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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Detrended fluctuation analysis ： ウィキペディア英語版 Detrended fluctuation analysis In stochastic processes, chaos theory and time series analysis, detrended fluctuation analysis (DFA) is a method for determining the statistical self-affinity of a signal. It is useful for analysing time series that appear to be long-memory processes (diverging correlation time, e.g. power-law decaying autocorrelation function) or 1/f noise. The obtained exponent is similar to the Hurst exponent, except that DFA may also be applied to signals whose underlying statistics (such as mean and variance) or dynamics are non-stationary (changing with time). It is related to measures based upon spectral techniques such as autocorrelation and Fourier transform. Peng et al. introduced DFA in 1994 in a paper that has been cited over 2000 times as of 2013 and represents an extension of the (ordinary) fluctuation analysis (FA), which is affected by non-stationarities. ==Calculation== Given a bounded time series $x\_t$, $t\; \backslash in\; \backslash mathbb$, integration or summation first converts this into an unbounded process $X\_t$: :$X\_t=\backslash sum\_^t\; (x\_i-\backslash langle\; x\backslash rangle)$ $X\_t$ is called cumulative sum or profile. This process converts, for example, an i.i.d. white noise process into a random walk. Next, $X\_t$ is divided into time windows $Y\_j$ of length $L$ samples, and a local least squares straight-line fit (the local trend) is calculated by minimising the squared error $E^2$ with respect to the slope and intercept parameters $a,\; b$: :$E^2\; =\; \backslash sum\_^L\; \backslash left(\; Y\_j\; -\; j\; a\; -\; b\; \backslash right)^2.$ Trends of higher order can be removed by higher order DFA, where the linear function $j\; a\; +b$ is replaced by a polynomial of order $n$. Next, the root-mean-square deviation from the trend, the fluctuation, is calculated over every window at every time scale: :$F(\; L\; )\; =\; \backslash left(\backslash frac\backslash sum\_^L\; \backslash left(\; Y\_j\; -\; j\; a\; -\; b\; \backslash right)^2\; \backslash right)^\}.$ This detrending followed by fluctuation measurement process is repeated over the whole signal at a range of different window sizes $L$, and a log-log graph of $L$ against $F(L)$ is constructed. A straight line on this log-log graph indicates statistical self-affinity expressed as $F(L)\; \backslash propto\; L^$. The scaling exponent $\backslash alpha$ is calculated as the slope of a straight line fit to the log-log graph of $L$ against $F(L)$ using least-squares. This exponent is a generalization of the Hurst exponent. Because the expected displacement in an uncorrelated random walk of length L grows like $\backslash sqrt$, an exponent of $\backslash tfrac$ would correspond to uncorrelated white noise. When the exponent is between 0 and 1, the result is Fractional Brownian motion, with the precise value giving information about the series self-correlations: * : anti-correlated * $\backslash alpha\; \backslash simeq\; 1/2$: uncorrelated, white noise * $\backslash alpha>1/2$: correlated * $\backslash alpha\backslash simeq\; 1$: 1/f-noise, pink noise * $\backslash alpha>1$: non-stationary, unbounded * $\backslash alpha\backslash simeq\; 3/2$: Brownian noise There are different orders of DFA. In the described case, linear fits ($n=1$) are applied to the profile, thus it is called DFA1. In general, DFA$n$, uses polynomial fits of order $n$. Due to the summation (integration) from $x\_i$ to $X\_t$, linear trends in the mean of the profile represent constant trends in the initial sequence, and DFA1 only removes such constant trends (steps) in the $x\_i$. In general DFA of order $n$ removes (polynomial) trends of order $n-1$. For linear trends in the mean of $x\_i$ at least DFA2 is needed. The Hurst R/S analysis removes constants trends in the original sequence and thus, in its detrending it is equivalent to DFA1. The DFA method was applied to many systems; e.g., DNA sequences, neuronal oscillations, speech pathology detection, and heartbeat fluctuation in different sleep stages. Effect of trends on DFA were studied in and relation to the power spectrum method is presented in. Since in the fluctuation function $F(L)$ the square(root) is used, DFA measures the scaling-behavior of the second moment-fluctuations, this means $\backslash alpha=\backslash alpha(2)$. The multifractal generalization (MF-DFA) uses a variable moment $q$ and provides $\backslash alpha(q)$. Kantelhardt et al. intended this scaling exponent as a generalization of the classical Hurst exponent. The classical Hurst exponent corresponds to the second moment for stationary cases $H=\backslash alpha(2)$ and to the second moment minus 1 for nonstationary cases $H=\backslash alpha(2)-1$.〔〔 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Detrended fluctuation analysis」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.