A View of Damped Trend as Incorporating a Tracking Signal into a State Space Model

Damped trend exponential smoothing has previously been established as an important forecasting method. Here, it is shown to have close links to simple exponential smoothing with a smoothed error tracking signal. A special case of damped trend exponential smoothing emerges from our analysis, one that is more parsimonious because it effectively relies on one less parameter. This special case is compared with its traditional counterpart in an application to the annual data from the M3 competition and is shown to be quite competitive.Exponential smoothing, monitoring forecasts, structural change, adjusting forecasts, state space models, damped trend

Incorporating a Tracking Signal into State Space Models for Exponential Smoothing

It is a common practice to complement a forecasting method such as simple exponential smoothing with a monitoring scheme to detect those situations where forecasts have failed to adapt to structural change. It will be suggested in this paper that the equations for simple exponential smoothing can be augmented by a common monitoring statistic to provide a method that automatically adapts to structural change without human intervention. It is shown that the resulting equations conform to those of damped trend corrected exponential smoothing. In a similar manner, exponential smoothing with drift, when augmented by the same monitoring statistic, produces equations that split the trend into long term and short term components.Forecasting, exponential smoothing, tracking signals.

Another Look at Measures of Forecast Accuracy

We discuss and compare measures of accuracy of univariate time series forecasts. The methods used in the M-competition and the M3-competition, and many of the measures recommended by previous authors on this topic, are found to be inadequate, and many of them are degenerate in commonly occurring situations. Instead, we propose that the mean absolute scaled error become the standard measure for comparing forecast accuracy across multiple time series.Forecast accuracy, Forecast evaluation, Forecast error measures, M-competition, Mean absolute scaled error.

Empirical Information Criteria for Time Series Forecasting Model Selection

In this paper, we propose a new Empirical Information Criterion (EIC) for model selection which penalizes the likelihood of the data by a function of the number of parameters in the model. It is designed to be used where there are a large number of time series to be forecast. However, a bootstrap version of the EIC can be used where there is a single time series to be forecast. The EIC provides a data-driven model selection tool that can be tuned to the particular forecasting task. We compare the EIC with other model selection criteria including Akaike's Information Criterion (AIC) and Schwarz's Bayesian Information Criterion (BIC). The comparisons show that for the M3 forecasting competition data, the EIC outperforms both the AIC and BIC, particularly for longer forecast horizons. We also compare the criteria on simulated data and find that the EIC does better than existing criteria in that case also.Exponential smoothing; forecasting; information criteria; M3 competition; model selection.

Welding, brazing, and soldering handbook

Handbook gives information on the selection and application of welding, brazing, and soldering techniques for joining various metals. Summary descriptions of processes, criteria for process selection, and advantages of different methods are given

Investigation of the chemical kinetics of an advanced high energy propellant system Quarterly progress report, 1 Jun. - 1 Sep. 1968

Reaction kinetics of high energy oxygen difluoride diborane propellant syste

Forecasting Compositional Time Series with Exponential Smoothing Methods

Compositional time series are formed from measurements of proportions that sum to one in each period of time. We might be interested in forecasting the proportion of home loans that have adjustable rates, the proportion of nonagricultural jobs in manufacturing, the proportion of a rock's geochemical composition that is a specific oxide, or the proportion of an election betting market choosing a particular candidate. A problem may involve many related time series of proportions. There could be several categories of nonagricultural jobs or several oxides in the geochemical composition of a rock that are of interest. In this paper we provide a statistical framework for forecasting these special kinds of time series. We build on the innovations state space framework underpinning the widely used methods of exponential smoothing. We couple this with a generalized logistic transformation to convert the measurements from the unit interval to the entire real line. The approach is illustrated with two applications: the proportion of new home loans in the U.S. that have adjustable rates; and four probabilities for specified candidates winning the 2008 democratic presidential nomination.compositional time series, innovations state space models, exponential smoothing, forecasting proportions

Differential Relationship between Physical Activity and Intake of Added Sugar and Nutrient-Dense Foods: A Cross-Sectional Analysis

A curvilinear relationship exists between physical activity (PA) and dietary energy intake (EI), which is reduced in moderately active when compared to inactive and highly active individuals, but the impact of PA on eating patterns remains poorly understood. Our goal was to establish the relationship between PA and intake of foods with varying energy and nutrient density. Data from the 2009–2010 United States National Health and Nutrition Examination Survey were used to include a Dietary Screener Questionnaire for estimated intakes of added sugar, fruits and vegetables, whole grains, fiber, and dairy. Participants (n = 4766; 49.7% women) were divided into sex-specific quintiles based on their habitual PA. After adjustment for age, body mass index, household income, and education, intakes were compared between PA quartiles, using the lowest activity quintile (Q1) as reference. Women in the second to fourth quintile (Q2-Q4) consumed less added sugar from sugary foods (+2 tsp/day) and from sweetened beverages (+2 tsp/day; all p \u3c 0.05 vs. Q1). In men, added sugar intake was elevated in the highest activity quintile (Q5: +3 ± 1 tsp/day, p = 0.007 vs. Q1). Fruit and vegetable intake increased (women: Q1-Q4 +0.3 ± 0.1 cup eq/day; p \u3c 0.001; men: Q1-Q3 +0.3 ± 0.1 cup eq/day, p = 0.002) and stagnated in higher quintiles. Dairy intake increased with PA only in men (Q5: +0.3 ± 0.1 cup eq/day, p \u3c 0.001 vs. Q1). Results demonstrate a differential relationship between habitual PA and dietary intakes, whereby moderate but not necessarily highest PA levels are associated with reduced added sugar and increased nutrient-dense food consumption. Future research should examine specific mechanisms of food choices at various PA levels to ensure dietary behaviors (i.e., increased sugary food intake) do not negate positive effects of PA

Exponential Smoothing Model Selection for Forecasting

Applications of exponential smoothing to forecast time series usually rely on three basic methods: simple exponential smoothing, trend corrected exponential smoothing and a seasonal variation thereof. A common approach to select the method appropriate to a particular time series is based on prediction validation on a withheld part of the sample using criteria such as the mean absolute percentage error. A second approach is to rely on the most appropriate general case of the three methods. For annual series this is trend corrected exponential smoothing: for sub-annual series it is the seasonal adaptation of trend corrected exponential smoothing. The rationale for this approach is that a general method automatically collapses to its nested counterparts when the pertinent conditions pertain in the data. A third approach may be based on an information criterion when maximum likelihood methods are used in conjunction with exponential smoothing to estimate the smoothing parameters. In this paper, such approaches for selecting the appropriate forecasting method are compared in a simulation study. They are also compared on real time series from the M3 forecasting competition. The results indicate that the information criterion approach appears to provide the best basis for an automated approach to method selection, provided that it is based on Akaike's information criterion.Model Selection; Exponential Smoothing; Information Criteria; Prediction; Forecast Validation