Application of autoregressive spectral analysis to missing data problems

Time series solutions for spectral analysis in missing data problems use reconstruction of the missing data, or a maximum likelihood approach that analyzes only the available measured data. Maximum likelihood estimation yields the most accurate spectra. An approximate maximum likelihood algorithm is presented that uses only previous observations falling in a finite interval to compute the likelihood, instead of all previous observations. The resulting nonlinear estimation algorithm requires no user-provided initial solution, is suited for order selection, and can give very accurate spectra even if less than 10% of the data remains.     

Finite sample properties of ARMA order selection

The cost of order selection is defined as the loss in model quality due to selection. It is the difference between the quality of the best of all available candidate models that have been estimated from a finite sample of N observations and the quality of the model that is actually selected. The order selection criterion itself has an influence on the cost because of the penalty factor for each additionally selected parameter. Also, the number of competitive candidate models for the selection is important. The number of candidates is, of itself, small for the nested and hierarchical autoregressive/moving average (ARMA) models. However, intentionally reducing the number of selection candidates can be beneficial in combined ARMA(p,q) models, where two separate model orders are involved: the AR order p and the MA order q. The selection cost can be diminished by creating a nested sequence of ARMA(r,r-1) models. Moreover, not evaluating every combination (p,q) of the orders considerably reduces the required computation time. The disadvantage may be that the true ARMA(p,q) model is no longer among the nested candidate models. However, in finite samples, this disadvantage is largely compensated for by the reduction in the cost of order selection by considering fewer candidates. Thus, the quality of the selected model remains acceptable with only hierarchically nested ARMA(r,r-1) models as candidates.     

Differential automatic zero-adjusting amplifier

A method is described for building a low-voltage-drift differential dc amplifier featuring automatic zero adjustment, a high input impedance, and a bandwidth of 10 kHz. This is achieved by an asymmetric two-step process between the input signal and ground. Bandwidth can be extended by the use of a second amplifier during the ground-sampling time. The amplifier can be made with standard electronic components. A major advantage of this method is that an existing amplifier can easily be converted into a low-voltage-drift amplifier by adding the essential elements of the described automatic zero-adjusting amplifier to its input stage. To illustrate the method a practical example is constructed featuring a drift of 0.2 microV/ degrees C.     

Automatic identification of time-series models from long autoregressive models

Identification is the selection of the model type and of the model order by using measured data of a process with unknown characteristics. If the observations themselves are used, it is possible to identify automatically a good time-series model for stochastic data. The selected model is an adequate representation of the statistically significant spectral details in the observed process. Sometimes, identification has to be based on many less than N characteristics of the data. The reduced statistics information is assumed to consist of a long autoregressive (AR) model. That AR model has to be used for the estimation of moving average (MA) and of combined ARMA models and for the selection of the best model orders. The accuracy of ARMA models is improved by using four different types of initial estimates in a first stage. After a second stage, it is possible to select automatically which initial estimates were most favorable in the present case by using the fit of the estimated ARMA models to the given long AR model. The same principle is used to select the best type of the time-series models and the best model order. No spectral information is lost in using only the long AR representation instead of all data. The quality of the model identified from a long AR model is comparable to that of the best time-series model that can be computed if all observations are available.     

Time-series analysis if data are randomly missing

Maximum-likelihood (ML) theory presents an elegant asymptotic solution for the estimation of the parameters of time-series models. Unfortunately, the performance of ML algorithms in finite samples is often disappointing, especially in missing-data problems. The likelihood function is symmetric with respect to the unit circle for the estimated zeros of time-series models. As a consequence, the unit circle is either a local maximum or a local minimum in the likelihood of moving-average (MA) models. This is a trap for nonlinear optimization algorithms that often converge to poor models, with estimated zeros precisely on the unit circle. With ML estimation, it is much easier to estimate a long autoregressive (AR) model with only poles. The parameters of that long AR model can then be used to estimate MA and autoregressive moving-average (ARMA) models for different model orders. The accuracy of the estimated AR, MA, and ARMA spectra is very good. The robustness is excellent as long as the AR order is less than 10 or 15. For still-higher AR orders until about 60, the possible convergence to a useful model will depend on the missing fraction and on the specific properties of the data at hand.     

A comparison of dilution adjustment methods for urinary enzymes.

Urinary dilution adjustment methods can be used to reduce the intra-individual variability in concentrations of metals and other substances in urine due to variability in urinary flow. In this study linear and non-linear dilution adjustments with urinary flow, creatinine (CREAT) and urinary density (UD) were compared for the urinary enzymes alanine amino peptidase (AAP), beta-galactosidase (beta GAL) and N-acetyl-beta, D-glucosaminidase (NAG). The most optimal dilution adjustment for AAP was: AAPadjusted = AAPmeasured/(CREATmeasured)0.824 The optimal dilution adjustment for beta GAL was: beta GALadjusted = beta GALmeasured/(CREATmeasured)0.878 For NAG the optimal dilution adjustment parameter was the conventional linear adjustment with SG. It could not be determined whether urinary dilution methods can be useful for population based reference intervals of urinary enzymes. If personal reference intervals can be calculated, urinary dilution adjustment methods may be useful by reduction of intraindividual variability.     

Error Correction of Rainfall-Runoff Models With the ARMAsel Program

Improved predictions can be based on recent observed differences or errors between the best available model predictions and the actual measured data. This is possible in the predicted amount of supplies, services, sewage, transportation, power, water, heat, or gas, as well as in the predicted level of rivers. As an example, physical modeling of the dynamics of a catchment area produces models with a limited forecasting accuracy for the discharge of rivers. The discrepancies between the model and the actually observed past discharges can be used as information for error correction. With a time-series model of the error signal, an improved discharge forecast can be made for the next few days. The best type and order of the forecasting time-series model can be automatically selected. Adaptive modeling in data assimilation calculates updates of the time-series model estimated from the error data of only the last few weeks. The use of variable updated models has advantages in periods with the largest discharges, which are most important in flood forecasting.     

ARMAsel for Detection and Correction of Outliers in Univariate Stochastic Data

For stationary random data, an automatic estimation algorithm can now select a time series model with a spectral accuracy close to the Cramer-Rao lower bound. The parameters of that selected time series model accurately represent the spectral density and the autocovariance function of the data. That is all the possible information for Gaussian data, as well as the most important information for arbitrarily distributed data. A single model type and order is selected from many candidate time series models by looking for the smallest prediction error. The single selected model precisely includes only the statistically significant details that are present in the data. The residuals of the automatically selected time series model reveal the location of outliers or other irregularities that may not be visible in the measured signal. The program requires no user interaction and can be incorporated into automatic measurement instruments and protocols.