Autokorélasi

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Autokorélasi ngarupakeun salah sahiji alat matematika anu remen digunakeun dina signal processing keur analisa fungsi atawa deret nilai, saperti domain waktu signals. Autokorélasi nyaeta cross-correlation hiji signal jeung signalna sorangan. Autokorélasi ilahar dipake keur manggihkeun pola "pengulangan" dina signal, saperti nangtukeun aya periode signal nu kaganggu atawa keur identifikasi frekuensi dasar signal nu teu mibanda komponen frekuensi, tapi mangaruhan kana frekuensi nu harmonis.

Daptar eusi

1 Harti
- 1.1 Statistik
- 1.2 Signal processing
2 Sipat
3 Autokorélasi dina analisa regresi
4 Pamakean
5 Tempo oge
6 Tumbu kaluar
7 Rujukan

[édit] Harti

Bedana defisi autokorélasi nu dipake gumantung kana widang elmu nu ditalungtik nu satemenna teu sarua. Dina sababaraha widang elmu istilahna diganti ku autocovariance.

[édit] Statistik

Dina statistik, fungsi autokorelasi (ACF) keur deret waktu diskrit atawa hiji proses X_t hartina korelasi antara proses dina titik nu beda tur beda waktu oge. Lamun X_t ngabogaan mean μ jeung varian σ² mangka definisi ACF nyaeta

$R(t,s) = \frac{E[(X_t - \mu)(X_s - \mu)]}{\sigma^2}$ .

numana E nyaeta nilai ekspektasi. Note that this is not well-defined for all time-series or processes since the variance may be zero (for a constant process) or infinity. If the function is well defined then this definition has the attractive property of being in the range [−1, 1] with 1 indicating perfect correlation and −1 indicating perfect anti-correlation.

If X_t is second-order stationary then the ACF depends only on the difference between t and s and can be expressed as a function of a single variable. This gives the more familiar form,

$R(k) = \frac{E[(X_i - \mu)(X_{i+k} - \mu)]}{\sigma^2}$

where k is lag (|t - s|). It is common practice in many disciplines to drop the normalisation by σ² and use the term autocorrelation interchangeably with autocovariance. For a sample time series of length n, X₁, X₂ ... X_n with known mean and variance then an estimate may be obtained from

$\hat{R}(k)=\frac{1}{(n-k) \sigma^2} \sum_{t=1}^{n-k} (f(t)-\mu)(f(t+k)-\mu)$ .

for $k \in \mathbb{N}$ .

If the true mean or variance for the process are not known then μ and σ²may be replaced by the standard formulae for sample mean and sample variance although this leads to a biased estimator ^[1].

[édit] Signal processing

In signal processing, the above definition is often used without the normalisation, that is, without subtracting the mean and dividing by the variance.

Given a signal f(t), the continuous autocorrelation R_ff(τ) is most often defined as the continuous cross-correlation integral of f(t) with itself, at lag τ.

$R_{ff}(\tau) = f^*(-\tau) \circ f(\tau) = \int_{-\infty}^{\infty} f(t+\tau)f^*(t)\, dt = \int_{-\infty}^{\infty} f(t)f^*(t-\tau)\, dt$

where f^* represents the complex conjugate and the circle represents convolution. For a real function, f^* = f.

The discrete autocorrelation R at lag j for a discrete signal x_n is

$R_{xx}(j) = \sum_n (x_n)(x^*_{n-j} )\,$

The above definitions work for signals that are square integrable, or square summable, that is, of finite energy. Signals that "last forever" are treated instead as random processes, in which case different definitions are needed, based on expected values. For wide-sense-stationary random processes, the autocorrelations are defined as

$R_{ff}(\tau) = E\left[f(t)f^*(t-\tau)\right]$

$R_{xx}(j) = E\left[x_n x^*_{n-j}\right]$

For processes that are not stationary, these will also be functions of t, or n.

Alternatively, signals that last forever can be treated by a short-time autocorrelation function analysis, using finite time integrals. (See short-time Fourier transform for a related process.)

Multi-dimensional autocorrelation is defined similarly. For example, in three dimensions the autocorrelation of a square-summable discrete signal would be

$R(j,k,\ell) = \sum_{n,q,r} (x_{n,q,r})(x_{n-j,q-k,r-\ell}).$

When mean values are subtracted from signals before computing an autocorrelation function, the resulting function is usually called an auto-covariance function.

[édit] Sipat

Saterusna arek dijelaskeun sipat autokorélasi dina hiji-dimensi, sabab satemenna loba sipat hiji-dimensi nu gampang keur dipake dina kasus multi-dimensi.

Sipat dasar autokorélasi nyaeta simetri, R(i) = R(−i), numana gampang dibuktikeun tina harti ieu. Dina kasus kontinyu, autokorélasi nyaeta even function

$R_f(-\tau) = R_f(\tau)\,$

numana f nyaeta fungsi riil sarta autokorélasi nyaeta fungsi Hermitian

$R_f(-\tau) = R_f^*(\tau)\,$

numana f nyaeta complex function.

Fungsi autokorélasi kontinyu reaches its peak at the origin, where it takes a real value, i.e. for any delay τ, $|R_f(\tau)| \leq R_f(0)$ . This is a consequence of the Cauchy-Schwarz inequality. The same result holds in the discrete case.

The autocorrelation of a periodic function is, itself, periodic with the very same period.

The autocorrelation of the sum of two completely uncorrelated functions (the cross-correlation is zero for all τ) is the sum of the autocorrelations of each function separately.

Since autocorrelation is a specific type of cross-correlation, it maintains all the properties of cross-correlation.

The autocorrelation of a white noise signal will have a strong peak at τ = 0 and will be close to 0 for all other τ. This shows that a sampled instance of a white noise signal is not statistically correlated to a sample instance of the same white noise signal at another time.

The Wiener-Khinchin theorem relates the autocorrelation function to the power spectral density via the Fourier transform:

$R(\tau) = \int_{-\infty}^\infty S(f) e^{j 2 \pi f \tau} \, df$

$S(f) = \int_{-\infty}^\infty R(\tau) e^{- j 2 \pi f \tau} \, d\tau.$

[édit] Autokorélasi dina analisa regresi

Dina regression analysis migunakeun time series, autokorélasi nyaeta sesa tina ieu masalah, sarta nuju ka arah bias nu luhur dina estimasi statistical significance tina koefisien estimasi, saperti dina T statistic. Tes standar keur ayana autokorélasi nyaeta Durbin-Watson statistic atawa, lamun ngajelaskeun variabel kaasup lagged dependent variable, Durbin's h statistic.

Respon keur autokorélasi kaasup differencing data sarta ayana struktur "lag" dina estimasi.

[édit] Pamakean

Salah sahiji pamakean autokorélasi nyaeta dina ngukur spektra optik jeung "pengukuran" durasi-pangpondok na light pulses nu dihasilkeun ku laser, make optical autocorrelators.

Dina optik, normalized autokorélasi sarta cross-correlations mere hasil degree of coherence dina medan elektromagentik.

Dina signal processing, autokorélasi bisa mere informasi ngeunaan ulangan kajadian saperti music beat atawa pulsar frequencies, nu ngaliwatan tapi teu bisa nangtukeun posisina.

[édit] Tempo oge

Cross-correlation

[édit] Tumbu kaluar

Citakan:MathWorld
Autocorrelation articles in Comp.DSP (DSP usenet group).

[édit] Rujukan

↑ Spectral analysis and time series, M.B. Priestley (London, New York : Academic Press, 1982)

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