{"id":5594,"date":"2018-07-04T04:57:09","date_gmt":"2018-07-04T02:57:09","guid":{"rendered":"http:\/\/www.sigterritoires.fr\/?p=5594"},"modified":"2018-10-17T13:01:13","modified_gmt":"2018-10-17T11:01:13","slug":"exploratory-data-analysis-for-geostatics-trend-analysis","status":"publish","type":"post","link":"https:\/\/www.sigterritoires.fr\/index.php\/en\/exploratory-data-analysis-for-geostatics-trend-analysis\/","title":{"rendered":"Exploratory data analysis for geostatics: trend analysis"},"content":{"rendered":"

Following\u00a0the article\u00a0\u00a0\u00a0Introduction to\u00a0<\/a>\u00a0 Exploratory\u00a0data\u00a0analysis for\u00a0geostatistics<\/a>\u00a0\u00a0\u00a0we\u00a0will\u00a0discuss\u00a0each\u00a0of the\u00a0available tools\u00a0to\u00a0perform the exploratory\u00a0analysis\u00a0of the spatial\u00a0data.\u00a0We\u00a0have\u00a0discussed\u00a0the\u00a0 histograms<\/a>\u00a0,\u00a0\u00a0\u00a0\u00a0 QQ-Plots<\/a>\u00a0, and \u00a0Vorono\u00ef<\/a> maps\u00a0.\u00a0Now, it is the turn for the data trends.<\/p>\n

Trend?<\/strong>\u00a0What is<\/strong>\u00a0it?<\/strong><\/p>\n

Firstly we have to know what we are looking for. Certainly, you are already\u00a0familiar\u00a0with the notion of\u00a0trend\u00a0in\u00a0the\u00a0temporal series\u00a0(the\u00a0trend\u00a0of\u00a0unemployment\u00a0is\u00a0decreasing\u00a0or\u00a0increasing,\u00a0etc\u00a0…).<\/p>\n

The topic here is spatial data\u00a0processing.\u00a0Therefore the notion of\u00a0trend\u00a0is\u00a0a\u00a0little\u00a0different\u00a0since we\u00a0do\u00a0not\u00a0have, a priori, the variable time to organize our\u00a0points\u00a0in\u00a0a temporal series.<\/p>\n

Let\u2019s use\u00a0the\u00a0example used\u00a0in the\u00a0previous\u00a0articles,\u00a0a\u00a0series\u00a0of\u00a0points\u00a0resulting from the use of a bathymetric\u00a0sensor.<\/p>\n

$\"\"$ <\/p>\n

Remember\u00a0what\u00a0we\u00a0said\u00a0in\u00a0a previous article\u00a0Introduction\u00a0on the basics of\u00a0geostatistical:\u00a0it\u00a0deals\u00a0with random phenomena\u00a0random\u00a0with\u00a0dependency.\u00a0Are we entitled to\u00a0assume\u00a0that in the considered area the\u00a0depths\u00a0are\u00a0distributed\u00a0in\u00a0our\u00a0study\u00a0area\u00a0so\u00a0randomly?\u00a0Of\u00a0course\u00a0not.\u00a0We know that depth will increase, in principle, as we move away from the coast. If we\u00a0draw\u00a0a\u00a0line off the coast\u00a0:<\/p>\n

$\"\"$ <\/p>\n

and that we\u00a0project\u00a0the\u00a0value\u00a0of\u00a0our\u00a0points on the plane\u00a0defined\u00a0by\u00a0this\u00a0line\u00a0:<\/p>\n

$\"\"$ <\/p>\n

we\u00a0notice\u00a0that indeed\u00a0, the\u00a0cloud\u00a0of\u00a0our\u00a0points\u00a0shows\u00a0a\u00a0relationship between the distance to the\u00a0coast\u00a0and the\u00a0depth\u00a0observed\u00a0.<\/p>\n

That is what\u00a0we\u00a0will call\u00a0a\u00a0trend\u00a0in\u00a0our\u00a0spatial data:\u00a0when\u00a0a\u00a0function is verified\u00a0between the\u00a0values\u00a0of the points and\u00a0a\u00a0given\u00a0direction\u00a0in the three\u00a0spatial\u00a0axes.<\/p>\n

What to do and<\/strong>\u00a0why<\/strong>.<\/p>\n

If\u00a0you apply\u00a0a\u00a0geostatistical method\u00a0(kriging, for\u00a0example) on\u00a0the\u00a0data\u00a0that shows a\u00a0trend\u00a0,\u00a0you\u00a0are not\u00a0respecting\u00a0the\u00a0basic\u00a0assumptions\u00a0of\u00a0geostatistics\u00a0.\u00a0The\u00a0phenomenon\u00a0that is\u00a0going\u00a0to stand out\u00a0concealing\u00a0any\u00a0other\u00a0phenomenon\u00a0is\u00a0the distance to the\u00a0flank.\u00a0You are\u00a0going\u00a0to mess around\u00a0with a\u00a0complicated tool\u00a0to\u00a0get\u00a0a\u00a0result\u00a0that will not be\u00a0better\u00a0than the one\u00a0you could get\u00a0with\u00a0a\u00a0deterministic method\u00a0\u00a0(IDW, Spline\u00a0, …) much simpler to\u00a0execute\u00a0.<\/p>\n

If you are interested in that trend, forget the geostatics and use the classic deterministic interpolating methods.<\/p>\n

However,\u00a0if\u00a0you are interested in researching whether there are more subtle phenomena that determine the depths:<\/p>\n

$\"\"$ <\/p>\n

The midpoint between the\u00a0two\u00a0black\u00a0bars\u00a0is explained\u00a0by the distance to the\u00a0coast.\u00a0But, the dispersion of\u00a0values\u00a0for\u00a0each\u00a0distance,\u00a0can be modelled or not? Here is where\u00a0geostatistics comes into play: \u00a0if\u00a0you\u00a0want to, not only,\u00a0forecast\u00a0the\u00a0values\u00a0as a function of\u00a0distance\u00a0to the coast, but also, as function of those\u00a0phenomena\u00a0invisible at first\u00a0sight.<\/p>\n

Trend<\/strong>\u00a0analysis<\/strong><\/p>\n

The Geostatistical Analyst\u00a0tool to analyse the trend can\u00a0be helpful to identify\u00a0global trends\u00a0in\u00a0the\u00a0entry data\u00a0set.
\nThis\u00a0tool\u00a0\u00a0\u00a0provides\u00a0a\u00a0three\u00a0\u2013\u00a0dimensional view of the\u00a0data.\u00a0The various locations\u00a0of the sampling\u00a0points\u00a0are\u00a0plotted\u00a0on the x, y plan. Besides, \u00a0for each\u00a0sampling\u00a0point,\u00a0the\u00a0value\u00a0is defined\u00a0by the height\u00a0\u00a0\u00a0in\u00a0the z dimension\u00a0\u00a0\u00a0.\u00a0The interest\u00a0of\u00a0the\u00a0trend\u00a0analysis\u00a0tool lies\u00a0in\u00a0the fact that the\u00a0values\u00a0are\u00a0next projected\u00a0on the x, z and y plans, z\u00a0depicts the\u00a0scatter\u00a0plots.<\/p>\n

$\"\"$ <\/p>\n

This can\u00a0be\u00a0considered\u00a0as\u00a0lateral views\u00a0through the\u00a0three-dimensional data.\u00a0Then the\u00a0polynomials\u00a0are\u00a0adjusted\u00a0through the\u00a0scatterplots\u00a0on the\u00a0projected\u00a0plans.<\/p>\n

By\u00a0applying\u00a0rotations on the\u00a0different\u00a0axes\u00a0you\u00a0can\u00a0observe the\u00a0result\u00a0live,\u00a0on the\u00a0points\u00a0cloud\u00a0projected\u00a0onto\u00a0each\u00a0plane\u00a0as well as the\u00a0adjustment\u00a0polynomial\u00a0calculated curve.<\/p>\n

$\"\"$ <\/p>\n

This\u00a0returns to off the coast straight line\u00a0of\u00a0our\u00a0example\u00a0and\u00a0turn it\u00a0in\u00a0all\u00a0the\u00a0directions in order to figure out\u00a0which\u00a0is the most\u00a0marked trend.<\/p>\n

By\u00a0rotating\u00a0the axes, we get to\u00a0see\u00a0both\u00a0adjustment curves.\u00a0The\u00a0blue curve\u00a0adjusts very\u00a0closely the\u00a0cloud\u00a0of blue dots, for\u00a0a given\u00a0orientation.\u00a0In\u00a0our\u00a0example, the\u00a0green curve,\u00a0perpendicular to the blue curve,\u00a0does not indicate\u00a0any\u00a0visible\u00a0trend.<\/p>\n

Trend<\/strong>,\u00a0autocorrelation<\/strong>\u00a0and<\/strong>\u00a0nugget effect<\/strong><\/p>\n

A\u00a0surface\u00a0can\u00a0be\u00a0composed\u00a0of\u00a0two main\u00a0components:\u00a0a\u00a0\u00a0\u00a0fixed global trend<\/strong>\u00a0\u00a0\u00a0 and\u00a0a\u00a0\u00a0\u00a0short range random<\/strong>\u00a0variation<\/strong>.<\/p>\n

The global trend is frequently called fixed average structure. Short range random variation (also called random error) can be modelled in two parts: spatial autocorrelation and nugget effect. These two notions are used when modelling the semi-variogram.<\/p>\n

If\u00a0you\u00a0identify a\u00a0global trend\u00a0in\u00a0your\u00a0data,\u00a0then\u00a0you\u00a0must\u00a0decide\u00a0how to\u00a0model it.\u00a0Whether you will use a deterministic or geostatic method to create a surface depends of your final objective. If\u00a0you\u00a0only want\u00a0to model\u00a0the global\u00a0trend\u00a0and\u00a0create\u00a0a\u00a0smooth\u00a0surface,\u00a0you\u00a0can\u00a0use\u00a0a local or global\u00a0interpolating\u00a0polynomial method\u00a0to\u00a0perform a\u00a0final interpolation.<\/p>\n

However, you can handle the trend with a geostatistical method. First of all, we have to delete the trend, by subtracting the trend at each point and by keeping the residue as inputs of the geostatistical method.<\/p>\n

Then, we model the residues (the remaining component) as short range random variations by using a Kriging method.<\/p>\n

When we obtain the result, we\u00a0must add the result\u00a0to\u00a0each\u00a0mesh, the\u00a0value\u00a0of the calculated\u00a0trend for\u00a0this\u00a0mesh\u00a0.<\/p>\n

The\u00a0main\u00a0reason\u00a0to\u00a0delete\u00a0a\u00a0trend\u00a0in\u00a0geostatistics\u00a0is\u00a0to\u00a0satisfy\u00a0the\u00a0stationary hypothesis.<\/p>\n

If\u00a0you\u00a0break down\u00a0your\u00a0data\u00a0in\u00a0trend\u00a0plus the\u00a0short\u00a0reach variation,\u00a0you\u00a0suppose\u00a0the\u00a0trend\u00a0is\u00a0fixed and that\u00a0short\u00a0range variation is\u00a0random. Here,\u00a0random\u00a0does\u00a0not\u00a0mean\u00a0\u201dunpredictable\u00a0\u00bb\u00a0but\u00a0rather\u00a0what\u00a0is\u00a0ruled by\u00a0the probability laws\u00a0that\u00a0include\u00a0dependence\u00a0on\u00a0neighbouring values otherwise\u00a0called\u00a0autocorrelation\u00a0.<\/p>\n

The final surface will be the\u00a0sum\u00a0of the fixed and\u00a0random\u00a0surfaces.\u00a0In\u00a0other\u00a0terms,\u00a0think\u00a0that\u00a0you\u00a0will\u00a0add\u00a0two\u00a0layers to\u00a0achieve\u00a0the\u00a0final result:\u00a0that\u00a0which\u00a0never\u00a0changes\u00a0(trend),\u00a0and\u00a0one\u00a0that changes\u00a0randomly\u00a0(autocorrelation).<\/p>\n

If\u00a0you\u00a0can\u00a0identify and quantify the\u00a0trend,\u00a0you\u00a0will get a\u00a0deeper\u00a0understanding\u00a0of\u00a0your\u00a0data\u00a0and, therefore,\u00a0you\u00a0will be able to make\u00a0better\u00a0decisions.\u00a0If\u00a0you\u00a0remove\u00a0the\u00a0trend,\u00a0you\u00a0will be\u00a0able\u00a0to\u00a0model\u00a0more accurately\u00a0the\u00a0random\u00a0variation\u00a0at\u00a0short\u00a0range\u00a0because the global\u00a0trend\u00a0\u00a0will\u00a0not\u00a0influence\u00a0your\u00a0spatial\u00a0analysis\u00a0.<\/p>\n

Examination<\/strong>\u00a0of the<\/strong>\u00a0trend<\/strong>\u00a0using global<\/strong>\u00a0trend<\/strong>\u00a0analysis<\/strong><\/p>\n

The\u00a0geostatistical analyst Trend Tool\u00a0projects\u00a0the points\u00a0in\u00a0two\u00a0directions (default,\u00a0north\u00a0and\u00a0east) on plans\u00a0perpendicular\u00a0to the\u00a0map\u00a0plan.<\/p>\n

A\u00a0polynomial curve is\u00a0adapted\u00a0to\u00a0each\u00a0projection<\/a>.\u00a0The entire\u00a0surface of the map\u00a0can\u00a0be\u00a0turned\u00a0in\u00a0anything\u00a0independent of the\u00a0direction,\u00a0which\u00a0also amends\u00a0the direction\u00a0represented\u00a0by the\u00a0projected plans.\u00a0If the\u00a0curve\u00a0through the\u00a0projected\u00a0points\u00a0is\u00a0flat,\u00a0no\u00a0trend\u00a0exists,\u00a0as\u00a0shown\u00a0in\u00a0the\u00a0green line\u00a0in\u00a0the plan\u00a0projected\u00a0to the left\u00a0in\u00a0the above image.<\/p>\n

If the\u00a0polynomial\u00a0is defined\u00a0as\u00a0a\u00a0\u00a0curved\u00a0model (up\u00a0or\u00a0down,\u00a0as\u00a0shown\u00a0with the\u00a0 blue line\u00a0in\u00a0the\u00a0projected\u00a0left plan\u00a0in\u00a0the\u00a0above diagram\u00a0),\u00a0this\u00a0suggests\u00a0a\u00a0trend\u00a0\u00a0in\u00a0the global\u00a0data.<\/p>\n

It\u00a0suggests\u00a0that a\u00a0second-\u00a0order\u00a0polynomial\u00a0can\u00a0be\u00a0adapted\u00a0to the\u00a0data.\u00a0Thanks\u00a0to the\u00a0refinement\u00a0granted by\u00a0the\u00a0trend\u00a0analysis tool,\u00a0we\u00a0can\u00a0identify\u00a0the real orientation\u00a0of the\u00a0trend.\u00a0In\u00a0this case,\u00a0the strongest influence\u00a0is\u00a0from\u00a0north-east\u00a0to\u00a0south-west.<\/p>\n

Trend order<\/strong><\/p>\n

Once you have\u00a0detected\u00a0a\u00a0trend,\u00a0you\u00a0still have to define its order (first,\u00a0second\u00a0,\u00a0third\u00a0, …).\u00a0Even\u00a0if\u00a0it\u00a0seems\u00a0complicated,\u00a0it is\u00a0very\u00a0simple.<\/p>\n

Take\u00a0a\u00a0sheet\u00a0of paper.<\/p>\n

Keep\u00a0it\u00a0right\u00a0and\u00a0incline it. \u00a0If\u00a0your\u00a0trend\u00a0looks\u00a0like\u00a0that,\u00a0you\u00a0have\u00a0a\u00a0first\u00a0order\u00a0trend.\u00a0At\u00a0the\u00a0tool\u00a0trend\u00a0geostatistical analyst level it\u00a0translates as\u00a0adjustments\u00a0in\u00a0two\u00a0straight lines\u00a0.<\/p>\n

Bend\u00a0the\u00a0sheet\u00a0so as\u00a0to\u00a0create\u00a0a\u00a0hollow\u00a0and\u00a0a\u00a0hump.\u00a0If\u00a0your\u00a0tendency\u00a0to\u00a0this\u00a0shape,\u00a0you\u00a0have\u00a0a\u00a0second order trend. At\u00a0the\u00a0geostatistical analyst tool trend\u00a0this\u00a0translates\u00a0into an adjustment\u00a0as two curves, each one with its own curvature.<\/p>\n

And\u00a0so\u00a0on.\u00a0You\u00a0simply\u00a0must\u00a0count\u00a0the\u00a0number\u00a0of\u00a0bends\u00a0you\u00a0find\u00a0and\u00a0add\u00a01, to\u00a0have\u00a0the order\u00a0of the\u00a0polynomial.<\/p>\n","protected":false},"excerpt":{"rendered":"

Following\u00a0the article\u00a0\u00a0\u00a0Introduction to\u00a0\u00a0Exploratory\u00a0data\u00a0analysis for\u00a0geostatistics\u00a0\u00a0\u00a0we\u00a0will\u00a0discuss\u00a0each\u00a0of the\u00a0available tools\u00a0to\u00a0perform the exploratory\u00a0analysis\u00a0of the spatial\u00a0data.\u00a0We\u00a0have\u00a0discussed\u00a0the\u00a0 histograms\u00a0,\u00a0\u00a0\u00a0\u00a0QQ-Plots\u00a0, and\u00a0Vorono\u00ef maps\u00a0.\u00a0Now, it is the turn for the data trends. Trend?\u00a0What is\u00a0it? Firstly we have to know what we are looking for. Certainly, you are…<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_bbp_topic_count":0,"_bbp_reply_count":0,"_bbp_total_topic_count":0,"_bbp_total_reply_count":0,"_bbp_voice_count":0,"_bbp_anonymous_reply_count":0,"_bbp_topic_count_hidden":0,"_bbp_reply_count_hidden":0,"_bbp_forum_subforum_count":0,"sfsi_plus_gutenberg_text_before_share":"","sfsi_plus_gutenberg_show_text_before_share":"","sfsi_plus_gutenberg_icon_type":"","sfsi_plus_gutenberg_icon_alignemt":"","sfsi_plus_gutenburg_max_per_row":"","_monsterinsights_skip_tracking":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_feature_clip_id":0,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_post_was_ever_published":false},"categories":[1260],"tags":[],"class_list":["post-5594","post","type-post","status-publish","format-standard","hentry","category-non-classe-en"],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t