Detrended fluctuation analysis (DFA) [NBTwiki.net]
The Neurophysiological Biomarker Toolbox (NBT)
, follow the
, become a
Most popular pages
Approximate time needed to complete the tutorial: 60 min.
The detrended fluctuation analysis (DFA) algorithm is a scaling analysis method used to estimate long-range temporal correlations of power-law form
. In other words, if a sequence of events has a non-random temporal structure with slowly decaying auto-correlations, DFA can quantify how slowly these correlations decay as indexed by the DFA power-law exponent. Typical classes of correlations are reflected in DFA exponents 'α' of:
Uncorrelated sequence: α ~ 0.5
Anti-correlated sequence: 0 & α & 0.5
Long-range temporal correlations: 0.5 & α &1
Strong correlations that are not of a power-law form: α & 1
The interest in long-range temporal correlations (LRTC) analysis is rooted in empirical as well as modeling results indicating that the dynamics of self-organizing multi-unit systems is characterized by fluctuations that tend to appear in a specific temporal ordering as indexed by power-law decaying correlations (i.e., very slowly decaying correlations). That a system is “self-organizing” means that, except for some “fairly slow” stochastic drive, most of the fluctuations are endogenous as opposed to being determined by strong external inputs. The physical background for the phenomenon of temporal autocorrelations that persist over many time scales is that activities make changes to the internal structure of these systems and this structure in turn provides a bias of when and where future activity can flow
. You may think of it as water flowing down a mountain slope carving a network of rivers that is ever changing but also sufficiently stable that it strongly influences where water can flow from day to day, month to month, year to year, etc. In other words, the dynamical spatial structures in complex self-organizing systems are the substrate of the memory that ensures a degree of correlation in activity over time.
Long-range temporal correlations have been found in the amplitude modulation of human brain waves , and have been shown to be impaired in disorders such as Alzheimer’s disease , epilepsy , and other disorders (See the reference list below).
The interpretation of DFA crucially depends on the pre-processing of the data you analyze. In our case (analysis of human brain waves (e.g. measured using electroencephalography (EEG) or magnetoencephalography (MEG)), the preprocessing consists of extracting the amplitude envelope of distinct oscillations, but it could equally well be, e.g., the phase-locking of oscillations in different brain areas as computed in a sliding window. In the latter case, DFA applied to the time series of synchronization values would provide you with information about the possibly non-random temporal order of synchronization between the two brain regions .
In this tutorial, you will explore the idea of ongoing neuronal oscillations holding a memory of their own dynamics on fairly long time scales, i.e., several orders of magnitude longer than the period of the oscillatory process. Moreover, we will investigate the influence of signal-to-noise ratio on the estimated correlations in simulated data.
This page is divided into two sections:
, which gives you a basic introduction to computing detrended fluctuation analysis.
, which is meant to give you a deeper understanding of DFA.
This tutorial has been made as supporting material for .
Please cite this paper if you use our DFA scripts in your studies.
Edit summary
Minor Changes
We here show you how you can perform detrended fluctuation analysis in Matlab using NBT.
In this test case we assume, that we want to find the DFA in the alpha frequency band (8–13 Hz), with a sampling frequency of 250 Hz. If you do not have any data you can simply generate a signal using random numbers (in the Matlab command window):
Signal = randn(300*250,23);
This will generate a 5 mins (i.e., 300 seconds) long signal with 23 channels.
Edit summary
Minor Changes
NBT has several functions to help removing artifacts, we will not go into details here. Only, we would like to emphasize that large-amplitude transient artifacts will strongly influence the temporal structure of the signal and, therefore, are better cut away. Importantly, the time series should be stitched together after cutting artifact segments away. Also note, that signals in NBT have time along the 1st dimension, and channels along 2nd dimension of the signal matrix (see example above).
Edit summary
Minor Changes
First, we use nbt_GetAmplitudeEnvelope to filter the signal and get the amplitude envelope, [Signal, SignalInfo] = nbt_GetAmplitudeEnvelope(Signal, SignalInfo, hp, lp, filter_order).
SignalInfo = nbt_I %this initialize an Info Object
SignalInfo.converted_sample_frequency = 250; %This sets the sampling frequency to 250 Hz.
AmplitudeEnvelope = nbt_GetAmplitudeEnvelope(Signal, SignalInfo, 8, 13, 2/8); % Extract the amplitude envelope using a FIR filter, and the hilbert transform.
The function nbt_Info defines an NBT info object, which contains information about your signal. Note the last parameter 2/8. This is the filter order (in seconds), which we set such that at least two oscillation cycles are covered by the filter window.
Edit summary
Minor Changes
Hereafter, it is easy to calculate the DFA exponents. From the function definition [DFAobject,DFA_exp] = nbt_doDFA(Signal, InfoObject, FitInterval, CalcInterval, DFA_Overlap, DFA_Plot, ChannelToPlot, res_logbin);, we see that we can find the DFA exponent and plot the fluctuation function by :
[DFAobject,DFA_exp] = nbt_doDFA(AmplitudeEnvelope, SignalInfo, [1 15], [0.8 25], 0.5, 1, 1, []);
Signal, and SignalInfo are the Signal and the associated information about this signal (see above). The two next parameters are FitInterval, and Calcinterval; they determine which interval we fit, and which we calculate. The DFA_overlap tells how much overlap we want of our windows (see above), in this case 50%. The plotting parameters DFA_plot and ChannelToPlot should be self-explanatory. The last parameter res_logbin is the resolution of the logarithmic binning, which by default is 10.
After the calculations are finished you can find the DFA values for all channels in the MarkerValues field.
DFAobject.MarkerValues
Edit summary
Minor Changes
Let us test that the DFA yields a power-law exponent of α~0.5 for random data. We define a “sampling frequency” of 300 Hz in order to interpret the time scales in units of seconds and generate “20 min of random activity”.
Signal = randn(20*60*Fs,1);
Before computing the DFA fluctuation function 'DFA_y' and how it scales with increasing window sizes, we have to define a time range of interest. The DFA is not accurate for very short windows nor for long windows, because of statistical variability. As a rule of thumb, you should only estimate correlations up to time scales of 10% of the amount of data available.
Note, the computation of 'nbt_doDFA.m' may take tens of seconds to complete!
SignalInfo = nbt_I
SignalInfo.converted_sample_frequency = Fs;
[DFAobject,DFA_exp] = nbt_doDFA(Signal, SignalInfo, [0.1 120], [0.08 1200], 0.5, 1, 1, []);
Did you get a DFA-exponent α close to 0.5?
Long-range temporal correlations imply that there are modulations of the fluctuations not only on the short time scales that tend to catch the attention of our visual system, but also on long time scales. Low-frequency fluctuations, however, are usually removed at data acquisition using a high-pass filter of around 0.1–1 Hz. Besides, neuronal systems typically do not generate much low-frequency activity compared to the amount of low-frequency noise in the laboratory. Therefore, we cannot use the “raw” data from the acquisition for LRTC analysis and need to pre-process the data before searching for correlations. Since we are interested in distinct oscillations, we use a bandpass filter and the Hilbert transform to extract the amplitude envelope of the narrow-band oscillations. The difference between analyzing modulations of the original signal versus the amplitude envelope of a filtered signal is explained in the figure below.
The frequency content of broadband signals and of the amplitude envelope of narrow-frequency-band neural oscillations. A, A signal from a single MEG channel (0.1–100 Hz) is shown at two time scales. Note from the upper trace that the high-amplitude 10-Hz oscillations are riding on slow fluctuations (~ 1 Hz). B, The slow fluctuations are unlikely to be of neural origin, as may be inferred from the power spectra showing the spectral density of the entire 20-min signal in A (thick line) and reference data from the same channel (thin line, see Materials and Methods). The neural signals dominate above but not below 1 Hz, with prominent peaks at 10 and 20 Hz (see arrows). C, The signal shown in A has been filtered at 10 Hz with a Morlet wavelet (passband 6.7–13.3 Hz). The thin lines are the real part and the amplitude envelopes (thick lines) are the modulus of the wavelet transform. D, The power spectrum of the amplitude envelope of the signals at 10 Hz exhibits a 1/f-type of power spectrum (circles) with α = 0.58 in the range from 0.01–1 Hz, thereby indicating that the fluctuations of the amplitude envelope of these oscillations are correlated at time scales of 1–100 s. On the contrary, the 10-Hz amplitude envelope of the reference data gave rise to a white-noise spectrum characteristic of a temporally uncorrelated process (dots). From (Linkenkaer-Hansen et al., 2004).
One should, however, always be cautious that the pre-processing of data does not introduce unwanted properties in the data. For example, band-pass filtering is based on a weighted sum of samples within a window that is a few oscillation cycles wide and therefore even a sequence of random numbers is expected to have temporal autocorrelations in the amplitude fluctuations on certain (short) time scales after filtering (!).
To see this, we narrow-band filter the random data at 8–13 Hz (the so-called “alpha-frequency band”) and compute the DFA on the amplitude envelope of the oscillations (i.e., 'abs(hilbert(Data_filtered))'). First we design a so-called “FIR filter”, which is a finite impulse response filter, i.e., only samples in a limited time window are integrated by the filter, namely the number of samples indicated by the so-called “order” of the filter. In the filter script ('nbt_filter_fir.m'), the filter order is given in units of seconds (to prevent an influence of sampling frequency). The filter order should be chosen to include at least two cycles of the low-frequency component included in the passband, which is 0.25 s for an 8-Hz oscillation:
highpass = 8;
% Highpass corner frequency.
lowpass = 13;
% Lowpass corner frequency.
fir_order = 0.25;
% Filter order in units of seconds.
[Data_filtered] = nbt_filter_fir(Signal, highpass, lowpass, Fs, 2/8);
To see that the filter worked as expected, plot the 'Data_filtered':
plot(Data_filtered,'r')
If you zoom in on a shorter time scale you will notice that the signal is oscillatory.
The amplitude envelope of the oscillations (see Figure on p. 5) may be determined by the absolute value of the so-called hilbert transform:
&& plot(abs(hilbert(Data_filtered)),'k')
You should now see oscillations that fluctuate erratically in amplitude!
[DFAobject,DFA_exp] = nbt_doDFA(abs(hilbert(Data_filtered)), SignalInfo, [0.1 120], [0.08 1200], 0.5, 1, 1, []);
As you can see, the DFA fluctuation function no longer increases linearly with increasing window size, which means that the correlations do not decay according to a simple power-law with just one exponent for all time scales. In order to find the time scales that are sufficiently long that autocorrelations are not affected by the integration by the filter, we have to find the time range where fitting the power law gives a good fit and DFA exponent α ~ 0.5
Re-compute the DFA, fitting first to the short-range time scales, e.g.:
[DFAobject,DFA_exp] = nbt_doDFA(abs(hilbert(Data_filtered)), SignalInfo, [0.1 1], [0.08 1200], 0.5, 1, 1, []);
and then to the long-range time scales, e.g.:
[DFAobject,DFA_exp] = nbt_doDFA(abs(hilbert(Data_filtered)), SignalInfo, [3 120], [0.08 1200], 0.5, 1, 1, []);
From these two DFA plots, you can see that at time scales from 100 ms to 1 s, the amplitude fluctuations of the 10 Hz filtered data are strongly correlated. In fact, the exponent is larger than 1, indicating that the correlations are not of a power-law form (only for 0.5 & α & 1). In the time range of 3–120 s, however, correlations are still absent as evidenced by α ~ 0.5. Depending on the properties of the bandpass filter, one may be able to achieve uncorrelated fluctuations after filtering at shorter time scales than 3 s. Nevertheless, the conclusion from the above is that for this specific 8–13-Hz filter it is safe to interpret correlations on time scales longer than 3 s as originating from the original signal and not introduced by the filter.
You have just studied the influence of filtering on autocorrelations. Let's turn to real MEG data and quantify the decay of long-range temporal correlations (LRTC) in the alpha-frequency band. Download the
Use the fitting range that you have just identified as having a proper null-hypothesis, i.e., when quantifying correlations in real data, you wish to test the time scales that do not contain correlations induced by the filters.
[RSM_filtered] = nbt_filter_fir(RSM, highpass, lowpass, Fs, 2/8);
[DFAobject,DFA_exp] = nbt_doDFA(abs(hilbert(RSM_filtered)), SignalInfo, [0.1 120], [0.08 1200], 0.5, 1, 1, []);
Long-range temporal correlations require a non-uniformity of fluctuations over time, or a so-called “patchy” time structure.
To see this, compare the time series of the filtered noise and the filtered MEG brain signal using the 'plot' command (you may want to write a 'whos' command to see your workspace variables):
% you should know how to code these commands by now...
It would be unfortunate if LRTC were present already in the laboratory noise. Check this by analyzing the empty-room recording 'RMS_noise' as you just analyzed the 'RSM'.
% you should know how to code these commands by now...
Narrow-band filtered MEG noise fortunately seems to have close to no long-range temporal autocorrelations. This raises another issue, however: If the signal-to-noise ratio is poor, can we then reliably quantify the temporal correlations of the underlying physiological process?
A crude way to elucidate the problem is to “artificially decrease the signal-to-noise ratio” of your neuronal data by adding an amplified version of the empty-room recording on top of the neuronal data and see if it influences the estimated LRTC (make sure you understand the data input!):
% you may be able to do this by now, otherwise ask...
What happened to the DFA-exponent compared to the original filtered RSM data without added noise?
This is a problem, because different conditions, animals or subject groups may often have different signal-to-noise ratio and, therefore, a change in LRTC potentially could reflect different oscillation amplitudes rather than a true effect on LRTC. This influence of signal-to-noise ratio on biomarker values, however, is a generic one that also applies to coherence, cross-frequency phase locking, graph analyses, etc.
Edit summary
Minor Changes
Peng et al., 1995.
Chaos, 5(1), pp.82-7, doi:10.141.
Hardstone et al., 2012.
Frontiers in Fractal Physiology, 3:450, doi: 10.3389/fphys..
Poil et al., 2012.
J. Neurosci., 32(29), pp., doi: 10.1523/JNEUROSCI.2.
Linkenkaer-Hansen et al., 2001.
J. Neurosci., 21(4), pp..
Montez et al., 2009.
PNAS, 106(5), pp..
Monto et al., 2007.
Cerebral Cortex, 7(6), pp., doi: 10.1093/cercor/bhl049.
Montez et al., 2006.
NeuroImage, 33(4), pp., doi: 10.1016/j.neuroimage..
Edit summary
Minor Changes
tutorial/detrended_fluctuation_analysis_dfa.txt & Last modified:
00:04 by Simon-Shlomo Poil<div id="click_content_aid_
<div id="favrite_content_aid_
每日新闻早报：印度法院禁止小米手机销售，Vivo的价值观是本分 ...
每日新闻早报：印度法院禁止小米手机销售，Vivo的价值观是本分 ...
07:44&&|&&作者：
&&|&&关键字：,,,,
1、印度法院下发禁令 禁止小米销售和进口手机。2、陌陌回应网易炮轰：唐岩将就恶意指控展开自辩。3、谷歌主席施密特：不要害怕人工智能。4、惠普计划2015年发布“革命性”操作系统Linux++。5、vivo创始人罕见直面媒体：我们不上市 不融资。6、黑莓CEO程守宗：正在制定进军中国市场计划。
1、据印度媒体报道，德里高等法院日前裁定，小米侵犯了爱立信的专利，并下发了禁令，要求小米停止在印度销售和进口手机。需要指出的是，存在争议的这些专利属于标准必要专利(SEP)，适用于“公平、合理、无歧视”原则(FRAND)。同时，也是爱立信起诉印度本土智能手机厂商Micromax、Gionee和Intex所涉及的专利。属于FRAND范畴的专利意味着这些专利十分必要，爱立信必须要以合理的价格授权给任何一家公司。业内人士称，很显然小米和爱立信并未对此达成一致。报道称，爱立信今年7月曾要求小米为这些专利支付费用，但小米并未回复。根据法院的裁定，小米不能向印度进口任何新手机，也不能推广或销售这些产品。这意味着，在解决专利纠纷前，小米可能不能在印度销售任何手机。2、陌陌周三向美国证券交易委员会(SEC)提交了更新后的IPO(首次公开招股)招股书文件，对网易“炮轰”该公司CEO唐岩一事作出了回应。2014年11月，陌陌联合创始人、董事长兼CEO唐岩收到了来自一家中国法律事务所的律师信，该事务所代表的是网易旗下网之易信息技术(北京)有限公司。唐岩曾于2003年12月到2011年9月供职于网之易。律师信称，由于唐岩在2011年7月创立陌陌，且陌陌在2011年8月发布了自己的应用，从时间上来说均处于唐岩仍为网之易员工期间，因此违反了他与网之易之间的雇佣协议，违背了他在职期间不得与网之易竞争的契约。该信要求唐岩立即向网之易作出书面道歉，并保留网之易未来采取进一步行动的权利。唐岩则认为，这一指控并无价值，并计划就此展开积极的自我辩护。3、埃里克·施密特（Eric Schmidt）想让你知道，机器人是你的朋友。这说得过去，考虑到作为谷歌的主席以及前任CEO，施密特致力于开发各种世界上最复杂的人工智能系统，从无人驾驶汽车到预测结果的搜索引擎。公司近期甚至拥有了自己的内部机器人实验室。施密特自己也承认，乘坐无人驾驶汽车不会是一个令人高兴的体验，但是他同时也认为，担心机器会造成失业或者取代人类统治世界是毫无根据的。4、据Technology Review报道，日前惠普宣布将于明年在一款新奇的计算机上发布一款“革命性”系统，这有望重振它的计算机业务并撼动整个行业。该公司的研究部门正在打造一款名为The
Machine的计算机，它将比现在市面上的计算机更加节能省电，同时拥有更强的性能。惠普计划用一种新型的计算机内存来代替当前计算机上使用的两种类型内存。当前的内存读取方式起源于上世纪40年代，由于数据需要在两种类型的内存之间来回穿梭，因此限制了计算机的性能。“这种计算方式从一开始就已经反映出问题，它阻碍了我们的发展。”The Machine的首席架构师科克·布瑞思尼克（Kirk Bresniker）表示。这个项目由惠普实验室负责运作，该部门的200名研究人员中有四分之三都参与其中。此前惠普首席执行官梅格·怀特曼（Meg
Whitman）加大了对这个项目的支持力度。5、手机界一直有个令人费解的难题，那就是vivo和oppo到底是什么关系？同一个源远：步步高；同一个纽带：步步高董事长段永平；相似的定位：年轻时尚；却总是告诉别人，我们一点关系都没有。昨天，在vivo新品X5Max发布会后，vivo创始人、CEO沈炜罕见露面，开始从后台走到前台。也许是面对媒体次数还不多，他并不像其他一些CEO对敏感问题含糊其辞，而是老老实实回答每一个问题。在1小时的交流中，他时时刻刻挂在嘴边的是公司的价值观“本分”。他对在场的媒体说，“vivo短时间内没有任何上市的想法，也不需要融资，大部分上市是想弄点钱，但是我们并不缺钱。”6、黑莓CEO程守宗(John
Chen)日前发表博客文章称，黑莓正在制定进军中国市场的计划。他说：“至于中国市场，可以确定的是，我们正在制定一项周全的计划，这需要考虑到许多因素，包括时间和长期战略等。出于安全担忧等因素，此事不能操之过急。另外，该计划不包括并购。”彭博社近日公布了程守宗在今年7月份接受采访时的一段录音，程守宗当时明确表示，黑莓不会接受中国公司的收购提议，美国、德国和英国等多国政府都是黑莓的客户，这些西方国家不可能同意这笔交易。
本文读者还喜欢
（你可匿名或登录后发表评论。没有帐号可，或使用和直接登录）
非著名编辑，自认为是个理性、有内涵的观察者，实则腹黑、心不黑，坚信阳光普照大地，只待涤荡眼前的黑暗。
扫一扫右边的二维码
关注超能网微信账号
你没看错真的是599元！OCZ ARC100 240GB降价了
主板中的白富美，华硕剑齿虎Z97 Mark S预订中
只要499元的盈通HD 7770，给力优惠周五截止
先马金牌500W，只要299元的DC2DC电源}