Probabilistic Demand Prediction Models

𝗪𝗵𝗮𝘁 𝗶𝗳 𝘆𝗼𝘂 𝗰𝗼𝘂𝗹𝗱 𝗳𝗼𝗿𝗲𝗰𝗮𝘀𝘁 𝘁𝗵𝗲 𝗳𝘂𝘁𝘂𝗿𝗲 𝗯𝘆 𝘁𝗿𝗲𝗮𝘁𝗶𝗻𝗴 𝗻𝘂𝗺𝗯𝗲𝗿𝘀 𝗹𝗶𝗸𝗲 𝘄𝗼𝗿𝗱𝘀? That's exactly what Amazon did with Chronos. They took T5 (yes, the language model) and taught it to "read" time series data. The trick? Tokenize continuous values into ~4096 discrete bins. Suddenly, forecasting becomes next-token prediction. 𝗘𝘃𝗼𝗹𝘂𝘁𝗶𝗼𝗻: 📍 Chronos (Feb 2024) — Original release 📍 Chronos-Bolt (Nov 2024) — ~250× faster inference 📍 Chronos 2.0 (Oct 2025) — Multivariate support 𝗛𝗼𝘄 𝗶𝘁 𝘄𝗼𝗿𝗸𝘀: 🔹 𝘛5 𝘌𝘯𝘤𝘰𝘥𝘦𝘳-𝘋𝘦𝘤𝘰𝘥𝘦𝘳 — Bidirectional encoder captures dependencies; autoregressive decoder generates multi-step forecasts 🔹 𝘛𝘰𝘬𝘦𝘯𝘪𝘻𝘢𝘵𝘪𝘰𝘯 — Mean-scale values → quantize into bins → regression becomes classification. Now you can use all the LLM tricks. 🔹 𝘗𝘳𝘰𝘣𝘢𝘣𝘪𝘭𝘪𝘴𝘵𝘪𝘤 𝘖𝘶𝘵𝘱𝘶𝘵 — Outputs a distribution over bins per timestep. Sample → get prediction intervals with calibrated uncertainty. 🔹 𝘊𝘩𝘳𝘰𝘯𝘰𝘴-𝘉𝘰𝘭𝘵 — One-shot decoding (all future timesteps in one forward pass). ~250× speedup + ~5% accuracy gain via knowledge distillation. 𝗣𝗿𝗲𝘁𝗿𝗮𝗶𝗻𝗶𝗻𝗴: • Large corpus: energy, traffic, economics, weather, web traffic • Heavy augmentation: scaling, jittering, warping 𝗪𝗵𝘆 𝗶𝘁 𝗺𝗮𝘁𝘁𝗲𝗿𝘀: ✅ Bridges time series & NLP—use mature LLM infrastructure ✅ Native probabilistic forecasting ✅ Chronos 2.0: multivariate + cross-variable learning ✅ Multiple sizes (Mini → Large) ✅ Apache-2.0 license 𝗣𝗲𝗿𝘀𝗼𝗻𝗮𝗹 𝗻𝗼𝘁𝗲: Syama Sundar Rangapuram is one of the co-authors on this work. He taught me ML during my grad school days and helped me out more than I can say. Seeing his work shape the field like this — super proud. 🙌 The LLM playbook works for time series. Who knew? #TimeSeries #MachineLearning #Forecasting #AI #FoundationModels

Lately, my team has published our research "Uncertainty Quantification of Spatiotemporal Travel Demand with Probabilistic Graph Neural Networks" in IEEE Transactions on Intelligent Transportation Systems! This work addresses a critical gap in travel demand prediction by introducing a framework of probabilistic graph neural networks (Prob-GNNs). Unlike previous approaches, Prob-GNNs not only offer deterministic forecasts but also quantify the inherent uncertainty in travel demand predictions. Our findings underscore the significance of incorporating randomness into deep learning models, revealing that probabilistic assumptions play a crucial role in accurately capturing demand uncertainty. I extend my heartfelt thanks to the esteemed collaborators: Qingyi Wang, Dingyi Zhuang, Haris N. Koutsopoulos, and Jinhua Zhao. See the IEEE Publication here: https://lnkd.in/gKskqDG3 The arXiv version here: https://lnkd.in/g_JnR2fz Massachusetts Institute of Technology MIT Mobility Initiative MIT School of Architecture and Planning (MIT SAP) Singapore-MIT Alliance for Research & Technology Centre University of Florida University of Florida College of Design, Construction and Planning #ResearchPublication #TravelDemandPrediction #ProbabilisticModels #TransportationInnovation #TransportationResearch #NeuralNetworks #TravelForecasting #DeepLearning #DataScience #TransportationModeling #ProbabilisticForecasting #SpatiotemporalAnalysis #ResearchFindings #TransportationPlanning #UncertaintyQuantification #TravelPatterns #MITResearch #FloridaResearch #IEEEJournal #TransportationTechnology #GraphNeuralNetworks #TravelBehavior #UrbanMobility

📈 Building a Probabilistic Forecast with Linear Regression 👇🏼 There are many ways to model uncertainty in forecasting; a common method is using prediction intervals for a parametric model, such as linear regression, and conformal prediction when using both parametric and non-parametric models. While those methods are powerful, in some cases, you want to estimate the probability of hitting some value in the future. This is where probabilistic forecasting comes in 🎯. 🔍 What is a probabilistic forecast? A probabilistic forecast provides a distribution of possible future outcomes, not just a single predicted value. Instead of saying “sales next month will be 10,500,” we can say, for example, “there’s a 90% chance sales will fall between 9,800 and 11,300.” This gives decision-makers a clearer picture of risk and uncertainty 💡. 🛠️ How to build a probabilistic forecast with linear regression Even a simple linear regression can generate rich uncertainty estimates. Here’s the workflow: 1️⃣ Fit a linear regression model Start by estimating the relationship between your target variable and predictors. The model produces coefficients along with their standard errors. 2️⃣ Extract the coefficient distributions Under standard assumptions, regression coefficients follow approximately normal distributions centered at their estimated values. These distributions represent our uncertainty about the true parameter values. 3️⃣ Simulate coefficient draws Generate many random samples from each coefficient’s distribution. For each draw, compute a forecast using the simulated coefficients. This gives you an ensemble of possible forecast paths, each representing one plausible future scenario. 4️⃣ Aggregate the forecast distribution Once you have many simulated forecasts: ➡️ Take the median or mean as your point forecast ➡️ Use percentiles (e.g., 5th & 95th, or 10th & 90th) to form prediction intervals These intervals capture both parameter uncertainty and the variability inherent in the data. Probabilistic forecasting is essential for planning under uncertainty. And the beauty is—you don’t need a complex model. Even linear regression can provide meaningful, simulation-based uncertainty estimates that elevate the quality of your insights. The screenshot below shows an example of creating a probabilistic forecast for US monthly natural gas demand. #timeseries #forecasting #datascience

Most demand forecasts are built on a single method chosen by habit. Simple moving average because it is familiar. Exponential smoothing because someone set it up years ago. The method stays even when the data changes. The problem is that no single forecasting method works best for every demand pattern. Stable demand with no trend behaves differently than demand with a clear upward trend. Seasonal products need a completely different approach than items with flat, irregular consumption. Using the wrong method does not just produce a less accurate forecast. It produces systematically biased safety stock levels, reorder points, and procurement timing. The Demand Forecasting Tool runs five methods simultaneously on your historical data: Simple Moving Average, Weighted Moving Average, Single Exponential Smoothing, Holt's Double Exponential Smoothing for trending data, and Holt-Winters Triple Exponential Smoothing for data with both trend and seasonality. For each method, it automatically optimizes the smoothing parameters to minimize error on your specific data rather than using defaults. It then scores all five methods against your history using three error metrics: MAPE, MAD, and MSE. The best-fit method is identified automatically and used to generate the forward forecast. The Safety Stock tab takes the forecast error directly from the best method and calculates safety stock and reorder point across four service level targets using the standard formula. Paste your data, set your lead time and service level, and get a defensible stocking recommendation in under two minutes. Link in the comments. #SupplyChain #DemandForecasting #InventoryManagement #ProcurementAnalytics #CPSM

My previous post discussed the pitfalls of applying AI/ML models to proxy demand signals in forecasts. Today’s post discusses randomness in demand. Here’s a pop quiz - Is a demand forecast: 1) a single time series of predicted demand, OR 2) Is it a statistical distribution? If you chose 1, I believe you would be in the majority of planners in how we apply forecasts in supply chains. However, we can intuitively agree that demand is composed of predictable and random components. Enter probabilistic forecasting - the ability to produce statistical demand distributions. But the big question is what is the utility of the complexity introduced by demand distributions? Quick sidebar - there is an entire family of demand patterns where you are better off not forecasting and just using replenishment models to “pull” signals. That is not the focus of my discussion here. Having led large demand planning teams, I have observed planners across e-commerce and consumer product supply chains. The one thing I have observed is that planners are not wired to think in probabilistic terms, especially around demand. It’s far more tempting to operate with a single demand prediction and make operational plans around it. So why do analytically sharp planners struggle with demand distributions? A demand distribution models a statistical distribution with probabilities around quantiles of the distribution. For example, the 90th percentile of demand (denoted P90) is one where there is only a 10% chance that demand is above it. Planners already struggle to align demand with S&OP/IBP stakeholders. It is natural that planners have little patience to deal with complex distributions that are unwieldy. And the natural response is to run with a single prediction - typically the mean forecast. What a tragedy then to invest in a sophisticated AI/ML-forecasting solution but only use mean forecasts that ignore the randomness in demand! So what is the fix? In my opinion, it is essential that we generate demand distributions - another story on how valid the distributions themselves are. But I would keep distributions out of the S&OP/IBP domain. S&OP should continue to focus on aligning the mean forecast along with any business overrides. Instead, I would move demand distributions to the domain of inventory strategy to model the trade-off between service levels and cost-to-serve. Smart teams have figured out inventory models that ingest demand distributions, business inputs, lifecycle policies, and costs to optimize inventory investments. This gives planners the room to have a conscious inventory strategy that is codified in policies, respect randomness in demand and explicitly target service levels and costs. In Summary, build agile and robust supply chains using probabilistic demand but be thoughtful on when and where to introduce probabilistic computation in your planning process!! Drop a comment on what has worked for you.