missing Seasonal Values

Missing observations in specific seasons:

Approach 1: Use seasonal average.

\hat{y}_t = \bar{y}_s

Approach 2: Interpolate using seasonal pattern.

\hat{y}_t = \text{Trend}_t + \bar{y}_s

Approach 3: Kalman filter with seasonal state space model.

Choice depends on: Amount of missing data, pattern complexity.

Seasonal Adjustment Quality Metrics

F-tests: Compare variance explained by seasonal component to residual variance.

M-statistics: Suite of diagnostics for seasonal adjustment quality.

Spectral diagnostics: Check if seasonal frequencies removed from adjusted series.

Revision history: Assess stability of adjustments over time.

Residual seasonality tests: Verify no remaining seasonal pattern after adjustment.

Business Applications

Retail: Plan inventory based on seasonal demand patterns.

Tourism: Staff hotels and attractions according to seasonal peaks.

Agriculture: Predict harvest yields accounting for planting seasons.

Energy: Forecast electricity/gas demand with seasonal temperature effects.

Finance: Adjust earnings for seasonal patterns (Q4 typically strongest).

HR: Anticipate seasonal hiring needs (retail holiday season).

Seasonal Averages for Outlier Detection

Comparison to seasonal norm:

\text{Deviation}_t = y_t - \bar{y}_s

Flag outliers: |\text{Deviation}_t| > k \sigma_s

where \sigma_s is standard deviation for season s.

Example: Unusually high sales in February (typically low season) may indicate special event.

Application: Quality control, anomaly detection.

Visualization Techniques

Seasonal plot:

Overlay multiple years, each season on x-axis.

Polar (circular) plot:

Angle represents season, radius represents value.

Heatmap:

Rows = years, columns = seasons, color = value.

Box plots by season:

Distribution of values for each season.

All aid in: Identifying seasonal patterns visually.

Challenges with Short Time Series

Few cycles: Seasonal averages unreliable.

Rule of thumb: Need at least 2-3 full cycles for stable seasonal estimates.

Solutions:

1. Use external seasonal indices (industry benchmarks)
2. Pool data across similar entities
3. Use domain knowledge to set seasonal pattern
4. Employ Bayesian methods with informative priors

Example: New product with only 1 year of data. Use category-level seasonal pattern.

Comparison to Moving Averages

Seasonal average: Average across cycles for same season.

Moving average: Average across consecutive observations.

Purpose:

• Seasonal average: Extract seasonal pattern
• Moving average: Smooth noise, extract trend

Complementary: Often used together in decomposition.

Example: Detrend with MA, compute seasonal averages from detrended series.

Seasonal Co-Movement

Cross-sectional seasonality:

Multiple series share seasonal pattern.

Example: All retail categories peak in December.

Factor models: Extract common seasonal factors.

Application: Forecasting related series, portfolio diversification.

Correlation by season: Correlation may vary across seasons.

Examples

Hints

1. For position p, iterate with range(p, len(series), period).
2. Average each collected seasonal group independently.

Requirements

• Group observations by their index modulo period.
• Compute the arithmetic mean of every seasonal group.
• Return exactly period floating-point values.

Constraints

• 1 <= period <= len(series).
• series contains numeric values.
• Time limit: 300 ms.

Starter Code

def seasonal_average(series: list, period: int) -> list:
    """
    Returns the average for each position in the seasonal cycle.
    """
    # Write code here
    pass

Test Cases