Estimation of the Conifer-Broadleaf Ratio in Mixed Forests Based on Time-Series Data

Most natural forests are mixed forests, a mixed broadleaf-conifer forest is essentially a heterogeneously mixed pixel in remote sensing images. Satellite missions rely on modeling to acquire regional or global vegetation parameter products. However, these retrieval models often assume homogeneous conditions at the pixel level, resulting in a decrease in the inversion accuracy, which is an issue for heterogeneous forests. Therefore, information on the canopy composition of a mixed forest is the basis for accurately retrieving vegetation parameters using remote sensing. Medium and high spatial resolution multispectral time-series data are important sources for canopy conifer-broadleaf ratio estimation because these data have a high frequency and wide coverage. This paper highlights a successful method for estimating the conifer-broadleaf ratio in a mixed forest with diverse tree species and complex canopy structures. Experiments were conducted in the Purple Mountain, Nanjing, Jiangsu Province of China, where we collected leaf area index (LAI) time-series and forest sample plot inventory data. Based on the Invertible Forest Reflectance Model (INFORM), we simulated the normalized difference vegetation index (NDVI) time-series of different conifer-broadleaf ratios. A time-series similarity analysis was performed to determine the typical separable conifer-broadleaf ratios. Fifteen Gaofen-1 (GF-1) satellite images of 2015 were acquired. The conifer-broadleaf ratio estimation was based on the GF-1 NDVI time-series and semi-supervised k-means cluster method, which yielded a high overall accuracy of 83.75%. This study demonstrates the feasibility of accurately estimating separable conifer-broadleaf ratios using field measurement data and GF-1 time series in mixed broadleaf-conifer forests

(Psycho-)Analysis of Benchmark Experiments

It is common knowledge that certain characteristics of data sets -- such as linear separability or sample size -- determine the performance of learning algorithms. In this paper we propose a formal framework for investigations on this relationship. The framework combines three, in their respective scientific discipline well-established, methods. Benchmark experiments are the method of choice in machine and statistical learning to compare algorithms with respect to a certain performance measure on particular data sets. To realize the interaction between data sets and algorithms, the data sets are characterized using statistical and information-theoretic measures; a common approach in the field of meta learning to decide which algorithms are suited to particular data sets. Finally, the performance ranking of algorithms on groups of data sets with similar characteristics is determined by means of recursively partitioning Bradley-Terry models, that are commonly used in psychology to study the preferences of human subjects. The result is a tree with splits in data set characteristics which significantly change the performances of the algorithms. The main advantage is the automatic detection of these important characteristics. The framework is introduced using a simple artificial example. Its real-word usage is demonstrated by means of an application example consisting of thirteen well-known data sets and six common learning algorithms. All resources to replicate the examples are available online

Generalized Separable Nonnegative Matrix Factorization

Nonnegative matrix factorization (NMF) is a linear dimensionality technique for nonnegative data with applications such as image analysis, text mining, audio source separation and hyperspectral unmixing. Given a data matrix $M$ and a factorization rank $r$, NMF looks for a nonnegative matrix $W$ with $r$ columns and a nonnegative matrix $H$ with $r$ rows such that $M \approx WH$. NMF is NP-hard to solve in general. However, it can be computed efficiently under the separability assumption which requires that the basis vectors appear as data points, that is, that there exists an index set $\mathcal{K}$ such that $W = M(:,\mathcal{K})$. In this paper, we generalize the separability assumption: We only require that for each rank-one factor $W(:,k)H(k,:)$ for $k=1,2,\dots,r$, either $W(:,k) = M(:,j)$ for some $j$ or $H(k,:) = M(i,:)$ for some $i$. We refer to the corresponding problem as generalized separable NMF (GS-NMF). We discuss some properties of GS-NMF and propose a convex optimization model which we solve using a fast gradient method. We also propose a heuristic algorithm inspired by the successive projection algorithm. To verify the effectiveness of our methods, we compare them with several state-of-the-art separable NMF algorithms on synthetic, document and image data sets.Comment: 31 pages, 12 figures, 4 tables. We have added discussions about the identifiability of the model, we have modified the first synthetic experiment, we have clarified some aspects of the contributio

General CMB and Primordial Bispectrum Estimation I: Mode Expansion, Map-Making and Measures of f_NL

We present a detailed implementation of two bispectrum estimation methods which can be applied to general non-separable primordial and CMB bispectra. The method exploits bispectrum mode decompositions on the domain of allowed wavenumber or multipole values. Concrete mode examples constructed from symmetrised tetrahedral polynomials are given, demonstrating rapid convergence for known bispectra. We use these modes to generate simulated CMB maps of high resolution (l > 2000) given an arbitrary primordial power spectrum and bispectrum or an arbitrary late-time CMB angular power spectrum and bispectrum. By extracting coefficients for the same separable basis functions from an observational map, we are able to present an efficient and general f_NL estimator for a given theoretical model. The estimator has two versions comparing theoretical and observed coefficients at either primordial or late times, thus encompassing a wider range of models, including secondary anisotropies, lensing and cosmic strings. We provide examples and validation of both f_NL estimation methods by direct comparison with simulations in a WMAP-realistic context. In addition, we show how the full bispectrum can be extracted from observational maps using these mode expansions, irrespective of the theoretical model under study. We also propose a universal definition of the bispectrum parameter F_NL for more consistent comparison between theoretical models. We obtain WMAP5 estimates of f_NL for the equilateral model from both our primordial and late-time estimators which are consistent with each other, as well as with results already published in the literature. These general bispectrum estimation methods should prove useful for the analysis of nonGaussianity in the Planck satellite data, as well as in other contexts.Comment: 41 pages, 17 figure

Robust Near-Separable Nonnegative Matrix Factorization Using Linear Optimization

Nonnegative matrix factorization (NMF) has been shown recently to be tractable under the separability assumption, under which all the columns of the input data matrix belong to the convex cone generated by only a few of these columns. Bittorf, Recht, R\'e and Tropp (`Factoring nonnegative matrices with linear programs', NIPS 2012) proposed a linear programming (LP) model, referred to as Hottopixx, which is robust under any small perturbation of the input matrix. However, Hottopixx has two important drawbacks: (i) the input matrix has to be normalized, and (ii) the factorization rank has to be known in advance. In this paper, we generalize Hottopixx in order to resolve these two drawbacks, that is, we propose a new LP model which does not require normalization and detects the factorization rank automatically. Moreover, the new LP model is more flexible, significantly more tolerant to noise, and can easily be adapted to handle outliers and other noise models. Finally, we show on several synthetic datasets that it outperforms Hottopixx while competing favorably with two state-of-the-art methods.Comment: 27 page; 4 figures. New Example, new experiment on the Swimmer data se