[MUSIC] In this lecture, you will study spatial interpolation, also known as spatial smoothing, and three different methods including trend surface analysis, inverse distance weighting, and kriging. For your better understanding, I would like to start with three categories of estimations which are, filtering, interpolation, and extrapolation, or prediction. The categorization is dependent on where the estimation is made. If the estimate is conducted at the point or time where we have an observation, it is filtering. If we find an estimate for a point or time within the range of observations where no observation is made, it is called interpolation. If finding an estimate beyond the range of observations, we call it extrapolation or prediction. The figure illustrates the three types of estimations, filtering, interpolation, and extrapolation on one-dimensional space, for example, on time dimension. We can extend the same concept to two-dimensional space, then we can imagine what spatial interpolation means. The given problem is a set of precipitation observations at weather stations in Texas. The observations were made at only marginal number of weather stations, two dimensional points. So we would like to estimate precipitation at other locations where no observations were made. This is a typical spatial interpolation problem. There are three different methods, as I mentioned here, which are trend surface analysis, inverse distance weighting, and kriging. Trend surface analysis is a kind of regression method. In which x,y coordinates are independent variables with respect to one target variable z, in this case. In the previous example, z is precipitation. Generally, we can use first, second, or even third order polynomial functions. Here a second polynomial is given as an example. Instead of polynomials of x, y other models such as negative exponential function is also often used for spatial interpolation of socio-economic data. The given equation is an example in which D stands for population density as a target variable, and r is independent variable, the distance from Central Business District, CBD. There are numerous variations of the given negative exponential function for socio-economic modeling. An outcome of trend surface analysis for precipitation data in Texas is given here. Which is based on first order polynomial and it clearly shows linear pattern of the estimated trend surface. Now you are looking at another trend surface based on second order polynomial. Where do you think you can we apply the method? As the name implies, it is definitely useful to see a global trend of a given spatial data. On the contrary, it would be a wrong choice to see local variation. For taking local variations into consideration, inverse distance weighting could be an alternative choice for spatial interpolation. IDW, which stands for inverse distance weighting, is a heuristic method of weighted averaging of surrounding observations. The key issue of the method is how to determine the weight? As the name goes, it is based on inverse of distance with a power. The larger the power K, the more local variation is considered in spatial interpolation. The outcomes of IDW with the same precipitation data are illustrated in the figure. You can see the local variation is more preserved while the power K is increased. However, you can also notice that IDW with a larger K is not representing the general trend very well. You have studied trend surface analysis, and IDW for spatial interpolation problems and their pros and cons. Another serious problem is that both methods would not present the best estimation. In other words, they are rather heuristic approaches than mathematical rigid ones. Kriging is the solution to the problem. The best estimation, mathematically rigid approach which is best linear unbiased estimation called as BLUE, B-L-U-E. It is considered as the de facto standard for spatial interpolation problem. The name kriging was named after D G Krige, a mining engineer and statistician in South Africa. Similarly to IDW, the estimation is formulated by a linear combination of observations and corresponding weights. So kriging is basically to find the best weight for the estimation. The weights can be computed from spatial interpolation. The given equation is defined as a variogram, which is a mathematical description to formulate spatial auto-correlation or dependency. In other words, how variance of observations are changed over distance between observations, which is h in the equation. With respect to given point, plots of variogram, a selected function is formulated by means of regression. It should be noted that we cannot choose arbitrary function for 15 points plot of variogram, only certain types of function can be used such as spherical, positive linear, exponential, Gaussian, and so on. Selection of function and estimation of coefficience is truly the key process of the kriging. High level of expertise is required for that. Now you are looking at the types of variogram, spherical, Gaussian, positive linear, and exponential. There are a variety of different kriging methods, each of them based on the assumption how the given space is categorized. For example, Ordinary kriging assumes that with respect to all the points small s in space large S, the expectation of property Z(s) is an unknown constant value. while simple kriging assumes a known constant for the expectation. On the other hand, universal kriging assumes that the expectation of property Z(s) has a trend or drift formulated by function f(s). Co-kriging takes other variables into consideration in addition to point location S. For example, imagine kriging for temperature which is clearly affected by elevation. In such cases, co-kriging should be the choice to deal with both point locations and elevation. Block Kriging is designed for spatial interpolation for blocks of areas, not for points. Now let's try to apply kriging to a real dataset, the precipitation data in Texas. The first step is to decide which type of kriging is the best for the given dataset. If we choose one out of a simple kriging, ordinary or universal kriging, what would be the best choice? The answer is universal kriging, because it has clear spatial trend that the rainfall decrease in the direction of the east to the west, which should be incorporated to the spatial interpolation. The next step is develop a variogram. And we chose a spherical function among different models of variogram. In fact, the selection of variogram model requires some expertise because it is essential step in kriging and there are numerous variations. By the way, three new terms, range, sill, and nugget are introduced here. Range represent the distance, where the model flattens out. It is the effective distance of spatial interpolation. If the distance is farther than the range, then no spatial interpolation can be considered. Sill is the variance value at the range, which implies the variance of the data set. Nugget is the y-intercept of the variogram, and it represents the local variability of data set, which can be interpreted as measurement error. Now you are looking at kriged surface based on the universal kriging, with the spherical variogram model. Kriging has quite a few superior features over other methods for spatial interpolation. First of all, it is a mathematical rigid method, as I mentioned, which make the estimates have characteristics of, number one, unbiasedness, and number two, minimum variance, which is a great characteristics. Second of all, it can present standard error of the estimate. In other words, you can have information over uncertainty of your estimation. In this lecture you study the concept of spatial interpolation and three different methods, trend surface analysis, inverse distance weighting and kriging. Kriging is the de facto standard in spatial interpolation. But it requires a series of steps to get the result, including selection of kriging types, selection of estimation of variogram model. In fact, building variogram model is the core step as well as a difficult step in kriging There are numerous applications in which spatially related data are collected and estimates of fill ins are required. Such as environmental science, natural resource mapping, and socio-economic analysis and so on. All right, this is the end of this lecture. See you all in the next lecture.