Search

There is considerable interest in whether weekly cycles are present in certain types of atmospheric and hydrologic data. Seven days is not a natural period for cyclic behaviour, but rather a human construct. Hence the presence of weekly cycles can help to confirm the presence and nature of anthropogenic climate change. Several statistical tests have been used to investigate the presence of such cycles. In this short note, a test previously used for ‘flatness’ of rank histograms in ensemble forecasting is shown to be useful for this task for certain types of data, with some advantages compared with existing techniques. Some disadvantages and extensions are also described.

When probability forecasts are made of a binary event, a commonly used measure for assessing the forecasts is the Brier score. One of its properties is that it is proper, meaning that its expected value cannot be improved by the forecaster issuing a probability other than his/her true belief. This property assumes that the occurrence or otherwise of the forecast event is recorded without error. This note investigates what forecast should be made in order to minimize the expected value of the Brier score when errors are present in the observations. Should it still be the forecaster's true belief or should it be something else, implying that the forecaster should hedge his/her forecast? The answer is that it depends on whether the forecaster can model the error mechanism or whether the error mechanism is unknown. It is shown that in the former case the forecaster's true belief of the probability of the event should still be forecast. However, in the case of an unknown error mechanism, the forecaster should attempt to forecast the probability that the erroneous observation indicates that the event has occurred, rather than the true probability of the event.

In weather and climate forecasting, hedging is said to occur whenever a forecaster's judgement and the forecast differ, and it is usually taken as evident that hedging is undesirable. Forecasts are often judged by computing a verification measure or score. A number of different scores is available in most circumstances, and to choose between them, various desirable properties of scores have been defined. It is generally accepted that it is undesirable to use a score for which hedging can improve the score or its expected value. Three ‘desirable’ properties of scores are linked to the idea that hedging should be avoided, namely propriety, equitability and consistency. It is fair to say that none of these properties is fully understood. The aim of this article is to provide some clarification and new insights, as well as some historical background. Nearly as many questions are raised as are answered. Copyright © 2008 Royal Meteorological Society

Empirical orthogonal functions (EOFs) are widely used in climate research to identify dominant patterns of variability and to reduce the dimensionality of climate data. EOFs, however, can be difficult to interpret. Rotated empirical orthogonal functions (REOFs) have been proposed as more physical entities with simpler patterns than EOFs. This study presents a new approach for finding climate patterns with simple structures that overcomes the problems encountered with rotation. The method achieves simplicity of the patterns by using the main properties of EOFs and REOFs simultaneously. Orthogonal patterns that maximise variance subject to a constraint that induces a form of simplicity are found. The simplified empirical orthogonal function (SEOF) patterns, being more 'local'. are constrained to have zero loadings outside the main centre of action. The method is applied to winter Northern Hemisphere (NH) monthly mean sea level pressure (SLP) reanalyses over the period 1948-2000. The 'simplified' leading patterns of variability are identified and compared to the leading patterns obtained from EOFs and REOFs. Copyright (C) 2005 Royal Meteorological Society.

Principal component analysis (PCA) is a dimension-reducing tool that replaces the variables in a multivariate data set by a smaller number of derived variables. Dimension reduction is often undertaken to help in interpreting the data set but, as each principal component usually involves all the original variables, interpretation of a PCA can still be difficult. One way to overcome this difficulty is to select a subset of the original variables and use this subset to approximate the principal components. This article reviews a number of techniques for choosing subsets of the variables and examines their merits in terms of preserving the information in the PCA, and in aiding interpretation of the main sources of variation in the data. Copyright © 2005 John Wiley & Sons, Ltd.

This article discusses dynamic factor analysis, a technique for estimating common trends in multivariate time series. Unlike more common time series techniques such as spectral analysis and ARIMA models, dynamic factor analysis can analyse short, non-stationary time series containing missing values. Typically, the parameters in dynamic factor analysis are estimated by direct optimization, which means that only small data sets can be analysed if computing time is not to become prohibitively long and the chances of obtaining sub-optimal estimates are to be avoided. This article shows how the parameters of dynamic factor analysis can be estimated using the EM algorithm, allowing larger data sets to be analysed. The technique is illustrated on a marine environmental data set. Copyright © 2003 John Wiley & Sons, Ltd.

When large multivariate datasets are analyzed, it is often desirable to reduce their dimensionality. Principal component analysis is one technique for doing this. It replaces the p original variables by a smaller number, q, of derived variables, the principal components, which are linear combinations of the original variables. Often, it is possible to retain most of the variability in the original variables with q very much smaller than p. Despite its apparent simplicity, principal component analysis has a number of subtleties, and it has many uses and extensions. A number of choices associated with the technique are briefly discussed, namely, covariance or correlation, how many components, and different normalization constraints, as well as confusion with factor analysis. Various uses and extensions are outlined. Keywords: dimension reduction; factor analysis; multivariate analysis; variance maximization

The words eigenvalue and eigenvector often appear in computer output for multivariate statistical techniques. An explanation is given of what is meant by these terms for some specific techniques and more generally. Keywords: matrices; multivariate analysis; optimization problems

Summary. During a clinical trial of a new treatment, a large number of variables are measured to monitor the safety of the treatment. It is important to detect outlying observations which may indicate that something abnormal is happening. To do this effectively, techniques are needed for finding multivariate outliers. Six techniques of this sort are described and illustrated on a typical laboratory safety data set. Their properties are investigated more thoroughly by means of a simulation study. The results show that some methods do better than others depending on whether or not the data set is multivariate normal, the dimension of the data set, the type of outlier, the proportion of outliers in a data set and the degree of contamination, i.e. 'outlyingness'. The results indicate that it is desirable to run a battery of multivariate methods on a particular data set in an attempt to highlight possible outliers.

An earlier paper (Jolliffe IT, Sarria-Dodd DE. 1994. International Journal of Climatology 14: 71–76) investigated the problem of deciding when the wet season has started in tropical and sub-tropical climates. In particular, methodology based on linear discriminant analysis was developed for using data from the current season to make the decision, rather than relying only on information from previous seasons. It was shown, for three stations in eastern Africa, that the methodology was potentially valuable. The present study is much larger, using data from 24 stations, covering a range of annual rainfall totals, in western Africa. It is confirmed that linear discriminant analysis can indeed be useful in detecting when the wet season has started, and hence in deciding when to plant crops. As well as being a larger analysis than that reported previously, the present study also extends the previous work by investigating an alternative definition of the start of the wet season and by including ‘date’ as a potential explanatory variable. Copyright © 2001 Royal Meteorological Society.

In clinical laboratory safety data, multivariate outlier detection methods may highlight a patient whose laboratory measurements do not follow the same pattern of relationships as the majority of patients, although their individual measurements are not found to be outlying when considered one at a time. Missing data problems are often dealt with by imputing a single value as an estimate of the missing value. The completed data set may then be analysed using traditional methods. A disadvantage of using single imputation is the underestimation of variability, with a corresponding distortion of power in hypothesis testing. Multiple imputation methods attempt to overcome this problem, and in this paper a study is described which considers the application of multivariate outlier detection methods to multiply imputed clinical laboratory safety data sets. Three different proportions of missing data are generated in laboratory data sets of dimensions 4, 7, 12 and 30, and a comparison of eight multiple imputation methods is carried out. Two outlier detection techniques, Mahalanobis distance and generalized principal component analysis, are applied to the multiply imputed data sets, and their performances are discussed. Measures are introduced for assessing the accuracy of the missing data results, depending on which method of analysis is used.

In this paper we examine time series modelling of surface pressure data, as measured by a barograph, at Herne Bay, England, during the years 1981‐1989. Autoregressive moving average (ARMA) models have been popular in many fields over the past 20 years, although applications in climatology have been rather less widespread than in some disciplines. Some recent examples are Milionis and Davies (Int. J. Climatol., 14, 569‐579) and Seleshi et al .( Int. J. Climatol., 14, 911‐923). We fit standard ARMA models to the pressure data separately for each of six 2-month natural seasons. Differences between the best fitting models for different seasons are discussed. Barograph data are recorded continuously, whereas ARMA models are fitted to discretely recorded data. The effect of different spacings between the fitted data on the models chosen is discussed briefly. Often, ARMA models can give a parsimonious and interpretable representation of a time series, but for many series the assumptions underlying such models are not fully satisfied, and more complex models may be considered. A specific feature of surface pressure data in the UK is that its behaviour is different at high and at low pressures: day-to-day changes are typically larger at low pressure levels than at higher levels. This means that standard assumptions used in fitting ARMA models are not valid, and two ways of overcoming this problem are investigated. Transformation of the data to better satisfy the usual assumptions is considered, as is the use of non-linear, specifically threshold autoregressive (TAR), models. © 1998 Royal Meteorological Society.

For daily rainfall the normalized rainfall curve (NRC) provides a plot of the cumulative percentage of rain days (y) against the cumulative percentage of rain amount (x). It is not immediately clear whether the equations that have been widely used to represent the relationship between xand ycorrespond to a valid probability distribution for daily rainfall amounts. We show that such distributions exist, but that they are truncated, i.e. not all positive values of rainfall have non-zero probability. In practice, the truncation may not be too important, but the form of the equations relating xand yis also determined for a number of well-known probability distributions that are not truncated. The fit of the corresponding NRCs to some published rainfall data is examined.

In many tropical and sub-tropical climates the year is divided into well-defined wet and dry seasons. To maximize agricultural production it is important that crops are planted as early as possible in a wet season, but deciding exactly when the wet season has started may be a problem. A decision can be made on the basis of previous years' data, or we may try to incorporate information from the current season. The latter approach is discussed in the present paper, and shown to be potentially of greater value than relying solely on historical data.

In a recent paper, Hamilton examined the distribution of very warm spells and very cold spells at Edgbaston, Birmingham, during the period 1920–1979. The present paper takes the analysis of these data further by modelling them with a simple statistical distribution, namely the geometric distribution. It is shown that this distribution provides a good fit to the lengths of both warm and cold spells, although there is clear seasonal variation in the mean of the distribution.

Factor analysis is a latent variable technique. It reduces the dimension of a data set by assuming that underlying the measured variables is a smaller number of unobservable factors. The measured variables are assumed to be linearly related to the factors, apart from an error term. There are many ways of estimating the loadings that relate the variables to the factors, and predicting the scores of the factors themselves. Solutions are typically nonunique, and are made as “simple” as possible by means of rotation of loadings. Much of factor analysis has been developed within psychology, but it is useful more widely. The technique is often used in an exploratory or descriptive manner, but confirmatory versions are also available. Keywords: factor loadings; factor scores; latent variables; principal components analysis; psychology; rotation; simple structure

Summary The χ2 goodness-of-fit test is often one of the first tests of hypotheses encountered by students. When some of the expected frequencies are small, classes need to be combined. A real-life example is given that illustrates a surprising sensitivity of the results of the test to the way in which such combinations are chosen.

Summary This short note shows that caution can be very necessary in use of the sample median.

A recent paper by Richman (1986) provides a very useful review of the ideas of rotation in principal component analysis, in the context of climatology. However, the paper also raises a number of points which are open to discussion. The present note comments on a number of these points, and aims to give a balanced view of the usefulness of principal component analysis, both with and without rotation, in climatology.

Quasi-periodic behaviour in meteorological or climatological series is a widespread phenomenon; various attempts have been made to explain such behaviour in physical terms, often invoking external influences in the explanations. While it is not disputed that external influences or feedback mechanisms may have a part to play, it is argued in the present note that a substantial part of quasi-cyclic behaviour can often be explained by a simpler mechanism, namely second-order dependence of a series on its own past. The plausibility of second-order dependence is discussed, and the way in which such dependence leads to quasi-periodic behaviour is described. Data on sunspots. Baltic ice area and a zonal circulation index are used to illustrate these ideas.