Prospects for Applied Mathematics and Data Analysis

Prediction intervals for temporally dependent data require both conditional quantile estimation and a calibration mechanism capable of responding to distribution shift. An adaptive rolling conformal quantile boosting (ARCQB) formulation is developed in which boosted quantile functions provide a nonlinear base interval and a sequential state variable controls the empirical conformal quantile. For target miscoverage 𝛼, the calibration state follows a projected stochastic recurrence, 𝛼𝑡+1 = ΠA{𝛼𝑡 + 𝛾(𝛼 − 𝑒𝑡 )}, where 𝑒𝑡 is the realized miss indicator. A telescoping identity links the time-averaged miss frequency to the state displacement and projection residuals; in the unprojected bounded case, the calibration error is 𝑂(𝑇−1). The interval width admits the exact decomposition 𝑤𝑡 = 𝑤(0) 𝑡 + 2𝑞𝑡 , separating predictive sharpness from conformal inflation. Numerical evaluation uses a strictly chronological one-hour-ahead design on hourly Beijing air-quality measurements. For nominal 90% coverage, raw boosted quantiles attain 83.18%, static conformal calibration 87.40%, and rolling conformal calibration 89.87%. ARCQB attains 90.05% with mean width 47.14 𝜇gm−3 and the lowest interval score, 72.86. Its maximum seasonal coverage deviation is 0.38 percentage points, compared with 7.99 points for the uncalibrated interval. The numerical behavior is therefore consistent with the feedback relation predicted by the calibration dynamics, while high-pollution regimes remain the principal source of conditional under-coverage.

Contiguous observation gaps are difficult to reconstruct when a count process is simultaneously periodic, weathersensitive, and subject to abrupt operating-state changes. A single isotropic penalty treats these components as if they had the same spectral roughness; weak regularization then leaves high-frequency coefficients unstable, whereas strong regularization suppresses legitimate diurnal peaks. This paper formulates hourly reconstruction as an anisotropic periodic inverse problem. The response is represented on a square-root scale by exogenous covariates, nested daily, weekly, and annual Fourier systems, and modulated daily harmonics. A block-diagonal Tikhonov operator assigns separate regularization levels to these mechanisms and weights the kth periodic mode by k4, the Fourier form of a squared-curvature penalty. The estimator is available in closed form. Conditions for uniqueness, a perturbation bound for reconstructed gaps, a generalized spectral-filter representation, and monotonicity of the effective dimension are derived. Numerical reconstruction on one year of hourly bicycle demand was evaluated under independent missingness, eight-hour blocks, and complete-day outages using 24 repeated masks per regime. The proposed estimator reduced RMSE relative to a periodic-profile reconstruction by 29.75%, 26.85%, and 29.24%, respectively. Against an isotropic ridge model using the same 139-dimensional basis, the anisotropic model was slightly better for random and full-day gaps, statistically indistinguishable for eight-hour blocks, and attained an effective dimension of 113.57, illustrating that the principal gain is controlled multiscale regularization rather than an increase in model size.

Harmonic Regression–Stochastic Residual Decomposition ofAtmospheric Carbon Dioxide Concentration: SpectralCharacterization and Forecast-Error StructureIka Hesti Agustin1,*1 Department of Mathematics, University of Jember, Jember, East Java, IndonesiaEmail: ikahesti.fmipa@unej.ac.idReceived: May 31, 2025 Revised: August 09, 2025 Accepted: October 14, 2025 ⋆ Corresponding authorFor a series yt =gt +εt combining a curving trend with a strong seasonal cycle, this paper develops and proves properties of an explicit alternative to seasonal differencing: gt =β0+β1t+β2t2+ΣKk =1[ak sin(2πkt/m)+bk cos(2πkt/m)] and εt ∼ ARMA(p,q)×(P,Q)m, stationary and invertible. Four results are proved: near-orthogonality of the harmonic regressors, with Var( ˆ ak) ≈ 2σ2 ε /n; spectral concentration of the periodogram at ωk = 2πk/m; stationarity of the fitted residual via its characteristic roots, guaranteeing aWold representation εt = Σj ψjat−j with Σj ψ2j < ∞; and a three-way decomposition MSE(h) = Bias(h)2+Var( ˆ gT+h)+σ2 a Σh−1 j=0 ψ2j, whose noise term is shown to converge under stationarity but to diverge linearly, as in the exact random-walk case, under integration. Every result is verified numerically. Applied to the Mauna Loa CO2 record (h = 12,24,36,60 months against a linear-trend and a directly differenced SARIMA(1,1,1)(1,1,1)12 benchmark), gt explains R2 = 0.998 of variance with a significant, HAC-robust quadratic coefficient; the SARIMA benchmark attains marginally lower error at every horizon, a gap a Diebold–Mariano test does not find significant (p = 0.275 and 0.862), even though the two models’ forecast variances are confirmed, against their own state-space output, to grow through different mechanisms – bounded for the proposed model, unbounded for SARIMA. The proposed model’s own error decomposition further shows parameter-estimation variance uniformly negligible, so its non-stochastic error is attributable almost entirely to trend-misspecification bias.