Analysis

1. Simple Analysis
2. Correlation
3. Advanced Analysis
4. Complex Analysis
5. Radial Basis Function Network (RBF)
6. Counter Prop Network
7. Hill climbing Algorithm and Cluster_2_value
8. Sensitivity Analysis

Simple Analysis

In SAMT are integrated some simple analysis methods like a statistic and a histogram function. The "STAT" calculates a set of statistical values of a grid like minimum, maximum, mean, standard deviation etc..

The "HIST" function draws a histogram with number of bins in parameter1 (or as default 250 bins). This function gives a good overview of a selected grid.

In addition there is a little "INFO" function that tells the number of rows, number of cols and the nodata values (in the line edit window).
The correlation of two grids is sometimes interesting. The method correlation calculates correlation of the grid in P1 and the grid in P2 and shows the correlation in P1 and the variance of map1 in P2 and variance of map2 in P3.

The is connected with a routine for a chart along a line. This means a line (click with the right mouse button and hold it, move it to the second position an give it release button) will appear in the map and a chart comes up with the values along this line. If you change the map you can use the same line in the changed map. This is sometimes useful to compare parts of different maps.

The same procedure allows to draw a power spectra or a semi variogram along the selected line. This procedures are located in the menu Analysis.

Correlation

The correlation is useful to compare two different maps. Firstly it calculates the covariance of the two maps:

cov(map1,map2)=sum((map1[i,j]-m1)*(map2[i,j]-m2))
with m1=mean(map1) and m2=mean(map2)

The covariance and the two variances of map1 and map2 were be used to calculate the correlation:

cov(map1,map2)/(sqrt(variance(map1))*sqrt(variance(map2))

The result in P2=variance(map1) and P3=variance(map2).

Advanced Analysis

The advanced analysis is connected with three dimensional views. There are special views to the data, like a simple elevation model, and a splatter model. This models have used the 3D visualization library vtk. In future we will use the new features of qt4.x The first idea was to include the 3D visualization into SAMT. But this has some disadvantages:

• The 3D-visualization could not be exchanged.
• The 2D-visualization can be done by qt itself. (you do not need additional software if you do not use 3D)
• Bug in visualization could damage SAMT.

In this version of SAMT the 3D-visualization is an extra process and runs independently from SAMT. The data transfer between 3D and SAMT uses a temporary hdf-file. This is fast and stable. Our intention is to develop a software for visualization of wind- or water- erosion, for dynamic change in sense of artifical live etc.. There is a lot to do.
The "SPLATTER" function calculates a Gaussian splatter from 10.000 randomly selected grid cells. This grid must be put in the parameter: P1, P2, P3. It is optional for a selected grid to produce a red colored splatter (for values better than 0.75) and added to the grayed splatter from P1, P2, P3. This technique can produce an overview of inputs in relation to results (maybe model output or results of an obersevation).
The "Elevation" function produces a 3D view of a selected grid. Parameters are a scale factor [1,99] and a smoothing factor [1,25].

Complex Analysis

Complex analysis functions include a cluster algorithm and a kohonen feature map. As an external representation is used the hasse diagram. That means the analysis function is performed by an external program that communicates over a socket interface with SAMT.
The implemented cluster algorithm is a simple method to cluster up to three grids and put it in a selected number of clusters (parameter1). The algorithm performs the following steps:

1. select n random points in the space - the so called cluster centers
2. calculate for every grid cell the euclidian distance to the centers
3. assign the grid cells to the centers for minimum distance
4. move the centers into its centare of gravity
5. terminate if the centers are stable

The cluster algorithm will produce a new grid called "cluster1". This grid contains the cluster number of the center.
A little modification of the algorithm is useable if additional parameter is used. This modification selects the initial class centers as points of a circle. This method is sometimes advantageous.

The second complex analysis function is more complicated. It is a kohonen feature map. The basic idea is that the cluster centers are modified not independent of the location of activation.

In the first step a map of nodes is created (for example 5*5 nodes). The nodes contain a vector (1..3 dimension) of inputs. This vector is set randomly. In the training phase a location is randomly selected. The best matching node is called the winner. This winner will be moved a bit in direction of the selected input. But not only the winner, also the neighbor of it will be moved. The strength of the movement depends on the distance of the node to the winner. A small distance means a big movement and a big distance a small movement. This special training method has a lot of interesting properties. A main feature is its good sensitivity in regions of the map that are frequent in the map. And nodes that are neighbours in the grid are also neighbors in the map. Neighbors in a grid mean not the real location, but the distance in the data vector. A kohonen map selects regions from a (up to three) grid with similar grid values. This can give good idea of to the underlying data structure.
The cluster and the kohonen map are not very helpful without the additional tool, the so called rbf-neural networks.

RBF neural networks

The radial basis function network is a special type of network. It is more robust than a feed forward network but it is not so powerful. The basic idea is to use a trained kohon map or a cluster as basis and build a linear combination of fixed parameters to the target. The parameters will be calculated using a householder transformation. For a detailed description see: Gershenfeld, N.: "The Nature of Mathematical Modelling", 2002 Cambridge UP. To use a rbf in SAMT is really simple. All what you need is a cluster (kohon) of 2 or 3 inputs and a target map. after training the kohon or cluster the rbf will be initialized. P1 holds the 0/1 for cluster/kohon, P2 hold the number of inputs 2/3. After this step a "rbf" appears in the model part of SAMT. Put the target map in P1 and call rbf using the run button. After a few seconds a new map "rbf1" appears and contains the result. Remark, the rbf model is automaticaly stored and can be used as a model without a new training of the inputs. For this reason please use the prepared model rbf_train and include it in your project. A trained rbf netowrk can now be used with new data sets to do real modeling tasks like the feed forward network. It is an alternative way and gives sometimes better training results than a feed forward network. It seems to be more sensitive against noise in data compared to the feed forward network but it depends on the concrete model. Please try it.

Counter Prop Network

A counter prop network is simplified version of an radial basis network. The basis is again a kohonen feature map or simpler a cluster. But this time a large number (more than 100) of inputs is preferable. This basis deals as classifier using the winner takes all rule. That means the best matching node is used. A simple training algorithm is used to adjust the output layer of the counter prop network.

Because of the winner takes all rule only one kohonen node is active and only one a_i is used to calculate the result. A large number of kohonen nodes can help to make this network more smooth. The best use of such a network would be to simulated a lot (50..200) different networks and overlay the result. This can produce very smooth results but it is very time consuming. The counter prop should be used as a simple and fast network that can be used as an additional method to the other networks.

Hill climbing algorithm and Cluster_2_value

The so called hill climbing algorithm is an alternative cluster algorithm. The number of clusters must be provided. The algorithm changes the members of a cluster to an other cluster if the variance will be smaller after the change. The algorithm stops if the number of iterations (P2) is reached or the difference of the variances are zero. The hill climbing procedure is very fast and can be checked alternatively to the k-means algorithms.

The Cluster_2_value is a simple but useful algorithm to assign a cluster number to a value. The value grid contains the real values of the target grid. To apply the cluster to the target a new map will be produced the contains the mean values of the target that contains to one cluster. The cluster map contains numbers of the cluster without any meaning:

With the values the map shows a real clustered elevation model:

Remark: The Cluster_2_value can be also used together with kohonen feature maps or k-means.