Time series recsys

Splitting a series into trend, seasonal and remainder parts that are added together: y = trend + seasonal + remainder. It fits when the size of seasonal swings stays about the same regardless of the level of the series.

Example. Weekly footfall that goes up by about 200 people on weekends whether the base is 1,000 or 2,000.

Related: Multiplicative decomposition, STL decomposition, Trend, Seasonality

A method for matrix factorisation that alternates between fixing item vectors and solving for user vectors in closed form, then the reverse. Each step is a set of independent least-squares problems, so it parallelises well and suits implicit feedback data.

Example. Fix the item factors, solve each user's vector; fix the user factors, solve each item's; repeat about 10 to 15 times.

Related: Matrix factorisation, Implicit feedback, Latent factors, Regularisation

Finding points or stretches in a series that differ sharply from expected behaviour. Common approaches compare actual values with forecasts and flag large residuals, or use isolation-based and reconstruction-based models.

Example. Flag a day when payment failures exceed the forecast by more than 3 standard deviations.

Related: Residual, Changepoint, Forecasting, Seasonality

A model in which the variance of today's shock depends on the size of recent shocks, allowing volatility to change over time. It was introduced for economic data where calm and turbulent periods alternate.

Example. After a large market move, the model predicts a larger expected move tomorrow.

Related: GARCH model, Volatility clustering, Heteroskedasticity, Autoregressive model

A widely used forecasting model that differences the series d times to make it stationary and then fits an ARMA(p, q) to the result. It is written ARIMA(p, d, q) and can handle trends but not seasonality directly.

Example. ARIMA(1,1,1) differences once, then uses one lagged value and one lagged shock.

Related: ARMA, Differencing, SARIMA, Box-Jenkins method

An ARIMA model extended with external regressors, so the forecast depends on past values, past shocks and outside variables. It suits cases such as demand driven by price or weather.

Example. Predict power demand with ARIMAX using temperature as the external input.

Related: ARIMA, SARIMAX, Exogenous variable, Autoregressive model

A model combining autoregressive and moving-average parts to describe a stationary series with few parameters. ARMA(p, q) uses p lagged values and q lagged shocks. It needs the series to be stationary, so differencing or transformation is usually applied first, and orders are chosen from correlation plots or information criteria.

Example. ARMA(1,1): y_t = c + 0.6*y_(t-1) + e_t + 0.3*e_(t-1).

Related: Autoregressive model, Moving average model, ARIMA, Stationarity

A hypothesis test for whether a series has a unit root. The null hypothesis is that it does, so a small p-value (for example below 0.05) is evidence that the series is stationary. It includes lagged differences to handle autocorrelation.

Example. An ADF p-value of 0.72 on daily sales suggests differencing is needed before modelling.

Related: Unit root, KPSS test, Stationarity, Differencing

The correlation of a series with a delayed copy of itself. At lag k it measures how strongly values k steps apart move together. Strong autocorrelation means the past helps predict the future, and it also breaks the independence assumption of ordinary regression.

Example. Daily temperatures have lag-1 autocorrelation of about 0.9: today's temperature is a good guide to tomorrow's.

Related: Autocorrelation function, Partial autocorrelation function, Lag, White noise

A plot or list of autocorrelations at lags 1, 2, 3 and so on. Slow decay suggests a trend, spikes at multiples of a period show seasonality, and a cut-off after lag q hints at a moving-average process of order q.

Example. Spikes at lags 7, 14 and 21 in daily data reveal a weekly pattern.

Related: Partial autocorrelation function, Correlogram, Autocorrelation, Moving average model

Evaluating a forecasting or trading method by replaying it on past data as if it were being used at that time, using only information available then. It estimates real-world accuracy, but is only as good as its care about leakage.

Example. Simulate the last 12 monthly forecasts using only earlier data each time and compare with actuals.

Related: Walk-forward validation, Look-ahead bias, Forecasting, Mean absolute percentage error

A training approach for implicit feedback that learns to score items a user interacted with above items they did not, by optimising pairwise preferences. Formula per pair: maximise log sigmoid(score(user, positive) - score(user, negative)).

Example. For a user who bought item A but not B, the model is trained so that score(A) exceeds score(B).

Related: Implicit feedback, Negative sampling, Pairwise ranking, Matrix factorisation

A classic three-step recipe for ARIMA modelling: identify the orders using autocorrelation plots, estimate the parameters, and check the residuals for leftover structure, repeating until they look like noise. It is the standard workflow taught for classical time-series modelling.

Example. Plot ACF and PACF, try ARIMA(1,1,0), test residuals with a Ljung-Box test, then adjust.

Related: ARIMA, Autocorrelation function, Ljung-Box test, Partial autocorrelation function

Columns derived from the timestamp, such as day of week, month, hour, holiday flags, month-end or festival indicators. They give models direct access to human schedules and are among the most useful inputs for demand forecasting.

Example. Add is_weekend, is_diwali_week and days_to_salary_day as inputs for an e-commerce demand model.

Related: Seasonality, Exogenous variable, Holiday effect, Lag feature

The first stage of a large recommender, which quickly narrows millions of items to a few hundred plausible ones using cheap methods such as embeddings and nearest-neighbour search. Recall matters here, since anything dropped cannot be shown.

Example. From 10 million videos, fetch the 500 whose embeddings are closest to the user's embedding.

Related: Ranking stage, Two-tower model, Re-ranking stage, Approximate nearest neighbour search

A moment at which the behaviour of a series shifts, for example in mean, trend or variance. Detecting them helps flexible trend models and monitoring systems adapt to new regimes.

Example. Daily active users flatten after an app redesign on a particular date.

Related: Structural break, Anomaly detection (time series), Trend, Prophet

The fraction of shown items that users click. Formula: CTR = clicks / impressions. It is a common signal for ranking and evaluating ads and recommendations, though optimising it alone can promote clickbait.

Example. An ad shown 20,000 times and clicked 500 times has CTR = 500 / 20,000 = 2.5 percent.

Related: Click-through rate prediction, Conversion rate, Ranking stage, Position bias

Estimating the probability that a user will click a given item in a given context, typically with a binary classifier trained on logs. Its output is used to rank ads and content, so well-calibrated probabilities matter.

Example. A model predicts 0.034 for a banner shown to a user on mobile in the evening.

Related: Click-through rate, Wide and Deep, DeepFM, Calibration

A situation in which several non-stationary series each wander, but some combination of them stays stable, meaning they share a long-run link. It is used in pairs trading and in error-correction models.

Example. Two similar stocks drift, but their price gap keeps returning to a stable level.

Related: Unit root, Vector autoregression, Stationarity, Random walk

A recommendation approach that predicts what a user will like from the behaviour of other users with similar tastes, without needing item descriptions. It works from patterns in a user-item interaction record.

Example. Two users both liked films A, B and C; one also liked D, so D is suggested to the other.

Related: User-based collaborative filtering, Item-based collaborative filtering, Matrix factorisation, User-item matrix

A recommendation approach that suggests items whose own attributes, such as genre, text or image features, match the user's past preferences. It needs no other users' data and handles new items, but tends to recommend more of the same.

Example. A user who watched three cricket documentaries is shown other documentaries tagged sports.

Related: Collaborative filtering, Hybrid recommender, Cold start, Item embedding

The share of users who complete a desired action such as purchase or sign-up after seeing or clicking an item. Formula: conversion rate = conversions / visitors (or clicks). It is a deeper business signal than clicks but rarer and delayed.

Example. 60 purchases from 2,000 clicks is a conversion rate of 3 percent.

Related: Click-through rate, Click-through rate prediction, Multi-task learning, Recommender system

A bar chart of autocorrelation values against lag, with confidence bands (often around plus or minus 1.96/sqrt(n)). Bars outside the band suggest real correlation. It is the standard first look at temporal structure.

Example. A correlogram of residuals with all bars inside the band suggests nothing systematic remains.

Related: Autocorrelation function, Partial autocorrelation function, Ljung-Box test, White noise

Estimating how a new recommendation or ranking method would have performed using only logs generated by an old method, correcting for the fact that the logs reflect the old method's choices. It reduces the need for costly online tests.

Example. Estimate a new ranker's click rate from last month's logs using inverse propensity weights.

Related: Inverse propensity scoring, Off-policy evaluation, Exposure bias, A/B testing

A forecasting method for intermittent demand that separately smooths the size of non-zero demands and the interval between them, then divides size by interval to get an expected demand per period.

Example. Average size 4 units and average gap 8 days give a forecast of 0.5 units per day.

Related: Intermittent demand, Exponential smoothing, Forecasting, Weighted absolute percentage error

Rises and falls in a series that occur without a fixed period, often tied to economic or business conditions and lasting more than a year. Unlike seasonality, their length varies, so they are harder to forecast.

Example. Real-estate demand swelling and slumping over multi-year credit cycles.

Related: Seasonality, Trend, Time series, Additive decomposition

A probabilistic forecasting method that trains one recurrent neural network across many related series and outputs the parameters of a probability distribution at each future step. Sharing across series helps those with little history.

Example. Train one network on 10,000 product series and forecast each with a negative binomial distribution.

Related: Probabilistic forecasting, Autoregressive model, Temporal fusion transformer, N-BEATS

A click-prediction model that combines a factorisation machine, for low-order feature interactions, and a deep neural network, for high-order ones, sharing the same embeddings and needing no manual feature engineering.

Example. Both branches read the same embeddings of user, item and context fields, and their outputs are summed.

Related: Factorisation machine, Wide and Deep, Click-through rate prediction, Embedding

Replacing each value by its change from the previous value, y_t - y_(t-1), to remove a trend and help make a series stationary. Applying it again gives the second difference. Over-differencing adds unwanted noise.

Example. Sales 100, 104, 109, 115 become changes 4, 5, 6 after one difference.

Related: Seasonal differencing, Stationarity, ARIMA, Unit root

A multi-step strategy that trains a separate model for each future step, so each predicts its horizon straight from known data. It avoids error feedback but needs more models and ignores links between horizons.

Example. Fit 14 different models, one per day ahead, for a 14-day forecast.

Related: Recursive forecasting, Multi-step forecasting, Forecast horizon, Lag feature

How different the items in a recommended list are from one another. Often measured as the average pairwise distance between item embeddings or attributes. Higher diversity lowers the risk that all items fail for the same reason.

Example. A list with 3 genres among 10 items is more diverse than one with 1 genre.

Related: Novelty, Serendipity, Re-ranking stage, Filter bubble

A constant average change per step added to a random walk, or the average slope used in a simple forecasting method. Forecasts extend a straight line from the last value using that average change.

Example. Sales grow on average by Rs 2,000 per month, so the forecast is last value plus 2,000 for each month ahead.

Related: Random walk, Trend, Naive forecast, Differencing

A method for measuring similarity between two sequences that may run at different speeds, by finding the alignment that stretches or compresses time to minimise the total difference. It is used for matching patterns in signals and speech.

Example. Two recordings of the same word, one spoken slowly, still match well under it.

Related: Time series, Autocorrelation, Anomaly detection (time series), Nearest neighbour

A state-space framework for exponential smoothing in which each of the error, trend and seasonal parts is labelled additive, multiplicative or absent. Choosing among the variants by information criterion gives automatic forecasting with proper prediction intervals.

Example. ETS(M,A,M) has multiplicative error, additive trend and multiplicative seasonality.

Related: Exponential smoothing, Holt-Winters method, State space model, Prediction interval

A variable that comes from outside the series being modelled and may help explain it, such as weather, price or a holiday flag. For forecasting, its future values must be known or predicted too.

Example. Public holiday indicators for the next 30 days are known in advance and can be used for retail footfall.

Related: SARIMAX, ARIMAX, Calendar features, Forecasting

A forecasting family that predicts with a weighted average of past values whose weights shrink geometrically with age, so recent points count most. Simple form: level_t = alpha * y_t + (1 - alpha) * level_(t-1).

Example. With alpha = 0.3, last level 100 and new value 110, the new level is 0.3*110 + 0.7*100 = 103.

Related: Holt's linear method, Holt-Winters method, ETS model, Moving average

A model that predicts from input features and all their pairwise interactions, where each interaction weight is a dot product of learned low-dimensional vectors. It handles sparse one-hot data such as user id, item id and context efficiently.

Example. Inputs: user=Asha, item=Film7, device=mobile; the model scores the weights of each and each pair of them.

Related: DeepFM, Wide and Deep, Click-through rate prediction, Latent factors

How far into the future predictions are made, measured in time steps. Accuracy usually declines as it grows, and the right horizon is set by the decision the forecast supports.

Example. A 14-day horizon for daily demand means predicting each of the next 14 days.

Related: Multi-step forecasting, Forecasting, Prediction interval, Direct forecasting

Adjusting forecasts made at different levels of a hierarchy so that they add up exactly, while using information from all levels. Methods include bottom-up, top-down and optimal combination approaches such as MinT.

Example. If city forecasts sum to 1,020 but the national forecast is 1,000, reconciliation moves both toward a consistent total.

Related: Hierarchical forecasting, Forecasting, Prediction interval, Multi-step forecasting

Predicting future values of a quantity from its past values and, often, other known factors. It supports planning such as stock levels, staffing and capacity. Results are best given with a range of uncertainty rather than one number.

Example. Predict next week's daily electricity demand for a city from the last 3 years of readings and weather forecasts.

Related: Forecast horizon, Prediction interval, Backtesting, Naive forecast

Sine and cosine columns of the time index, at the seasonal frequency and its multiples, used as inputs to model smooth repeating patterns. More pairs allow more complex shapes. They handle long or multiple seasonal periods economically.

Example. For yearly seasonality in daily data, add sin(2*pi*k*t/365.25) and cos(2*pi*k*t/365.25) for k = 1 to 3.

Related: Seasonality, Prophet, Calendar features, Exogenous variable

A matrix factorisation method popularised by the Netflix Prize that learns user and item vectors by gradient descent on the squared error of observed ratings only, with regularisation. It is not a true singular value decomposition despite the name.

Example. Predict a rating as global mean + user bias + item bias + dot product of the two factor vectors.

Related: Matrix factorisation, Latent factors, Explicit feedback, Regularisation

A generalisation of ARCH in which today's variance depends on recent squared shocks and on recent variances. It fits volatility that is persistent, and is standard in finance for risk measures.

Example. GARCH(1,1): variance_t = w + a*shock_(t-1)^2 + b*variance_(t-1).

Related: ARCH model, Volatility clustering, Heteroskedasticity, Forecasting

A statistical test of whether past values of one series improve the prediction of another beyond what its own past gives. It shows predictive usefulness, not true cause and effect.

Example. Past ad spend improving forecasts of weekly sales after using past sales is evidence that ad spend Granger-causes sales.

Related: Vector autoregression, Autocorrelation, Cointegration, Causal inference

A recurrent neural network method for session-based recommendation using gated recurrent units, which reads the sequence of items clicked in a session and predicts the next one, trained with ranking-based losses and session-parallel mini-batches.

Example. Feed clicks [phone, case, charger] into the network and it outputs scores for every possible next item.

Related: Session-based recommendation, Sequential recommendation, SASRec, Recurrent neural network

Forecasting a collection of series that add up in a structure, such as store to city to state, so forecasts at different levels can be produced and must agree. Separately made forecasts often do not sum correctly.

Example. Forecast demand for each of 500 stores, each city and the whole country.

Related: Forecast reconciliation, Forecasting, Probabilistic forecasting, Multi-step forecasting

A systematic rise or fall in a series around special dates, which may spread to days before and after. Models handle it with holiday flags, windows around them, or dedicated components.

Example. Grocery orders rise 40 percent in the two days before Diwali and dip on the day itself.

Related: Calendar features, Prophet, Seasonality, Exogenous variable

An extension of exponential smoothing that tracks both a level and a trend, each with its own smoothing parameter, so forecasts can rise or fall along a line. It suits data with a trend but no seasonality.

Example. If level is 200 and trend +5 per month, the 3-month-ahead forecast is 215.

Related: Exponential smoothing, Holt-Winters method, Trend, ETS model

An extension that smooths level, trend and seasonal pattern together, in additive or multiplicative form. It is a strong, simple baseline for series with both trend and repeating seasonality. Its parameters are usually found by minimising squared forecast error on past data.

Example. Forecast monthly retail demand with a 12-month seasonal cycle using three smoothing parameters.

Related: Holt's linear method, Exponential smoothing, Seasonality, ETS model

A recommender that combines several approaches, typically collaborative and content-based, to offset the weaknesses of each. Combination can be by blending scores, switching by situation, or feeding both signals to one model.

Example. Use content features for brand-new items and collaborative signals once enough interactions exist.

Related: Collaborative filtering, Content-based filtering, Cold start, Wide and Deep

Demand that is zero in many periods and occasionally non-zero, common for spare parts and slow-moving items. Ordinary methods behave poorly, so specialised methods model both how often demand occurs and how large it is when it does.

Example. A shop sells a rare spare part on only 6 of 60 days, in quantities of 1 to 3.

Related: Croston's method, Forecasting, Weighted absolute percentage error, Probabilistic forecasting

A correction method that weights each logged observation by one divided by the probability that it was shown or chosen. It gives an unbiased estimate of performance under a different policy, but can have high variance when probabilities are tiny.

Example. A click on an item shown with probability 0.1 counts as 10 clicks in the estimate.

Related: Propensity score, Off-policy evaluation, Position bias, Exposure bias

A method that recommends items similar to ones a user already liked, where similarity is measured from co-occurring interactions across all users. Item similarities change slowly, so they can be precomputed, which made it popular in large systems.

Example. People who bought a phone case often bought a screen guard, so the guard is suggested to a phone-case buyer.

Related: User-based collaborative filtering, Collaborative filtering, Cosine similarity, Item2Vec

A recursive algorithm that estimates the hidden state of a linear system with Gaussian noise by combining a prediction with each new noisy measurement, weighted by their uncertainties. It is used in tracking, navigation and forecasting.

Example. A phone estimates its position by mixing GPS readings (noisy) with a motion model (smooth).

Related: State space model, Time series, Forecasting, Residual

A hypothesis test whose null hypothesis is that a series is stationary around a level or trend. It is used alongside the ADF test, since the two have opposite nulls, and agreement between them gives more reliable conclusions.

Example. ADF fails to reject a unit root and KPSS rejects stationarity: both say the series needs differencing.

Related: Augmented Dickey-Fuller test, Stationarity, Unit root, Differencing

A widely used ranking algorithm that combines gradient boosted trees with LambdaRank gradients, which scale pairwise updates by how much swapping two items would change a list metric such as NDCG. It is a strong baseline for search and recommendation ranking.

Example. Implementations exist in libraries such as XGBoost and LightGBM with a ranking objective.

Related: Learning to rank, RankNet, NDCG, Gradient boosting

Hidden numeric attributes learned from data, such as tendency toward action versus romance, that describe users and items so that their match can be computed as a simple product. They are not named in advance and are found by the model.

Example. A user vector [0.8, -0.2] and film vector [0.9, 0.1] score 0.8*0.9 + (-0.2)*0.1 = 0.70.

Related: Matrix factorisation, Item embedding, User embedding, Collaborative filtering

Machine learning methods that train a model to order items by relevance, using labelled or logged preferences. Approaches are grouped as pointwise, pairwise and listwise depending on what the loss looks at.

Example. Train on search logs where clicked results should be ranked above skipped ones.

Related: Pointwise ranking, Pairwise ranking, Listwise ranking, LambdaMART

A learning-to-rank approach that defines the loss over a whole ordered list, aligning training more closely with list-based metrics such as NDCG. It is more complex, since sorting is not differentiable.

Example. Optimise a smooth surrogate of NDCG computed over all 20 results for a query.

Related: Pointwise ranking, Pairwise ranking, NDCG, LambdaMART

A hypothesis test of whether a group of autocorrelations of residuals are jointly zero. A small p-value means leftover structure remains and the model is inadequate. It is usually applied to model residuals as a final check that a fitted model has captured the available structure.

Example. Testing 10 lags of residuals gives p = 0.31, so no evidence of remaining autocorrelation.

Related: White noise, Residual, Box-Jenkins method, Autocorrelation

The large set of items that each receive few interactions, contrasted with a small head of very popular items. Recommenders tend to favour the head, so helping users reach the tail is a common goal.

Example. In a catalogue of 100,000 books, 500 bestsellers may account for most of the purchases.

Related: Popularity bias, Novelty, Item coverage, Cold start

A smoothed series made by averaging the most recent k values at each point. It reduces random noise and reveals the trend, but lags behind real changes and gives equal weight to all k values.

Example. Daily sales 10, 14, 12, 16 with k = 3 gives averages 12 and 14 for the last two days.

Related: Exponential smoothing, Rolling window feature, Trend, Moving average model

A model that predicts the current value from a constant plus the current and q previous random shocks, each with a weight. Not to be confused with smoothing by averaging. Formula for MA(1): y_t = c + e_t + theta * e_(t-1).

Example. With theta = 0.5, yesterday's surprise of +4 raises today's value by 2.

Related: Autoregressive model, ARMA, Autocorrelation function, White noise

A design that passes items through successive stages, usually retrieval, ranking and re-ranking, each with fewer items and costlier models. It balances quality with latency and compute at large catalogue sizes.

Example. 10 million items to 1,000 to 100 to the 20 shown, in under 200 milliseconds.

Related: Candidate generation, Ranking stage, Re-ranking stage, Latency

Predicting several future time steps at once rather than only the next one. Strategies include recursive, direct and multi-output methods, each with different error behaviour. Predicting further ahead is harder, so error normally grows with the horizon and the strategy chosen matters.

Example. Predict sales for each of the next 28 days from the last 90.

Related: Recursive forecasting, Direct forecasting, Forecast horizon, Lag feature

Splitting a series into trend, seasonal and remainder parts that are multiplied: y = trend x seasonal x remainder. It fits when seasonal swings grow in proportion to the level, as often happens with sales or passenger counts.

Example. December sales that are always 30 percent above the yearly average, whether that average is Rs 10 lakh or Rs 20 lakh.

Related: Additive decomposition, STL decomposition, Log transformation, Seasonality

A neural forecasting architecture made only of stacked fully connected blocks, each producing a backward estimate that is subtracted from the input and a forward forecast that is added to the total. It performed strongly on standard forecasting benchmarks without series-specific engineering.

Example. Forecast monthly series with a stack of blocks that separately learn trend and seasonal shapes.

Related: DeepAR, Temporal fusion transformer, Forecasting, Trend

The simplest forecast: predict that the next value equals the last observed one. It is the baseline any serious method must beat, and is surprisingly hard to beat on random-walk-like series.

Example. If today's demand was 480 units, the naive forecast for tomorrow is 480.

Related: Seasonal naive forecast, Random walk, Mean absolute scaled error, Backtesting

How unfamiliar or unpopular the recommended items are to the user or in general. A common measure is the average of negative log popularity of items, so rarer items score higher.

Example. Recommending a niche indie film the user has never heard of, instead of a blockbuster.

Related: Diversity, Serendipity, Popularity bias, Long tail

A learning-to-rank approach whose loss compares two items at a time and encourages the more relevant one to score higher. It focuses on relative order, though it treats all mistakes in pairs equally regardless of list position.

Example. For results A (relevant) and B (not), penalise the model if score(B) is not below score(A).

Related: Pointwise ranking, Listwise ranking, RankNet, Bayesian personalised ranking

The correlation between a series and its lag-k copy after removing the effect of the shorter lags in between. A cut-off after lag p suggests an autoregressive process of order p, so it is used to choose model order.

Example. PACF with significant spikes at lags 1 and 2 only suggests an AR(2) model.

Related: Autocorrelation function, Autoregressive model, Box-Jenkins method, Lag

A Transformer-based forecasting model that splits each series into patches, treats each as a token, and models each channel independently. Patching shortens the sequence and lets attention cover longer history.

Example. A 512-step history cut into patches of 16 steps yields 32 tokens for attention.

Related: Temporal fusion transformer, N-BEATS, Attention mechanism, Multi-step forecasting

A learning-to-rank approach that treats each item independently, predicting its relevance score or label with regression or classification, then sorts by prediction. It is simple but ignores the relative order among items.

Example. Predict a 0 to 4 relevance for each search result separately, then sort.

Related: Pairwise ranking, Listwise ranking, Learning to rank, Regression

The tendency of recommenders to over-recommend already popular items, because they have more data, which makes them more popular still. It hurts discovery and smaller sellers, and is countered by re-weighting or re-ranking.

Example. A top-10 list filled with the same bestsellers shown to nearly every user.

Related: Long tail, Novelty, Feedback loop, Exposure bias

A range that is expected to contain a future observation with a stated probability, such as 95 percent. It is wider than a confidence interval for the mean because it includes both model uncertainty and random noise, and it widens with the horizon.

Example. Forecast 1,000 orders with 95 percent interval 900 to 1,120.

Related: Quantile forecast, Probabilistic forecasting, Forecast horizon, Confidence interval

Producing a full probability distribution or set of quantiles for future values rather than a single number. It tells decision-makers how uncertain the outlook is, which matters for inventory, energy and risk.

Example. Instead of 'demand will be 500', output that demand has a 10 percent chance of exceeding 620.

Related: Quantile forecast, Prediction interval, DeepAR, Pinball loss

An open-source forecasting library that fits an additive model of trend, several seasonalities and holiday effects, and is designed to be easy to use with missing data and changepoints. It is popular for business series with strong calendar patterns.

Example. Model daily orders with weekly and yearly seasonality and a list of Indian festival dates as holidays.

Related: Additive decomposition, Changepoint, Holiday effect, Seasonality

A prediction of a chosen quantile of the future distribution, such as the 10th, 50th or 90th percentile, instead of the average. It lets planners handle asymmetric costs, such as stocking to a 90th percentile to avoid stock-outs.

Example. The 90th percentile demand forecast is 1,200 units: demand is expected to be at or below it 90 percent of the time.

Related: Pinball loss, Prediction interval, Probabilistic forecasting, Forecasting

A series in which each value equals the previous value plus a random step: y_t = y_(t-1) + e_t. It has no fixed mean, unpredictable direction, and its best forecast is simply the last observed value.

Example. A stock price modelled as yesterday's price plus a random daily change.

Related: Unit root, Naive forecast, Drift, Differencing

The stage after candidate generation in which a heavier model scores each shortlisted item using rich features and orders them by predicted usefulness. Because it only handles hundreds of items, it can afford complex models.

Example. A gradient boosted model scores 500 candidates on predicted click and watch time and sorts them.

Related: Candidate generation, Re-ranking stage, Learning to rank, Click-through rate prediction

A neural pairwise ranking method that models the probability that item i should rank above item j as a sigmoid of the difference in their scores, and trains with cross-entropy on preference pairs.

Example. P(i above j) = 1 / (1 + exp(-(s_i - s_j))); with scores 2.0 and 1.0 it is 0.73.

Related: Pairwise ranking, LambdaMART, Learning to rank, Cross-entropy loss

A final adjustment of an ordered list to satisfy goals beyond raw score, such as diversity, freshness, fairness, business rules or removing duplicates. It works on a short list, so it can use expensive logic.

Example. Ensure no more than 2 of the top 10 items come from the same seller.

Related: Ranking stage, Diversity, Serendipity, Learning to rank

A system that suggests items such as products, videos or songs to each user, based on what is known about the user, the items and past behaviour. It helps people find relevant things in huge catalogues and drives much of the engagement in modern apps.

Example. A shopping app showing 20 suggested products on the home page for each user.

Related: Collaborative filtering, Content-based filtering, Ranking stage, Candidate generation

A multi-step strategy that predicts one step ahead and feeds each prediction back as an input for the next step. It needs only one model, but errors can pile up as predictions are reused.

Example. Predict day 1, use it as a lag to predict day 2, and so on to day 14.

Related: Direct forecasting, Multi-step forecasting, Lag feature, Forecast horizon

A summary such as the mean, standard deviation or maximum computed over the most recent fixed number of observations, recalculated as time moves. It gives a model smoothed recent context and is built with a shift to avoid leakage.

Example. A 7-day rolling mean of orders, shifted by one day, used to predict tomorrow's orders.

Related: Lag feature, Moving average, Look-ahead bias, Exponential smoothing

An extension of ARIMA with extra seasonal terms, written ARIMA(p,d,q)(P,D,Q)m, where m is the seasonal period. It applies seasonal differencing and seasonal lag terms in addition to ordinary ones. Orders are usually chosen from correlation plots, tests and information criteria, and residuals are checked for leftover pattern afterwards.

Example. Monthly data with yearly cycle could use SARIMA(1,1,1)(1,1,1)12.

Related: ARIMA, Seasonal differencing, SARIMAX, Seasonality

A seasonal ARIMA model that also accepts external explanatory variables, called exogenous regressors. It lets known drivers such as price, holidays or promotions improve forecasts, provided their future values are known.

Example. Forecast daily sales with SARIMAX using festival flags and planned discount percentages as external inputs.

Related: SARIMA, ARIMAX, Exogenous variable, State space model

Subtracting the value from one full season earlier, y_t - y_(t-m) with m as the period, to remove a repeating seasonal pattern. For monthly data with yearly seasonality, m = 12.

Example. March 2026 sales minus March 2025 sales for each year in a monthly series.

Related: Differencing, Seasonality, SARIMA, Stationarity

A baseline that predicts each value as the value from the same point in the previous season. It is a strong benchmark for data with clear repeating patterns. Beating it is a minimum requirement for any seasonal model, and it needs no fitting.

Example. For Monday's demand, use last Monday's demand.

Related: Naive forecast, Seasonality, Mean absolute scaled error, Holt-Winters method

A pattern that repeats at a fixed, known period such as a day, week or year, caused by calendar or habit effects. Recognising it lets a forecaster separate predictable repeats from real change.

Example. Ice-cream sales peak every summer; food-delivery orders peak around 8 pm every evening.

Related: Trend, Seasonal differencing, Fourier terms, STL decomposition

Recommendation that models the ordered history of a user's interactions to predict what they will want next, since recent items and their order carry information. Models include recurrent networks and Transformers.

Example. After a user watches episodes 1 and 2 of a series, episode 3 should rank first.

Related: Session-based recommendation, SASRec, BERT4Rec, Item embedding

The quality of recommendations that are both unexpected and useful, delighting a user with something they would not have found or thought to look for. It is hard to measure, and usually defined relative to a baseline of obvious recommendations.

Example. A cricket fan gets a well-liked kabaddi documentary that they enjoy.

Related: Novelty, Diversity, Filter bubble, Beyond-accuracy metrics

Recommending the next item using only the sequence of actions within the current visit, without needing a user profile. It suits anonymous users or short-lived interest, and is typically modelled with sequence models.

Example. An anonymous visitor viewed three trekking shoes; the system suggests socks and a rain jacket.

Related: Sequential recommendation, GRU4Rec, Cold start, Item embedding

A property of a series whose statistical behaviour does not change over time: constant average, constant spread, and correlations depending only on the gap between points. Many classical models assume it, so series are often transformed to achieve it.

Example. Daily percentage changes in a stock index hover around zero, while the index level itself keeps drifting, so only the changes are roughly stationary.

Related: Differencing, Unit root, Augmented Dickey-Fuller test, Autocorrelation

A robust method that splits a series into trend, seasonal and remainder using repeated local smoothing (loess). It handles any seasonal period, allows the seasonal shape to change slowly, and resists outliers.

Example. Decompose 5 years of monthly data with period 12 to see whether the annual pattern is getting stronger.

Related: Additive decomposition, Seasonality, Trend, Residual

A stronger condition requiring that the joint probability distribution of any group of values stays the same when the whole group is shifted in time. It is hard to verify in practice, so weak stationarity is used instead.

Example. Independent, identically distributed noise is strictly stationary.

Related: Weak stationarity, Stationarity, White noise, Random walk

A lasting change in the relationship or parameters governing a series, caused by events such as policy changes or a pandemic. Models fitted across a break can be badly wrong, so tests such as the Chow test are used.

Example. Retail sales before and after a GST rate change follow different patterns.

Related: Changepoint, Stationarity, Forecasting, Backtesting

A neural forecasting architecture that combines recurrent layers, attention and gating to handle several series, known future inputs, static attributes and multiple horizons. It outputs quantile forecasts and offers some interpretation of variable importance.

Example. Forecast store-level sales using price, holiday flags and store type, with quantile outputs.

Related: Probabilistic forecasting, Quantile forecast, DeepAR, Attention mechanism

A sequence of measurements recorded in time order, usually at regular intervals such as every hour or day. Because neighbouring values tend to be related, the order carries information, and ordinary methods that shuffle data can give misleading results.

Example. Daily UPI transaction counts for a bank over 3 years form a series of about 1,095 values.

Related: Trend, Seasonality, Stationarity, Forecasting

A version of cross-validation where every validation block comes strictly after its training block, with training windows that expand or slide. Random splits are avoided because they let the model see the future. In scikit-learn it is TimeSeriesSplit.

Example. TimeSeriesSplit(n_splits=5) on 600 rows gives five train/validation pairs, each later in time than the last.

Related: Walk-forward validation, Backtesting, Look-ahead bias, Cross-validation

The slow, long-term rise, fall or level movement in a series after short-term ups and downs are ignored. It can be roughly straight or curved, and may change direction over time.

Example. Monthly smartphone sales in India rising from 10 lakh to 14 lakh units over four years.

Related: Seasonality, Additive decomposition, Differencing, Stationarity

A property of a series, such as a random walk, where shocks never fade and the level wanders without returning to a fixed mean. Its presence means the series is not stationary and needs differencing.

Example. A series with y_t = y_(t-1) + shock has a unit root: today's level is yesterday's plus a random step.

Related: Random walk, Augmented Dickey-Fuller test, Differencing, Stationarity

A method that finds users most similar to the target user and predicts ratings as a similarity-weighted average of those neighbours' ratings. It is intuitive but costly with very many users, and similarity changes as behaviour changes.

Example. Find the 30 users whose past ratings correlate most with yours and average their ratings for a film you have not seen.

Related: Item-based collaborative filtering, Collaborative filtering, Cosine similarity, Pearson correlation

A table with users as rows, items as columns and each cell holding a rating, click, purchase or blank. It is usually extremely sparse because most users interact with a tiny fraction of items.

Example. 1 million users and 100,000 items give 10^11 cells, of which perhaps 50 million are filled.

Related: Collaborative filtering, Sparsity, Matrix factorisation, Explicit feedback

A multivariate model in which each series is predicted from past values of itself and of all the other series in the group. It captures how several related series influence one another over time.

Example. Model daily petrol price, transport demand and inflation together, each depending on the last 2 days of all three.

Related: Autoregressive model, Granger causality, Cointegration, Exogenous variable

The observed tendency of large changes to be followed by large changes and small ones by small ones, regardless of direction. It is common in financial returns and is the reason models with time-varying variance are used.

Example. Nifty daily returns show calm weeks and turbulent weeks, rather than uniform swings.

Related: GARCH model, ARCH model, Heteroskedasticity, Autocorrelation

The practical version of stationarity that requires only a constant mean, a constant variance, and covariance between two values that depends solely on the lag between them. It is what most classical time-series methods actually assume.

Example. A series with mean 50, variance 4 and lag-1 covariance 2 throughout is weakly stationary.

Related: Stationarity, Autocorrelation, ARMA, Strict stationarity

The sum of absolute errors divided by the sum of actuals, giving a single percentage weighted by volume. It is widely used in retail because large items count more, and it does not explode for small ones.

Example. Errors of 10, 5 and 20 units on actual sales of 100, 50 and 200 units give WAPE = 35/350 = 10 percent.

Related: Mean absolute percentage error, Symmetric MAPE, Forecasting, Intermittent demand

A series of uncorrelated random values with constant mean and variance, so nothing about the past helps predict the next value. Model residuals should look like it; if they do not, the model has missed some structure.

Example. Random draws from a normal distribution with mean 0 and variance 1, one per day.

Related: Residual, Autocorrelation, Ljung-Box test, Random walk

A recommendation architecture from Google that jointly trains a wide linear model, which memorises specific feature combinations, and a deep neural network, which generalises to unseen ones. Their outputs are added before the final prediction.

Example. Memorise 'installed app A then impressions of app B', while the deep part generalises to similar apps.

Related: DeepFM, Factorisation machine, Click-through rate prediction, Hybrid recommender