Efficient updating of kriging estimates and variances


01-Mar-2020 00:51

The semivariogram then is the sum of squared differences between values separated by a distance .As an aside, contrast this with the formulation for variance, Here, is the number of data points, is the sample mean, and is a data point.Our kriging function takes the data set def krige( P, model, hs, bw, u, N ): ''' Input (P) ndarray, data (model) modeling function - spherical - exponential - gaussian (hs) kriging distances (bw) kriging bandwidth (u) unsampled point (N) number of neighboring points to consider ''' # covariance function covfct = cvmodel( P, model, hs, bw ) # mean of the variable mu = np.mean( P[:,2] ) # distance between u and each data point in P d = np.sqrt( ( P[:,0]-u[0] )**2.0 ( P[:,1]-u[1] )**2.0 ) # add these distances to P P = np.vstack(( P. T # sort P by these distances # take the first N of them P = P[d.argsort()[: N # apply the covariance model to the distances k = covfct( P[:,3] ) # cast as a matrix k = np.matrix( k ).

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The semivariogram then is the sum of squared differences between values separated by a distance .

As an aside, contrast this with the formulation for variance, Here, is the number of data points, is the sample mean, and is a data point.

Our kriging function takes the data set def krige( P, model, hs, bw, u, N ): ''' Input (P) ndarray, data (model) modeling function - spherical - exponential - gaussian (hs) kriging distances (bw) kriging bandwidth (u) unsampled point (N) number of neighboring points to consider ''' # covariance function covfct = cvmodel( P, model, hs, bw ) # mean of the variable mu = np.mean( P[:,2] ) # distance between u and each data point in P d = np.sqrt( ( P[:,0]-u[0] )**2.0 ( P[:,1]-u[1] )**2.0 ) # add these distances to P P = np.vstack(( P. T # sort P by these distances # take the first N of them P = P[d.argsort()[: N]] # apply the covariance model to the distances k = covfct( P[:,3] ) # cast as a matrix k = np.matrix( k ).

T # form a matrix of distances between existing data points K = squareform( pdist( P[:,:2] ) ) # apply the covariance model to these distances K = covfct( K.ravel() ) # re-cast as a Num Py array -- thanks M. K = np.array( K ) # reshape into an array K = K.reshape(N, N) # cast as a matrix K = np.matrix( K ) # calculate the kriging weights weights = np.linalg.inv( K ) * k weights = np.array( weights ) # calculate the residuals residuals = P[:,2] - mu # calculate the estimation estimation = np.dot( weights.

This Kalman filter was first described and partially developed in technical papers by Swerling (1958), Kalman (1960) and Kalman and Bucy (1961). The fact that the MIT engineers were able to pack such good software (one of the very first applications of the Kalman filter) into such a tiny computer is truly remarkable. In fact, some of the special case linear filter's equations appeared in these papers by Stratonovich that were published before summer 1960, when Kalman met with Stratonovich during a conference in Moscow.

The Apollo computer used 2k of magnetic core RAM and 36k wire rope [...]. Kalman filters have been vital in the implementation of the navigation systems of U. Navy nuclear ballistic missile submarines, and in the guidance and navigation systems of cruise missiles such as the U. Kalman filter uses a system's dynamic model (e.g., physical laws of motion), known control inputs to that system, and multiple sequential measurements (such as from sensors) to form an estimate of the system's varying quantities (its state) that is better than the estimate obtained by using only one measurement alone.

The semivariogram is given by, Here, is distance specified by the user, and and are two points that are separated spatially by .

The term is the number of points we have that are separated by the distance .

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In statistics and control theory, Kalman filtering, also known as linear quadratic estimation (LQE), is an algorithm that uses a series of measurements observed over time, containing statistical noise and other inaccuracies, and produces estimates of unknown variables that tend to be more accurate than those based on a single measurement alone, by estimating a joint probability distribution over the variables for each timeframe. Kálmán, one of the primary developers of its theory.Furthermore, the Kalman filter is a widely applied concept in time series analysis used in fields such as signal processing and econometrics.



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