{"id":178861,"date":"2012-05-07T23:01:18","date_gmt":"2012-05-08T06:01:18","guid":{"rendered":"http:\/\/www.esri.com\/arcgis-blog\/?post_type=blog&#038;p=178861"},"modified":"2018-12-18T11:36:03","modified_gmt":"2018-12-18T19:36:03","slug":"dealing-with-extreme-values-in-kriging","status":"publish","type":"blog","link":"https:\/\/www.esri.com\/arcgis-blog\/products\/arcgis-desktop\/analytics\/dealing-with-extreme-values-in-kriging","title":{"rendered":"Dealing with extreme values in kriging"},"author":5071,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","format":"standard","meta":{"_acf_changed":false,"_searchwp_excluded":""},"categories":[23341],"tags":[36351,24321,34131,39491],"industry":[],"product":[36991],"class_list":["post-178861","blog","type-blog","status-publish","format-standard","hentry","category-analytics","tag-arcgis-geostatistical-analyst","tag-geoprocessing","tag-interpolation","tag-kriging","product-arcgis-desktop"],"acf":{"short_description":"Introduction\r\nOne of the most common problems we have when attempting to interpolate data using kriging is the presence of outliers in the data.  ","flexible_content":[{"acf_fc_layout":"content","content":"<h4>Introduction<\/h4>\n<p>One of the most common problems we have when attempting to interpolate data using\u00a0<a href=\"http:\/\/help.arcgis.com\/en\/arcgisdesktop\/10.0\/help\/index.html#\/Kriging_in_Geostatistical_Analyst\/003100000032000000\/\">kriging<\/a>\u00a0is the presence of\u00a0<a href=\"http:\/\/help.arcgis.com\/en\/arcgisdesktop\/10.0\/help\/index.html#\/\/003100000010000000.htm\">outliers<\/a>\u00a0in the data.\u00a0 An outlier is a data value that is either very large or very small compared to the rest of the data.\u00a0 Outliers often result from malfunctions in the monitoring equipment or typos during data entry, such as accidentally removing a decimal.\u00a0 These erroneous data points should be manually corrected or removed before attempting to interpolate.\u00a0 However, not all outliers are the result of machine or human error.\u00a0 Some outliers are valid values, and this blog will demonstrate how to deal with this kind of outlier.<span id=\"more-13675\"><\/span><\/p>\n<h4>Description of the data<\/h4>\n<p>The data used in this blog comes from measurements of heavy metal concentrations of mosses in Austria in 1995.\u00a0 Various heavy metals were measured in milligrams of heavy metal per kilogram of moss, and here we will focus on molybdenum.\u00a0 As the following graphic shows, lower molybdenum concentrations were found in the north and higher concentrations in the south.\u00a0 However, note that two locations had molybdenum concentrations that were much higher than the rest of the data (7.66 and 1.81 mg\/kg).<\/p>\n<div id=\"attachment_13695\" class=\"wp-caption aligncenter\">\n<p><a href=\"http:\/\/blogs.esri.com\/esri\/arcgis\/files\/2012\/05\/HeavyMetals_Outliers.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-13695 \" src=\"http:\/\/blogs.esri.com\/esri\/arcgis\/files\/2012\/05\/HeavyMetals_Outliers.png\" alt=\"Molybdenum concentrations in Austria, 1995\" width=\"595\" height=\"312\" \/><\/a><\/p>\n<p class=\"wp-caption-text\">Molybdenum concentrations in Austria, 1995<\/p>\n<\/div>\n<p>While the large concentrations appear to be outliers, other heavy metals were also found in high concentrations at these locations, so they are unlikely to be from machine or human error.\u00a0 To create an accurate prediction map for molybdenum concentrations, the measurements at these two locations cannot simply be deleted or ignored.<\/p>\n<h4>The problem with outliers<\/h4>\n<p>It is very difficult to build a valid kriging model when some values are several times larger than all other values, and this is where many people typically get stuck.\u00a0 They try to find a\u00a0<a href=\"http:\/\/help.arcgis.com\/en\/arcgisdesktop\/10.0\/help\/index.html#\/Understanding_a_semivariogram_The_range_sill_and_nugget\/0031000000mq000000\/\">semivariogram<\/a>\u00a0to fit all their data, but the outliers exert so much influence on the estimated semivariogram that no combination of semivariogram parameters seems to fit the data.\u00a0 For example, with this molybdenum data, the\u00a0<a href=\"http:\/\/help.arcgis.com\/en\/arcgisdesktop\/10.0\/help\/index.html#\/A_quick_tour_of_Geostatistical_Analyst\/003100000004000000\/\">Geostatistical Wizard<\/a>\u00a0suggests using the semivariogram pictured below.\u00a0 Notice that the semivariogram (dark blue curve) is nearly flat, and the empirical semivariances (blue crosses) go up and down erratically.\u00a0 This indicates that the semivariogram is not acceptable for any serious analysis.<\/p>\n<div id=\"attachment_13697\" class=\"wp-caption aligncenter\">\n<p><a href=\"http:\/\/blogs.esri.com\/esri\/arcgis\/files\/2012\/05\/semivariogram1.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-13697 \" src=\"http:\/\/blogs.esri.com\/esri\/arcgis\/files\/2012\/05\/semivariogram1.png\" alt=\"This semivariogram does not fit the data.\" width=\"574\" height=\"440\" \/><\/a><\/p>\n<p class=\"wp-caption-text\">This semivariogram does not fit the data.<\/p>\n<\/div>\n<h4>A possible solution<\/h4>\n<p>While there is no single method that is guaranteed to work, a potential solution is to split the kriging process into two steps.\u00a0 To understand how this works, you need to understand that the kriging process already implicitly has two steps:<\/p>\n<ol>\n<li><em>Modeling<\/em>\u00a0\u2013 Build a semivariogram to model the spatial relationships between points.\u00a0 This step quantifies the correlation between data points based on their distance apart.<\/li>\n<li><em>Prediction<\/em>\u00a0\u2013 Use the semivariogram and a dataset to make predictions at new locations.<\/li>\n<\/ol>\n<p>Almost always, the same dataset is used for both modeling and prediction, but the trick for dealing with outliers is that they do not have to be the same. When you\u2019re confronted with a dataset that has outliers that you cannot ignore (as with these molybdenum concentrations), a common approach is to explicitly remove the outliers for the modeling, then use the whole dataset (outliers included) in the prediction.\u00a0 This workflow is effective because the modeling is not corrupted by the outliers, but the prediction surface still accounts for the extreme values.<\/p>\n<h4>Modeling steps<\/h4>\n<ol>\n<li style=\"list-style-type: none;\">\n<ol>\n<li><a href=\"http:\/\/help.arcgis.com\/en\/arcgisdesktop\/10.0\/help\/index.html#\/\/001700000071000000\">Select<\/a>\u00a0all points in the dataset except the outliers.\u00a0 For this demonstration, I selected all data except the two outliers described above.<\/li>\n<li>Use the Geostatistical Wizard to build a semivariogram to model the spatial relationships between the non-outliers.\u00a0 I chose the semivariogram pictured below.\u00a0 Compare this semivariogram to the one pictured above.\u00a0 The removal of only two points stabilizes the semivariogram such that it passes through the empirical semivariances almost perfectly.<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<div id=\"attachment_13698\" class=\"wp-caption aligncenter\">\n<p><a href=\"http:\/\/blogs.esri.com\/esri\/arcgis\/files\/2012\/05\/semivariogram.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-13698\" src=\"http:\/\/blogs.esri.com\/esri\/arcgis\/files\/2012\/05\/semivariogram.png\" alt=\"Good Semivariogram\" width=\"574\" height=\"440\" \/><\/a><\/p>\n<p class=\"wp-caption-text\">This semivariogram fits the data very well.<\/p>\n<\/div>\n<ol>\n<li>Click Finish, then click OK on the Method Report window.\u00a0 The geostatistical layer is added to ArcMap\u2019s Table of Contents, and I\u2019ve given it the name \u201cModel Step\u201d.\u00a0 This geostatistical layer can also be persisted to disk with the\u00a0<a href=\"http:\/\/help.arcgis.com\/en\/arcgisdesktop\/10.0\/help\/index.html#\/\/001700000070000000\">Save to Layer File<\/a>\u00a0geoprocessing tool.<\/li>\n<li>This completes the workflow for the modeling step.<\/li>\n<\/ol>\n<h4>Prediction steps<\/h4>\n<ol>\n<li>Use the\u00a0<a href=\"http:\/\/help.arcgis.com\/en\/arcgisdesktop\/10.0\/help\/index.html#\/\/003000000015000000.htm\">Create Geostatistical Layer<\/a>\u00a0geoprocessing tool to use the model above to make predictions using the entire dataset (outliers included).\u00a0 As shown in the graphic below, use the \u201cModel Step\u201d geostatistical layer as the\u00a0<em>Input geostatistical model source<\/em>\u00a0(if you persisted the geostatistical layer to a layer file, you can also specify this layer file as input).\u00a0 For\u00a0<em>Input dataset(s)<\/em>, specify the feature class and field of all the molybdenum measurements (outliers included).\u00a0 Give the\u00a0<em>Output geostatistical layer\u00a0<\/em>a name (I\u2019ve named it \u201cPrediction Step\u201d), and press OK to run the tool.\n<div id=\"attachment_13699\" class=\"wp-caption aligncenter\">\n<p><a href=\"http:\/\/blogs.esri.com\/esri\/arcgis\/files\/2012\/05\/CGL.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-13699 \" src=\"http:\/\/blogs.esri.com\/esri\/arcgis\/files\/2012\/05\/CGL.png\" alt=\"Create Geostatistical Layer geoprocessing tool\" width=\"523\" height=\"356\" \/><\/a><\/p>\n<p class=\"wp-caption-text\">Create Geostatistical Layer geoprocessing tool<\/p>\n<\/div>\n<\/li>\n<li>The \u201cPrediction Step\u201d geostatistical layer is added to ArcMap\u2019s Table of Contents.\u00a0 Again, this geostatistical layer can be persisted to disk using the\u00a0<a href=\"http:\/\/help.arcgis.com\/en\/arcgisdesktop\/10.0\/help\/index.html#\/\/001700000070000000\">Save to Layer File<\/a>\u00a0geoprocessing tool.<\/li>\n<li>This completes the workflow for creating a prediction surface for molybdenum concentrations in Austrian mosses.<\/li>\n<\/ol>\n<h4>Results<\/h4>\n<p>The prediction surface for molybdenum concentrations in Austrian mosses is shown below.\u00a0 As expected, low molybdenum concentrations are predicted in the north and high concentrations in the south.\u00a0 The largest predicted values are in the area around the outliers.<\/p>\n<div id=\"attachment_13700\" class=\"wp-caption aligncenter\">\n<p><a href=\"http:\/\/blogs.esri.com\/esri\/arcgis\/files\/2012\/05\/HeavyMetals_PredSurface.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-13700\" src=\"http:\/\/blogs.esri.com\/esri\/arcgis\/files\/2012\/05\/HeavyMetals_PredSurface.png\" alt=\"Predicted molybdenum concentrations\" width=\"594\" height=\"309\" \/><\/a><\/p>\n<p class=\"wp-caption-text\">Predicted molybdenum concentrations<\/p>\n<\/div>\n<h4>Comparison<\/h4>\n<p>As a final sanity check, we can compare the predicted molybdenum concentrations that include the outliers (Prediction Step) to the predictions that do not include the outliers (Model Step).\u00a0 Recall that both surfaces were created with the same semivariogram that did not include the outliers.<\/p>\n<p>The graphic below shows the percent difference between these two surfaces.\u00a0 In order to calculate the percent difference, both geostatistical layers must be\u00a0<a href=\"http:\/\/help.arcgis.com\/en\/arcgisdesktop\/10.0\/help\/index.html#\/\/003000000017000000.htm\">converted to raster<\/a>\u00a0in order to perform the required\u00a0<a href=\"http:\/\/help.arcgis.com\/en\/arcgisdesktop\/10.0\/help\/index.html#\/\/009z000000z7000000.htm\">map algebra<\/a>.\u00a0 Notice that the predictions are identical in all areas except around the two outliers.\u00a0 This shows that the predictions in areas far from the outliers were not influenced by the extreme values, and it also shows that including the outliers elicits significantly higher predictions near the areas with the large molybdenum concentrations.\u00a0 This is exactly what we were hoping to see, and it provides us with confidence that our predictions accurately represent molybdenum concentrations in Austrian mosses.<\/p>\n<div id=\"attachment_13701\" class=\"wp-caption aligncenter\">\n<p><a href=\"http:\/\/blogs.esri.com\/esri\/arcgis\/files\/2012\/05\/HeavyMetals_PercentDiff.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-13701\" src=\"http:\/\/blogs.esri.com\/esri\/arcgis\/files\/2012\/05\/HeavyMetals_PercentDiff.png\" alt=\"Percent difference in predicted molybdenum concentrations\" width=\"595\" height=\"312\" \/><\/a><\/p>\n<p class=\"wp-caption-text\">Percent difference in predicted molybdenum concentrations<\/p>\n<\/div>\n<h4>Data reference<\/h4>\n<p>Krivoruchko K. (2011) Spatial Statistical Data Analysis for GIS Users. Esri Press, 928 p.<\/p>\n<p><strong><em>This post was contributed by Eric Krause, a product engineer on the analysis and geoprocessing team.<\/em><\/strong><\/p>\n"}],"authors":[{"ID":5071,"user_firstname":"Eric","user_lastname":"Krause","nickname":"Eric Krause","user_nicename":"eric6346","display_name":"Eric Krause","user_email":"EKrause@esri.com","user_url":"","user_registered":"2018-03-02 00:16:42","user_description":"Eric Krause is a Product Engineer on the Spatial Statistics and Geostatistical Analyst teams.  He has worked at Esri since 2010 and specializes in geostatistical interpolation, spatial statistics, and general spatial analysis.","user_avatar":"<img data-del=\"avatar\" src='https:\/\/www.esri.com\/arcgis-blog\/app\/uploads\/2024\/05\/EK-headshot-213x200.jpg' class='avatar pp-user-avatar avatar-96 photo ' height='96' width='96'\/>"}],"related_articles":"","card_image":false,"wide_image":false},"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v25.9 (Yoast SEO v25.9) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Dealing with extreme values in kriging<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.esri.com\/arcgis-blog\/products\/arcgis-desktop\/analytics\/dealing-with-extreme-values-in-kriging\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta 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