Uddin Demand Modeling

• Uddin Demand is a methodology to estimate latent demand by compensating for censored data using techniques like Gaussian Process regression and AR(p) models.
• It leverages non-stationary covariance kernels and changepoint detection to accurately capture nonlinear trends and structural discontinuities in demand patterns.
• Empirical evaluations show that this approach significantly reduces prediction errors in airline bookings and queueing systems compared to traditional methods.

Uddin Demand refers to the field and methodology surrounding the estimation, modeling, and inference of latent or unsatisfied demand in systems where observed data is censored or truncated due to system constraints, limits, or truncation processes. This problem is central in operational research domains such as airline revenue management, queueing systems with censored demand, and economic time series where system or market constraints cap observable demand, leaving underlying or "true" demand only partially revealed in data.

1. Conceptual Foundation of Demand Unconstraining

In many real-world systems, demand is not directly observable beyond imposed upper limits or due to boundary effects (e.g., booking caps in airlines, buffer overflows in queues, or unfilled job openings in labor markets). Consider observed demand data $D = D_T \cup D_C$, with $D_T$ as unconstrained (true) observations and $D_C$ as censored (constrained to a cap). The primary objective is the recovery or imputation of the underlying, "would-be" true demand $D_U = \{\hat d_i\}$, forming the reconstructed data $\widehat D = D_T \cup D_U$ (Price et al., 2017).

For queueing contexts, if $Q(t)$ is the queue length and $L(t)$ is the cumulative censored demand, the Skorokhod "reflection" principle gives

$Q(t) = Q(0) + Z(t) + L(t) \geq 0,$

where $Z(t)$ is the net input (supply minus demand), and $L(t)$ increases only when $Q(t) = 0$ (Fendick, 2011). The unobserved $I_n = L(n) - L(n-1)$ represents censored (unsatisfied) demand per period.

2. Traditional and Gaussian Process Approaches

Classical single-class demand unconstraining techniques include:

• Naïve Exclusion: Ignoring censored data points; leads to bias when censoring is substantial.
• Expectation–Maximisation/Pickup-Detruncation (EM/PD): Replaces censored observations with conditional expectations assuming (approximately) Gaussian demand distributions.
• Double Exponential Smoothing (DES): Linear extrapolation of cumulative bookings, assuming stationary, linear trends.

These approaches become inadequate when demand exhibits non-Gaussianity, strong nonlinear growth, or substantial heteroscedasticity. For airline bookings, EM/PD and DES can mis-estimate when demand trends are nonlinear, variances are heterogeneous, or censoring windows are extended (Price et al., 2017).

The Gaussian Process (GP) regression method introduced by Uddin and coauthors generalizes unconstraining via a probabilistic modeling framework:

• Treat the latent demand rate as a stochastic function $f(x)$,
• Model daily observed bookings $y_i$ as samples from a Poisson process with log-rate $\lambda_i = \log[1 + e^{f(x_i)}]$,
• Place a GP prior $f(x) \sim GP(0, k(x,x';\theta_c))$ with a non-stationary covariance kernel $k(\cdot, \cdot)$,
• Use the Laplace approximation to deal with non-Gaussian posteriors generated by the Poisson likelihood.

The GP kernel is innovatively constructed to handle key features: nonlinearity, heteroscedasticity, and changepoints.

3. Non-Stationary Covariance Structures and Changepoints

The demand process in question often manifests nonlinear temporal dynamics and abrupt changepoints. The variable-degree polynomial kernel modeled as

$k(x, x') = \sigma^2 \,[\,x^T x' + c\,]^p$

with learnable $p$ allows the GP to adapt to nonlinearity in the demand curve and avoid collapse of the prediction far from observed data (Price et al., 2017). Positive definiteness is enforced via a diagonal shift.

For regimes with observable structural discontinuities (changepoints), a segmented kernel is introduced:

$$k(x,x') = \begin{cases} \sigma_1^2 [x^T x' + c_1]^{p_1} & \text{if } x, x' < x_c \ \sigma_2^2 [x^T x' + c_2]^{p_2} & \text{if } x, x' \geq x_c \ 0 & \text{otherwise} \end{cases}$$

where $x_c$ encapsulates the changepoint, and parameters are learned separately on each segment.

This GP-based approach yields flexible extrapolation and accommodates complex, realistic demand patterns, outperforming legacy methods particularly in the presence of nonlinear and extended censoring.

4. Queueing Systems: Sharp and Graduated Censored Demand

In queueing theory, estimation of censored demand revolves around reconstructing the unobservable $I_n$ where the queue is at (or near) empty—canonical in economic time series (e.g., labor markets) or service systems. The autoregressive net-input queue model in (Fendick, 2011) posits the net input increments $\Delta Z_n$ as AR($p$) processes:

$$X_n = \Delta Z_n - p = \phi_1 X_{n-1} + \cdots + \phi_p X_{n-p} + \epsilon_n,\quad \epsilon_n \sim \mathcal{N}(0, \sigma^2).$$

Censoring occurs at a boundary (usually zero) but can be "graduated," i.e., demand begins to truncate before the boundary. The “graduated censoring” model introduces a parameter $\omega \geq 1$ to control the aggressiveness of pre-zero truncation, adapting the conditional queue-length distribution $F_n^{(\omega)}(q|z)$ to more flexibly model real-world queue depletion and censoring (Fendick, 2011).

Estimation proceeds by reconstructing unobserved increments and censored demands via the Distribution-Minimization (DM) algorithm, a stochastic EM-like procedure that iteratively imputes (draws) the latent increments conditional on observed queues, then re-estimates process parameters.

5. Inference and Estimation: Methodologies and Performance

Gaussian Process Inference

Inference in the GP unconstraining framework uses the Laplace approximation for non-Gaussian posteriors. Hyperparameters are optimized by marginal likelihood maximization, either via gradients (for the Gaussian noise case) or via grid-based quadrature in more complex or multimodal likelihoods. The predictive posterior for unconstrained booking days is given by

$f_*|y \approx N(\mu_*, \Sigma_*),$

with $\mu_*$ and $\Sigma_*$ determined via kernel projections and the effective Hessian evaluated at the mode (Price et al., 2017).

Daily unconstrained demand $\hat{y}_*$ is predicted via:

$$\hat \lambda(x_*) = \log[1 + e^{\mu_*}], \quad \hat y_* \approx \hat \lambda(x_*),$$

and cumulative unconstrained demand via summation.

Queueing and AR(p) Models

For queue models, parameter estimation assimilates detrended and normalized time series, selection of lag length $M$ and regressor dimension $K$ (via $R^2$), and recovery of censored demand by iterative imputation. The best three-dimensional AR(11) fit on U.S. nonfarm job openings data achieves $R^2 = 0.78$, with estimated mean drift $p \approx -0.029$, variance $\theta \approx 0.162$, and autocorrelation parameter $\tau \approx 0.006$. During economic downturns, the model estimates unsatisfied monthly hires peaking at approximately 400,000 (Fendick, 2011).

6. Empirical Evaluation and Practical Considerations

Experimental comparisons for airline unconstraining (Double Poisson Process data, 45/90 convex-type curves, 20 days of constraint) summarize as follows (Price et al., 2017):

Method	E1 (Mean %)	E2 (Daily Err)	E3 (Cum. Err)
EM	1.10 %	–	29.08
PD	0.86 %	–	29.20
DES	8.15 %	13.88	30.81
EM-Daily	2.30 %	12.27	24.58
PD-Daily	0.49 %	10.94	22.36
GP	1.46 %	9.56	18.65

GP regression yields dramatically lower E2 and E3 errors, especially under complex data regimes with nonlinear trends and extended censoring. The changepoint kernel further improves reconstruction of demand profiles with abrupt changes.

For queueing and macroeconomic applications, the DM algorithm effectively reconstructs unsatisfied demand time series, revealing countercyclical spikes in estimated latent demand during economic recessions, with conditional uncertainties quantified via analytic variance formulas (Fendick, 2011).

7. Limitations, Assumptions, and Practical Recommendations

The discussed methodologies have several assumptions and limitations:

• Gaussian process models require careful kernel selection and regularization to avoid overfitting or spurious extrapolation, especially under limited unconstrained data.
• In AR($p$) queueing models, real arrivals and departures are integer counts rather than Gaussian increments—Gaussianity is an approximation.
• Stationarity and structural invariance are assumed for the net-input process, which may break down under regime shifts or shocks (e.g., sudden policy changes).
• The DM algorithm is not a full maximum likelihood estimator, limiting large-sample inference for parameter standard errors.
• Boundary modeling (sharp vs. graduated censoring) may substantially affect the inferred unsatisfied demand, especially in contexts where demand suppression shadows the nominal constraint boundary.

Recommended practice involves preprocessing (detrending, normalization), model selection guided by out-of-sample $R^2$, and robustness checks against alternative censoring boundary models. Quantification of reconstructed latent demand should report conditional uncertainties and, when possible, cross-validate with alternative estimation approaches.

---

Uddin Demand thus encompasses a set of statistical and probabilistic methods oriented towards reconstructing censored or latent demand in operational systems, with advances driven by non-stationary GP regression (Price et al., 2017), AR($p$) queueing theory, and stochastic imputation frameworks (Fendick, 2011). These methodologies are indispensable in domains where observed data fails to fully reveal system demand due to capacity, policy, or process-imposed constraints.