Wilfredo Palma | Pontificia Universidad Catolica de Chile (original) (raw)

Papers by Wilfredo Palma

Research paper thumbnail of Analysis of the correlation structure of square time series

Journal of Time Series Analysis, Jul 1, 2004

This paper analyses the asymptotic behaviour of the autocorrelation structure exhibited by square... more This paper analyses the asymptotic behaviour of the autocorrelation structure exhibited by squares of time series with a Wold expansion where the input error is a sequence of random variables with mean zero and finite kurtosis. Two important cases are discussed: (i) when the errors are independent and, (ii) when the errors are uncorrelated but their squares are correlated. Both situations are addressed when the process exhibits short or long memory. Consequences of these results on certain models widely used in many disciplines are also discussed.

Research paper thumbnail of Minimum distance estimation of ARFIMA processes

Computational Statistics & Data Analysis, Feb 1, 2013

This paper proposes a new minimum distance methodology for the estimation of ARFIMA processes wit... more This paper proposes a new minimum distance methodology for the estimation of ARFIMA processes with Gaussian and non-Gaussian errors. The main advantage of this method is that it allows for a computationally efficient estimation when the long-memory parameter is in the interval d ∈ (− 1 2 , 1 2). Previous minimum distance estimation techniques are usually limited to the range d ∈ (− 1 2 , 1 4), leaving outside the very important case of strong long memory with d ∈ [ 1 4 , 1 2). It is shown that the new estimator satisfies a central limit theorem and Monte Carlo experiments indicate that the proposed estimator performs very well even for small sample sizes. The methodology is illustrated with three applications. The first two examples involve real-life time series while the third application illustrates that the proposed methodology is a sound alternative for dealing with incomplete time series.

Research paper thumbnail of Miscellanea. Statistical analysis of incomplete long-range dependent data

Biometrika, Dec 1, 1999

This paper addresses both theoretical and methodological issues related to the prediction of long... more This paper addresses both theoretical and methodological issues related to the prediction of long-memory models with incomplete data. Estimates and forecasts are calculated by means of state space models and the influence of data gaps on the performance of short and long run predictions is investigated. These techniques are illustrated with a statistical analysis of the minimum water levels of the Nile river, a time series exhibiting strong dependency.

Research paper thumbnail of Estimation of seasonal fractionally integrated processes

Computational Statistics & Data Analysis, 2006

This paper discusses the estimation of fractionally integrated processes with seasonal components... more This paper discusses the estimation of fractionally integrated processes with seasonal components. In order to estimate the fractional parameters, we propose several estimators obtained from the regression of the log-periodogram on different bandwidths selected around and/or between the seasonal frequencies. For comparison purposes, the semi-parametric method introduced in Geweke and Porter-Hudak (1983) and Porter-Hudak (1990) and the maximum-likelihood estimates (ML) are also considered. As indicated by the Monte Carlo simulations, the performance of the estimators proposed is good even for small sample sizes.

Research paper thumbnail of Red de Análisis Estocástico y Aplicaciones (Sistemas abiertos, energía y dinámica de la información)

Research paper thumbnail of Simultaneous variable selection and structural identification for time‐varying coefficient models

Journal of Time Series Analysis, 2021

Research paper thumbnail of An irregular discrete time series model to identify residuals with autocorrelation in astronomical light curves

Monthly Notices of the Royal Astronomical Society, 2018

Time series observations are ubiquitous in astronomy and are generated, for example, to distingui... more Time series observations are ubiquitous in astronomy and are generated, for example, to distinguish between different types of supernovae to detect and characterize extrasolar planets and to classify variable stars. These time series are usually modelled using a parametric and/or physical model that assumes independent and homoscedastic errors, but in many cases, these assumptions are not accurate and there remains a temporal dependence structure on the errors. This can occur, for example, when the proposed model cannot explain all the variability of the data or when the parameters of the model are not properly estimated. In this work, we define an autoregressive model for irregular discrete-time series based on the discrete time representation of the continuous autoregressive model of order 1. We show that the model is ergodic and stationary. We further propose a maximum likelihood estimation procedure and assess the finite sample performance by Monte Carlo simulations. We implement the model on real and simulated data from Gaussian as well as other distributions, showing that the model can flexibly adapt to different data distributions. We apply the irregular autoregressive model to the residuals of a transit of an extrasolar planet to illustrate errors that remain with temporal structure. We also apply this model to residuals of an harmonic fit of light curves from variable stars to illustrate how the model can be used to detect incorrect parameter estimation.

Research paper thumbnail of Estimation of Long-Memory Time Series Models: a Survey of Different Likelihood-Based Methods

Advances in Econometrics

Since the seminal works by Granger and Joyeux (1980) and Hosking (1981), estimations of long-memo... more Since the seminal works by Granger and Joyeux (1980) and Hosking (1981), estimations of long-memory time series models have been receiving considerable attention and a number of parameter estimation procedures have been proposed. This paper gives an overview of this plethora of methodologies with special focus on likelihood-based techniques. Broadly speaking, likelihood-based techniques can be classified into the following categories: the exact maximum likelihood (ML) estimation (Sowell, 1992; Dahlhaus, 1989), ML estimates based on autoregressive approximations (Granger & Joyeux, 1980; Li & McLeod, 1986), Whittle estimates (Fox & Taqqu, 1986; Giraitis & Surgailis, 1990), Whittle estimates with autoregressive truncation (Beran, 1994a), approximate estimates based on the Durbin-Levinson algorithm (Haslett & Raftery, 1989), state-space-based maximum likelihood estimates for ARFIMA models (

Research paper thumbnail of On the eigenstructure of generalized fractional processes

Statistics & Probability Letters, 2003

This work establishes bounds for the eigenvalues of the covariance matrix from a general class of... more This work establishes bounds for the eigenvalues of the covariance matrix from a general class of stationary processes. These results are applied to the statistical analysis of the large sample behavior of estimates and testing procedures of generalized long memory models, including Seasonal ARFIMA and k-factor GARMA processes, among others.

Research paper thumbnail of Analysis of the correlation structure of square time series

Journal of Time Series Analysis, 2004

This paper analyses the asymptotic behaviour of the autocorrelation structure exhibited by square... more This paper analyses the asymptotic behaviour of the autocorrelation structure exhibited by squares of time series with a Wold expansion where the input error is a sequence of random variables with mean zero and finite kurtosis. Two important cases are discussed: (i) when the errors are independent and, (ii) when the errors are uncorrelated but their squares are correlated. Both situations are addressed when the process exhibits short or long memory. Consequences of these results on certain models widely used in many disciplines are also discussed.

Research paper thumbnail of Efficient Estimation of Seasonal Long-Range-Dependent Processes

Journal of Time Series Analysis, 2005

This paper studies asymptotic properties of the exact maximum likelihood estimates (MLE) for a ge... more This paper studies asymptotic properties of the exact maximum likelihood estimates (MLE) for a general class of Gaussian seasonal long-range-dependent processes. This class includes the commonly used Gegenbauer and seasonal autoregressive fractionally integrated moving average processes. By means of an approximation of the spectral density, the exact MLE of this class are shown to be consistent, asymptotically normal and efficient. Finite sample performance of these estimates is examined by Monte Carlo simulations and it is shown that the estimates behave very well even for moderate sample sizes. The estimation methodology is illustrated by a real-life Internet traffic example.

Research paper thumbnail of A Class of Antipersistent Processes

Journal of Time Series Analysis, 2007

We introduce a class of stationary processes characterized by the behavior of their infinite movi... more We introduce a class of stationary processes characterized by the behavior of their infinite moving average parameters. We establish the asymptotic behavior of the covariance function and the behavior around zero of the spectral density of these processes, showing their antipersistent character. Then, we discuss the existence of an infinite autoregressive representation for this family of processes, and we present some consequences for fractional autoregressive moving average models.

Research paper thumbnail of On the sample mean of locally stationary long-memory processes

Journal of Statistical Planning and Inference, 2010

Some asymptotic statistical properties of the sample mean of a class locally stationary long-memo... more Some asymptotic statistical properties of the sample mean of a class locally stationary long-memory process are studied in this paper. Conditions for consistency are investigated and precise convergence rates of the variance of the sample mean are established for a class of time-varying long-memory parameter functions. A central limit theorem for the sample mean is also established. Furthermore, the calculation of the variance of the sample mean is illustrated through several numerical and simulation experiments.

Research paper thumbnail of Estimating seasonal long-memory processes: a Monte Carlo study

Journal of Statistical Computation and Simulation, 2006

This paper discusses extensions of the popular methods proposed by Geweke and Porter-Hudak [Gewek... more This paper discusses extensions of the popular methods proposed by Geweke and Porter-Hudak [Geweke, J. and Porter-Hudak, S., 1983, The estimation and application of long memory times series models.

Research paper thumbnail of Estimation of Tropical Sea Level Anomaly by an Improved Kalman Filter

Journal of Physical Oceanography, 1996

Research paper thumbnail of A ground‐level ozone forecasting model for Santiago, Chile

Journal of Forecasting, 2002

A physically based model for ground‐level ozone forecasting is evaluated for Santiago, Chile. The... more A physically based model for ground‐level ozone forecasting is evaluated for Santiago, Chile. The model predicts the daily peak ozone concentration, with the daily rise of air temperature as input variable; weekends and rainy days appear as interventions. This model was used to analyse historical data, using the Linear Transfer Function/Finite Impulse Response (LTF/FIR) formalism; the Simultaneous Transfer Function (STF) method was used to analyse several monitoring stations together. Model evaluation showed a good forecasting performance across stations—for low and high ozone impacts—with power of detection (POD) values between 70 and 100%, Heidke's Skill Scores between 40% and 70% and low false alarm rates (FAR). The model consistently outperforms a pure persistence forecast. Model performance was not sensitive to different implementation options. The model performance degrades for two‐ and three‐days ahead forecast, but is still acceptable for the purpose of developing an env...

Research paper thumbnail of Estimation and forecasting of long-memory processes with missing values

Journal of Forecasting, 1997

This paper addresses the issues of maximum likelihood estimation and forecasting of a long-memory... more This paper addresses the issues of maximum likelihood estimation and forecasting of a long-memory time series with missing values. A state-space representation of the underlying long-memory process is proposed. By incorporating this representation with the Kalman ®lter, the proposed method allows not only for an ecient estimation of an ARFIMA model but also for the estimation of future values under the presence of missing data. This procedure is illustrated through an analysis of a foreign exchange data set. An investment scheme is developed which demonstrates the usefulness of the proposed approach.

Research paper thumbnail of Assessing influence in Gaussian long-memory models

Computational Statistics & Data Analysis, 2008

A statistical methodology for detecting influential observations in long-memory models is propose... more A statistical methodology for detecting influential observations in long-memory models is proposed. The identification of these influential points is carried out by case-deletion techniques. In particular, a Kullback-Leibler divergence is considered to measure the effect of a subset of observations on predictors and smoothers. These techniques are illustrated with an analysis of the River Nile data where the proposed methods are compared to other well-known approaches such as the Cook and the Mahalanobis distances.

Research paper thumbnail of Data analysis using regression models with missing observations and long-memory: an application study

Computational Statistics & Data Analysis, 2006

The objective of this work is to propose a statistical methodology to handle regression data exhi... more The objective of this work is to propose a statistical methodology to handle regression data exhibiting long memory errors and missing values. This type of data appears very often in many areas, including hydrology and environmental sciences, among others. A generalized linear model is proposed to deal with this problem and an estimation strategy is developed that combines both classical and Bayesian approaches. The estimation methodology proposed is illustrated with an application to air pollution data which shows the impact of the long memory in the statistical inference and of the missing values on the computations. From a Bayesian standpoint, genuine priors are considered for the parameters of the model which are justified within the context of the air pollution model derivation.

Research paper thumbnail of Minimum distance estimation of ARFIMA processes

Computational Statistics & Data Analysis, 2013

This paper proposes a new minimum distance methodology for the estimation of ARFIMA processes wit... more This paper proposes a new minimum distance methodology for the estimation of ARFIMA processes with Gaussian and non-Gaussian errors. The main advantage of this method is that it allows for a computationally efficient estimation when the long-memory parameter is in the interval d ∈ (− 1 2 , 1 2). Previous minimum distance estimation techniques are usually limited to the range d ∈ (− 1 2 , 1 4), leaving outside the very important case of strong long memory with d ∈ [ 1 4 , 1 2). It is shown that the new estimator satisfies a central limit theorem and Monte Carlo experiments indicate that the proposed estimator performs very well even for small sample sizes. The methodology is illustrated with three applications. The first two examples involve real-life time series while the third application illustrates that the proposed methodology is a sound alternative for dealing with incomplete time series.

Research paper thumbnail of Analysis of the correlation structure of square time series

Journal of Time Series Analysis, Jul 1, 2004

This paper analyses the asymptotic behaviour of the autocorrelation structure exhibited by square... more This paper analyses the asymptotic behaviour of the autocorrelation structure exhibited by squares of time series with a Wold expansion where the input error is a sequence of random variables with mean zero and finite kurtosis. Two important cases are discussed: (i) when the errors are independent and, (ii) when the errors are uncorrelated but their squares are correlated. Both situations are addressed when the process exhibits short or long memory. Consequences of these results on certain models widely used in many disciplines are also discussed.

Research paper thumbnail of Minimum distance estimation of ARFIMA processes

Computational Statistics & Data Analysis, Feb 1, 2013

This paper proposes a new minimum distance methodology for the estimation of ARFIMA processes wit... more This paper proposes a new minimum distance methodology for the estimation of ARFIMA processes with Gaussian and non-Gaussian errors. The main advantage of this method is that it allows for a computationally efficient estimation when the long-memory parameter is in the interval d ∈ (− 1 2 , 1 2). Previous minimum distance estimation techniques are usually limited to the range d ∈ (− 1 2 , 1 4), leaving outside the very important case of strong long memory with d ∈ [ 1 4 , 1 2). It is shown that the new estimator satisfies a central limit theorem and Monte Carlo experiments indicate that the proposed estimator performs very well even for small sample sizes. The methodology is illustrated with three applications. The first two examples involve real-life time series while the third application illustrates that the proposed methodology is a sound alternative for dealing with incomplete time series.

Research paper thumbnail of Miscellanea. Statistical analysis of incomplete long-range dependent data

Biometrika, Dec 1, 1999

This paper addresses both theoretical and methodological issues related to the prediction of long... more This paper addresses both theoretical and methodological issues related to the prediction of long-memory models with incomplete data. Estimates and forecasts are calculated by means of state space models and the influence of data gaps on the performance of short and long run predictions is investigated. These techniques are illustrated with a statistical analysis of the minimum water levels of the Nile river, a time series exhibiting strong dependency.

Research paper thumbnail of Estimation of seasonal fractionally integrated processes

Computational Statistics & Data Analysis, 2006

This paper discusses the estimation of fractionally integrated processes with seasonal components... more This paper discusses the estimation of fractionally integrated processes with seasonal components. In order to estimate the fractional parameters, we propose several estimators obtained from the regression of the log-periodogram on different bandwidths selected around and/or between the seasonal frequencies. For comparison purposes, the semi-parametric method introduced in Geweke and Porter-Hudak (1983) and Porter-Hudak (1990) and the maximum-likelihood estimates (ML) are also considered. As indicated by the Monte Carlo simulations, the performance of the estimators proposed is good even for small sample sizes.

Research paper thumbnail of Red de Análisis Estocástico y Aplicaciones (Sistemas abiertos, energía y dinámica de la información)

Research paper thumbnail of Simultaneous variable selection and structural identification for time‐varying coefficient models

Journal of Time Series Analysis, 2021

Research paper thumbnail of An irregular discrete time series model to identify residuals with autocorrelation in astronomical light curves

Monthly Notices of the Royal Astronomical Society, 2018

Time series observations are ubiquitous in astronomy and are generated, for example, to distingui... more Time series observations are ubiquitous in astronomy and are generated, for example, to distinguish between different types of supernovae to detect and characterize extrasolar planets and to classify variable stars. These time series are usually modelled using a parametric and/or physical model that assumes independent and homoscedastic errors, but in many cases, these assumptions are not accurate and there remains a temporal dependence structure on the errors. This can occur, for example, when the proposed model cannot explain all the variability of the data or when the parameters of the model are not properly estimated. In this work, we define an autoregressive model for irregular discrete-time series based on the discrete time representation of the continuous autoregressive model of order 1. We show that the model is ergodic and stationary. We further propose a maximum likelihood estimation procedure and assess the finite sample performance by Monte Carlo simulations. We implement the model on real and simulated data from Gaussian as well as other distributions, showing that the model can flexibly adapt to different data distributions. We apply the irregular autoregressive model to the residuals of a transit of an extrasolar planet to illustrate errors that remain with temporal structure. We also apply this model to residuals of an harmonic fit of light curves from variable stars to illustrate how the model can be used to detect incorrect parameter estimation.

Research paper thumbnail of Estimation of Long-Memory Time Series Models: a Survey of Different Likelihood-Based Methods

Advances in Econometrics

Since the seminal works by Granger and Joyeux (1980) and Hosking (1981), estimations of long-memo... more Since the seminal works by Granger and Joyeux (1980) and Hosking (1981), estimations of long-memory time series models have been receiving considerable attention and a number of parameter estimation procedures have been proposed. This paper gives an overview of this plethora of methodologies with special focus on likelihood-based techniques. Broadly speaking, likelihood-based techniques can be classified into the following categories: the exact maximum likelihood (ML) estimation (Sowell, 1992; Dahlhaus, 1989), ML estimates based on autoregressive approximations (Granger & Joyeux, 1980; Li & McLeod, 1986), Whittle estimates (Fox & Taqqu, 1986; Giraitis & Surgailis, 1990), Whittle estimates with autoregressive truncation (Beran, 1994a), approximate estimates based on the Durbin-Levinson algorithm (Haslett & Raftery, 1989), state-space-based maximum likelihood estimates for ARFIMA models (

Research paper thumbnail of On the eigenstructure of generalized fractional processes

Statistics & Probability Letters, 2003

This work establishes bounds for the eigenvalues of the covariance matrix from a general class of... more This work establishes bounds for the eigenvalues of the covariance matrix from a general class of stationary processes. These results are applied to the statistical analysis of the large sample behavior of estimates and testing procedures of generalized long memory models, including Seasonal ARFIMA and k-factor GARMA processes, among others.

Research paper thumbnail of Analysis of the correlation structure of square time series

Journal of Time Series Analysis, 2004

This paper analyses the asymptotic behaviour of the autocorrelation structure exhibited by square... more This paper analyses the asymptotic behaviour of the autocorrelation structure exhibited by squares of time series with a Wold expansion where the input error is a sequence of random variables with mean zero and finite kurtosis. Two important cases are discussed: (i) when the errors are independent and, (ii) when the errors are uncorrelated but their squares are correlated. Both situations are addressed when the process exhibits short or long memory. Consequences of these results on certain models widely used in many disciplines are also discussed.

Research paper thumbnail of Efficient Estimation of Seasonal Long-Range-Dependent Processes

Journal of Time Series Analysis, 2005

This paper studies asymptotic properties of the exact maximum likelihood estimates (MLE) for a ge... more This paper studies asymptotic properties of the exact maximum likelihood estimates (MLE) for a general class of Gaussian seasonal long-range-dependent processes. This class includes the commonly used Gegenbauer and seasonal autoregressive fractionally integrated moving average processes. By means of an approximation of the spectral density, the exact MLE of this class are shown to be consistent, asymptotically normal and efficient. Finite sample performance of these estimates is examined by Monte Carlo simulations and it is shown that the estimates behave very well even for moderate sample sizes. The estimation methodology is illustrated by a real-life Internet traffic example.

Research paper thumbnail of A Class of Antipersistent Processes

Journal of Time Series Analysis, 2007

We introduce a class of stationary processes characterized by the behavior of their infinite movi... more We introduce a class of stationary processes characterized by the behavior of their infinite moving average parameters. We establish the asymptotic behavior of the covariance function and the behavior around zero of the spectral density of these processes, showing their antipersistent character. Then, we discuss the existence of an infinite autoregressive representation for this family of processes, and we present some consequences for fractional autoregressive moving average models.

Research paper thumbnail of On the sample mean of locally stationary long-memory processes

Journal of Statistical Planning and Inference, 2010

Some asymptotic statistical properties of the sample mean of a class locally stationary long-memo... more Some asymptotic statistical properties of the sample mean of a class locally stationary long-memory process are studied in this paper. Conditions for consistency are investigated and precise convergence rates of the variance of the sample mean are established for a class of time-varying long-memory parameter functions. A central limit theorem for the sample mean is also established. Furthermore, the calculation of the variance of the sample mean is illustrated through several numerical and simulation experiments.

Research paper thumbnail of Estimating seasonal long-memory processes: a Monte Carlo study

Journal of Statistical Computation and Simulation, 2006

This paper discusses extensions of the popular methods proposed by Geweke and Porter-Hudak [Gewek... more This paper discusses extensions of the popular methods proposed by Geweke and Porter-Hudak [Geweke, J. and Porter-Hudak, S., 1983, The estimation and application of long memory times series models.

Research paper thumbnail of Estimation of Tropical Sea Level Anomaly by an Improved Kalman Filter

Journal of Physical Oceanography, 1996

Research paper thumbnail of A ground‐level ozone forecasting model for Santiago, Chile

Journal of Forecasting, 2002

A physically based model for ground‐level ozone forecasting is evaluated for Santiago, Chile. The... more A physically based model for ground‐level ozone forecasting is evaluated for Santiago, Chile. The model predicts the daily peak ozone concentration, with the daily rise of air temperature as input variable; weekends and rainy days appear as interventions. This model was used to analyse historical data, using the Linear Transfer Function/Finite Impulse Response (LTF/FIR) formalism; the Simultaneous Transfer Function (STF) method was used to analyse several monitoring stations together. Model evaluation showed a good forecasting performance across stations—for low and high ozone impacts—with power of detection (POD) values between 70 and 100%, Heidke's Skill Scores between 40% and 70% and low false alarm rates (FAR). The model consistently outperforms a pure persistence forecast. Model performance was not sensitive to different implementation options. The model performance degrades for two‐ and three‐days ahead forecast, but is still acceptable for the purpose of developing an env...

Research paper thumbnail of Estimation and forecasting of long-memory processes with missing values

Journal of Forecasting, 1997

This paper addresses the issues of maximum likelihood estimation and forecasting of a long-memory... more This paper addresses the issues of maximum likelihood estimation and forecasting of a long-memory time series with missing values. A state-space representation of the underlying long-memory process is proposed. By incorporating this representation with the Kalman ®lter, the proposed method allows not only for an ecient estimation of an ARFIMA model but also for the estimation of future values under the presence of missing data. This procedure is illustrated through an analysis of a foreign exchange data set. An investment scheme is developed which demonstrates the usefulness of the proposed approach.

Research paper thumbnail of Assessing influence in Gaussian long-memory models

Computational Statistics & Data Analysis, 2008

A statistical methodology for detecting influential observations in long-memory models is propose... more A statistical methodology for detecting influential observations in long-memory models is proposed. The identification of these influential points is carried out by case-deletion techniques. In particular, a Kullback-Leibler divergence is considered to measure the effect of a subset of observations on predictors and smoothers. These techniques are illustrated with an analysis of the River Nile data where the proposed methods are compared to other well-known approaches such as the Cook and the Mahalanobis distances.

Research paper thumbnail of Data analysis using regression models with missing observations and long-memory: an application study

Computational Statistics & Data Analysis, 2006

The objective of this work is to propose a statistical methodology to handle regression data exhi... more The objective of this work is to propose a statistical methodology to handle regression data exhibiting long memory errors and missing values. This type of data appears very often in many areas, including hydrology and environmental sciences, among others. A generalized linear model is proposed to deal with this problem and an estimation strategy is developed that combines both classical and Bayesian approaches. The estimation methodology proposed is illustrated with an application to air pollution data which shows the impact of the long memory in the statistical inference and of the missing values on the computations. From a Bayesian standpoint, genuine priors are considered for the parameters of the model which are justified within the context of the air pollution model derivation.

Research paper thumbnail of Minimum distance estimation of ARFIMA processes

Computational Statistics & Data Analysis, 2013

This paper proposes a new minimum distance methodology for the estimation of ARFIMA processes wit... more This paper proposes a new minimum distance methodology for the estimation of ARFIMA processes with Gaussian and non-Gaussian errors. The main advantage of this method is that it allows for a computationally efficient estimation when the long-memory parameter is in the interval d ∈ (− 1 2 , 1 2). Previous minimum distance estimation techniques are usually limited to the range d ∈ (− 1 2 , 1 4), leaving outside the very important case of strong long memory with d ∈ [ 1 4 , 1 2). It is shown that the new estimator satisfies a central limit theorem and Monte Carlo experiments indicate that the proposed estimator performs very well even for small sample sizes. The methodology is illustrated with three applications. The first two examples involve real-life time series while the third application illustrates that the proposed methodology is a sound alternative for dealing with incomplete time series.