The Stochastic Forecasting Engine: 4 Competing Models per SKU

Demand forecasting is the foundation of any inventory management. Yet the vast majority of CIS retailers still use one of two approaches: either "order the same as last month" or a moving average in Excel. Both ignore seasonality, trends, and nonlinear demand patterns. Our StochasticProjector engine works fundamentally differently: for every SKU, 4 competing mathematical models are fitted, and the one that best describes that specific product wins.

Four Models

For each product in each period, the system fits parameters for four models:

1. Polynomial (polynomial degree 1). A simple trend line through the data. Best suited for stable products with steady demand — everyday essentials, hygiene products, consumables. Formula: y = a·x + b, where x is the period number.

2. Sinusoidal. Fits sin/cos functions with angular frequency ω = 2π/T, where T is the cycle length (12 for monthly data, 52 for weekly). Captures seasonal waves: cold remedies spiking in weeks 44-48, allergy products peaking in weeks 14-22, sunscreen with a summer maximum. Formula: y = a + b·sin(ωx) + c·cos(ωx).

3. Exponential. Performs the fit in log-space: ln(y) = a·x + b, then back-transforms. Best for new products with accelerating growth or declining products at end-of-life.

4. Logarithmic. Log-transforms the X axis (time): y = a·ln(x) + b. Captures the "tapering growth" pattern — a product that surged initially and is now leveling off.

None of these are neural networks. None require GPU training. They are classical statistical models, and they work — not because they're sophisticated, but because they're matched to the right data.

Winner Selection: Weighted R² with Exponential Decay

The naive approach is to pick the model with the lowest average error. But this gives equal weight to year-old data and last week's data. In real retail, demand patterns change: a product that was seasonal last year may have gone flat due to a new competitor. A product with exponential growth hit its ceiling.

We use weighted coefficient of determination R² with exponential recency weighting. The half-life is approximately length/5 of the time series. If a product has 50 weeks of history, the half-life is 10 weeks: data from 10 weeks ago carries half the weight of today's, 20 weeks ago — a quarter.

Weight formula for point at index i with series length N:

• •w(i) = exp(-ln(2) · (N - i) / half_life)
• •half_life ≈ N / 5

Weighted R² is calculated as: 1 - (Σ w_i · (y_i - ŷ_i)²) / (Σ w_i · (y_i - ȳ_w)²), where ȳ_w is the weighted mean. The model with the highest weighted R² wins for that SKU.

P10/P50/P90 Confidence Bands

A forecast of "500 units next month" is nearly useless for a buyer. What's needed is a range: "between 380 and 620 units, with 500 most likely."

We generate three scenarios based on a Poisson model:

• •P50 = model prediction (median)
• •P10 = prediction − 1.96 × √(λ + σ²_residual) — optimistic consumption scenario
• •P90 = prediction + 1.96 × √(λ + σ²_residual) — pessimistic scenario

Where λ is the Poisson rate parameter (estimated from data), and σ_residual is the standard deviation of model errors on historical data. The Poisson distribution is chosen deliberately: retail demand is count data (0, 1, 2, ... units, never 3.7).

Practical application: a store manager orders to P50 for regular products, P90 for critical items (where stockouts mean lost customers), P10 for low-margin items (where overstock is the bigger risk).

Demand Cliff Detection

In real retail, demand doesn't taper gracefully to 0.3 units per week. It stops. A product is either selling or it isn't. The built-in cliff detector fires when both conditions are met:

• •Forecast drops to 20% or less of the previous period's value
• •Absolute predicted value ≤ 3 units

When triggered, the forecast is truncated to zero. Without this, the system would generate phantom low-level forecasts leading to unnecessary small orders — each consuming buyer time, logistics resources, and freezing capital.

Stockout Prediction

Based on the forecast and current balance, the system calculates three potential stockout dates:

• •Optimistic — under P10 scenario (low consumption)
• •Neutral — under P50 (most likely consumption)
• •Pessimistic — under P90 (high consumption)

The calculation accounts for trend: if the linear regression slope shows growing demand, the projection accelerates. Days of stock are recalculated with weekend and holiday adjustments using the company's global seasonality coefficient (DEFAULT_WEEKEND_SALES_MULTIPLIER = 0.9).

Maximum stock days are capped at MAX_STOCK_DAYS = 365 — if the product will last a year or more at current velocity, further forecast precision has no practical value.

Seasonality Integration

The forecasting engine doesn't operate in a vacuum. For each SKU, seasonality coefficients are available from the classification engine:

• •monthly_seasonality — 12 coefficients by month
• •weekly_seasonality — 52 coefficients by ISO week
• •weekly_day_seasonality — 7 coefficients by day of week

The base forecast is modulated by seasonal coefficients: if the current week's coefficient is 1.3, the forecast increases by 30%. If the weekend coefficient is 0.9, Saturday and Sunday forecasts decrease by 10%.

Period Support

The engine works with three period types:

Period choice affects parameters: the sinusoidal model uses ω = 2π/12 for monthly data and ω = 2π/52 for weekly. Daily-to-weekly aggregation is handled by aggregate_sales(), which also handles date duplicates and gaps.

Results on Real Data

On a retail chain of 8 locations (1,000 SKUs, 78 categories, 72 suppliers), the StochasticProjector demonstrated:

• •P50 forecast accuracy: 85-92% (MAPE 8-15%)
• •Average computation time: <2 seconds per SKU for a 52-week horizon
• •Share of SKUs with R² > 0.7: 68% (good model quality)
• •Share of SKUs with cliff detection: 12% (119 products with ceased demand)

Most frequent model winner: polynomial (42%), sinusoidal (31%), logarithmic (18%), exponential (9%). The sinusoidal model dominates seasonal categories with pronounced seasonality — cold remedies, allergy products, vitamins.

Key Takeaways

• •Model competition beats a single model. Different products follow different patterns. A system using only ARIMA or only exponential smoothing will be good for some of the assortment and poor for the rest.
• •Data freshness matters more than volume. Exponential decay weighting with half_life = N/5 means the system adapts to changes in weeks, not quarters.
• •Range matters more than point. P10/P50/P90 enables risk-aware decisions rather than hoping a single forecast is right.
• •Cliffs must be detected explicitly. Without cliff detection, the system wastes capital on phantom orders for products that have stopped selling.