Detrended fluctuation analysis (DFA) is a scaling analysis method used to estimate long-range power-law correlation exponents in noisy signals. Many noisy signals in real systems display trends, so that the scaling results obtained from the DFA method become difficult to analyze. We systematically study the effects of three types of trends -linear, periodic, and power-law trends, and offer examples where these trends are likely to occur in real data. We compare the difference between the scaling results for artificially generated correlated noise and correlated noise with a trend, and study how trends lead to the appearance of crossovers in the scaling behavior. We find that crossovers result from the competition between the scaling of the noise and the "apparent" scaling of the trend. We study how the characteristics of these crossovers depend on (i) the slope of the linear trend; (ii) the amplitude and period of the periodic trend; (iii) the amplitude and power of the power-law trend and (iv) the length as well as the correlation properties of the noise. Surprisingly, we find that the crossovers in the scaling of noisy signals with trends also follow scaling laws -i.e. long-range power-law dependence of the position of the crossover on the parameters of the trends. We show that the DFA result of noise with a trend can be exactly determined by the superposition of the separate results of the DFA on the noise and on the trend, assuming that the noise and the trend are not correlated. If this superposition rule is not followed, this is an indication that the noise and the superimposed trend are not independent, so that removing the trend could lead to changes in the correlation properties of the noise. In addition, we show how to use DFA appropriately to minimize the effects of trends, and how to recognize if a crossover indicates indeed a transition from one type to a different type of underlying correlation, or the crossover is due to a trend without any transition in the dynamical properties of the noise.

Detrended fluctuation analysis (DFA) is a scaling analysis method used to quantify long-range power-law correlations in signals. Many physical and biological signals are "noisy", heterogeneous and exhibit different types of nonstationarities, which can affect the correlation properties of these signals. We systematically study the effects of three types of nonstationarities often encountered in real data. Specifically, we consider nonstationary sequences formed in three ways: (i) stitching together segments of data obtained from discontinuous experimental recordings, or removing some noisy and unreliable parts from continuous recordings and stitching together the remaining parts -a "cutting" procedure commonly used in preparing data prior to signal analysis; (ii) adding to a signal with known correlations a tunable concentration of random outliers or spikes with different amplitude, and (iii) generating a signal comprised of segments with different properties -e.g. different standard deviations or different correlation exponents. We compare the difference between the scaling results obtained for stationary correlated signals and correlated signals with these three types of nonstationarities. We find that introducing nonstationarities to stationary correlated signals leads to the appearance of crossovers in the scaling behavior and we study how the characteristics of these crossovers depend on: (a) the fraction and size of the parts cut out from the signal; (b) the concentration of spikes and their amplitudes; (c) the proportion between segments with different standard deviations or different correlations; and (d) the correlation properties of the stationary signal. We show how to develop strategies for pre-processing "raw" data prior to analysis, which will minimize the effects of nonstationarities on the scaling properties of the data and how to interpret the results of DFA for complex signals with different local characteristics.

Detrended fluctuation analysis (DFA) and detrended moving average (DMA) are two scaling analysis methods designed to quantify correlations in noisy non-stationary signals. We systematically study the performance of different variants of the DMA method when applied to artificially generated long-range power-law correlated signals with an a-priori known scaling exponent α0 and compare them with the DFA method. We find that the scaling results obtained from different variants of the DMA method strongly depend on the type of the moving average filter. Further, we investigate the optimal scaling regime where the DFA and DMA methods accurately quantify the scaling exponent α0, and how this regime depends on the correlations in the signal. Finally, we develop a three-dimensional representation to determine how the stability of the scaling curves obtained from the DFA and DMA methods depends on the scale of analysis, the order of detrending, and the order of the moving average we use, as well as on the type of correlations in the signal.PACS numbers: 95.75.Wx, 95.75.Pq,

Carbon (C) and nitrogen (N) are the primary elements involved in the growth and development of plants. The C:N ratio is an indicator of nitrogen use efficiency (NUE) and an input parameter for some ecological and ecosystem models. However, knowledge remains limited about the convergent or divergent variation in the C:N ratios among different plant organs (e.g., leaf, branch, trunk, and root) and how evolution and environment affect the coefficient shifts. Using systematic measurements of the leaf–branch–trunk–root of 2,139 species from tropical to cold‐temperate forests, we comprehensively evaluated variation in C:N ratio in different organs in different taxa and forest types. The ratios showed convergence in the direction of change but divergence in the rate of change. Plants evolved toward lower C:N ratios in the leaf and branch, with N playing a more important role than C. The C:N ratio of plant organs (except for the leaf) was constrained by phylogeny, but not strongly. Both the change of C:N during evolution and its spatial variation (lower C:N ratio at midlatitudes) help develop the adaptive growth hypothesis. That is, plants with a higher C:N ratio promote NUE under strong N‐limited conditions to ensure survival priority, whereas plants with a lower C:N ratio under less N‐limited environments benefit growth priority. In nature, larger proportion of species with a high C:N ratio enabled communities to inhabit more N‐limited conditions. Our results provide new insights on the evolution and drivers of C:N ratio among different plant organs, as well as provide a quantitative basis to optimize land surface process models.