Time Series and Forecasting · 172

Glossary

The technical terms of this track, with short definitions (and the Persian term where available). Each term links to the lesson where it is introduced.

$2 \times m$ moving average: The average of two neighbouring $m$ -MAs for even $m$ . It is centred at $t$ , and with $m$ equal to the period it removes the seasonal pattern. Introduced in: Lesson 1
adaptive conformal inference (ACI): Conformal prediction with feedback: after each outcome, $\alpha_{t+1} = \alpha_t + \gamma(\alpha - \mathrm{err}_t)$ . It guarantees the long-run average coverage even when the series changes. Introduced in: Lesson 10
ADF test: The augmented Dickey–Fuller test. Its null hypothesis is a unit root; a small p-value supports stationarity. Introduced in: Lesson 2
airline model: The seasonal ARIMA $(0,1,1)(0,1,1)_{12}$ : $(1 - B)(1 - B^{12})y_t = (1 + \theta B)(1 + \Theta B^{12})\varepsilon_t$ . Introduced in: Lesson 6
anomaly: A value or pattern that does not fit the normal behaviour of the series. Introduced in: Lesson 11
anomaly score and threshold: A detector computes a score $s_t$ for each time and raises an alarm when $s_t > \lambda$ for a threshold $\lambda$ . Introduced in: Lesson 11
AR polynomial: $\phi(z) = 1 - \phi_1 z - \dots - \phi_p z^p$ , so that the AR model reads $\phi(B)y_t = c + \varepsilon_t$ . The model is stationary when all roots of $\phi(z)$ lie outside the unit circle. Introduced in: Lesson 5
AR(1) process: $y_t = \phi\, y_{t-1} + \varepsilon_t$ . With $|\phi| < 1$ it is stationary and its ACF is $\rho(k) = \phi^k$ . Introduced in: Lesson 2
ARIMA( $p, d, q$ ): A model in which the $d$ -times differenced series $\nabla^d y_t$ is a stationary, invertible ARMA( $p, q$ ): $\phi(B)(1 - B)^d y_t = c + \theta(B)\varepsilon_t$ . Introduced in: Lesson 6
ARMA( $p, q$ ): A model with an AR part and an MA part: $\phi(B)y_t = c + \theta(B)\varepsilon_t$ . Introduced in: Lesson 5
autocorrelation function (ACF): $\rho(k) = \gamma(k)/\gamma(0)$ , the correlation between values $k$ steps apart. The sample ACF $\hat\rho(k)$ divides the sample autocovariance by $n$ . Introduced in: Lesson 2
autocovariance function: $\gamma(k) = \operatorname{Cov}(y_t, y_{t+k})$ for a stationary process, as a function of the lag $k$ . Introduced in: Lesson 2
autoregressive model AR( $p$ ): $y_t = c + \phi_1 y_{t-1} + \dots + \phi_p y_{t-p} + \varepsilon_t$ : a regression of the series on its own $p$ past values. Introduced in: Lesson 5
average run length (ARL): The mean number of steps until an alarm. $\mathrm{ARL}_0$ (no change) should be large; $\mathrm{ARL}_1$ (after a shift) should be small. Introduced in: Lesson 11
back-transformation: Mapping forecasts of a transformed series (for example $\log y$ ) back to the original scale. $\exp$ of the forecast mean gives the median, not the mean. Introduced in: Lesson 6
backshift operator: The operator $B$ with $B y_t = y_{t-1}$ . Polynomials in $B$ write differences and ARMA models in short form. Introduced in: Lesson 2
backtest (rolling-origin evaluation): Time series cross-validation: at many forecast origins $t_1 < \dots < t_K$ , fit on the past only, forecast the next values and record the errors. Introduced in: Lesson 3
benchmark contamination: Overlap between a model's pretraining data and a benchmark's test data. It turns a "zero-shot" score into an in-domain score. Introduced in: Lesson 9
benchmark forecasts (baselines): Forecasts that need no fitting: mean, naive (last value), seasonal naive (value one season earlier) and drift (line through the first and last value). A new method must beat them. Introduced in: Lesson 3
Box–Cox transform: For $y_t > 0$ , $w_t = (y_t^\lambda - 1)/\lambda$ if $\lambda \ne 0$ and $w_t = \log y_t$ if $\lambda = 0$ . It is used to make the variance stable and to turn multiplicative patterns into additive ones. Introduced in: Lesson 1
Box–Jenkins workflow: The iterative cycle identify, estimate, check, forecast: transform, difference, choose orders from the ACF/PACF and AICc, fit by maximum likelihood, check the residuals, then forecast. Introduced in: Lesson 6
breakdown point: The share of data that can be moved to infinity before an estimate also goes to infinity: 0 for the mean and standard deviation, 50 % for the median and the MAD. Introduced in: Lesson 11
calibration: Forecast probabilities match observed frequencies: a central 80 % interval contains about 80 % of the outcomes. Introduced in: Lesson 10
causal dilated convolution: $y_t = \sum_{j=0}^{k-1} w_j\,x_{t-dj}$ : a convolution that reads only present and past inputs, with gaps of size $d$ between them. Introduced in: Lesson 9
causal process: A stationary solution that depends only on present and past shocks: $y_t = \mu + \sum_{j \ge 0}\psi_j\varepsilon_{t-j}$ . Introduced in: Lesson 5
change point: A time after which the level, variance, trend or season changes and stays changed. Introduced in: Lesson 11
channel independence: Applying the same univariate network to each variable of a multivariate series separately, with shared weights and no information between channels. Introduced in: Lesson 9
cointegration: Two series with a unit root are cointegrated if some combination $y_t - \beta x_t$ is stationary. Then they share a common stochastic trend. Introduced in: Lesson 2
companion matrix: The $p \times p$ matrix that writes an AR( $p$ ) as a first-order vector recursion. Its eigenvalues are the reciprocals $1/z$ of the roots of $\phi(z)$ . Introduced in: Lesson 5
conditional sum of squares: A cheap estimate for ARMA models: minimise $\sum_t e_t^2$ of the recursively computed one-step errors, with $e_t = 0$ before the start. Introduced in: Lesson 5
conformal prediction: A method that sets interval widths from the errors on a calibration set, with a finite-sample coverage guarantee when the data are exchangeable. Introduced in: Lesson 10
conformalized quantile regression (CQR): Conformal prediction applied to the interval of two quantile models. The width of the interval can change with $x$ . Introduced in: Lesson 10
continuous ranked probability score (CRPS): $\operatorname{CRPS}(F, y) = \int(F(x) - \mathbf 1\{x \ge y\})^2\,dx$ , a strictly proper score in the units of $y$ . For a point forecast it equals the absolute error. Introduced in: Lesson 10
coverage (nominal and empirical): The nominal coverage of a central $(1 - \alpha)$ interval is $1 - \alpha$ ; its empirical coverage is the share of outcomes inside it in a backtest. Introduced in: Lesson 10
critical ratio: In the newsvendor problem, $\tau = c_u/(c_u + c_o)$ with under-stock cost $c_u$ and over-stock cost $c_o$ . The best order is the $\tau$ -quantile of demand. Introduced in: Lesson 10
CUSUM: A detector for small lasting shifts that adds up evidence: $C_t = \max(0, C_{t-1} + z_t - k)$ , with an alarm when $C_t > h_c$ . Here $k$ is the reference value and $h_c$ the decision threshold. Introduced in: Lesson 11
cycle: A rise and fall without a fixed period, such as a business cycle or the sunspot cycle. It is not seasonal. Introduced in: Lesson 1
cyclic encoding: The features $(\sin(2\pi k/m), \cos(2\pi k/m))$ for season $k$ of $m$ . They put the seasons on a circle, so December is close to January. Introduced in: Lesson 8
damped trend method: Holt's method with a damping parameter $0 < \phi < 1$ : the forecast $\ell_t + (\phi + \dots + \phi^h)b_t$ becomes flat at long horizons. Introduced in: Lesson 4
data vintage: The version of a revised data series that was available at a given date. A fair backtest uses the vintage that existed at each origin. Introduced in: Lesson 3
decomposition (additive and multiplicative): Writing a series as parts: additive $y_t = T_t + S_t + R_t$ when the seasonal swing has a fixed size, multiplicative $y_t = T_t \cdot S_t \cdot R_t$ when the swing grows with the level. Introduced in: Lesson 1
DeepAR: An autoregressive recurrent network (Salinas et al.) that outputs the parameters of a distribution for the next value. It is trained with teacher forcing and forecasts by sampling paths. Introduced in: Lesson 9
differencing: Taking changes: $\nabla y_t = (1 - B)y_t = y_t - y_{t-1}$ . It removes a unit root or a linear trend. Introduced in: Lesson 2
diffuse prior: A start of the filter with a very large covariance $P_{0\mid 0}$ , used when nothing is known about the initial state. Introduced in: Lesson 7
direct strategy: One model $f_h$ per horizon $h$ , trained on the $h$ -step target with inputs known at the origin. Introduced in: Lesson 8
DLinear: A forecaster that splits the context into a moving-average trend and a remainder and applies one linear layer to each. It is a strong linear baseline. Introduced in: Lesson 9
drift: The constant $c$ in a model with $d = 1$ : the average change per step. The random walk with drift forecasts $y_n + hc$ . Introduced in: Lesson 6
dynamic harmonic regression: A regression with ARIMA errors whose covariates are Fourier terms $\sin(2\pi kt/m)$ and $\cos(2\pi kt/m)$ , $k = 1, \dots, K$ . It handles long or non-integer seasonal periods. Introduced in: Lesson 6
embargo (gap) and purging: An embargo leaves a gap between training and test data; purging removes training samples whose labels overlap the test period. Both stop leakage through overlapping targets. Introduced in: Lesson 3
ETS model: An exponential smoothing method written as a statistical model, named by its Error, Trend and Seasonal parts, for example ETS(A,N,N) for SES. It adds a likelihood, model selection and prediction intervals. Introduced in: Lesson 4
exchangeable: Random variables whose joint distribution does not change under any permutation. I.i.d. data are exchangeable; an autocorrelated or trending series is not. Introduced in: Lesson 10
exogenous variable (covariate): An input other than the series itself that helps to forecast it. Some covariates are known in advance (calendar, holidays, planned promotions); others are only observed (weather). Introduced in: Lesson 1
expanding and sliding window: Two ways to choose the training data in a backtest: an expanding window always starts at $y_1$ ; a sliding window keeps a fixed length and drops the oldest values. Introduced in: Lesson 3
exposure bias: The mismatch between training on true inputs and predicting on the model's own outputs. Introduced in: Lesson 9
extended and unscented Kalman filter: Filters for nonlinear models: the EKF linearises the model with Jacobians; the UKF pushes sigma points through the nonlinear functions. Introduced in: Lesson 7
false-alarm budget: The number of false alarms per day (or per year) that the users accept. The threshold is chosen from it. Introduced in: Lesson 11
forecast error: $e_{t+h} = y_{t+h} - \hat y_{t+h\mid t}$ , measured on data the method did not see. Introduced in: Lesson 3
forecast horizon: The number of steps ahead $h$ . The forecast of $y_{t+h}$ made at origin $t$ is written $\hat y_{t+h\mid t}$ . Introduced in: Lesson 3
forecast origin: The time $t$ at which a forecast is made. Only $y_1, \dots, y_t$ are known there. Introduced in: Lesson 3
forecast reconciliation: Adjusting the base forecasts of a hierarchy (all series $y_t = S b_t$ with the summing matrix $S$ ) so that they add up, that is, they are coherent. Examples are bottom-up $\tilde y = S\hat b$ and MinT. Introduced in: Lesson 8
Fourier terms: Pairs $\sin(2\pi jt/m)$ , $\cos(2\pi jt/m)$ for $j = 1, \dots, K$ , which give a smooth seasonal shape with $2K$ columns. Introduced in: Lesson 8
frequency (sampling): The fixed spacing between the values of a regularly sampled series: one value every hour, day, week, month or quarter. Introduced in: Lesson 1
GARCH(1,1): A model for a changing shock variance: $\varepsilon_t = \sigma_t z_t$ with $\sigma_t^2 = \omega + a_1\varepsilon_{t-1}^2 + b_1\sigma_{t-1}^2$ . It describes volatility clustering in financial returns. Introduced in: Lesson 6
Granger causality: Series $x$ Granger-causes $y$ if the past of $x$ improves the forecast of $y$ beyond the past of $y$ alone. It is a statement about prediction, not about causation. Introduced in: Lesson 5
Holt's linear trend method: Exponential smoothing with a level $\ell_t$ and a slope $b_t$ . The forecast is the straight line $\ell_t + h\,b_t$ . Introduced in: Lesson 4
Holt-Winters method: Holt's method plus seasonal states $s_t$ with period $m$ . In the additive version the season is added; in the multiplicative version it multiplies the level. Introduced in: Lesson 4
information set: $\mathcal I_t$ : everything known at origin $t$ , namely past values, past observed covariates, known-in-advance covariates and static features. Introduced in: Lesson 8
innovation (Kalman filter): The surprise $\nu_t = y_t - H\hat x_{t\mid t-1}$ , with covariance $S_t = HP_{t\mid t-1}H^\top + R$ . Introduced in: Lesson 7
innovation (shock): The part $\varepsilon_t$ of $y_t$ that the past cannot predict: the one-step prediction error of the true model. Introduced in: Lesson 5
innovations state-space model: A state-space model in which one error $\varepsilon_t$ drives both the observation and the state equations, for example $y_t = \ell_{t-1} + \varepsilon_t$ , $\ell_t = \ell_{t-1} + \alpha\varepsilon_t$ . Introduced in: Lesson 4
invertible: An MA or ARMA model whose MA polynomial has all roots outside the unit circle, so that the shocks can be recovered from the past data. For MA(1) this means $|\theta| < 1$ . Introduced in: Lesson 5
isolation forest: An outlier detector made of random trees with random splits. Anomalies are isolated after few splits, so a short average path means a high score. Introduced in: Lesson 11
Joseph form: The covariance update $P_{t\mid t} = (I - K_tH)P_{t\mid t-1}(I - K_tH)^\top + K_tRK_t^\top$ . It stays symmetric and positive semidefinite in floating point. Introduced in: Lesson 7
Kalman filter: A recursion that computes the exact mean and covariance of the state given $y_1, \dots, y_t$ in a linear Gaussian state-space model, with a predict step and an update step. Introduced in: Lesson 7
Kalman gain: The matrix $K_t = P_{t\mid t-1}H^\top S_t^{-1}$ that says how far the update moves the estimate toward the new observation: near 1 for a precise measurement, near 0 for a noisy one. Introduced in: Lesson 7
KPSS test: The Kwiatkowski–Phillips–Schmidt–Shin test. Its null hypothesis is stationarity; a small p-value says the series needs differencing. Introduced in: Lesson 2
lag: The number of steps $k$ between two values $y_t$ and $y_{t-k}$ . A lag plot shows $y_t$ against $y_{t-k}$ . Introduced in: Lesson 1
lag feature: A feature equal to a past value of the series. For the direct $h$ -step model only lags $k \ge h$ are known at the origin. Introduced in: Lesson 8
leakage: Use of information that was not available at the forecast origin, by the model or any step before it. It makes backtest errors smaller than real errors. Introduced in: Lesson 3
level: The running estimate $\ell_t$ of the current height of the series in exponential smoothing and state-space models. Introduced in: Lesson 4
linear Gaussian state-space model: $x_t = F x_{t-1} + w_t$ , $w_t \sim \mathcal N(0, Q)$ , and $y_t = H x_t + v_t$ , $v_t \sim \mathcal N(0, R)$ , with independent noises. Introduced in: Lesson 7
Ljung–Box test: A test of "no autocorrelation up to lag $\ell$ " with $Q = n(n+2)\sum_{k=1}^{\ell}\hat\rho(k)^2/(n-k)$ , compared with a $\chi^2$ distribution. It is used to check residuals. Introduced in: Lesson 2
local and global model: A local model has its own parameters for each series; a global model has one set of parameters trained on the rows of all series (cross-learning). Introduced in: Lesson 8
local level model: $\ell_t = \ell_{t-1} + \eta_t$ (a random walk) and $y_t = \ell_t + \varepsilon_t$ . Its steady-state Kalman filter is simple exponential smoothing. Introduced in: Lesson 7
local linear trend model: A level and a slope that both move as random walks. It is the stochastic version of Holt's linear method. Introduced in: Lesson 7
Loess: Local regression: at each point it fits a low-degree polynomial to the nearby points, with more weight on the closest ones. Introduced in: Lesson 1
lookback (context length): The number $L$ of past values that a model reads at the forecast origin. Introduced in: Lesson 8
LSTM and GRU: Recurrent cells with gates. The LSTM keeps a hidden state and a cell state controlled by forget, input and output gates; the GRU is a lighter variant with two gates. Introduced in: Lesson 9
MA polynomial: $\theta(z) = 1 + \theta_1 z + \dots + \theta_q z^q$ . This track uses the plus sign; some books use a minus sign. Introduced in: Lesson 5
MAE and RMSE: Mean absolute error $\frac1J\sum_j|e_j|$ and root mean squared error $\sqrt{\frac1J\sum_j e_j^2}$ . Both are in the units of $y$ ; the best constant forecast is the median for MAE and the mean for RMSE. Introduced in: Lesson 3
MAPE and sMAPE: Percentage errors: MAPE $= \frac{100}{J}\sum_j|e_j|/|y_j|$ , and the symmetric version sMAPE. They are undefined or huge when $y$ is zero or near zero. Introduced in: Lesson 3
MASE: Mean absolute scaled error: the MAE divided by the in-sample MAE $s$ of the one-step seasonal naive (or naive) forecast. MASE < 1 means better than that benchmark in-sample. Introduced in: Lesson 3
masking and swamping: Two failures of a non-robust scale: masking (outliers inflate the standard deviation and are no longer flagged) and swamping (outliers shift the mean, so normal values get flagged). Introduced in: Lesson 11
matrix profile and discord: The matrix profile stores, for every window, the z-normalised distance to its nearest non-overlapping window. A discord is a window whose nearest neighbour is far away: a collective anomaly. Introduced in: Lesson 11
mean scaling: Dividing each window by the mean absolute value of the context. It keeps zeros at zero; DeepAR and Chronos use it. Introduced in: Lesson 9
median absolute deviation (MAD): $\operatorname{MAD}(r) = \operatorname{med}_i|r_i - \operatorname{med}(r)|$ . Multiplied by 1.4826 it estimates the standard deviation of normal data, and it is robust to outliers. Introduced in: Lesson 11
moving average (centred and trailing): The mean of $k$ neighbouring values. A centred moving average of order $k$ uses values on both sides of $t$ ; a trailing one uses only $y_t, \dots, y_{t-k+1}$ and lags behind a trend. Introduced in: Lesson 1
moving-average model MA( $q$ ): $y_t = \mu + \varepsilon_t + \theta_1\varepsilon_{t-1} + \dots + \theta_q\varepsilon_{t-q}$ : a weighted sum of the last $q + 1$ shocks. Its ACF cuts off after lag $q$ . Introduced in: Lesson 5
multi-output (MIMO) strategy: One model that predicts the whole vector of $H$ future values at once. Introduced in: Lesson 8
N-BEATS: A deep stack of fully connected blocks; each block outputs a backcast (removed from its input) and a partial forecast, and the forecasts are added. Introduced in: Lesson 9
naive forecast: Every future value equals the last observed value: $\hat y_{t+h\mid t} = y_t$ . It is the optimal forecast for a random walk. Introduced in: Lesson 3
nonconformity score: A number that says how badly a point fits the model, for example the absolute residual $|y_i - \hat y_i|$ . Introduced in: Lesson 10
over-differencing: Differencing a series that is already stationary. It adds a non-invertible MA part, for example $\nabla\varepsilon_t$ has $\rho(1) = -0.5$ , and makes forecasts worse. Introduced in: Lesson 2
panel data: Many related series of the same kind, for example the daily sales of 10 000 products. Introduced in: Lesson 1
partial autocorrelation function (PACF): The correlation $\phi_{kk}$ between $y_t$ and $y_{t-k}$ after removing the linear effect of the values in between. It equals the last coefficient of the best linear predictor with $k$ lags. Introduced in: Lesson 2
particle filter: A filter that represents the state distribution by many weighted samples: simulate, weight by the likelihood, resample. It works for any model. Introduced in: Lesson 7
PatchTST: A transformer forecaster that cuts each channel's window into patches of $P$ values; each patch is one token. Introduced in: Lesson 9
PELT and binary segmentation: Offline change-point methods that minimise a segment cost plus a penalty per change point. Binary segmentation splits greedily; PELT finds the exact optimum with pruning. Introduced in: Lesson 11
periodogram: The sample estimate of the spectral density: $I(\omega_j) = \frac1n\bigl|\sum_t (y_t - \bar y)e^{-2\pi i\omega_j t}\bigr|^2$ at the Fourier frequencies $\omega_j = j/n$ . It shows how the variance is spread over frequencies. Introduced in: Lesson 2
pinball loss (quantile loss): $\rho_\tau(u) = \tau u$ for $u \ge 0$ and $(\tau - 1)u$ for $u < 0$ , with $u = y - q$ . Its expected value is minimised by the $\tau$ -quantile. Introduced in: Lesson 10
point adjustment: An evaluation that counts all points of a true anomaly segment as detected if one point is flagged. It makes even random detectors look good, so honest evaluation avoids it. Introduced in: Lesson 11
point, contextual and collective anomaly: A point anomaly is one value far from all normal values; a contextual anomaly is normal in general but not at this time; a collective anomaly is a group of values that is unusual together. Introduced in: Lesson 11
prediction interval: An interval that should contain a future value with a given probability. With normal errors the 95 % interval is $\hat y_{n+h\mid n} \pm 1.96\,\hat\sigma_h$ . Introduced in: Lesson 3
prediction-error decomposition: Writing the Gaussian log-likelihood as a sum over the independent innovations $\nu_t$ with covariances $S_t$ . One filter pass gives the exact likelihood. Introduced in: Lesson 7
probabilistic forecast: A forecast of the predictive distribution of $y_{t+h}$ given the data up to $t$ : a full distribution, quantiles, an interval or sample paths. Introduced in: Lesson 10
probability integral transform (PIT): The value $u_t = F_t(y_t)$ of the forecast CDF at the outcome. Probabilistically calibrated forecasts give uniform PIT values on $[0, 1]$ . Introduced in: Lesson 10
process noise and observation noise: The noise $w_t$ in the state equation (covariance $Q$ ) and the noise $v_t$ in the observation equation (covariance $R$ ). Introduced in: Lesson 7
proper scoring rule: A score $S(F, y)$ whose expected value under the true distribution $G$ is smallest at $F = G$ (strictly proper if only there). It rewards honest forecasts. Introduced in: Lesson 10
psi-weights ( $\psi$ -weights): The coefficients of the MA( $\infty$ ) form $y_t = \mu + \sum_{j\ge0}\psi_j\varepsilon_{t-j}$ , with $\psi(z) = \theta(z)/\phi(z)$ . They give the response to one shock and the forecast variance. Introduced in: Lesson 5
quantile: The $\tau$ -quantile of a distribution is the value with probability $\tau$ below it, $F^{-1}(\tau)$ . Introduced in: Lesson 10
quantile crossing: Separately trained quantile models that give $\hat q_{\tau_1}(x) > \hat q_{\tau_2}(x)$ for $\tau_1 < \tau_2$ . Sorting the predicted quantiles fixes it. Introduced in: Lesson 10
quantile regression: Fitting a model $\hat q_\tau(x)$ by minimising the mean pinball loss. The best model outputs the conditional $\tau$ -quantile of $y$ given $x$ . Introduced in: Lesson 10
quasi-cycle: An oscillation with a typical period but random amplitude and phase. An AR(2) with complex roots produces it, and its ACF is a damped cosine. Introduced in: Lesson 5
random walk: $y_t = y_{t-1} + \varepsilon_t$ : the running sum of white noise. Every shock stays forever and $\operatorname{Var}(y_t) = t\sigma^2$ . With a constant $c$ added it is a random walk with drift. Introduced in: Lesson 2
Rauch–Tung–Striebel (RTS) smoother: A backward pass after the Kalman filter that estimates each $x_t$ from all observations $y_1, \dots, y_n$ . Smoothed estimates must not be used for forecasts or in a backtest. Introduced in: Lesson 7
receptive field: The span of inputs that one output can see. For stacked causal convolutions with kernel size $k$ and dilations $d_i$ it is $r = 1 + (k - 1)\sum_i d_i$ , so doubling dilations make it grow exponentially with depth. Introduced in: Lesson 9
recursive strategy: One one-step model whose forecasts are fed back as inputs to reach longer horizons. Errors can accumulate. Introduced in: Lesson 8
reduction to supervised learning: Turning a series into a table: each window gives one row, with past values as features and a later value as the target. Introduced in: Lesson 8
regression with ARIMA errors: $y_t = \beta_0 + \beta_1 x_{1,t} + \dots + \beta_k x_{k,t} + \eta_t$ with $\eta_t \sim$ ARIMA: covariates plus autocorrelated errors (also called dynamic regression or ARIMAX). Introduced in: Lesson 6
remainder: What is left after removing trend and season, written $R_t$ : noise and unusual events. Introduced in: Lesson 1
residual: The one-step error on the training data, $e_t = y_t - \hat y_{t\mid t-1}$ , with parameters fitted on all the training data (in-sample). Introduced in: Lesson 3
reversible instance normalisation (RevIN): Standardise each input window with its own mean and standard deviation, forecast on that scale, then undo the scaling at the output. Introduced in: Lesson 9
RMSSE: Root mean squared scaled error $\sqrt{\frac1J\sum_j e_j^2/s_2}$ , where $s_2$ is the in-sample mean of $(y_t - y_{t-m})^2$ . It was used in the M5 competition. Introduced in: Lesson 3
robust z-score: $z_t = (r_t - \operatorname{med}(r))/(1.4826\cdot\operatorname{MAD}(r))$ , a standardised residual that outliers hardly change. Introduced in: Lesson 11
rolling-mean feature: The mean of the last $w$ values that are known at the forecast origin. Without a shift it leaks the target. Introduced in: Lesson 8
sample path: One possible future $y^{(i)}_{t+1}, \dots, y^{(i)}_{t+H}$ , drawn step by step from a generative model. Many paths describe the joint uncertainty over horizons. Introduced in: Lesson 9
scale-dependent metric: A metric in the units of the data, such as MAE or RMSE. It cannot be averaged over series with different units or sizes. Introduced in: Lesson 3
seasonal adjustment: Removing the estimated seasonal component from a series, so that a change from one period to the next is not just the season. Introduced in: Lesson 1
seasonal ARIMA (SARIMA): An ARIMA $(p, d, q)(P, D, Q)_m$ that multiplies non-seasonal polynomials in $B$ with seasonal polynomials in $B^m$ and uses $D$ seasonal differences. Introduced in: Lesson 6
seasonal component: A pattern $S_t$ that repeats with a fixed, known period $m$ from the calendar or the clock, for example $m = 12$ for monthly data with a yearly pattern. Introduced in: Lesson 1
seasonal difference: $\nabla_m y_t = (1 - B^m)y_t = y_t - y_{t-m}$ , the change from the same season one period earlier. Introduced in: Lesson 2
seasonal naive forecast: Every future value equals the observed value one season earlier, for example next March equals the last March. Introduced in: Lesson 3
seasonal period: The number of steps $m$ after which the seasonal pattern repeats: 12 for monthly data, 7 for daily data with a weekly pattern, 4 for quarterly data. Introduced in: Lesson 1
sharpness: How concentrated the predictive distributions are, for example how narrow the intervals are. The goal is to maximise sharpness subject to calibration. Introduced in: Lesson 10
signal-to-noise ratio: In the local level model, $q = Q/R$ . It alone fixes the steady-state gain, and so the SES parameter $\alpha$ . Introduced in: Lesson 7
simple exponential smoothing (SES): The forecast is the level $\ell_t = \alpha y_t + (1 - \alpha)\ell_{t-1}$ , a weighted average of the past with weights $\alpha(1-\alpha)^j$ that fade exponentially. The forecast is flat. Introduced in: Lesson 4
smoothing parameter: A number in $(0, 1]$ that sets how fast old values are forgotten: $\alpha$ for the level, $\beta^*$ for the trend, $\gamma$ for the season. Introduced in: Lesson 4
split conformal prediction: Compute scores $s_i = |y_i - \hat y_i|$ on $n$ calibration points, take $\hat q$ as the $\lceil(n + 1)(1 - \alpha)\rceil$ -th smallest, and report $\hat y \pm \hat q$ . Coverage is at least $1 - \alpha$ . Introduced in: Lesson 10
spurious regression: A regression of one random walk on another, independent one that looks significant far more often than the test level says (Granger and Newbold, 1974). Introduced in: Lesson 2
standardised innovation: $\nu_t/\sqrt{S_t}$ from a Kalman filter. It is $\mathcal N(0, 1)$ when the model is right, so it is a natural anomaly score. Introduced in: Lesson 11
state-space model: A model with a hidden state $x_t$ that moves by a state equation and an observation $y_t$ made from the state by an observation equation. Introduced in: Lesson 7
stationary (weakly stationary): A process with constant mean, finite constant variance, and a covariance $\operatorname{Cov}(y_t, y_{t+k})$ that depends only on the lag $k$ . Strict stationarity asks that all joint distributions do not change under a time shift. Introduced in: Lesson 2
STL: Seasonal and Trend decomposition using Loess (Cleveland et al., 1990). It repeats Loess smoothing steps, lets the season change slowly, and can down-weight outliers. Introduced in: Lesson 1
strength of trend and seasonality: $F_T = \max(0, 1 - \operatorname{Var}(R_t)/\operatorname{Var}(T_t + R_t))$ and $F_S = \max(0, 1 - \operatorname{Var}(R_t)/\operatorname{Var}(S_t + R_t))$ . Values near 1 mean a strong component. Introduced in: Lesson 1
teacher forcing: Training an autoregressive model with the true previous values as inputs at every step. It equals maximum likelihood for the one-step model. Introduced in: Lesson 9
temporal convolutional network (TCN): A stack of causal dilated convolution blocks with residual connections, used for sequence modelling and forecasting. Introduced in: Lesson 9
Temporal Fusion Transformer (TFT): A forecaster that combines LSTM layers, attention, gating and variable selection for static, known-future and observed covariates, and outputs quantiles. Introduced in: Lesson 9
time series: A sequence of values $y_1, y_2, \dots, y_n$ observed in time order. The index $t$ is time and $n$ is the length. Introduced in: Lesson 1
time-series foundation model: A large network pretrained on many series from many domains and then used on new series, often zero-shot. Examples are Chronos, TimesFM and Moirai. Introduced in: Lesson 9
transition and observation matrix: In a linear state-space model, $F$ moves the state one step forward and $H$ maps the state to the expected observation. Introduced in: Lesson 7
trend (trend-cycle): The slow, long-term movement of the level of a series, written $T_t$ . The trend-cycle also includes slow cycles without a fixed period. Introduced in: Lesson 1
unit root: A root of the AR polynomial at $z = 1$ , as in the random walk ( $\phi = 1$ ). A series with a unit root is not stationary; one difference removes it. Introduced in: Lesson 2
univariate and multivariate series: A univariate series measures one quantity over time; a multivariate series measures several quantities at the same times. Introduced in: Lesson 1
vector autoregression VAR( $p$ ): An AR model for a vector of $K$ series: $\mathbf y_t = \mathbf c + \Phi_1\mathbf y_{t-1} + \dots + \Phi_p\mathbf y_{t-p} + \boldsymbol\varepsilon_t$ with $K \times K$ matrices $\Phi_i$ . Introduced in: Lesson 5
WAPE: Weighted absolute percentage error $\sum_j|e_j| / \sum_j|y_j|$ : the MAE divided by the mean of $|y|$ . Introduced in: Lesson 3
weighted quantile loss (WQL): The pinball losses over a set of quantile levels (for example $\tau = 0.1, \dots, 0.9$ ), summed over a test set and divided by $\sum_t|y_t|$ . It is a scale-free approximation of the CRPS. Introduced in: Lesson 10
white noise: A process $\varepsilon_t$ with mean 0, constant variance $\sigma^2$ and no correlation between different times. Gaussian white noise is i.i.d. $\mathcal N(0, \sigma^2)$ . Introduced in: Lesson 2
white-noise band: The band $\pm 1.96/\sqrt n$ on an ACF or PACF plot. For white noise about 95 % of the sample values fall inside it. Introduced in: Lesson 2
Wold decomposition: Every zero-mean covariance-stationary process is an MA( $\infty$ ) of its one-step prediction errors plus a perfectly predictable part. This is why ARMA models are general. Introduced in: Lesson 5
Yule–Walker equations: The linear equations that link the coefficients of the best linear predictor (or of an AR model) to the autocorrelations. Their matrix is a symmetric Toeplitz matrix of $\rho(k)$ values. Introduced in: Lesson 2
Yule–Walker estimator: Estimates of the AR coefficients from the Yule–Walker equations with the sample ACF in place of the true ACF. With the $1/n$ sample ACF the fitted model is always stationary. Introduced in: Lesson 5
zero-shot forecasting: Forecasting a new series with a pretrained model without any gradient update on that series; the model reads only its context. Introduced in: Lesson 9