Getting Started with iAR

In iAR: Irregularly Observed Autoregressive Models

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Overview

The iAR package provides autoregressive models for time series observed at irregularly spaced time points, a common situation in astronomy, finance, and environmental science. It implements three complementary model classes:

| Class | Coefficient | Autocorrelation | Reference | |-------|-------------|-----------------|-----------| | iAR | Scalar $\phi \in (0,1)$ | Positive only | Eyheramendy et al. (2018) | | CiAR | Vector $(\phi_R, \phi_I)$, $\|\phi\| < 1$ | Positive and negative | Elorrieta et al. (2019) | | BiAR | Vector $(\phi_R, \phi_I)$, $\|\phi\| < 1$ | Bivariate, cross-correlated | Elorrieta et al. (2021) |

All three share the same workflow: construct → simulate → estimate → forecast / interpolate.

library(iAR)

---

Generating Irregular Observation Times

Real astronomical or environmental data is rarely sampled on a regular grid. The gentime() function generates realistic irregular time grids from several distributions. The first argument x defaults to NULL, so the utilities object is created internally and you can call gentime() directly without any setup step.

set.seed(2847)

# x defaults to NULL — no setup needed
times <- gentime(n = 100)@times

cat("Number of observations:", length(times), "\n")
cat("Time span: [", round(min(times), 1), ",", round(max(times), 1), "]\n")
cat("Mean gap between observations:", round(mean(diff(times)), 2), "\n")

The default distribution ("expmixture") mixes two exponential distributions, mimicking data sets where most observations are closely spaced but occasional long gaps occur. Alternative distributions include "uniform", "exponential", and "gamma".

gaps <- diff(times)
hist(log1p(gaps),
     main = "Inter-observation gaps (log1p scale)",
     xlab = "log(1 + gap)", col = "steelblue", border = "white",
     breaks = 20)

---

iAR: Univariate Irregular Autoregressive Model

Simulation

Construct an iAR object with the desired coefficient, then call sim().

set.seed(2847)
times <- gentime(n = 100)@times

# Build the model: family = "norm", phi = 0.9
m_iar <- iAR(family = "norm", times = times, coef = 0.9)
m_iar <- sim(m_iar)

head(m_iar@series)

plot(m_iar, type = "o",main = "Simulated iAR series (phi = 0.9)",ylab="Values",xlab="Time",pch=20)

Parameter Estimation

Two estimators are available:

• loglik() — direct MLE; supports all three families ("norm", "t", "gamma").
• kalman() — Kalman-filter MLE; available for all three model classes (iAR, CiAR, BiAR).

Before estimation, standardise the series (zero mean, unit variance) and supply a vector of error standard deviations. For simulated data with no measurement noise, zeros suffice.

y       <- m_iar@series
y_std   <- y / sd(y)           # unit variance
m_iar@series     <- y_std
m_iar@series_esd <- rep(0, length(times))

Using loglik()

m_loglik <- loglik(m_iar)
cat("loglik estimate: phi =", round(m_loglik@coef, 4), "\n")
cat("Log-likelihood: ", round(m_loglik@loglik, 4), "\n")

Using kalman()

m_kalman <- kalman(m_iar)
cat("kalman estimate: phi =", round(m_kalman@coef, 4), "\n")
cat("Kalman log-lik: ", round(m_kalman@kalmanlik, 4), "\n")

Both estimators should recover a value close to the true $\phi = 0.9$.

Standard Errors via the Hessian

Set hessian = TRUE to obtain standard errors and p-values.

m_h <- iAR(family = "norm", times = times, coef = 0.9, hessian = TRUE)
m_h@series     <- y_std
m_h@series_esd <- rep(0, length(times))
m_h <- loglik(m_h)

# Summary contains the coefficient, SE, p-value, information criteria, and residual diagnostics
summary(m_h)

Forecasting

After estimation, use forecast() to predict tAhead time units into the future.

m_fc <- forecast(m_kalman, tAhead = 10)
cat("Forecast (10 time units ahead):", round(m_fc@forecast, 4), "\n")

plot_forecast(m_fc,ylab="Values",xlab="Time",type="o",pch=20)

Interpolation (Imputing Missing Values)

interpolation() imputes NA values in the series using the fitted model.

# Introduce a missing value at position 10
m_interp           <- m_kalman
m_interp@series[10] <- NA

m_interp <- interpolation(m_interp)
cat("Interpolated value at position 10:",
    round(m_interp@interpolated_values, 4), "\n")
cat("Original value at position 10    :",
    round(y_std[10], 4), "\n")

---

CiAR: Complex Irregular Autoregressive Model

CiAR extends iAR to allow negative autocorrelation via a complex coefficient $\phi = \phi_R + i\,\phi_I$. The stationarity condition is $|\phi| = \sqrt{\phi_R^2 + \phi_I^2} < 1$.

Simulation and Estimation

set.seed(2847)
times <- gentime(n = 100)@times

# phi = (0.9, 0): purely real CiAR — same as iAR but more general
m_ciar <- CiAR(times = times, coef = c(0.9, 0))
m_ciar <- sim(m_ciar)

# Standardise
y_c           <- m_ciar@series
y_c_std       <- y_c / sd(y_c)
m_ciar@series     <- y_c_std
m_ciar@series_esd <- rep(0, length(times))

m_ciar <- kalman(m_ciar)
cat("CiAR estimate: phiR =", round(m_ciar@coef[1], 4),
    " phiI =", round(m_ciar@coef[2], 4), "\n")
cat("Modulus |phi| =", round(sqrt(sum(m_ciar@coef^2)), 4), "\n")

Forecasting

m_ciar <- forecast(m_ciar, tAhead = 10)
cat("CiAR forecast:", round(m_ciar@forecast, 4), "\n")

---

BiAR: Bivariate Irregular Autoregressive Model

BiAR models two time series observed at the same irregular time points with a shared autoregressive structure and cross-correlation $\rho$.

Simulation

set.seed(2847)
times <- gentime(n = 100)@times

# coef = c(phiR, phiI), rho = cross-series correlation
m_biar <- BiAR(times = times, coef = c(0.9, 0.3), rho = 0.9)
m_biar <- sim(m_biar)

head(m_biar@series)   # matrix: column 1 = series 1, column 2 = series 2

Estimation

n       <- length(times)
y_b     <- m_biar@series
y_b_std <- y_b / matrix(apply(y_b, 2, sd), nrow = n, ncol = 2, byrow = TRUE)

m_biar@series     <- y_b_std
m_biar@series_esd <- matrix(0, n, 2)

m_biar <- kalman(m_biar)
cat("BiAR estimate: phiR =", round(m_biar@coef[1], 4),
    " phiI =", round(m_biar@coef[2], 4), "\n")

Forecasting

m_biar <- forecast(m_biar, tAhead = 10)
cat("BiAR forecast (series 1):", round(m_biar@forecast[1], 4), "\n")
cat("BiAR forecast (series 2):", round(m_biar@forecast[2], 4), "\n")

---

Real Data Example: AGN Light Curve

The package includes the agn dataset — 237 K-band flux measurements of the active galactic nucleus MCG-6-30-15, taken between 2006 and 2011.

data(agn)
head(agn)

plot(agn$t, agn$m, xlab = "Heliocentric JD - 2450000",
     ylab = "Flux [10^-15 erg/s/cm^2/A]",
     main = "AGN MCG-6-30-15 (K band)",type="o",pch=20)

Fitting an iAR Model

# Standardise: centre and scale by SD (measurement errors available)
y_agn   <- agn$m
y_agn_c <- (y_agn - mean(y_agn)) / sd(y_agn)
esd_agn <- agn$merr / sd(y_agn)

m_agn <- iAR(
  family     = "norm",
  times      = agn$t,
  series     = y_agn_c,
  series_esd = esd_agn
)

m_agn <- kalman(m_agn)
cat("Estimated phi:", round(m_agn@coef, 4), "\n")
cat("Kalman log-lik:", round(m_agn@kalmanlik, 4), "\n")

Fitted Values and Forecast

m_agn <- fit(m_agn)
plot_fit(m_agn, xlab = "Heliocentric JD - 2450000",
     ylab = "Flux [10^-15 erg/s/cm^2/A]",type="o",pch=20)

m_agn <- forecast(m_agn, tAhead = 1)
cat("One-step-ahead forecast:", round(m_agn@forecast, 4), "\n")

---

Utility: Pairing Two Irregular Time Series

When two series are observed at different (but overlapping) time points, pairingits() matches them within a tolerance window before fitting a BiAR model.

data(cvnovag)
data(cvnovar)

# Pair the G-band and R-band CV Nova light curves
paired <- pairingits(data1 = cvnovag, data2 = cvnovar, tol = 0.01)

cat("Paired observations:", nrow(paired@paired), "\n")
head(paired@paired)

---

Summary

The table below collects the key functions for each step of the modelling workflow.

| Step | iAR | CiAR | BiAR | |------|-----|------|------| | Construct | iAR(family, times, coef) | CiAR(times, coef) | BiAR(times, series, coef, rho) | | Simulate | sim(m) | sim(m) | sim(m) | | Estimate (direct MLE) | loglik(m) | — | — | | Estimate (Kalman) | kalman(m) | kalman(m) | kalman(m) | | Forecast | forecast(m, tAhead) | forecast(m, tAhead) | forecast(m, tAhead) | | Interpolate | interpolation(m) | interpolation(m) | interpolation(m) | | Plot fitted | plot_fit(m) | plot_fit(m) | plot_fit(m) | | Plot forecast | plot_forecast(m) | plot_forecast(m) | plot_forecast(m) | | Generate times | gentime(n = ...) | — | — | | Pair two series | pairingits(data1, data2) | — | — |

References

Eyheramendy, S., Elorrieta, F., & Palma, W. (2018). An irregular discrete time series model to identify residuals with autocorrelation in astronomical light curves. Monthly Notices of the Royal Astronomical Society, 481(4), 4990–5003.

Elorrieta, F., Eyheramendy, S., & Palma, W. (2019). Discrete semi-parametric model for the investigation of variable stars. Astronomy & Astrophysics, 627, A120.

Elorrieta, F., Eyheramendy, S., Palma, W., & Ojeda, C. (2021). A bivariate irregular autoregressive model. Monthly Notices of the Royal Astronomical Society, 505(1), 1105–1116.

iAR documentation built on Oct. 27, 2026, 5:11 p.m.