Autocorrelated Errors Research Articles

Improved environmental mapping and validation using bagging models with spatially clustered data

Spatially clustered sampling may result in non-independent data that pose challenges for environmental mapping applications. Two outstanding challenges resulting from the use of spatially clustered data for predictive geospatial modelling with machine learning approaches are biased model training and validation. These issues can be severe for popular bagging models such as Random Forest, yet one or both are often ignored or are handled using sub-optimal approaches. We propose to address these challenges using information on both the spatial autocorrelation of map errors and the spatial sampling intensity. This is achieved by applying the residual spatial covariance as a weighting function for the bagging procedure and for the calculation of weighted validation statistics. Using this approach, the full feature space of the sample data is retained during model training and validation. The utility of covariance weighting for these purposes is investigated through extensive simulation with a range of sample clustering configurations. Results are benchmarked against existing approaches. Covariance weighting improved model performance across a range of clustering scenarios but appeared to produce the greatest improvements for highly clustered data. Covariance-weighted validation demonstrated low bias across a broad range of clustering scenarios compared to existing spatial methods. Findings also suggest, though, that conditional Gaussian simulation approaches may perform well when the proportion of clustered data is very high. Covariance weighting is straightforward to implement, computationally efficient, and scales to different sample sizes and spatial extents.

Process incapability index for autocorrelated data in the presence of measurement errors

Process incapability index is a useful tool for measuring the performance of a process to maintain the quality of products. In this article, we discuss some statistical properties of the estimator of process incapability index when process observations are autocorrelated and contaminated by measurement errors. In many industry, it has been proven that observed quality characteristics for consecutive items are autocorrelated. Moreover, some amount of measurement errors is always present in the observed data. Therefore, it is important to analyze the statistical properties of the estimator under the combined effect of autocorrelation and measurement error for commenting approximately on the ability of a manufacturing process.

Uncovering the spatio-temporal impact of the COVID-19 pandemic on shared e-scooter usage: A spatial panel model

This study examines the spatio-temporal effects of the COVID-19 pandemic on shared e-scooter usage by leveraging two years (2019 and 2020) of daily shared micromobility data from Austin, Texas. We employed a series of random effects spatial-autoregressive model with a spatially autocorrelated error (SAC) to examine the differences and similarities in determinants of e-scooter usage during regular and pandemic periods and to identify factors contributing to the changesin e-scooter use during the Pandemic. Model results provided strong evidence of spatial autocorrelation in the e-scooter trip data and found a spatial negative spillover effect in the 2020 model. The key findings are: i) while the daily e-scooter trips reduced, the average trip distance and the average trip duration increased during the Pandemic; ii) the central part of Austin city experienced a major decrease in e-scooter usage during the Pandemic compared to other parts of Austin; iii) areas with low median income and higher number of available e-scooter devices experienced a smaller decrease in daily total e-scooter trips, trip distance, and trip duration during the Pandemic while the opposite result was found in areas with higher public transportation services. The results of this study provide policymakers with a timely understanding of the changes in shared e-scooter usage during the Pandemic, which can help redesign and revive the shared micromobility market in the post-pandemic era.

Forecast Families: A New Method to Systematically Evaluate the Benefits of Improving the Skill of an Existing Forecast

A growing number of studies have investigated how forecast skill, i.e., predictive power, translates into forecast value, i.e., usefulness, for improving forecast-informed decisions. The relationship between skill and value is widely understood to be complex and case-specific, yet few methods enable its systematic exploration using realistic forecast errors. This paper addresses this gap by proposing a single-parameter linear scaling method to generate families of synthetic forecasts with the desired skill improvements on an existing hindcast (a retrospective forecast of already-observed events). The method is applicable to any quantity for which a deterministic or ensemble hindcast is available, and generates a set of forecasts with different skill but strictly proportional errors. This like-for-like comparison preserves the autocorrelation and cross-correlations of errors, and opens the door for thorough, yet easily interpretable, explorations of the relationship between skill and value of a realistic forecast. We apply this new method to seasonal precipitation hindcasts (produced by the fifth generation of the Seasonal forecasting System of the European Centre for Medium-range Weather Forecasts, ECMWF-SEAS5) in order to explore their value for improving the management of a water supply system in the UK. The application showed that although value generally increases with skill, the skill–value relationship is not necessarily linear, and it strongly depends on operational preferences and hydrological conditions (wet or dry years). It also suggests that the forecast families methodology can help water managers and forecast developers identify when a skill increase would be most valuable. This has the potential to foster productive two-way conversations between forecast producers and users.

Improved prediction of tree species richness and interpretability of environmental drivers using a machine learning approach

Biodiversity is in decline globally and predicting species diversity is critically important if current trends are to be reversed. Tree species richness (TSR) has long been a key measure of biodiversity, but considerable uncertainties exist in current models, particularly given the classic statistical assumptions and poor ecological interpretability of machine learning outcomes. Here, we test several ecologically interpretable machine learning approaches to predict TSR and interpret the driving environmental factors in the continental United States. We develop two artificial neural networks (ANN) and one random forest (RF) model to predict TSR using Forest Inventory and Analysis data and 20 environmental covariates and compare them to a classic generalized linear model (GLM). Models were evaluated on an independent, unseen testing dataset using R2 and Mean Absolute Error (MAE) and residual spatial autocorrelation analysis. An Interpretable Machine Learning approach, SHapley Additive exPlanations (SHAP), was adopted to explain the major environmental factors driving TSR. Compared to a baseline GLM (R2 = 0.7; MAE = 4.7), the ANN and RF models achieved R2 greater than 0.9 and MAE<3.1. Additionally, the ANN and RF models produced less spatially clustered TSR residuals than the GLM. SHAP analysis suggested that TSR is best predicted by Aridity Index, Forest Area, Altitude, Mean Precipitation of the Driest Quarter and Mean Annual Temperature. SHAP further revealed a non-linear relationship of environmental covariates with TSR and complex interactions that were not revealed by the GLM. The study highlights the need for conservation efforts of forest areas and reducing precipitation-related physiological stress on tree species in low forested but arid regions. The machine learning approach used here is transferrable for studies of biodiversity for other organisms or prediction of TSR under future climatic scenarios.

On bootstrapping tests of equal forecast accuracy for nested models

AbstractThe asymptotic distributions of the recursive out‐of‐sample forecast accuracy test statistics depend on stochastic integrals of Brownian motion when the models under comparison are nested. This often complicates their implementation in practice because the computation of their asymptotic critical values is burdensome. Hansen and Timmermann (2015, Econometrica) propose a Wald approximation of the commonly used recursive ‐statistic and provide a simple characterization of the exact density of its asymptotic distribution. However, this characterization holds only when the larger model has one extra predictor or the forecast errors are homoscedastic. No such closed‐form characterization is readily available when the nesting involves more than one predictor and heteroscedasticity or serial correlation is present. We first show through Monte Carlo experiments that both the recursive ‐test and its Wald approximation have poor finite‐sample properties, especially when the forecast horizon is greater than one and forecast errors exhibit serial correlation. We then propose a hybrid bootstrap method consisting of a moving block bootstrap and a residual‐based bootstrap for both statistics and establish its validity. Simulations show that the hybrid bootstrap has good finite‐sample performance, even in multi‐step ahead forecasts with more than one predictor, and with heteroscedastic or autocorrelated forecast errors. The bootstrap method is illustrated on forecasting core inflation and GDP growth.

Improving air pollutant prediction in Henan Province, China, by enhancing the concentration prediction accuracy using autocorrelation errors and an Informer deep learning model

Air pollution is an important issue affecting sustainable development in China, and accurate air quality prediction has become an important means of air pollution control. At present, traditional methods, such as deterministic and statistical approaches, have large prediction errors and cannot provide effective information to prevent the negative effects of air pollution. Therefore, few existing methods could obtain accurate air pollutant time series predictions. To this end, a deep learning-based air pollutant prediction method, namely, the autocorrelation error-Informer (AE-Informer) model, is proposed in this study. The model implements the AE based on the Informer model. The AE-Informer model is used to predict the hourly concentrations of multiple air pollutants, including PM10, PM2.5, NO2, and O3. The experimental results show that the mean absolute error (MAE) and root mean square error (RMSE) values of AE-Informer in multivariate prediction are 3% less than those of the Informer model; thus, the prediction error is effectively reduced. In addition, a stacking ensemble model is proposed to supplement the missing air pollutant time series data. This study uses Henan Province in China as an example to test the validity of the proposed methodology.

Uncertainty of spatial averages and totals of natural resource maps

AbstractGlobal, continental and regional maps of concentrations, stocks and fluxes of natural resources provide baseline data to assess how ecosystems respond to human disturbance and global warming. They are also used as input to numerous modelling efforts. But these maps suffer from multiple error sources and, hence, it is good practice to report estimates of the associated map uncertainty so that users can evaluate their fitness for use.We explain why quantification of uncertainty of spatial aggregates is more complex than uncertainty quantification at point support because it must account for spatial autocorrelation of the map errors. Unfortunately, this is not done in a number of recent high‐profile studies. We describe how spatial autocorrelation of map errors can be accounted for with block kriging, a method that requires geostatistical expertise. Next, we propose a new, model‐based approach that avoids the numerical complexity of block kriging and is feasible for large‐scale studies where maps are typically made using machine learning. Our approach relies on Monte Carlo integration to derive the uncertainty of the spatial average or total from point support prediction errors. We account for spatial autocorrelation of the map error by geostatistical modelling of the standardized map error.We show that the uncertainty strongly depends on the spatial autocorrelation of the map errors. In a first case study, we used block kriging to show that the uncertainty of the predicted topsoil organic carbon in France decreases when the support increases. In a second case study, we estimated the uncertainty of spatial aggregates of a machine learning map of the above‐ground biomass in Western Africa using Monte Carlo integration. We found that this uncertainty was small because of the weak spatial autocorrelation of the standardized map errors.We present a tool to get realistic estimates of the uncertainty of spatial averages and totals of natural resource maps. The method presented in this paper is essential for parties that need to evaluate whether differences in aggregated environmental variables or natural resources between regions or over time are statistically significant.

Evaluating store location and department composition based on spatial heterogeneity in sales potential

In this paper, we extend a retail location evaluation model with the possibility to include the effect of department size adaptation at the store level. We relate department-level store sales to a store's competitive and demographic environment, thereby providing richer insights into the drivers of department sales than a model of just aggregate sales. Further, we accommodate heterogeneity in consumer characteristics over space by using zip code level data and unobserved spatial effects in department sales by including spatially autocorrelated error terms.Using spatial panel data for 30 clothing stores belonging to one Dutch retail chain, we demonstrate how to use the modeling approach to analyze and predict sales performance of new and existing stores. We show that the predictive performance of our model is superior to that of a benchmark model that does not include spatial autocorrelation.

Effects of climate and price on soybean production: Empirical analysis based on panel data of 116 prefecture-level Chinese cities.

This paper uses panel data from 116 prefecture-level cities in China from 2003 to 2019 to study the impact of price and climate factors on soybean planting area and yield per unit area in China. We adopt the panel instrumental variable method to control the endogeneity of the price in the regression and allow possible spatial autocorrelation errors. According to the research results, price is the primary factor affecting soybean production. For every 1% increase in soybean prices, the soybean planting area increases by 1.650%, and the per unit yield decreases by 0.898%. As for fertilizer prices, for every 1% increase in fertilizer prices, the soybean planting area will decrease by 2.616%, and the yield per unit area will increase by 0.819%. At the same time, climate change will also significantly affect soybean production. For every 1 cm increase in precipitation in April and May, the soybean planting area will increase by 0.233% and decrease by 0.172%, respectively. The precipitation increase in June and July can also significantly promote soybean yield. The results demonstrate that because soybean is a shade-loving crop, the increase of growing degree days will hinder the progress of soybean yield.

The Efficiency of the Proposed Smoothing Method over the Classical Cubic Smoothing Spline Regression Model with Autocorrelated Residual

Spline smoothing is a technique used to filter out noise in time series observations when predicting nonparametric regression models. Its performance depends on the choice of the smoothing parameter. Most of the existing smoothing methods applied to time series data tend to over fit in the presence of autocorrelated errors. This study aims to determine the optimum performance value, goodness of fit and model overfitting properties of the proposed Smoothing Method (PSM), Generalized Maximum Likelihood (GML), Generalized Cross-Validation (GCV), and Unbiased Risk (UBR) smoothing parameter selection methods. A Monte Carlo experiment of 1,000 trials was carried out at three different sample sizes (20, 60, and 100) and three levels of autocorrelation (0.2, 05, and 0.8). The four smoothing methods' performances were estimated and compared using the Predictive Mean Squared Error (PMSE) criterion. The findings of the study revealed that: for a time series observation with autocorrelated errors, provides the best-fit smoothing method for the model, the PSM does not over-fit data at all the autocorrelation levels considered ( the optimum value of the PSM was at the weighted value of 0.04 when there is autocorrelation in the error term, PSM performed better than the GCV, GML, and UBR smoothing methods were considered at all-time series sizes (T = 20, 60 and 100). For the real-life data employed in the study, PSM proved to be the most efficient among the GCV, GML, PSM, and UBR smoothing methods compared. The study concluded that the PSM method provides the best fit as a smoothing method, works well at autocorrelation levels (ρ=0.2, 0.5, and 0.8), and does not over fit time-series observations. The study recommended that the proposed smoothing is appropriate for time series observations with autocorrelation in the error term and econometrics real-life data. This study can be applied to; non – parametric regression, non – parametric forecasting, spatial, survival, and econometrics observations.

A new ridge estimator for linear regression model with some challenging behavior of error term

Ridge regression is a variant of linear regression that aims to circumvent the issue of collinearity among predictors. The ridge parameter k has an important role in the bias-variance tradeoff. In this article, we introduce a new approach to select the ridge parameter to deal with the multicollinearity problem with different behavior of the error term. The proposed ridge estimator is a function of the number of predictors and the standard error of the regression model. An extensive simulation study is conducted to assess the performance of the estimators for the linear regression model with different error terms, which include normally distributed, non-normal and heteroscedastic or autocorrelated errors. Based upon the criterion of mean square error (MSE), it is found that the new proposed estimator outperforms OLS, commonly used and closely related estimators. Further, the application of the proposed estimator is provided on the COVID-19 data of India.

Is there regional convergence between Morocco and its OECD partner countries in terms of well-being?

Morocco wants its 12 regions to play the role as the main lever of its public policies to initiate harmonized spatial multidimensional development. In the context of this goal and Morocco's openness over the past two decades to bilateral and multilateral cooperation in an effort toward regional integration, this article studies the convergence of 389 regions in 36 countries (Morocco and 35 of its partner member countries of the Organization for Economic Co-operation and Development (OECD)) between 2000 and 2019 in terms of well-being. To this end, we considered the territorial dimension of β-convergence models for well-being and its four domains (economic, social, environmental, and governance). Then, we adapted the absolute β-convergence model taking into account the existence of spatial heterogeneity according to five specifications of spatial models. Thus, apart from environmental domain, we showed that β-convergence of regions is significant for well-being and three of its domains (economic, social, and governance). These convergences are made by a spatially autocorrelated error model (SEM). However, the speed and period of convergence are relatively low for social domain, partly explaining the very exacerbated tensions at the territorial level. The fastest convergence was achieved in governance domain, followed by economic domain. This suggests that emerging countries must pay particular attention to national public action in favor of social cohesion at the territorial level. The lack of convergence in environmental domain calls for common actions for all countries at the supranational level to protect the commons at the territorial level.

On some techniques of selecting spline smoothing parameters for a correlated dataset with autocorrelation structure in the residual

Residuals are minimized in a correlated dataset by selecting a smoothing parameter with optimum performance in the smoothing spline. The selection methods utilized in this study include Generalized Maximum Likelihood (GML), Generalized Cross-Validation (GCV), Unbiased Risk (UBR), and the Proposed Smoothing Method (PSM). The aim of this study is to compare the smoothing parameter selection ability of the four parameter selection methods for a correlated dataset with autocorrelation structure in the error term. To achieve this purpose, a Monte-Carlo simulation was conducted by utilizing program written in R-4.2.2. The performance of the parameter selection methods were evaluated using predictive Mean Squared Error (PMSE). Findings from the study indicated that GCV and GML were mostly affected by the presence of auto correlation in the residual and therefore had an asymptotically similar behavioural pattern. The estimators conformed to the asymptotic properties of the smoothing parameter selection methods considered; this is noticed in all the sample sizes and at all the smoothing parameters. The result also showed that; the most consistent and efficient among the four spline smoothing parameter selection methods considered in this study based on sample size and performance in the presence of autocorrelated residual error is the proposed smoothing method (PSM) because it does not undersmooth relative to the other smoothing method especially for small sample and medium sample size of 50 and 100.

Modeling compatible taper and stem volume of pure Scots pine stands in Northeastern Turkey

Compatible taper and volume equations for pure and natural Scots pine stands in the northeastern part of Turkey (Ardahan Province) were developed using a nonlinear mixed-effects modeling approach. Experimental data were obtained from 137 felled sample trees in different diameter and height classes. The most successful model (Jiang et al. 2005) explained 98.3% of the variance in stem diameter estimation and the RMSE, ME, MAE, AIC and BIC value obtained using this model were 1.955 cm, 0.043 cm, 1.300 cm, 9783.8 and 9812.6, respectively. Considering the criterion values of AIC, BIC and -2LnL, the model with random-effects in two parameters (b1 and b3) was the most successful for Scots pine. While the mixed model including random parameters did not completely solved the problem of the autocorrelation of errors in this study, the use of the autoregressive error structure AR(1) eliminated the autocorrelation in the residuals. In addition, the best estimation results among 20 different calibration options were obtained using the option of measuring two tree diameters at d1.30 and d5.30 with validation data. The most successful model explained 99.18% of the total variance in stem volume estimation in Scots pine.

The Efficiency of the Proposed Smoothing Method over the Classical Cubic Smoothing Spline Regression Model with Autocorrelated Residual

Spline smoothing is a technique used to filter out noise in time series observations when predicting nonparametric regression models. Its performance depends on the choice of the smoothing parameter. Most of the existing smoothing methods applied to time series data tend to overfit in the presence of autocorrelated errors. This study aims to determine the optimum performance value, goodness of fit and model overfitting properties of the proposed Smoothing Method (PSM), Generalized Maximum Likelihood (GML), Generalized Cross-Validation (GCV), and Unbiased Risk (UBR) smoothing parameter selection methods. A Monte Carlo experiment of 1,000 trials was carried out at three different sample sizes (20, 60, and 100) and three levels of autocorrelation (0.2, 05, and 0.8). The four smoothing methods' performances were estimated and compared using the Predictive Mean Squared Error (PMSE) criterion. The findings of the study revealed that: for a time series observation with autocorrelated errors, provides the best-fit smoothing method for the model, the PSM does not over-fit data at all the autocorrelation levels considered ( the optimum value of the PSM was at the weighted value of 0.04 when there is autocorrelation in the error term, PSM performed better than the GCV, GML, and UBR smoothing methods were considered at all-time series sizes (T = 20, 60 and 100). For the real-life data employed in the study, PSM proved to be the most efficient among the GCV, GML, PSM, and UBR smoothing methods compared. The study concluded that the PSM method provides the best fit as a smoothing method, works well at autocorrelation levels (ρ=0.2, 0.5, and 0.8), and does not over fit time-series observations. The study recommended that the proposed smoothing is appropriate for time series observations with autocorrelation in the error term and econometrics real-life data. This study can be applied to; non – parametric regression, non – parametric forecasting, spatial, survival, and econometrics observations.

Incorporating IMERG satellite precipitation uncertainty into seasonal and peak streamflow predictions using the Hillslope Link hydrological model

In global applications and data sparse regions, which comprise most of the earth, hydrologic model-based flood monitoring relies on precipitation data from satellite multisensor precipitation products or numerical weather forecasts. However, these products often exhibit substantial errors during the meteorological conditions that lead to flooding, including extreme rainfall. The propagation of precipitation forcing errors to predicted runoff and streamflow is scale-dependent and requires an understanding of the autocorrelation structure of precipitation errors, since error autocorrelation impacts the accumulation of precipitation errors over space and time in hydrologic models. Previous efforts to account for satellite precipitation uncertainty in hydrologic models have demonstrated the potential for improving streamflow estimates; however, these efforts use satellite precipitation error models that rely heavily on ground reference data such as rain gages or weather radar and do not characterize the nonstationarity of precipitation error autocorrelation structures. This work evaluates a new method, the Space-Time Rainfall Error and Autocorrelation Model (STREAM), which stochastically generates possible true precipitation fields, as input to the Hillslope Link Model to generate ensemble streamflow estimates. Unlike previous error models, STREAM represents the nonstationary and anisotropic autocorrelation structure of satellite precipitation error and does not use any ground reference to do so. Ensemble streamflow predictions are compared with streamflow generated using satellite precipitation fields as well as a radar-gage precipitation dataset during peak flow events. Results demonstrate that this approach to accounting for precipitation uncertainty effectively characterizes the uncertainty in streamflow estimates and reduces the error of predicted streamflow. Streamflow ensembles forced by STREAM improve streamflow prediction nearly to the level obtained using ground-reference forcing data across basin sizes.

SMALL AREA ESTIMATION OF MEAN YEARS SCHOOL IN KABUPATEN BOGOR USING SEMIPARAMETRIC P-SPLINE

The Fay-Herriot model, generally uses the EBLUP (Empirical Best Linear Unbiased Prediction) method, is less flexible due to the assumption of linearity. The P-Spline semiparametric model is a modification of the Fay-Herriot model which can accommodate the presence of two components, linear and nonlinear predictors. This paper also deals spatial dependence among the random area effects so that a model with spatially autocorrelated errors will be implemented, known as the SEBLUP (Spatial Empirical Best Linear Unbiased Prediction) method. Using data from SUSENAS, PODES, and some publication from BPS, the main objective of this study is to estimate the mean years school at kecamatan level in Kabupaten Bogor using EBLUP, Semiparametric P-Spline approach and SEBLUP method. The results show that based on the RRMSE value, the cubic P-Spline model with three knots predicts the mean years school better than EBLUP. Meanwhile, the addition of spatial effects into the small area estimation has not been able to improve the estimated value of the P-Spline semiparametric approach.

WITHDRAWN: Incorporating IMERG satellite precipitation uncertainty into seasonal and peak streamflow predictions using the Hillslope Link hydrological model

In global applications and data sparse regions, which comprise most of the earth, hydrologic model-based flood monitoring relies on precipitation data from satellite multisensor precipitation products or numerical weather forecasts. However, these products often exhibit substantial errors during the meteorological conditions that lead to flooding, including extreme rainfall. The propagation of precipitation forcing errors to predicted runoff and streamflow is scale-dependent and requires an understanding of the autocorrelation structure of precipitation errors, since error autocorrelation impacts the accumulation of precipitation errors over space and time in hydrologic models. Previous efforts to account for satellite precipitation uncertainty in hydrologic models have demonstrated the potential for improving streamflow estimates; however, these efforts use satellite precipitation error models that rely heavily on ground reference data such as rain gages or weather radar and do not characterize the nonstationarity of precipitation error autocorrelation structures. This work evaluates a new method, the Space-Time Rainfall Error and Autocorrelation Model (STREAM), which stochastically generates possible true precipitation fields, as input to the Hillslope Link Model to generate ensemble streamflow estimates. Unlike previous error models, STREAM represents the nonstationary and anisotropic autocorrelation structure of satellite precipitation error and does not use any ground reference to do so. Ensemble streamflow predictions are compared with streamflow generated using satellite precipitation fields as well as a radar-gage precipitation dataset during peak flow events. Results demonstrate that this approach to accounting for precipitation uncertainty effectively characterizes the uncertainty in streamflow estimates and reduces the error of predicted streamflow. Streamflow ensembles forced by STREAM improve streamflow prediction nearly to the level obtained using ground-reference forcing data across basin sizes.

The relationship between ensemble representations of facial information

People accurately evaluate various types of facial information (gaze direction, facial expression, facial identity, and gender) of multiple faces. Considering such varieties, summarizing abilities of facial information might vary depending on its type because it is either changeable (e.g., gaze direction and expression) or invariant (e.g., identity and gender). The current study investigated the relationship between the averaging abilities of facial information using an individual difference approach and a dual-task paradigm to understand the effect of facial information type on the ensemble coding of facial information. We conducted two online experiments on the relationship between the averaging abilities of facial expressions and gaze direction (Experiment 1) and those of facial expressions and gender (Experiment 2). Participants were asked to estimate the average of each piece of facial information in the first and second blocks (single task), respectively, and both sequentially in the third and fourth blocks (dual task). We found that most of the error autocorrelations of facial information were high, indicating high measurement reliability. Participants’ abilities to average facial expressions were correlated with those to average gaze directions, but not with those to average gender information. That is, the ensemble processing of facial expressions is related to gaze directions, but not genders. These results suggest that ensemble representations of facial information regarding changeable properties differ from those of invariant ones.