Estimation of forest variables using airborne laser scanning

Airborne laser scanning can provide three-dimensional measurements of the forest canopy with high efficiency and precision. There are presently a large number of airborne laser scanning instruments in operation. The aims of the studies reported in this thesis were, to develop and validate methods for estimation of forest variables using laser data, and to investigate the influence of laser system parameters on the estimates. All studies were carried out in hemi-boreal forest at a test area in southwestern Sweden (lat. 58°30’N, long. 13°40’ E). Forest variables were estimated using regression models. On plot level, the Root Mean Square Error (RMSE) for mean tree height estimations ranged between 6% and 11% of the average value for different datasets and methods. The RMSE for stem volume estimations ranged between 19% and 26% of the average value for different datasets and methods. On stand level (area 0.64 ha), the RMSE was 3% and 11% of the average value for mean tree height and stem volume estimations, respectively. A simulation model was used to investigate the effect of different scanning angles on laser measurement of tree height and canopy closure. The effect of different scanning angles was different within different simulated forest types, e.g., different tree species. High resolution laser data were used for detection of individual trees. In total, 71% of the field measurements were detected representing 91% of the total stem volume. Height and crown diameter of the detected trees could be estimated with a RMSE of 0.63 m and 0.61 m, respectively. The magnitude of the height estimation errors was similar to what is usually achieved using field inventory. Using different laser footprint diameters (0.26 to 3.68 m) gave similar estimation accuracies. The tree species Norway spruce (Picea abies L. Karst.) and Scots pine (Pinus sylvestris L.) were discriminated at individual tree level with an accuracy of 95%. The results in this thesis show that airborne laser scanners are useful as forest inventory tools. Forest variables can be estimated on tree level, plot level and stand level with similar accuracies as traditional field inventories

Fringe trees, Crump-Mode-Jagers branching processes and mm-ary search trees

This survey studies asymptotics of random fringe trees and extended fringe trees in random trees that can be constructed as family trees of a Crump-Mode-Jagers branching process, stopped at a suitable time. This includes random recursive trees, preferential attachment trees, fragmentation trees, binary search trees and (more generally) $m$-ary search trees, as well as some other classes of random trees. We begin with general results, mainly due to Aldous (1991) and Jagers and Nerman (1984). The general results are applied to fringe trees and extended fringe trees for several particular types of random trees, where the theory is developed in detail. In particular, we consider fringe trees of $m$-ary search trees in detail; this seems to be new. Various applications are given, including degree distribution, protected nodes and maximal clades for various types of random trees. Again, we emphasise results for $m$-ary search trees, and give for example new results on protected nodes in $m$-ary search trees. A separate section surveys results on height, saturation level, typical depth and total path length, due to Devroye (1986), Biggins (1995, 1997) and others. This survey contains well-known basic results together with some additional general results as well as many new examples and applications for various classes of random trees

Normal limit laws for vertex degrees in randomly grown hooking networks and bipolar networks

We consider two types of random networks grown in blocks. Hooking networks are grown from a set of graphs as blocks, each with a labelled vertex called a hook. At each step in the growth of the network, a vertex called a latch is chosen from the hooking network and a copy of one of the blocks is attached by fusing its hook with the latch. Bipolar networks are grown from a set of directed graphs as blocks, each with a single source and a single sink. At each step in the growth of the network, an arc is chosen and is replaced with a copy of one of the blocks. Using P\'olya urns, we prove normal limit laws for the degree distributions of both networks. We extend previous results by allowing for more than one block in the growth of the networks and by studying arbitrarily large degrees.Comment: 28 pages, 6 figure

Interactive Visual Analysis of Networked Systems: Workflows for Two Industrial Domains

We report on a first study of interactive visual analysis of networked systems. Working with ABB Corporate Research and Ericsson Research, we have created workflows which demonstrate the potential of visualization in the domains of industrial automation and telecommunications. By a workflow in this context, we mean a sequence of visualizations and the actions for generating them. Visualizations can be any images that represent properties of the data sets analyzed, and actions typically either change the selection of data visualized or change the visualization by choice of technique or change of parameters

Limit Laws for Functions of Fringe trees for Binary Search Trees and Recursive Trees

We prove limit theorems for sums of functions of subtrees of binary search trees and random recursive trees. In particular, we give simple new proofs of the fact that the number of fringe trees of size $k=k_n$ in the binary search tree and the random recursive tree (of total size $n$) asymptotically has a Poisson distribution if $k\rightarrow\infty$, and that the distribution is asymptotically normal for $k=o(\sqrt{n})$. Furthermore, we prove similar results for the number of subtrees of size $k$ with some required property $P$, for example the number of copies of a certain fixed subtree $T$. Using the Cram\'er-Wold device, we show also that these random numbers for different fixed subtrees converge jointly to a multivariate normal distribution. As an application of the general results, we obtain a normal limit law for the number of $\ell$-protected nodes in a binary search tree or random recursive tree. The proofs use a new version of a representation by Devroye, and Stein's method (for both normal and Poisson approximation) together with certain couplings

Asymptotic distribution of two-protected nodes in ternary search trees

We study protected nodes in $m$-ary search trees, by putting them in context of generalised P\'olya urns. We show that the number of two-protected nodes (the nodes that are neither leaves nor parents of leaves) in a random ternary search tree is asymptotically normal. The methods apply in principle to $m$-ary search trees with larger $m$ as well, although the size of the matrices used in the calculations grow rapidly with $m$; we conjecture that the method yields an asymptotically normal distribution for all $m\leq 26$. The one-protected nodes, and their complement, i.e., the leaves, are easier to analyze. By using a simpler P\'olya urn (that is similar to the one that has earlier been used to study the total number of nodes in $m$-ary search trees), we prove normal limit laws for the number of one-protected nodes and the number of leaves for all $m\leq 26$