TG3102 GEOSTATISTIK II 19 August 2019 1 Kegiatan Perkuliahan • Kehadiran minimal 80 % pertemuan • Tugas 3-4 pertemuan • Kuliah 11-12 pertemuan Evaluasi • Tugas 30 % • UTS 35 % • UAS 35 % Topics 1. 2. 3. 4. What is geostatistics The added value of geostatistics Geostatistical computing Project for statistical computing 3 1 What is "geostatistics"? • What is "statistics"? • What then is "geo"-statistics? 4 1.1 What is "statistics"? • The term "statistics" has two common meanings, which we want to clearly separate: descriptive and inferential statistics. • But to understand the difference between descriptive and inferential statistics, we must first be clear on the difference between populations and samples. 5 Populations and samples • A population is a set of well-defined objects – We must be able to say, for every object, if it is in the population or not – We must be able, in principle, to find every individual of the population A geographic example of a population is all pixels in a multi-spectral satellite image • A sample is a subset of a population – We must be able to say, for every object in the population, if it is in the sample or not – Sampling is the process of selecting a sample from a population Continuing the example, a sample from this population could be a set of pixels from known ground truth points 6 What do we mean by "statistics"? • Two common uses of the word: • Descriptive statistics: numerical summaries of samples – i.e., what was observed – Note the ‘sample’ may be exhaustive, i.e., identical to the population • Inferential statistics: from samples to populations – i.e., what could have been or will be observed in a larger population Example: Descriptive "The adjustments of 14 GPS control points for this orthorectification ranged from 3.63 to 8.36 m with an arithmetic mean of 5.145 m" Inferential "The mean adjustment for any set of GPS points taken under specified conditions and used for orthorectification is no less than 4.3 and no more than 6.1 m; this statement has a 5% probability of being wrong" 7 Inference 8 A concise definition of inferential statistics • Statistics: "The determination of the probable from the possible" – Davis, Statistics and Data Analysis in Geology, p. 6 • ...which implies the rigorous definition and then quantification of "probable" – Probable causes of past events or observations – Probable occurrence of future events or observations • Then, a definition of inferential statistics: Observations ⇒ Inferences 9 • As humans, we infer constantly from the evidence around us. For example, I observe a person who can not walk a straight line, whose speech is slurred, and who smells strongly of alcohol - these are observable facts. I infer that the person is drunk - yet I didn’t see the person drink alcohol, and I haven’t analyzed the person’s blood alcohol level. Often we do not realize the line between observable facts and inferences • In statistical inference we quantify the degree of probability that our inference is true 10 Why use statistical analysis? • Descriptive: we want to summarize some data in a shorter form • Inferential: We are trying to understand some process and maybe predict based on this understanding – So we need to model it, i.e. make a conceptual or mathematical representation, from which we infer the process – But how do we know if the model is "correct"? • Are we imagining relations where there are none? • Are there true relations we haven’t found? – Statistical analysis gives us a way to quantify the confidence we can have in our inferences 11 • The most common example of geostatistical inference is the prediction of some attribute at an unsampled point, based on some set of sampled points. • In the next two slides we show an example from the Meuse river floodplain in the southern Netherlands. The copper (Cu) content of soil samples has been measured at 155 points (left figure); from this we can predict at all points in the area of interest (right figure). 12 Google Maps Photo credit: www.binnenvaartinbee… on Panoramio Example of Inference Copper (Cu) content of soil samples 14 1.2 What is “geo”-statistics? • Geostatistics is statistics on a population with known location, i.e. coordinates: – In one dimension (along a line or curve) – In two dimensions (in a map or image) – In three dimensions (in a volume) • The most common application of geostatistics is in 2D (maps). Key point: Every observation (sample point) has both: – coordinates (where it is located); and – attributes (what it is). 15 • Let’s first look at a data set that is not geo-statistical – It is a list of soil samples (without their locations) with the lead (Pb) concentration. – The column Pb is the attribute of interest. Observation_ID 1 2 3 4 5 6 7 8 9 10 11 12 Pb 77.36 77.88 30.8 56.4 66.4 72.4 60 141 52.4 41.6 46 56.4 16 • Now we look at a data set that is geo-statistical. – These are soil samples taken in the Jura mountains of Switzerland, and their lead content; but this time with their coordinates – The columns E and N are the coordinates, i.e. the spatial reference; the column Pb is the attribute Observation_ID E N Pb 1 2.386 3.077 77.36 2 2.544 1.972 77.88 3 2.807 3.347 30.80 4 4.308 1.933 56.40 5 4.383 1.081 66.40 6 3.244 4.519 72.40 7 3.925 8 2.116 3.785 3.498 60.00 141.00 9 1.842 52.40 10 1.709 0.989 1.843 11 3.800 4.578 46.00 12 2.699 1.199 56.40 41.60 17 • On the figure (next slide) you will see: – A coordinate system (shown by the over-printed grid lines) – The locations of 256 sample points - where a soil sample was taken – The attribute value at each sample point - symbolized by the relative size of the symbol at each point- in this case the amount of lead (Pb) in the soil sample • This is called a post-plot ("posting" the value of each sample) or a bubble plot (the size of each "bubble" is proportional to its attribute value). 18 Post-plot of Pb values, Swiss Jura 19 • So, we know that each sample has a location. What is so special about that? After all, the attribute information is the same. What is the value-added of knowing the location? What new possibilities for analysis does this imply? 20 2 The added value of geostatistics • The location of a sample is an intrinsic part of its definition • All data sets from a given area are implicitly related by their coordinates – So they can be displayed and related in a GIS • Values at sample points can not be assumed to be independent: there is often evidence that nearby points tend to have similar values of attributes. – – – – That is, there may be a spatial structure to the data Classical statistics assumes independence of samples But, if there is a spatial structure, this is not true! This has major implications for sampling design and statistical inference • Data values may be related to their coordinates → spatial trend 21 • Let’s look again at the post-plot, this time to see if we can discover evidence of spatial dependence that is, points that are close to each other have similar attribute values. 22 Post-plot of Pb values, Swiss Jura • Do large circles (representing high Pb concentrations) seem to form clusters? • Do small circles (representing low Pb concentrations) seem to form clusters? • What is the approximate radius of the clusters? 23 3 Feature and geographic spaces • The word space is used in mathematics to refer to any set of variables that form metric axes and which therefore allow us to compute a distance between points in that space. • If these variables represent geographic coordinates, we have a geographic space. • If these variables represent attributes, we have a feature space. 24 • It is important to understand these two uses of the word space, because we often want to contrast an analysis in feature space (not taking spatial position into account) with an analysis in geographic space (considering spatial position as the key element in the analysis). • Let's see an example of a feature space. This concept should be familiar from non-spatial statistics, although the term “feature space” may be new to you. 25 Scatterplot of a 2D feature space This is a visualization of a 2D feature space using a scatterplot. The points are individual iris flowers measured by Edgar Anderson in 1935 and published in The irises of the Gaspe Peninsula, Bulletin of the American Iris Society, 59, 2-5. 26 3.1 Feature space • This “space” is not geographic space, but rather a mathematical space formed by any set of variables: – Axes are the range of each variable – Coordinates are values of variables, possibly transformed or combined – The observations are related in this “space”, e.g. the “distance” between them can be calculated – We often plot variables in this space, e.g. scatterplots in 2D or 3D • Note: Feature space is sometimes referred to as attribute space. 27 • You are probably quite familiar with feature space from your study of non-spatial statistics. • Even with one variable, we have a unit of measure; this forms a 1D or univariate feature space. • Most common are two variables which we want to relate with correlation or regression analysis; this is a bivariate feature space. • In multivariate analysis the feature space has more than two dimensions. 28 • Multivariate feature spaces can have many dimensions; we can only see three at a time. • The next slide is a visualization of a 3D feature space using a 3D-scatterplot. Again the observations are from the Anderson Iris data. 29 Scatterplot of a 3D feature space Anderson Iris data 30 • So, feature space is perhaps a new term but not a new concept if you've followed a statistics course with univariate, bivariate and multivariate analysis. • What then is geographic space? Simply put, it is a mathematical space where the axes are map coordinates that relate points to some reference location on or in the Earth. – These coordinates are often in some geographic coordinate system that was designed to give each location on (part of) the Earth a unique identification; a common example is the Universal Transverse Mercator (UTM) grid. – However, a local coordinate system can be used, as long as there is a clear relation between locations and coordinates. 31 3.2 Geographic space • Axes are 1D lines; they almost always have the same units of measure (e.g. meters, kilometers…) • One-dimensional: coordinates are on a line with respect to some origin (0): (x1) = x • Two-dimensional: coordinates are on a grid with respect to some origin (0,0): (x1,x2) = (x,y) = (E,N) • Three-dimensional: coordinates are grid and elevation from a reference elevation: (x1,x2,x3) = (x,y,z) = (E,N,H) • Note: latitude-longitude coordinates do not have equal distances in the two dimensions; they should be transformed to metric (grid) coordinates for geo-statistical analysis. 32
