Optimal designs for estimating linear and quadratic contrasts with three level factors, the case N ≡ 0 mod 3

The purpose of this paper is to find and construct optimal designs for estimating the standardized linear and quadratic contrasts in fractional factorials with k factors, each at 3 levels, when the number of runs or assemblies is N. The case N=3m is examined, the notion of Balanced Arrays BA(N, k, 3, 2) or BA(N, k) for short, is introduced and the optimal BA(N, k) is specified. It is shown that for N=9m the orthogonal array OA(N, k, 3, 2) or OA(N, k) for short, is the φ-optimal design. If N=9m+3 and N=9m+6 the optimal designs are BA(N, k) which are specified for every value of N and k. In the case N=9m+3 and k=3 the optimal BA(N, k) are constructed by augmenting OA(N, k, 3, 2) by three rows which are specified. If the OA(N, k, 3, 2) does not exist, algorithms are developed to construct the optimal BA(N, k). For N=9m+6 and k=3 the optimal BA(N, k) are constructed by augmenting OA(N, k) by six rows, which are specified, otherwise algorithms are developed. Under optimal BA(N, k), the estimators of linear and quadratic contrasts are uncorrelated. The cases N=12,15,21,24,30,33 are examined in detail and optimal BA(N, k) are presented for different values of the number k of factors. © 2017 Informa UK Limited, trading as Taylor & Francis Group

Optimal designs for estimating linear and quadratic contrasts with three level factors, the case N ≡ 0 mod 3

The purpose of this paper is to find and construct optimal designs for estimating the standardized linear and quadratic contrasts in fractional factorials with k factors, each at 3 levels, when the number of runs or assemblies is N. The case N=3m is examined, the notion of Balanced Arrays BA(N, k, 3, 2) or BA(N, k) for short, is introduced and the optimal BA(N, k) is specified. It is shown that for N=9m the orthogonal array OA(N, k, 3, 2) or OA(N, k) for short, is the φ-optimal design. If N=9m+3 and N=9m+6 the optimal designs are BA(N, k) which are specified for every value of N and k. In the case N=9m+3 and k=3 the optimal BA(N, k) are constructed by augmenting OA(N, k, 3, 2) by three rows which are specified. If the OA(N, k, 3, 2) does not exist, algorithms are developed to construct the optimal BA(N, k). For N=9m+6 and k=3 the optimal BA(N, k) are constructed by augmenting OA(N, k) by six rows, which are specified, otherwise algorithms are developed. Under optimal BA(N, k), the estimators of linear and quadratic contrasts are uncorrelated. The cases N=12,15,21,24,30,33 are examined in detail and optimal BA(N, k) are presented for different values of the number k of factors

Optimality results on orthogonal arrays plus p runs for s m factorial experiments

Fractional factorial, Optimal designs, Orthogonal array, Type 1 criteria,

Monthly mean pressure reconstructions for Europe for the 1780-1995 period

Monthly grid-point pressure data are reconstructed from station records of pressure for Europe since 1780. The region encompasses 35-70°N to 30°W-40°E. The reconstructions are based on a principal components regression technique, which relates surface pressure patterns to those of the station pressure data. The relationships are derived over a calibration period (1936-1995) and the results tested with independent data (the verification period, 1881-1935). The reconstructions are of excellent quality, although this is slightly lower for regions with poor station coverage in the early years, particularly during the summer months. The reconstructions are compared with other monthly mean pressure maps produced by Lamb and Johnson (1966) for the years 1780-1872 and by Kington (1980, 1988) for 1781-1785. Both of these map series show systematic biases relative to the present reconstructions

Monthly mean pressure reconstruction for the Late Maunder Minimum Period (AD 1675-1715)

The Late Maunder Minimum (LMM; 1675-1715) delineates a period with marked climate variability within the Little Ice Age in Europe. Gridded monthly mean surface pressure fields were reconstructed for this period for the eastern North Atlantic-European region (25°W-30°E and 35-70°N). These were based on continuous information drawn from proxy and instrumental data taken from several European data sites. The data include indexed temperature and rainfall values, sea ice conditions from northern Iceland and the Western Baltic. In addition, limited instrumental data, such as air temperature from central England (CET) and Paris, reduced mean sea level pressure (SLP) at Paris, and monthly mean wind direction in the Oresund (Denmark) are used. The reconstructions are based on a canonical correlation analysis (CCA), with the standardized station data as predictors and the SLP pressure fields as predictand. The CCA-based model was performed using data from the twentieth century. The period 1901-1960 was used to calibrate the statistical model, and the remaining 30 years (1961-1990) for the validation of the reconstructed monthly pressure fields. Assuming stationarity of the statistical relationships, the calibrated CCA model was then used to predict the monthly LMM SLP fields. The verification results illustrated that the regression equations developed for the majority of grid points contain good predictive skill. Nevertheless, there are seasonal and geographical limitations for which valid spatial SLP patterns can be reconstructed. Backward elimination techniques indicated that Paris station air pressure and temperature, CET, and the wind direction in the Oresund are the most important predictors, together sharing more than 65% of the total variance. The reconstructions are compared with additional data and subjectively reconstructed monthly pressure charts for the years 1675-1704. It is shown that there are differences between the two approaches. However, for extreme years the reconstructions are in good agreement