A methodology to model the number of completeness errors using count data regression models

Transcription

1 Malta, January 2015 INTERNATIONAL WORKSHOP ON SPATIAL DATA AND MAP QUALITY A methodology to model the number of completeness errors using count data regression models José Rodríguez-Avi 1 & Francisco Javier Ariza-López 2 1 Dpto. Estadística e Investigación Operativa, jravi@ujaen.es 2 Dpto. Ingeniería Cartográfica, Geodésica y Fotogrametría, fjariza@ujaen.es Universidad de Jaén Paraje de las Lagunillas S/N E Jaén 1

2 Completeness errors: They are very important in a spatial data set. Completeness is related with the quality of the survey and with the oldness of the data (is affected by time). Following ISO we can distinguish: Omissions: An element (such as a bridge, a building, a cross, a power plant) is in the terrain but it does not appear in the map) and 2 7 Commissions: An element appears in the map but it does not exist in the terrain The greater the number of these errors is, the worse is the quality of the product. We also want to obtain additional information about the product, related to structural aspects, and to investigate the existence of relationships between errors and structural aspects. 7 Omission of a pool 2 Omission of a building 2

3 We are used to employ linear regression model when the response variable is continuous IN OUR CASE: The number of errors is not a continuous but a count data random variable and it should be modelled by discrete models: Poisson, Negative Binomial, Waring, and so on. Additionally, structural aspects of the spatial data set may be taken into account as exogenous co variables in order to explain the number of errors: Use of Count Data Regression Models Omissions 3

4 Count Data Regression Models We start with a dependent count data variable, (for instance, number of omission errors in a tile) We add some structural information for each tile (covariates) We define a cell (of tiles) as the set of tiles that have the same covariates values: All the tiles in the same cell are indistinguishable in terms of covariates Appling a count data regression model we propose a residual discrete distribution for each cell, in a way that parameters of the discrete distribution depend on covariates. We propose several models, and we choose the best model (in any sense) Once the best model is selected, we can: Determine which covariates are related to and how this relation is Obtain the parameters of the residual distribution for any cell and calculate any probability. 4

5 Methodological procedure: Count Data Regression Models Let,, be the set of covariates. The distribution of the response variable given, has a discrete distribution with q parameters and whose mean is a function of the s: where,,, are coefficients to be estimated. The rest of distribution s parameter are estimated independently of covariates. In consequence the number of parameters to be estimated are (number of covariates) + 1 (Intercept) + 1 5

6 Methodological procedure: Count Data Regression Models The coefficient shows the relation, if any, between the independent variable and the dependent variable. Using properties of the MLE method, we can make a test of hypothesis for each dividing the coefficient by its standard deviation (the Wald test). If the coefficient can be consider as equal to 0 we conclude that the corresponding variable has not relation with Y. In other case, both variables are related and the coefficient s sign indicates if such relation is positive (if X increases, Y increases) or negative (If X increases, Y decreases). 6

7 Count Data Regression Models Statistical models: Residual density Equidispersion: Poisson Regression model (PRM). exp Overdispersion: Negative Binomial Regression model (NBRM). Γ Γ! Overdispersion: Generalized Waring Regression model (GWRM). P Y X Γ ρ Γ k ρ Γ Γ k Γ ρ! Γ y Γ k y Γ k ρ y y! 7

8 Count Data Regression Models Statistical models: Estimation of parameters Equidispersion: Poisson Regression model (PRM). Overdispersion: Negative Binomial Regression model (NBRM). Overdispersion: Generalized Waring Regression model (GWRM). 1 8

9 Count Data Regression Models Statistical models: Residual variance Equidispersion: Poisson Regression model (PRM). Overdispersion: Negative Binomial Regression model (NBRM). 1 Overdispersion: Generalized Waring Regression model (GWRM)

10 Count Data Regression Models Models selection Akaike Information Criteria (AIC) 2 ln 2 which is based on the fitted log likelihood function by the ML method, with a penalization related to the number of parameters of the model,. The model with the lower AIC is preferred. In consequence, once fitted all the models, we select the one which has the lower AIC. The AIC has only a comparative value. It does not signify anything by itself 10

11 Count Data Regression Models Models selection To probe if a set of variables must be included or excluded in a model we propose employing the Likelihood Ratio Test, (LRT) in its asymptotic version. If we denote model 1 the one with less covariates and Model 2 the one with more covariates, the LRT is given by: 2 where and are the corresponding likelihood values for the estimated model 1 and 2. The value is asymptotically distributed as a with degrees of freedom (where f is the difference between the number of estimated parameters in both models) 11

12 Count Data Regression Models Software The three models can be fitted using the statistical package R: the glm function of the stats package for the PRM, the glm.nbin function of the MASS package for the NBRM the GWRM package for the GWRM. In all cases, these programs provide Parameters estimation and Wald Test, log likelihood AIC values. 12

13 The Data for the Analysis This study is based on an actual case where a published GDS was assessed by means of a field survey. The GDS is called MTA10v (from Mapa Topográfico de Andalucía E10k vectorial ) which is the official cartography of Andalusia (Spain). It is a map series produced between 1987 and 2007 by the Instituto Cartográfico de Andalucía (nowadays Instituto de Estadística y Cartografía de Andalucía). The MTA10v is a topographic vector database derived from a topographic paper map designed at the beginning of the eighties. The MTA10v is the base for the different thematic datasets and maps of the Regional Government of Andalusia. It has a complete territorial coverage on a semidetailed scale (1:10000) and is updated in a four year cycle on a sheets basis. It is composed of 2745 sheets obtained by manual photogrammetric restitution of flights at 1:25000 scale and updated with flights at 1:20000 scale. 13

14 The Data for the Analysis Because of the large area (87000 km 2 ) covered by the MTA10v the region is divided into four quadrants (ICA, 2002b). So through the cyclic updating strategy each year yields an updating of a fourth of the region. The MTA10v is in the 30N UTM projected coordinate system, and referenced to the ED50 datum. The declared positional accuracy of the MTA10v is RMSE 3m (Corral and García, 2000). An independent accuracy study (Ariza López 2005) based on a random sample of 930 points surveyed by means of a differential GPS fast static survey informs us that positional accuracy is of 10.65m at 95% confidence level. 14

15 The Data for the Analysis General view of the content of the MTA 15

16 The Data for the Analysis Count of elements per layer in the MTA10v series Layer Count of total elements in the spatial data base (n) Edifications Buildings, storage (except water), shed, unique building, antenna, spotlight, transformer building), residential blocks, railway stations, port, airport, airfield, heliport, electrical substation, pumping station, fence, cave, etc. Hydrography River, stream, runway, tide, reservoir, sea, lake, etc. Energy Infrastructure Electric turret, transformer, power line, pipeline, etc. Water Infrastructure Canal, ditch, water supply, water systems, dam, water reservoir, water tank, water treatment plant, pool, well, fountain, spring water, siphon, etc. Communication routes Street, road, crossing, highway, knots, footpath, firewall, railway (all types), tunnel, bridge, cable cars, funicular, chairlift, lift, etc. Vegetation Overstory, parterre, garden, golf course, etc. Total

17 The Data for the Analysis Density map (count of elements per tiles of 1 1 km 2 ) 17

18 Spatial distribution of the sample of 192 clusters 18

19 Application We count completeness errors (omissions and commissions of features) in the 192 analysis items (spatial tiles of 1 1 km 2 ). Each item is a sampled cluster in which both the MTA10v and the ground truth were exhaustively revised and cross compared, and the differences in presences and absences were exhaustively counted The dependent random variables : and : Omissions 19

20 Covariates: Application Province. Categorical variable with 8 different values, 7 dummy variables, Cadiz (CA), Cordoba (CO), Granada (GR), Huelva (HU), Jaén (JA), Málaga (MA) and Seville (SE), with Almeria as the reference value. Urban. Discriminates between Urban (1) or Rural (0). Littoral. Discriminates between a Littoral (1) or Interior (0). Features. Quantitative variable that takes into account the number of features per tile. We introduce this variable as ln(features), and its effect is an offset. Density. Quantitative variable that takes into account the density of features per hectare/kilometre. In our case, due to the field-surveyed tiles are not exactly of 1 1 km 2 is introduced as correction factor. 20

21 Omissions 18 Application

22 Application Omissions Model Y Province +Urban+ Littoral + Ln(Features)+Density Mean expression: ln ln. Variables Model loglikelihood AIC Number of parameters Poisson Regression Model 1148, Province + Urban+ Littoral + ln(features)+density Negative Binomial Regression Model Generalized Waring Regression Model

23 Coefficients for the GWRM covariates estimates Standard dev. z p value (Intercept) Cadiz (province) Cordoba (province) Granada (province) Huelva (province) Jaen (province) Malaga (province) Seville (province) Urban Littoral log(features) Density

24 ln ln

25 Provided that the remaining covariates are equal, the effect of each qualitative covariate may be analysed in terms of Odd ratios, which are defined as: / using the Odd ratios we can observe that: The mean of the error number for Littoral tiles is 94.72% higher than the mean of the error number of interior tiles: / Urban tiles have a mean of the number of errors 124,25% higher than Rustic tiles: / For continuous covariates and also provided that the remaining covariates remain equal: If the number of features in a tile increases by 1%, the mean increases by % If Density in a tile increases by 1, the mean decreases by 3.28% 25

26 Count of errors estimated for each tile 26

27 Probability map for the case of having 10 or more grown features per tile 27

28 Probability maps for the case of having 10% or more grown elements per tile 28

29 Evolution of variance and partition of the variance versus mean (absolute values 29

30 Evolution of variance and partition of the variance versus mean (% values) 30

31 Nomogram showing the relation between the count of errors, probability and % of population 31

32 Frequency of the number of commission errors. 32

33 Summary of models fits (Dependent variable: Commissions) Variables Model Poisson Regression Model loglikelihood AIC Number of parameters Province + Urban+ Littoral + ln(features)+density Negative Binomial Regression Model Generalized Waring Regression Model

34 Covariates Estimate Std. Error value Pr (Intercept) Cadiz (province) Cordoba (province) Granada (province) Huelva (province) Jaen (province) Malaga (province) Sevilla (province) Urban Littoral log(features) e 06 Density