5779:
1702:
5381:
36:
5774:{\displaystyle {\begin{bmatrix}\gamma _{1}\\\gamma _{2}\\\gamma _{3}\\\vdots \\\gamma _{p}\\\end{bmatrix}}={\begin{bmatrix}\gamma _{0}&\gamma _{-1}&\gamma _{-2}&\cdots \\\gamma _{1}&\gamma _{0}&\gamma _{-1}&\cdots \\\gamma _{2}&\gamma _{1}&\gamma _{0}&\cdots \\\vdots &\vdots &\vdots &\ddots \\\gamma _{p-1}&\gamma _{p-2}&\gamma _{p-3}&\cdots \\\end{bmatrix}}{\begin{bmatrix}\varphi _{1}\\\varphi _{2}\\\varphi _{3}\\\vdots \\\varphi _{p}\\\end{bmatrix}}}
10749:
10729:
6942:
6949:
3106:
3443:
5110:. There is a direct correspondence between these parameters and the covariance function of the process, and this correspondence can be inverted to determine the parameters from the autocorrelation function (which is itself obtained from the covariances). This is done using the YuleâWalker equations.
9050:
one period prior to the one now being forecast is not known, so its expected valueâthe predicted value arising from the previous forecasting stepâis used instead. Then for future periods the same procedure is used, each time using one more forecast value on the right side of the predictive equation
8511:
The process is non-stationary when the poles are on or outside the unit circle, or equivalently when the characteristic roots are on or inside the unit circle. The process is stable when the poles are strictly within the unit circle (roots strictly outside the unit circle), or equivalently when the
6763:
Estimation of autocovariances or autocorrelations. Here each of these terms is estimated separately, using conventional estimates. There are different ways of doing this and the choice between these affects the properties of the estimation scheme. For example, negative estimates of the variance can
6932:
values in the series; in the second, the likelihood function considered is that corresponding to the unconditional joint distribution of all the values in the observed series. Substantial differences in the results of these approaches can occur if the observed series is short, or if the process is
6792:
Formulation as an extended form of ordinary least squares prediction problem. Here two sets of prediction equations are combined into a single estimation scheme and a single set of normal equations. One set is the set of forward-prediction equations and the other is a corresponding set of backward
6915:
future values of the same series. This way of estimating the AR parameters is due to John Parker Burg, and is called the Burg method: Burg and later authors called these particular estimates "maximum entropy estimates", but the reasoning behind this applies to the use of any set of estimated AR
8769:
7417:
2919:
9062:
There are four sources of uncertainty regarding predictions obtained in this manner: (1) uncertainty as to whether the autoregressive model is the correct model; (2) uncertainty about the accuracy of the forecasted values that are used as lagged values in the right side of the autoregressive
125:(ARIMA) models of time series, which have a more complicated stochastic structure; it is also a special case of the vector autoregressive model (VAR), which consists of a system of more than one interlocking stochastic difference equation in more than one evolving random variable.
2861:
2539:
6928:. Two distinct variants of maximum likelihood are available: in one (broadly equivalent to the forward prediction least squares scheme) the likelihood function considered is that corresponding to the conditional distribution of later values in the series given the initial
7945:
104:
is a representation of a type of random process; as such, it can be used to describe certain time-varying processes in nature, economics, behavior, etc. The autoregressive model specifies that the output variable depends linearly on its own previous values and on a
6741:
4699:
2336:
8245:
5241:
6916:
parameters. Compared to the estimation scheme using only the forward prediction equations, different estimates of the autocovariances are produced, and the estimates have different stability properties. Burg estimates are particularly associated with
2702:
4935:
3306:
7161:
5950:
3740:
6611:
3321:
8072:
9276:
3101:{\displaystyle \Phi (\omega )={\frac {1}{\sqrt {2\pi }}}\,\sum _{n=-\infty }^{\infty }B_{n}e^{-i\omega n}={\frac {1}{\sqrt {2\pi }}}\,\left({\frac {\sigma _{\varepsilon }^{2}}{1+\varphi ^{2}-2\varphi \cos(\omega )}}\right).}
6896:
8586:
7238:
6398:
5070:
4817:
461:
8976:
6472:
3857:
273:
4331:
4549:
668:
6174:
7731:
6788:
for this problem can be seen to correspond to an approximation of the matrix form of the YuleâWalker equations in which each appearance of an autocovariance of the same lag is replaced by a slightly different
1709:
The simplest AR process is AR(0), which has no dependence between the terms. Only the error/innovation/noise term contributes to the output of the process, so in the figure, AR(0) corresponds to white noise.
1493:
7487:
becomes nearer 1, there is stronger power at low frequencies, i.e. larger time lags. This is then a low-pass filter, when applied to full spectrum light, everything except for the red light will be filtered.
6310:
2720:
1319:
1605:
2418:
2240:
819:
7815:
8575:
8334:
1957:
7023:
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8133:
6240:
7827:
3585:
9038:
to equal its expected value, and the expected value of the unobserved error term is zero). The output of the autoregressive equation is the forecast for the first unobserved period. Next, use
6618:
322:
5848:
1248:
1145:
6016:
4554:
1022:
9536:
6051:
523:
2016:
2211:
6519:
8142:
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The partial autocorrelation of an AR(p) process equals zero at lags larger than p, so the appropriate maximum lag p is the one after which the partial autocorrelations are all zero.
4223:
1888:
is negative, then the process favors changes in sign between terms of the process. The output oscillates. This can be likened to edge detection or detection of change in direction.
9273:
5319:
2569:
7222:
4108:
5132:
2374:
583:
4159:
9089:
4059:
8471:
8410:
8370:
7521:
7454:
4436:
9027:
4368:
3941:
2900:
2596:
1984:
1380:
1353:
1172:
904:
846:
353:
3480:
2172:
2083:
2604:
5352:
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1645:
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6090:
5100:
3914:
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2043:
1886:
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approaches 1, the output gets a larger contribution from the previous term relative to the noise. This results in a "smoothing" or integration of the output, similar to a
1672:
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4403:
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109:
term (an imperfectly predictable term); thus the model is in the form of a stochastic difference equation (or recurrence relation) which should not be confused with a
4456:
4197:. It is therefore sometimes useful to understand the properties of the AR(1) model cast in an equivalent form. In this form, the AR(1) model, with process parameter
3618:
2403:
5860:
8505:
8002:
7975:
4003:
3972:
3887:
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3500:
3212:
3185:
3136:
2136:
1523:
1103:
1076:
1049:
985:
958:
931:
877:
699:
172:
10840:
2235:
8436:
3438:{\displaystyle \Phi (\omega )={\frac {1}{\sqrt {2\pi }}}\,{\frac {\sigma _{\varepsilon }^{2}}{1-\varphi ^{2}}}\,{\frac {\gamma }{\pi (\gamma ^{2}+\omega ^{2})}}}
9748:
4179:
3630:
6526:
2138:
depends on time lag t, so that the variance of the series diverges to infinity as t goes to infinity, and is therefore not weak sense stationary.) Assuming
8010:
9327:
9300:
11375:
9871:
8764:{\displaystyle S(f)={\frac {\sigma _{Z}^{2}}{1+\varphi _{1}^{2}+\varphi _{2}^{2}-2\varphi _{1}(1-\varphi _{2})\cos(2\pi f)-2\varphi _{2}\cos(4\pi f)}}}
7412:{\displaystyle S(f)={\frac {\sigma _{Z}^{2}}{|1-\varphi _{1}e^{-2\pi if}|^{2}}}={\frac {\sigma _{Z}^{2}}{1+\varphi _{1}^{2}-2\varphi _{1}\cos 2\pi f}}}
6802:
11199:
10465:
6327:
4981:
4707:
372:
9309:
8890:
6404:
752:
In an AR process, a one-time shock affects values of the evolving variable infinitely far into the future. For example, consider the AR(1) model
3782:
1705:
AR(0); AR(1) with AR parameter 0.3; AR(1) with AR parameter 0.9; AR(2) with AR parameters 0.3 and 0.3; and AR(2) with AR parameters 0.9 and â0.8
11802:
10795:
188:
9046:
period for which data is not yet available; again the autoregressive equation is used to make the forecast, with one difference: the value of
4231:
1899:
11332:
11312:
4465:
592:
6098:
9664:
9451:(Edited by D. G. Childers), NATO Advanced Study Institute of Signal Processing with emphasis on Underwater Acoustics. IEEE Press, New York.
7632:
9462:
9063:
equation; (3) uncertainty about the true values of the autoregressive coefficients; and (4) uncertainty about the value of the error term
11716:
1404:
9559:
2856:{\displaystyle B_{n}=\operatorname {E} (X_{t+n}X_{t})-\mu ^{2}={\frac {\sigma _{\varepsilon }^{2}}{1-\varphi ^{2}}}\,\,\varphi ^{|n|}.}
11633:
6247:
1259:
11317:
2534:{\displaystyle {\textrm {var}}(X_{t})=\operatorname {E} (X_{t}^{2})-\mu ^{2}={\frac {\sigma _{\varepsilon }^{2}}{1-\varphi ^{2}}},}
122:
9606:
11643:
11327:
9981:
1531:
118:
755:
11685:
11582:
9864:
9834:
7742:
8515:
8256:
11872:
11862:
11708:
11400:
11385:
9360:
6759:) model, by replacing the theoretical covariances with estimated values. Some of these variants can be described as follows:
1253:(where the constant term has been suppressed by assuming that the variable has been measured as deviations from its mean) as
9413:
128:
Contrary to the moving-average (MA) model, the autoregressive model is not always stationary as it may contain a unit root.
11772:
11736:
10654:
9635:
7940:{\displaystyle z_{1},z_{2}={\frac {1}{2\varphi _{2}}}\left(\varphi _{1}\pm {\sqrt {\varphi _{1}^{2}+4\varphi _{2}}}\right)}
6966:
8981:
have been estimated, the autoregression can be used to forecast an arbitrary number of periods into the future. First use
7550:
6736:{\displaystyle \rho _{2}=\gamma _{2}/\gamma _{0}={\frac {\varphi _{1}^{2}-\varphi _{2}^{2}+\varphi _{2}}{1-\varphi _{2}}}}
2341:
12040:
11777:
10755:
10306:
10043:
8084:
6917:
6195:
9718:
9693:
3535:
134:
are called autoregressive, but they are not a classical autoregressive model in this sense because they are not linear.
11689:
10887:
10788:
9257:
7537:
4694:{\displaystyle X_{t+n}=\theta ^{n}X_{t}+(1-\theta ^{n})\mu +\Sigma _{i=1}^{n}\left(\theta ^{n-i}\epsilon _{t+i}\right)}
2046:
539:
281:
8869:. Since the AR model is a special case of the vector autoregressive model, the computation of the impulse response in
5787:
1200:
1108:
11842:
10567:
10194:
10001:
9857:
9786:
9740:
9396:
9227:
9213:
9178:
5958:
79:
57:
9298:"On a Method of Investigating Periodicities in Disturbed Series, with Special Reference to Wolfer's Sunspot Numbers"
1194:
is affected by shocks occurring infinitely far into the past. This can also be seen by rewriting the autoregression
50:
11887:
11693:
11677:
11592:
11420:
11390:
10812:
10522:
9336:
4947:
2331:{\displaystyle \operatorname {E} (X_{t})=\varphi \operatorname {E} (X_{t-1})+\operatorname {E} (\varepsilon _{t}),}
6092:
of the autocorrelation function. The full autocorrelation function can then be derived by recursively calculating
1834:
are positive, the output will resemble a low pass filter, with the high frequency part of the noise decreased. If
990:
11792:
11757:
11726:
11721:
11157:
11074:
6021:
476:
8240:{\displaystyle f^{*}={\frac {1}{2\pi }}\cos ^{-1}\left({\frac {\varphi _{1}}{2{\sqrt {-\varphi _{2}}}}}\right),}
1989:
11731:
11360:
11355:
11162:
11059:
10709:
10649:
10247:
9142:
4965:
2177:
8870:
6481:
12045:
11822:
11658:
11557:
11542:
11081:
10954:
10870:
10781:
10242:
9931:
9324:
9297:
8845:
8815:
8811:
8805:
6925:
6771:
problem in which an ordinary least squares prediction problem is constructed, basing prediction of values of
6755:
The above equations (the YuleâWalker equations) provide several routes to estimating the parameters of an AR(
5236:{\displaystyle \gamma _{m}=\sum _{k=1}^{p}\varphi _{k}\gamma _{m-k}+\sigma _{\varepsilon }^{2}\delta _{m,0},}
4200:
11817:
11697:
8004:
are the reciprocals of the characteristic roots, as well as the eigenvalues of the temporal update matrix:
5297:
2547:
1674:
are the coefficients in the autoregression. The formula is valid only if all the roots have multiplicity 1.
11827:
10684:
10081:
10038:
9991:
9986:
9173:. Gwilym M. Jenkins, Gregory C. Reinsel (3rd ed.). Englewood Cliffs, N.J.: Prentice Hall. p. 54.
7177:
4064:
11832:
11468:
6956:
545:
11430:
11014:
10959:
10875:
10735:
10031:
9957:
4121:
11762:
9066:
8985:
to refer to the first period for which data is not yet available; substitute the known preceding values
4031:
12071:
11767:
11752:
11395:
11365:
10932:
10830:
10359:
10294:
9895:
9840:
9508:
9117:
9103:
increases because of the use of an increasing number of estimated values for the right-side variables.
8861:
of a system is the change in an evolving variable in response to a change in the value of a shock term
8443:
8382:
8342:
7493:
7426:
5123:
4408:
4194:
2697:{\displaystyle {\textrm {var}}(X_{t})=\varphi ^{2}{\textrm {var}}(X_{t-1})+\sigma _{\varepsilon }^{2},}
9005:
8339:
The terms involving square roots are all real in the case of complex poles since they exist only when
8077:
AR(2) processes can be split into three groups depending on the characteristics of their roots/poles:
4930:{\displaystyle \operatorname {Var} (X_{t+n}|X_{t})=\sigma ^{2}{\frac {1-\theta ^{2n}}{1-\theta ^{2}}}}
4336:
3919:
2873:
2574:
1962:
1358:
1331:
1150:
882:
824:
331:
11847:
11648:
11562:
11547:
11478:
11054:
10937:
10835:
10760:
10618:
10257:
10088:
9911:
9509:"Autoregressive spectral estimation by application of the Burg algorithm to irregularly sampled data"
9147:
8832:: the Bayesian statistics and probabilistic programming framework supports autoregressive modes with
3451:
2141:
2052:
532:
2913:
of the autocovariance function. In discrete terms this will be the discrete-time
Fourier transform:
12066:
11681:
11567:
11069:
11044:
10989:
10659:
9916:
9127:
8842:
supports parameter inference and model selection for the AR-1 process with time-varying parameters.
8780:
6768:
6056:
5355:
5324:
3138:, which is manifested as the cosine term in the denominator. If we assume that the sampling time (
1621:
1391:
1325:
44:
17:
7463:
6066:
5078:
3892:
3752:
3301:{\displaystyle B(t)\approx {\frac {\sigma _{\varepsilon }^{2}}{1-\varphi ^{2}}}\,\,\varphi ^{|t|}}
2021:
1864:
1837:
1810:
1783:
1780:
For an AR(2) process, the previous two terms and the noise term contribute to the output. If both
1650:
704:
11982:
11972:
11787:
11663:
11445:
11370:
11184:
11049:
10905:
10860:
10704:
10689:
10342:
10337:
10237:
10105:
9886:
8823:
7156:{\displaystyle S(f)={\frac {\sigma _{Z}^{2}}{|1-\sum _{k=1}^{p}\varphi _{k}e^{-i2\pi fk}|^{2}}}.}
5266:
1685:
Each real root contributes a component to the autocorrelation function that decays exponentially.
529:
9656:
9091:
for the period being predicted. Each of the last three can be quantified and combined to give a
4373:
3141:
2088:
1688:
Similarly, each pair of complex conjugate roots contributes an exponentially damped oscillation.
11924:
11852:
11277:
11267:
11111:
10664:
10424:
10143:
10138:
9477:
7527:. This similarly acts as a high-pass filter, everything except for blue light will be filtered.
6784:
previous values of the same series. This can be thought of as a forward-prediction scheme. The
5945:{\displaystyle \gamma _{0}=\sum _{k=1}^{p}\varphi _{k}\gamma _{-k}+\sigma _{\varepsilon }^{2},}
4961:
61:
4008:
1756:
1736:
1716:
11947:
11929:
11909:
11904:
11623:
11455:
11435:
11282:
11225:
11064:
10974:
10694:
10679:
10644:
10332:
10232:
10100:
4441:
4111:
3975:
3590:
2382:
110:
10562:
9556:
12022:
11977:
11967:
11653:
11628:
11597:
11577:
11415:
11337:
11189:
10714:
10669:
10115:
9906:
9901:
9801:
9577:
9520:
9122:
9112:
8476:
7980:
7953:
3981:
3950:
3865:
3508:
3485:
3190:
3170:
3114:
2114:
1501:
1081:
1054:
1027:
963:
936:
909:
855:
677:
145:
131:
114:
3735:{\displaystyle X_{t}=\varphi ^{N}X_{t-N}+\sum _{k=0}^{N-1}\varphi ^{k}\varepsilon _{t-k}.}
2220:
8:
12017:
11857:
11782:
11587:
11347:
11257:
11147:
10289:
10267:
10016:
10011:
9969:
9921:
9092:
8415:
3312:
1733:, only the previous term in the process and the noise term contribute to the output. If
467:
9805:
9524:
8808:'s Econometrics Toolbox and System Identification Toolbox includes autoregressive models
6606:{\displaystyle \rho _{1}=\gamma _{1}/\gamma _{0}={\frac {\varphi _{1}}{1-\varphi _{2}}}}
11987:
11952:
11867:
11837:
11668:
11607:
11602:
11425:
11262:
10927:
10865:
10804:
10674:
10252:
9775:
9598:
9137:
4164:
2707:
and then by noticing that the quantity above is a stable fixed point of this relation.
2085:
since it is obtained as the output of a stable filter whose input is white noise. (If
1615:
1175:
360:
9351:
Theodoridis, Sergios (2015-04-10). "Chapter 1. Probability and
Stochastic Processes".
9274:"Understanding Autoregressive Model for Time Series as a Deterministic Dynamic System"
8135:, the process has a pair of complex-conjugate poles, creating a mid-frequency peak at:
2866:
It can be seen that the autocovariance function decays with a decay time (also called
1328:
on the right side is carried out, the polynomial in the backshift operator applied to
12007:
11220:
11137:
11106:
10999:
10979:
10969:
10825:
10820:
10740:
10728:
10532:
10184:
10055:
10048:
9782:
9392:
9356:
9253:
9219:
9209:
9184:
9174:
7619:
4182:
2910:
11812:
11463:
8067:{\displaystyle {\begin{bmatrix}\varphi _{1}&\varphi _{2}\\1&0\end{bmatrix}}}
12027:
11914:
11797:
11673:
11410:
11167:
11142:
11091:
10942:
10895:
10485:
10475:
10282:
10076:
10026:
10021:
9964:
9952:
9528:
9384:
8858:
6785:
3944:
2906:
11019:
11992:
11892:
11877:
11638:
11572:
11250:
11194:
11177:
10922:
10598:
10542:
10364:
10006:
9926:
9563:
9331:
9304:
9280:
8250:
with bandwidth about the peak inversely proportional to the moduli of the poles:
1774:
11807:
11039:
9627:
6891:{\displaystyle X_{t}=\sum _{i=1}^{p}\varphi _{i}X_{t+i}+\varepsilon _{t}^{*}\,.}
11997:
11962:
11882:
11488:
11235:
11152:
11121:
11116:
11096:
11086:
11029:
11004:
10984:
10949:
10917:
10900:
10572:
10537:
10527:
10352:
10110:
9936:
2711:
11024:
9447:
Burg, John Parker (1968); "A new analysis technique for time series data", in
6793:
prediction equations, relating to the backward representation of the AR model:
12060:
11899:
11440:
11272:
11230:
11172:
10994:
10910:
10850:
10517:
10497:
10414:
10093:
9714:
9685:
9388:
2867:
2217:
by the very definition of weak sense stationarity. If the mean is denoted by
9532:
9223:
9188:
6393:{\displaystyle \gamma _{1}=\varphi _{1}\gamma _{0}+\varphi _{2}\gamma _{-1}}
5065:{\displaystyle X_{t}=\sum _{i=1}^{p}\varphi _{i}X_{t-i}+\varepsilon _{t}.\,}
4812:{\displaystyle \operatorname {E} (X_{t+n}|X_{t})=\mu \left+X_{t}\theta ^{n}}
1185:
values infinitely far into the future from when they occur, any given value
456:{\displaystyle X_{t}=\sum _{i=1}^{p}\varphi _{i}B^{i}X_{t}+\varepsilon _{t}}
11957:
11919:
11473:
11405:
11294:
11289:
11101:
11034:
11009:
10845:
10603:
10434:
9849:
9495:
Proceedings of the 37th
Meeting of the Society of Exploration Geophysicists
8971:{\displaystyle X_{t}=\sum _{i=1}^{p}\varphi _{i}X_{t-i}+\varepsilon _{t}\,}
7541:
7536:
The behavior of an AR(2) process is determined entirely by the roots of it
6467:{\displaystyle \gamma _{2}=\varphi _{1}\gamma _{1}+\varphi _{2}\gamma _{0}}
9828:
3852:{\displaystyle X_{t}=\sum _{k=0}^{\infty }\varphi ^{k}\varepsilon _{t-k}.}
12002:
11537:
11521:
11516:
11511:
11501:
11304:
11245:
11240:
11204:
10964:
10855:
10699:
10470:
10379:
10374:
9996:
9974:
9203:
9168:
8473:
it acts as a high-pass filter on the white noise with a spectral peak at
7623:
671:
356:
268:{\displaystyle X_{t}=\sum _{i=1}^{p}\varphi _{i}X_{t-i}+\varepsilon _{t}}
8412:
it acts as a low-pass filter on the white noise with a spectral peak at
4326:{\displaystyle X_{t+1}=X_{t}+(1-\theta )(\mu -X_{t})+\varepsilon _{t+1}}
4118:
growth or decay). In this case, the solution can be found analytically:
12012:
11552:
11496:
11380:
11333:
Generalized autoregressive conditional heteroskedasticity (GARCH) model
10773:
10593:
10552:
10547:
10460:
10369:
10277:
10189:
10169:
9844:
7524:
5368:, the set of equations can be solved by representing the equations for
4544:{\displaystyle X_{t+1}=\theta X_{t}+(1-\theta )\mu +\varepsilon _{t+1}}
663:{\displaystyle \Phi (z):=\textstyle 1-\sum _{i=1}^{p}\varphi _{i}z^{i}}
106:
9461:
Brockwell, Peter J.; Dahlhaus, Rainer; Trindade, A. Alexandre (2005).
9099:-step-ahead predictions; the confidence interval will become wider as
6169:{\displaystyle \rho (\tau )=\sum _{k=1}^{p}\varphi _{k}\rho (k-\tau )}
1355:
has an infinite orderâthat is, an infinite number of lagged values of
11506:
10588:
10557:
10455:
10299:
10262:
10199:
10153:
10148:
10133:
9741:"statsmodels.tsa.ar_model.AutoReg â statsmodels 0.12.2 documentation"
9132:
7726:{\displaystyle H_{z}=(1-\varphi _{1}z^{-1}-\varphi _{2}z^{-2})^{-1}.}
7457:
5119:
1753:
is close to 0, then the process still looks like white noise, but as
5361:
Because the last part of an individual equation is non-zero only if
10490:
10322:
2409:
528:
An autoregressive model can thus be viewed as the output of an all-
1488:{\displaystyle \rho (\tau )=\sum _{k=1}^{p}a_{k}y_{k}^{-|\tau |},}
10613:
10450:
10404:
10327:
10227:
10222:
10174:
9463:"Modified Burg Algorithms for Multivariate Subset Autoregression"
6941:
4188:
589:) model to be weak-sense stationary, the roots of the polynomial
538:
Some parameter constraints are necessary for the model to remain
9250:
Time series analysis and its applications : with R examples
10628:
10608:
10480:
10272:
9686:"The Time Series Analysis (TSA) toolbox for Octave and MatlabÂŽ"
4193:
The AR(1) model is the discrete-time analogy of the continuous
1701:
9493:
Burg, John Parker (1967) "Maximum
Entropy Spectral Analysis",
9002:
into the autoregressive equation while setting the error term
6305:{\displaystyle \rho _{1}=\gamma _{1}/\gamma _{0}=\varphi _{1}}
4960:
There are many ways to estimate the coefficients, such as the
4438:
is a white-noise process with zero mean and constant variance
3111:
This expression is periodic due to the discrete nature of the
1986:
is a white noise process with zero mean and constant variance
1314:{\displaystyle X_{t}={\frac {1}{\phi (B)}}\varepsilon _{t}\,.}
466:
so that, moving the summation term to the left side and using
10429:
10409:
10399:
10394:
10389:
10384:
10347:
10179:
9059:
right-side values are predicted values from preceding steps.
8829:
7456:
there is a single spectral peak at f=0, often referred to as
6948:
117:, it is a special case and key component of the more general
10419:
9507:
Bos, Robert; De Waele, Stijn; Broersen, Piet M. T. (2002).
1600:{\displaystyle \phi (B)=1-\sum _{k=1}^{p}\varphi _{k}B^{k}}
11313:
Autoregressive conditional heteroskedasticity (ARCH) model
5321:
is the standard deviation of the input noise process, and
814:{\displaystyle X_{t}=\varphi _{1}X_{t-1}+\varepsilon _{t}}
9460:
9353:
Machine
Learning: A Bayesian and Optimization Perspective
8774:
7810:{\displaystyle 1-\varphi _{1}z^{-1}-\varphi _{2}z^{-2}=0}
10841:
Independent and identically distributed random variables
9208:. David S. Stoffer. New York: Springer. pp. 90â91.
8580:
The full PSD function can be expressed in real form as:
8570:{\displaystyle -1\leq \varphi _{2}\leq 1-|\varphi _{1}|}
8329:{\displaystyle |z_{1}|=|z_{2}|={\sqrt {-\varphi _{2}}}.}
3482:
is the angular frequency associated with the decay time
1952:{\displaystyle X_{t}=\varphi X_{t-1}+\varepsilon _{t}\,}
8822:
contains several estimation functions for uni-variate,
1647:
is the function defining the autoregression, and where
92:
In statistics, econometrics, and signal processing, an
11318:
Autoregressive integrated moving average (ARIMA) model
8019:
7736:
It follows that the poles are values of z satisfying:
5701:
5468:
5390:
611:
9069:
9008:
8893:
8589:
8518:
8479:
8446:
8418:
8385:
8345:
8259:
8145:
8087:
8013:
7983:
7956:
7830:
7745:
7635:
7553:
7496:
7466:
7429:
7241:
7180:
7034:
7018:{\displaystyle \mathrm {Var} (Z_{t})=\sigma _{Z}^{2}}
6969:
6805:
6621:
6529:
6484:
6407:
6330:
6250:
6198:
6101:
6069:
6024:
5961:
5863:
5790:
5384:
5327:
5300:
5269:
5135:
5081:
4984:
4827:
4710:
4557:
4468:
4444:
4411:
4376:
4339:
4234:
4203:
4167:
4124:
4067:
4034:
4011:
3984:
3953:
3922:
3895:
3868:
3785:
3755:
3633:
3593:
3538:
3511:
3488:
3454:
3324:
3223:
3193:
3173:
3144:
3117:
2922:
2876:
2723:
2607:
2577:
2550:
2421:
2385:
2344:
2243:
2223:
2180:
2144:
2117:
2091:
2055:
2024:
1992:
1965:
1902:
1867:
1840:
1813:
1786:
1759:
1739:
1719:
1653:
1624:
1534:
1504:
1407:
1361:
1334:
1262:
1203:
1178:
then the effect diminishes toward zero in the limit.
1153:
1111:
1084:
1057:
1030:
993:
966:
939:
912:
885:
858:
827:
758:
707:
680:
595:
548:
479:
375:
334:
284:
191:
148:
9513:
7608:{\displaystyle 1-\varphi _{1}B-\varphi _{2}B^{2}=0,}
9836:
Econometrics lecture (topic: Autoregressive models)
9506:
9414:"The Yule Walker Equations for the AR Coefficients"
9170:
Time series analysis : forecasting and control
8802:
function to fit various models including AR models.
8128:{\displaystyle \varphi _{1}^{2}+4\varphi _{2}<0}
6322:The YuleâWalker equations for an AR(2) process are
6235:{\displaystyle \gamma _{1}=\varphi _{1}\gamma _{0}}
4955:
1147:. Continuing this process shows that the effect of
9774:
9083:
9021:
8970:
8763:
8569:
8499:
8465:
8430:
8404:
8364:
8328:
8239:
8127:
8066:
7996:
7969:
7939:
7809:
7725:
7607:
7515:
7479:
7448:
7411:
7216:
7155:
7017:
6890:
6735:
6605:
6513:
6466:
6392:
6304:
6234:
6168:
6084:
6045:
6010:
5944:
5842:
5773:
5346:
5313:
5282:
5235:
5094:
5064:
4929:
4811:
4693:
4543:
4450:
4430:
4397:
4362:
4325:
4217:
4173:
4153:
4102:
4053:
4017:
3997:
3966:
3935:
3916:kernel plus the constant mean. If the white noise
3908:
3881:
3851:
3768:
3734:
3620:in the defining equation. Continuing this process
3612:
3580:{\displaystyle \varphi X_{t-2}+\varepsilon _{t-1}}
3579:
3524:
3494:
3474:
3437:
3300:
3206:
3179:
3159:
3130:
3100:
2894:
2855:
2696:
2590:
2563:
2533:
2397:
2368:
2330:
2229:
2205:
2166:
2130:
2103:
2077:
2037:
2010:
1978:
1951:
1880:
1853:
1826:
1799:
1765:
1745:
1725:
1666:
1639:
1599:
1517:
1487:
1374:
1347:
1313:
1242:
1166:
1139:
1097:
1070:
1043:
1016:
979:
952:
925:
898:
871:
840:
813:
736:
693:
662:
577:
542:. For example, processes in the AR(1) model with
517:
455:
347:
316:
267:
166:
9815:Time Series and System Analysis with Applications
747:
317:{\displaystyle \varphi _{1},\ldots ,\varphi _{p}}
12058:
11200:Stochastic chains with memory of variable length
9378:
7523:there is a minimum at f=0, often referred to as
6924:Other possible approaches to estimation include
6059:. The AR parameters are determined by the first
5843:{\displaystyle \{\varphi _{m};m=1,2,\dots ,p\}.}
4005:will be approximately normally distributed when
3974:is also a Gaussian process. In other cases, the
3187:), then we can use a continuum approximation to
1243:{\displaystyle \phi (B)X_{t}=\varepsilon _{t}\,}
1140:{\displaystyle \varphi _{1}^{2}\varepsilon _{1}}
9796:Percival, Donald B.; Walden, Andrew T. (1993).
9579:astsa: Applied Statistical Time Series Analysis
9310:Philosophical Transactions of the Royal Society
6011:{\displaystyle \{\varphi _{m};m=1,2,\dots ,p\}}
9795:
9657:"Autoregressive Model - MATLAB & Simulink"
9247:
6750:
6055:An alternative formulation is in terms of the
4189:Explicit mean/difference form of AR(1) process
585:are not stationary. More generally, for an AR(
10789:
9865:
9575:
9379:Von Storch, Hans; Zwiers, Francis W. (2001).
1681:) process is a sum of decaying exponentials.
359:. This can be equivalently written using the
9879:
9813:Pandit, Sudhakar M.; Wu, Shien-Ming (1983).
9576:Stoffer, David; Poison, Nicky (2023-01-09),
9287:, June 2017, number 15, June 2017, pages 7-9
6005:
5962:
5834:
5791:
4425:
4412:
1891:
1692:
1385:
1017:{\displaystyle \varphi _{1}\varepsilon _{1}}
9798:Spectral Analysis for Physical Applications
9350:
9325:"On Periodicity in Series of Related Terms"
9248:Shumway, Robert H.; Stoffer, David (2010).
8876:
8375:Otherwise the process has real roots, and:
6046:{\displaystyle \sigma _{\varepsilon }^{2}.}
4941:
518:{\displaystyle \phi X_{t}=\varepsilon _{t}}
174:indicates an autoregressive model of order
11328:Autoregressiveâmoving-average (ARMA) model
10796:
10782:
9872:
9858:
9557:"Fit Autoregressive Models to Time Series"
8884:Once the parameters of the autoregression
5375:in matrix form, thus getting the equation
2011:{\displaystyle \sigma _{\varepsilon }^{2}}
1382:appear on the right side of the equation.
27:Representation of a type of random process
9454:
9205:Time series analysis and its applications
9080:
8967:
6884:
5061:
4359:
4211:
3393:
3355:
3277:
3276:
3025:
2953:
2829:
2828:
2206:{\displaystyle \operatorname {E} (X_{t})}
1948:
1307:
1239:
80:Learn how and when to remove this message
10803:
9812:
9381:Statistical analysis in climate research
9374:
9372:
6947:
6940:
6514:{\displaystyle \gamma _{-k}=\gamma _{k}}
5113:
1700:
123:autoregressive integrated moving average
43:This article includes a list of general
9721:from the original on September 28, 2020
9355:. Academic Press, 2015. pp. 9â51.
9201:
4218:{\displaystyle \theta \in \mathbb {R} }
3167:) is much smaller than the decay time (
1174:never ends, although if the process is
14:
12059:
11634:Doob's martingale convergence theorems
8871:vector autoregression#impulse response
8775:Implementations in statistics packages
5314:{\displaystyle \sigma _{\varepsilon }}
5126:, are the following set of equations.
2564:{\displaystyle \sigma _{\varepsilon }}
1677:The autocorrelation function of an AR(
11386:Constant elasticity of variance (CEV)
11376:ChanâKarolyiâLongstaffâSanders (CKLS)
10777:
9853:
9777:Time Series Techniques for Economists
9772:
9487:
9369:
7540:, which is expressed in terms of the
7217:{\displaystyle S(f)=\sigma _{Z}^{2}.}
5118:The YuleâWalker equations, named for
4103:{\displaystyle X_{t}=\varphi X_{t-1}}
3532:can be derived by first substituting
1713:For an AR(1) process with a positive
10710:Generative adversarial network (GAN)
9441:
7618:or equivalently by the poles of its
2369:{\displaystyle \mu =\varphi \mu +0,}
578:{\displaystyle |\varphi _{1}|\geq 1}
29:
9500:
9166:
9029:equal to zero (because we forecast
8852:
8826:and adaptive autoregressive models.
6918:maximum entropy spectral estimation
6615:Using the recursion formula yields
5290:is the autocovariance function of X
4154:{\displaystyle X_{t}=a\varphi ^{t}}
2598:. This can be shown by noting that
535:filter whose input is white noise.
24:
11873:Skorokhod's representation theorem
11654:Law of large numbers (weak/strong)
9569:
9084:{\displaystyle \varepsilon _{t}\,}
8865:periods earlier, as a function of
6977:
6974:
6971:
4711:
4629:
4054:{\displaystyle \varepsilon _{t}=0}
3889:is white noise convolved with the
3815:
3325:
3145:
2973:
2968:
2923:
2737:
2448:
2303:
2272:
2244:
2181:
2045:has been dropped.) The process is
596:
49:it lacks sufficient corresponding
25:
12083:
11843:Martingale representation theorem
9822:
9411:
9285:Predictive Analytics and Futurism
8512:coefficients are in the triangle
8466:{\displaystyle \varphi _{1}<0}
8405:{\displaystyle \varphi _{1}>0}
8365:{\displaystyle \varphi _{2}<0}
7516:{\displaystyle \varphi _{1}<0}
7449:{\displaystyle \varphi _{1}>0}
4975:) model is given by the equation
4968:(through YuleâWalker equations).
4551:and then deriving (by induction)
4431:{\displaystyle \{\epsilon _{t}\}}
11888:Stochastic differential equation
11778:Doob's optional stopping theorem
11773:DoobâMeyer decomposition theorem
10748:
10747:
10727:
9337:Proceedings of the Royal Society
9022:{\displaystyle \varepsilon _{t}}
8848:: implementation in statsmodels.
6523:Using the first equation yields
4956:Calculation of the AR parameters
4948:Partial autocorrelation function
4363:{\displaystyle |\theta |<1\,}
3936:{\displaystyle \varepsilon _{t}}
2895:{\displaystyle \tau =1-\varphi }
2591:{\displaystyle \varepsilon _{t}}
1979:{\displaystyle \varepsilon _{t}}
1525:are the roots of the polynomial
1375:{\displaystyle \varepsilon _{t}}
1348:{\displaystyle \varepsilon _{t}}
1167:{\displaystyle \varepsilon _{1}}
899:{\displaystyle \varepsilon _{1}}
841:{\displaystyle \varepsilon _{t}}
348:{\displaystyle \varepsilon _{t}}
34:
11758:Convergence of random variables
11644:FisherâTippettâGnedenko theorem
9751:from the original on 2021-02-28
9733:
9707:
9696:from the original on 2012-05-11
9678:
9667:from the original on 2022-02-16
9649:
9638:from the original on 2022-02-16
9628:"System Identification Toolbox"
9620:
9609:from the original on 2023-04-16
9591:
9550:
9539:from the original on 2023-04-16
9430:from the original on 2018-07-13
9230:from the original on 2023-04-16
6179:Examples for some Low-order AR(
3475:{\displaystyle \gamma =1/\tau }
2213:is identical for all values of
2167:{\displaystyle |\varphi |<1}
2078:{\displaystyle |\varphi |<1}
1398:) process can be expressed as
11356:Binomial options pricing model
10660:Recurrent neural network (RNN)
10650:Differentiable neural computer
9800:. Cambridge University Press.
9781:. Cambridge University Press.
9405:
9383:. Cambridge University Press.
9344:
9317:
9290:
9272:Lai, Dihui; and Lu, Bingfeng;
9266:
9241:
9195:
9160:
8755:
8743:
8718:
8706:
8697:
8678:
8599:
8593:
8563:
8548:
8299:
8284:
8276:
8261:
7708:
7649:
7320:
7276:
7251:
7245:
7190:
7184:
7137:
7069:
7044:
7038:
6994:
6981:
6963:) process with noise variance
6163:
6151:
6111:
6105:
6079:
6073:
4868:
4854:
4834:
4751:
4737:
4717:
4619:
4600:
4516:
4504:
4392:
4386:
4349:
4341:
4301:
4282:
4279:
4267:
3505:An alternative expression for
3429:
3403:
3334:
3328:
3292:
3284:
3233:
3227:
3085:
3079:
2932:
2926:
2844:
2836:
2772:
2743:
2670:
2651:
2628:
2615:
2472:
2454:
2442:
2429:
2322:
2309:
2297:
2278:
2263:
2250:
2200:
2187:
2154:
2146:
2065:
2057:
1634:
1628:
1544:
1538:
1476:
1468:
1417:
1411:
1291:
1285:
1213:
1207:
1024:. Then by the AR equation for
906:. Then by the AR equation for
748:Intertemporal effect of shocks
724:
709:
605:
599:
565:
550:
489:
483:
161:
155:
13:
1:
11823:Kolmogorov continuity theorem
11659:Law of the iterated logarithm
10705:Variational autoencoder (VAE)
10665:Long short-term memory (LSTM)
9932:Computational learning theory
9766:
9341:, Ser. A, Vol. 131, 518â532.
9314:, Ser. A, Vol. 226, 267â298.]
6926:maximum likelihood estimation
6018:are known, can be solved for
5347:{\displaystyle \delta _{m,0}}
2571:is the standard deviation of
1896:An AR(1) process is given by:
1640:{\displaystyle \phi (\cdot )}
137:
119:autoregressiveâmoving-average
11828:Kolmogorov extension theorem
11507:Generalized queueing network
11015:Interacting particle systems
10685:Convolutional neural network
9829:AutoRegression Analysis (AR)
7480:{\displaystyle \varphi _{1}}
6764:be produced by some choices.
6085:{\displaystyle \rho (\tau )}
5784:which can be solved for all
5095:{\displaystyle \varphi _{i}}
3909:{\displaystyle \varphi ^{k}}
3769:{\displaystyle \varphi ^{N}}
2038:{\displaystyle \varphi _{1}}
1881:{\displaystyle \varphi _{2}}
1854:{\displaystyle \varphi _{1}}
1827:{\displaystyle \varphi _{2}}
1800:{\displaystyle \varphi _{1}}
1667:{\displaystyle \varphi _{k}}
737:{\displaystyle |z_{i}|>1}
674:, i.e., each (complex) root
7:
10960:Continuous-time random walk
10680:Multilayer perceptron (MLP)
9202:Shumway, Robert H. (2000).
9106:
6936:
6933:close to non-stationarity.
6751:Estimation of AR parameters
5850:The remaining equation for
5283:{\displaystyle \gamma _{m}}
1181:Because each shock affects
10:
12088:
11968:Extreme value theory (EVT)
11768:Doob decomposition theorem
11060:OrnsteinâUhlenbeck process
10831:Chinese restaurant process
10756:Artificial neural networks
10670:Gated recurrent unit (GRU)
9896:Differentiable programming
9773:Mills, Terence C. (1990).
9497:, Oklahoma City, Oklahoma.
9252:(3rd ed.). Springer.
9167:Box, George E. P. (1994).
9143:OrnsteinâUhlenbeck process
9118:Linear difference equation
7622:, which is defined in the
5075:It is based on parameters
4945:
4398:{\displaystyle \mu :=E(X)}
4195:Ornstein-Uhlenbeck process
3315:for the spectral density:
3160:{\displaystyle \Delta t=1}
2104:{\displaystyle \varphi =1}
2018:. (Note: The subscript on
12036:
11940:
11848:Optional stopping theorem
11745:
11707:
11649:Large deviation principle
11616:
11530:
11487:
11454:
11401:HeathâJarrowâMorton (HJM)
11346:
11338:Moving-average (MA) model
11323:Autoregressive (AR) model
11303:
11213:
11148:Hidden Markov model (HMM)
11130:
11082:SchrammâLoewner evolution
10886:
10811:
10723:
10637:
10581:
10510:
10443:
10315:
10215:
10208:
10162:
10126:
10089:Artificial neural network
10069:
9945:
9912:Automatic differentiation
9885:
9715:"christophmark/bayesloop"
9476:: 197â213. Archived from
9148:Infinite impulse response
6902:Here predicted values of
1892:Example: An AR(1) process
1386:Characteristic polynomial
533:infinite impulse response
115:moving-average (MA) model
11763:DolĂŠans-Dade exponential
11593:Progressively measurable
11391:CoxâIngersollâRoss (CIR)
9917:Neuromorphic engineering
9880:Differentiable computing
9817:. John Wiley & Sons.
9449:Modern Spectrum Analysis
9389:10.1017/CBO9780511612336
9323:Walker, Gilbert (1931)
9153:
9128:Linear predictive coding
7531:
7227:
7171:For white noise (AR(0))
7166:
6769:least squares regression
6057:autocorrelation function
5356:Kronecker delta function
4942:Choosing the maximum lag
4181:is an unknown constant (
4018:{\displaystyle \varphi }
3776:will approach zero and:
1766:{\displaystyle \varphi }
1746:{\displaystyle \varphi }
1726:{\displaystyle \varphi }
1392:autocorrelation function
11983:Mathematical statistics
11973:Large deviations theory
11803:Infinitesimal generator
11664:Maximal ergodic theorem
11583:Piecewise-deterministic
11185:Random dynamical system
11050:Markov additive process
10690:Residual neural network
10106:Artificial Intelligence
9533:10.1109/TIM.2002.808031
8880:-step-ahead forecasting
7538:characteristic equation
4451:{\displaystyle \sigma }
4405:is the model mean, and
3613:{\displaystyle X_{t-1}}
2398:{\displaystyle \mu =0.}
821:. A non-zero value for
64:more precise citations.
11818:KarhunenâLoève theorem
11753:CameronâMartin formula
11717:BurkholderâDavisâGundy
11112:Variance gamma process
9599:"Econometrics Toolbox"
9421:stat.wharton.upenn.edu
9085:
9023:
8972:
8927:
8765:
8571:
8501:
8467:
8432:
8406:
8366:
8330:
8241:
8129:
8068:
7998:
7971:
7941:
7811:
7727:
7609:
7517:
7481:
7450:
7413:
7218:
7157:
7099:
7019:
6957:power spectral density
6952:
6945:
6911:would be based on the
6892:
6839:
6737:
6607:
6515:
6468:
6394:
6306:
6236:
6170:
6137:
6086:
6047:
6012:
5946:
5897:
5844:
5775:
5348:
5315:
5284:
5237:
5169:
5096:
5066:
5018:
4962:ordinary least squares
4931:
4813:
4695:
4545:
4452:
4432:
4399:
4364:
4327:
4219:
4175:
4155:
4104:
4055:
4019:
3999:
3968:
3937:
3910:
3883:
3853:
3819:
3770:
3749:approaching infinity,
3736:
3702:
3614:
3581:
3526:
3496:
3476:
3439:
3302:
3208:
3181:
3161:
3132:
3102:
2977:
2896:
2857:
2698:
2592:
2565:
2535:
2399:
2370:
2332:
2231:
2207:
2168:
2132:
2105:
2079:
2039:
2012:
1980:
1953:
1882:
1855:
1828:
1801:
1767:
1747:
1727:
1706:
1668:
1641:
1601:
1576:
1519:
1489:
1443:
1376:
1349:
1315:
1244:
1168:
1141:
1099:
1072:
1045:
1018:
981:
954:
927:
900:
873:
842:
815:
738:
695:
664:
638:
579:
519:
457:
409:
349:
318:
269:
225:
182:) model is defined as
168:
11948:Actuarial mathematics
11910:Uniform integrability
11905:Stratonovich integral
11833:LĂŠvyâProkhorov metric
11737:MarcinkiewiczâZygmund
11624:Central limit theorem
11226:Gaussian random field
11055:McKeanâVlasov process
10975:Dyson Brownian motion
10836:GaltonâWatson process
10645:Neural Turing machine
10233:Human image synthesis
9296:Yule, G. Udny (1927)
9086:
9024:
8973:
8907:
8766:
8572:
8502:
8500:{\displaystyle f=1/2}
8468:
8433:
8407:
8367:
8331:
8242:
8130:
8069:
7999:
7997:{\displaystyle z_{2}}
7972:
7970:{\displaystyle z_{1}}
7942:
7812:
7728:
7610:
7518:
7482:
7451:
7414:
7219:
7158:
7079:
7020:
6951:
6944:
6893:
6819:
6738:
6608:
6516:
6469:
6395:
6307:
6237:
6171:
6117:
6087:
6048:
6013:
5947:
5877:
5845:
5776:
5349:
5316:
5285:
5238:
5149:
5114:YuleâWalker equations
5097:
5067:
4998:
4932:
4814:
4696:
4546:
4462:By rewriting this as
4453:
4433:
4400:
4365:
4328:
4220:
4176:
4156:
4112:geometric progression
4105:
4056:
4020:
4000:
3998:{\displaystyle X_{t}}
3976:central limit theorem
3969:
3967:{\displaystyle X_{t}}
3938:
3911:
3884:
3882:{\displaystyle X_{t}}
3854:
3799:
3771:
3737:
3676:
3615:
3582:
3527:
3525:{\displaystyle X_{t}}
3497:
3495:{\displaystyle \tau }
3477:
3440:
3303:
3209:
3207:{\displaystyle B_{n}}
3182:
3180:{\displaystyle \tau }
3162:
3133:
3131:{\displaystyle X_{j}}
3103:
2954:
2897:
2858:
2699:
2593:
2566:
2536:
2400:
2371:
2333:
2232:
2208:
2169:
2133:
2131:{\displaystyle X_{t}}
2111:then the variance of
2106:
2080:
2047:weak-sense stationary
2040:
2013:
1981:
1954:
1883:
1856:
1829:
1802:
1768:
1748:
1728:
1704:
1669:
1642:
1602:
1556:
1520:
1518:{\displaystyle y_{k}}
1490:
1423:
1377:
1350:
1316:
1245:
1169:
1142:
1100:
1098:{\displaystyle X_{3}}
1073:
1071:{\displaystyle X_{2}}
1046:
1044:{\displaystyle X_{3}}
1019:
982:
980:{\displaystyle X_{2}}
955:
953:{\displaystyle X_{1}}
928:
926:{\displaystyle X_{2}}
901:
874:
872:{\displaystyle X_{1}}
843:
816:
739:
696:
694:{\displaystyle z_{i}}
670:must lie outside the
665:
618:
580:
540:weak-sense stationary
520:
458:
389:
350:
319:
270:
205:
169:
167:{\displaystyle AR(p)}
132:Large language models
111:differential equation
12023:Time series analysis
11978:Mathematical finance
11863:Reflection principle
11190:Regenerative process
10990:FlemingâViot process
10805:Stochastic processes
10736:Computer programming
10715:Graph neural network
10290:Text-to-video models
10268:Text-to-image models
10116:Large language model
10101:Scientific computing
9907:Statistical manifold
9902:Information geometry
9717:. December 7, 2021.
9123:Predictive analytics
9113:Moving average model
9067:
9006:
8891:
8787:package includes an
8587:
8516:
8477:
8444:
8416:
8383:
8343:
8257:
8143:
8085:
8011:
7981:
7954:
7828:
7743:
7633:
7551:
7494:
7464:
7427:
7239:
7178:
7032:
6967:
6803:
6619:
6527:
6482:
6405:
6328:
6248:
6196:
6099:
6067:
6022:
5959:
5861:
5788:
5382:
5325:
5298:
5267:
5133:
5079:
4982:
4825:
4708:
4701:, one can show that
4555:
4466:
4442:
4409:
4374:
4337:
4232:
4201:
4165:
4122:
4065:
4032:
4009:
3982:
3951:
3920:
3893:
3866:
3783:
3753:
3631:
3591:
3536:
3509:
3486:
3452:
3322:
3221:
3191:
3171:
3142:
3115:
2920:
2874:
2721:
2605:
2575:
2548:
2419:
2383:
2342:
2241:
2230:{\displaystyle \mu }
2221:
2178:
2142:
2115:
2089:
2053:
2022:
1990:
1963:
1900:
1865:
1838:
1811:
1784:
1757:
1737:
1717:
1651:
1622:
1532:
1502:
1405:
1359:
1332:
1260:
1201:
1151:
1109:
1082:
1055:
1028:
991:
964:
937:
910:
883:
856:
825:
756:
744:(see pages 89,92 ).
705:
678:
593:
546:
477:
373:
332:
282:
189:
146:
113:. Together with the
12018:Stochastic analysis
11858:Quadratic variation
11853:Prokhorov's theorem
11788:FeynmanâKac formula
11258:Markov random field
10906:Birthâdeath process
10082:In-context learning
9922:Pattern recognition
9806:1993sapa.book.....P
9745:www.statsmodels.org
9729:– via GitHub.
9525:2002ITIM...51.1289B
9093:confidence interval
8661:
8643:
8621:
8431:{\displaystyle f=0}
8102:
7913:
7374:
7352:
7273:
7210:
7066:
7014:
6883:
6698:
6680:
6039:
5938:
5213:
4648:
3372:
3255:
3046:
2807:
2690:
2507:
2471:
2007:
1481:
1326:polynomial division
1126:
468:polynomial notation
11988:Probability theory
11868:Skorokhod integral
11838:Malliavin calculus
11421:Korn-Kreer-Lenssen
11305:Time series models
11268:PitmanâYor process
10675:Echo state network
10563:JĂźrgen Schmidhuber
10258:Facial recognition
10253:Speech recognition
10163:Software libraries
9562:2016-01-28 at the
9330:2011-06-07 at the
9303:2011-05-14 at the
9279:2023-03-24 at the
9138:Levinson recursion
9081:
9019:
8968:
8761:
8647:
8629:
8607:
8567:
8497:
8463:
8428:
8402:
8362:
8326:
8237:
8125:
8088:
8064:
8058:
7994:
7967:
7937:
7899:
7807:
7723:
7605:
7513:
7477:
7446:
7409:
7360:
7338:
7259:
7214:
7196:
7153:
7052:
7015:
7000:
6953:
6946:
6888:
6869:
6733:
6684:
6666:
6603:
6511:
6464:
6390:
6302:
6232:
6166:
6082:
6043:
6025:
6008:
5942:
5924:
5840:
5771:
5765:
5690:
5454:
5344:
5311:
5280:
5233:
5199:
5092:
5062:
4927:
4809:
4691:
4628:
4541:
4448:
4428:
4395:
4360:
4323:
4215:
4171:
4151:
4100:
4051:
4015:
3995:
3964:
3933:
3906:
3879:
3849:
3766:
3732:
3610:
3577:
3522:
3492:
3472:
3435:
3358:
3313:Lorentzian profile
3298:
3241:
3204:
3177:
3157:
3128:
3098:
3032:
2892:
2853:
2793:
2694:
2676:
2588:
2561:
2531:
2493:
2457:
2395:
2366:
2328:
2227:
2203:
2164:
2128:
2101:
2075:
2035:
2008:
1993:
1976:
1949:
1878:
1861:is positive while
1851:
1824:
1797:
1763:
1743:
1723:
1707:
1664:
1637:
1616:backshift operator
1597:
1515:
1485:
1454:
1372:
1345:
1311:
1240:
1164:
1137:
1112:
1095:
1068:
1041:
1014:
977:
950:
923:
896:
869:
838:
811:
734:
691:
660:
659:
575:
515:
453:
361:backshift operator
345:
328:of the model, and
314:
265:
164:
12072:Signal processing
12054:
12053:
12008:Signal processing
11727:Doob's upcrossing
11722:Doob's martingale
11686:EngelbertâSchmidt
11629:Donsker's theorem
11563:Feller-continuous
11431:RendlemanâBartter
11221:Dirichlet process
11138:Branching process
11107:Telegraph process
11000:Geometric process
10980:Empirical process
10970:Diffusion process
10826:Branching process
10821:Bernoulli process
10771:
10770:
10533:Stephen Grossberg
10506:
10505:
9661:www.mathworks.com
9632:www.mathworks.com
9603:www.mathworks.com
9470:Statistica Sinica
9362:978-0-12-801522-3
9055:predictions, all
8759:
8321:
8228:
8225:
8172:
7930:
7877:
7620:transfer function
7407:
7331:
7148:
6767:Formulation as a
6731:
6601:
4966:method of moments
4925:
4183:initial condition
4174:{\displaystyle a}
4025:is close to one.
3433:
3391:
3353:
3352:
3274:
3089:
3023:
3022:
2951:
2950:
2911:Fourier transform
2826:
2648:
2612:
2526:
2426:
2237:, it follows from
1295:
90:
89:
82:
16:(Redirected from
12079:
12028:Machine learning
11915:Usual hypotheses
11798:Girsanov theorem
11783:Dynkin's formula
11548:Continuous paths
11456:Actuarial models
11396:GarmanâKohlhagen
11366:BlackâKarasinski
11361:BlackâDermanâToy
11348:Financial models
11214:Fields and other
11143:Gaussian process
11092:Sigma-martingale
10896:Additive process
10798:
10791:
10784:
10775:
10774:
10761:Machine learning
10751:
10750:
10731:
10486:Action selection
10476:Self-driving car
10283:Stable Diffusion
10248:Speech synthesis
10213:
10212:
10077:Machine learning
9953:Gradient descent
9874:
9867:
9860:
9851:
9850:
9837:
9818:
9809:
9792:
9780:
9760:
9759:
9757:
9756:
9737:
9731:
9730:
9728:
9726:
9711:
9705:
9704:
9702:
9701:
9682:
9676:
9675:
9673:
9672:
9653:
9647:
9646:
9644:
9643:
9624:
9618:
9617:
9615:
9614:
9595:
9589:
9588:
9587:
9586:
9573:
9567:
9554:
9548:
9547:
9545:
9544:
9504:
9498:
9491:
9485:
9484:
9482:
9467:
9458:
9452:
9445:
9439:
9438:
9436:
9435:
9429:
9418:
9409:
9403:
9402:
9376:
9367:
9366:
9348:
9342:
9321:
9315:
9294:
9288:
9270:
9264:
9263:
9245:
9239:
9238:
9236:
9235:
9199:
9193:
9192:
9164:
9090:
9088:
9087:
9082:
9079:
9078:
9042:to refer to the
9028:
9026:
9025:
9020:
9018:
9017:
8977:
8975:
8974:
8969:
8966:
8965:
8953:
8952:
8937:
8936:
8926:
8921:
8903:
8902:
8859:impulse response
8853:Impulse response
8770:
8768:
8767:
8762:
8760:
8758:
8736:
8735:
8696:
8695:
8677:
8676:
8660:
8655:
8642:
8637:
8620:
8615:
8606:
8576:
8574:
8573:
8568:
8566:
8561:
8560:
8551:
8537:
8536:
8506:
8504:
8503:
8498:
8493:
8472:
8470:
8469:
8464:
8456:
8455:
8437:
8435:
8434:
8429:
8411:
8409:
8408:
8403:
8395:
8394:
8371:
8369:
8368:
8363:
8355:
8354:
8335:
8333:
8332:
8327:
8322:
8320:
8319:
8307:
8302:
8297:
8296:
8287:
8279:
8274:
8273:
8264:
8246:
8244:
8243:
8238:
8233:
8229:
8227:
8226:
8224:
8223:
8211:
8205:
8204:
8195:
8186:
8185:
8173:
8171:
8160:
8155:
8154:
8134:
8132:
8131:
8126:
8118:
8117:
8101:
8096:
8073:
8071:
8070:
8065:
8063:
8062:
8043:
8042:
8031:
8030:
8003:
8001:
8000:
7995:
7993:
7992:
7976:
7974:
7973:
7968:
7966:
7965:
7946:
7944:
7943:
7938:
7936:
7932:
7931:
7929:
7928:
7912:
7907:
7898:
7893:
7892:
7878:
7876:
7875:
7874:
7858:
7853:
7852:
7840:
7839:
7816:
7814:
7813:
7808:
7800:
7799:
7787:
7786:
7774:
7773:
7761:
7760:
7732:
7730:
7729:
7724:
7719:
7718:
7706:
7705:
7693:
7692:
7680:
7679:
7667:
7666:
7645:
7644:
7614:
7612:
7611:
7606:
7595:
7594:
7585:
7584:
7569:
7568:
7522:
7520:
7519:
7514:
7506:
7505:
7486:
7484:
7483:
7478:
7476:
7475:
7455:
7453:
7452:
7447:
7439:
7438:
7418:
7416:
7415:
7410:
7408:
7406:
7390:
7389:
7373:
7368:
7351:
7346:
7337:
7332:
7330:
7329:
7328:
7323:
7317:
7316:
7295:
7294:
7279:
7272:
7267:
7258:
7223:
7221:
7220:
7215:
7209:
7204:
7162:
7160:
7159:
7154:
7149:
7147:
7146:
7145:
7140:
7134:
7133:
7109:
7108:
7098:
7093:
7072:
7065:
7060:
7051:
7024:
7022:
7021:
7016:
7013:
7008:
6993:
6992:
6980:
6897:
6895:
6894:
6889:
6882:
6877:
6865:
6864:
6849:
6848:
6838:
6833:
6815:
6814:
6786:normal equations
6742:
6740:
6739:
6734:
6732:
6730:
6729:
6728:
6712:
6711:
6710:
6697:
6692:
6679:
6674:
6664:
6659:
6658:
6649:
6644:
6643:
6631:
6630:
6612:
6610:
6609:
6604:
6602:
6600:
6599:
6598:
6582:
6581:
6572:
6567:
6566:
6557:
6552:
6551:
6539:
6538:
6520:
6518:
6517:
6512:
6510:
6509:
6497:
6496:
6473:
6471:
6470:
6465:
6463:
6462:
6453:
6452:
6440:
6439:
6430:
6429:
6417:
6416:
6399:
6397:
6396:
6391:
6389:
6388:
6376:
6375:
6363:
6362:
6353:
6352:
6340:
6339:
6311:
6309:
6308:
6303:
6301:
6300:
6288:
6287:
6278:
6273:
6272:
6260:
6259:
6241:
6239:
6238:
6233:
6231:
6230:
6221:
6220:
6208:
6207:
6175:
6173:
6172:
6167:
6147:
6146:
6136:
6131:
6091:
6089:
6088:
6083:
6052:
6050:
6049:
6044:
6038:
6033:
6017:
6015:
6014:
6009:
5974:
5973:
5951:
5949:
5948:
5943:
5937:
5932:
5920:
5919:
5907:
5906:
5896:
5891:
5873:
5872:
5849:
5847:
5846:
5841:
5803:
5802:
5780:
5778:
5777:
5772:
5770:
5769:
5762:
5761:
5741:
5740:
5727:
5726:
5713:
5712:
5695:
5694:
5682:
5681:
5664:
5663:
5646:
5645:
5599:
5598:
5587:
5586:
5575:
5574:
5556:
5555:
5541:
5540:
5529:
5528:
5510:
5509:
5495:
5494:
5480:
5479:
5459:
5458:
5451:
5450:
5430:
5429:
5416:
5415:
5402:
5401:
5374:
5367:
5353:
5351:
5350:
5345:
5343:
5342:
5320:
5318:
5317:
5312:
5310:
5309:
5289:
5287:
5286:
5281:
5279:
5278:
5263:equations. Here
5262:
5255:
5242:
5240:
5239:
5234:
5229:
5228:
5212:
5207:
5195:
5194:
5179:
5178:
5168:
5163:
5145:
5144:
5101:
5099:
5098:
5093:
5091:
5090:
5071:
5069:
5068:
5063:
5057:
5056:
5044:
5043:
5028:
5027:
5017:
5012:
4994:
4993:
4936:
4934:
4933:
4928:
4926:
4924:
4923:
4922:
4906:
4905:
4904:
4885:
4883:
4882:
4867:
4866:
4857:
4852:
4851:
4818:
4816:
4815:
4810:
4808:
4807:
4798:
4797:
4785:
4781:
4780:
4779:
4750:
4749:
4740:
4735:
4734:
4700:
4698:
4697:
4692:
4690:
4686:
4685:
4684:
4669:
4668:
4647:
4642:
4618:
4617:
4596:
4595:
4586:
4585:
4573:
4572:
4550:
4548:
4547:
4542:
4540:
4539:
4500:
4499:
4484:
4483:
4457:
4455:
4454:
4449:
4437:
4435:
4434:
4429:
4424:
4423:
4404:
4402:
4401:
4396:
4369:
4367:
4366:
4361:
4352:
4344:
4332:
4330:
4329:
4324:
4322:
4321:
4300:
4299:
4263:
4262:
4250:
4249:
4224:
4222:
4221:
4216:
4214:
4180:
4178:
4177:
4172:
4160:
4158:
4157:
4152:
4150:
4149:
4134:
4133:
4109:
4107:
4106:
4101:
4099:
4098:
4077:
4076:
4060:
4058:
4057:
4052:
4044:
4043:
4024:
4022:
4021:
4016:
4004:
4002:
4001:
3996:
3994:
3993:
3973:
3971:
3970:
3965:
3963:
3962:
3945:Gaussian process
3942:
3940:
3939:
3934:
3932:
3931:
3915:
3913:
3912:
3907:
3905:
3904:
3888:
3886:
3885:
3880:
3878:
3877:
3862:It is seen that
3858:
3856:
3855:
3850:
3845:
3844:
3829:
3828:
3818:
3813:
3795:
3794:
3775:
3773:
3772:
3767:
3765:
3764:
3741:
3739:
3738:
3733:
3728:
3727:
3712:
3711:
3701:
3690:
3672:
3671:
3656:
3655:
3643:
3642:
3619:
3617:
3616:
3611:
3609:
3608:
3586:
3584:
3583:
3578:
3576:
3575:
3557:
3556:
3531:
3529:
3528:
3523:
3521:
3520:
3501:
3499:
3498:
3493:
3481:
3479:
3478:
3473:
3468:
3444:
3442:
3441:
3436:
3434:
3432:
3428:
3427:
3415:
3414:
3395:
3392:
3390:
3389:
3388:
3371:
3366:
3357:
3354:
3345:
3341:
3307:
3305:
3304:
3299:
3297:
3296:
3295:
3287:
3275:
3273:
3272:
3271:
3254:
3249:
3240:
3213:
3211:
3210:
3205:
3203:
3202:
3186:
3184:
3183:
3178:
3166:
3164:
3163:
3158:
3137:
3135:
3134:
3129:
3127:
3126:
3107:
3105:
3104:
3099:
3094:
3090:
3088:
3063:
3062:
3045:
3040:
3031:
3024:
3015:
3011:
3006:
3005:
2987:
2986:
2976:
2971:
2952:
2943:
2939:
2909:function is the
2907:spectral density
2901:
2899:
2898:
2893:
2862:
2860:
2859:
2854:
2849:
2848:
2847:
2839:
2827:
2825:
2824:
2823:
2806:
2801:
2792:
2787:
2786:
2771:
2770:
2761:
2760:
2733:
2732:
2703:
2701:
2700:
2695:
2689:
2684:
2669:
2668:
2650:
2649:
2646:
2643:
2642:
2627:
2626:
2614:
2613:
2610:
2597:
2595:
2594:
2589:
2587:
2586:
2570:
2568:
2567:
2562:
2560:
2559:
2540:
2538:
2537:
2532:
2527:
2525:
2524:
2523:
2506:
2501:
2492:
2487:
2486:
2470:
2465:
2441:
2440:
2428:
2427:
2424:
2404:
2402:
2401:
2396:
2375:
2373:
2372:
2367:
2337:
2335:
2334:
2329:
2321:
2320:
2296:
2295:
2262:
2261:
2236:
2234:
2233:
2228:
2212:
2210:
2209:
2204:
2199:
2198:
2173:
2171:
2170:
2165:
2157:
2149:
2137:
2135:
2134:
2129:
2127:
2126:
2110:
2108:
2107:
2102:
2084:
2082:
2081:
2076:
2068:
2060:
2044:
2042:
2041:
2036:
2034:
2033:
2017:
2015:
2014:
2009:
2006:
2001:
1985:
1983:
1982:
1977:
1975:
1974:
1958:
1956:
1955:
1950:
1947:
1946:
1934:
1933:
1912:
1911:
1887:
1885:
1884:
1879:
1877:
1876:
1860:
1858:
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1852:
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1849:
1833:
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1825:
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1806:
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1803:
1798:
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1772:
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1769:
1764:
1752:
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1749:
1744:
1732:
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1729:
1724:
1673:
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1670:
1665:
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1646:
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1471:
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1370:
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1320:
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1312:
1306:
1305:
1296:
1294:
1277:
1272:
1271:
1249:
1247:
1246:
1241:
1238:
1237:
1225:
1224:
1173:
1171:
1170:
1165:
1163:
1162:
1146:
1144:
1143:
1138:
1136:
1135:
1125:
1120:
1104:
1102:
1101:
1096:
1094:
1093:
1077:
1075:
1074:
1069:
1067:
1066:
1050:
1048:
1047:
1042:
1040:
1039:
1023:
1021:
1020:
1015:
1013:
1012:
1003:
1002:
986:
984:
983:
978:
976:
975:
959:
957:
956:
951:
949:
948:
932:
930:
929:
924:
922:
921:
905:
903:
902:
897:
895:
894:
878:
876:
875:
870:
868:
867:
847:
845:
844:
839:
837:
836:
820:
818:
817:
812:
810:
809:
797:
796:
781:
780:
768:
767:
743:
741:
740:
735:
727:
722:
721:
712:
700:
698:
697:
692:
690:
689:
669:
667:
666:
661:
658:
657:
648:
647:
637:
632:
584:
582:
581:
576:
568:
563:
562:
553:
524:
522:
521:
516:
514:
513:
501:
500:
462:
460:
459:
454:
452:
451:
439:
438:
429:
428:
419:
418:
408:
403:
385:
384:
354:
352:
351:
346:
344:
343:
323:
321:
320:
315:
313:
312:
294:
293:
274:
272:
271:
266:
264:
263:
251:
250:
235:
234:
224:
219:
201:
200:
173:
171:
170:
165:
85:
78:
74:
71:
65:
60:this article by
51:inline citations
38:
37:
30:
21:
12087:
12086:
12082:
12081:
12080:
12078:
12077:
12076:
12067:Autocorrelation
12057:
12056:
12055:
12050:
12032:
11993:Queueing theory
11936:
11878:Skorokhod space
11741:
11732:KunitaâWatanabe
11703:
11669:Sanov's theorem
11639:Ergodic theorem
11612:
11608:Time-reversible
11526:
11489:Queueing models
11483:
11479:SparreâAnderson
11469:CramĂŠrâLundberg
11450:
11436:SABR volatility
11342:
11299:
11251:Boolean network
11209:
11195:Renewal process
11126:
11075:Non-homogeneous
11065:Poisson process
10955:Contact process
10918:Brownian motion
10888:Continuous time
10882:
10876:Maximal entropy
10807:
10802:
10772:
10767:
10719:
10633:
10599:Google DeepMind
10577:
10543:Geoffrey Hinton
10502:
10439:
10365:Project Debater
10311:
10209:Implementations
10204:
10158:
10122:
10065:
10007:Backpropagation
9941:
9927:Tensor calculus
9881:
9878:
9835:
9825:
9789:
9769:
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9708:
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9684:
9683:
9679:
9670:
9668:
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9654:
9650:
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9639:
9626:
9625:
9621:
9612:
9610:
9597:
9596:
9592:
9584:
9582:
9574:
9570:
9564:Wayback Machine
9555:
9551:
9542:
9540:
9505:
9501:
9492:
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9480:
9465:
9459:
9455:
9446:
9442:
9433:
9431:
9427:
9416:
9410:
9406:
9399:
9377:
9370:
9363:
9349:
9345:
9332:Wayback Machine
9322:
9318:
9305:Wayback Machine
9295:
9291:
9281:Wayback Machine
9271:
9267:
9260:
9246:
9242:
9233:
9231:
9216:
9200:
9196:
9181:
9165:
9161:
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9109:
9074:
9070:
9068:
9065:
9064:
9037:
9013:
9009:
9007:
9004:
9003:
8993:
8961:
8957:
8942:
8938:
8932:
8928:
8922:
8911:
8898:
8894:
8892:
8889:
8888:
8882:
8855:
8794:R, the package
8777:
8731:
8727:
8691:
8687:
8672:
8668:
8656:
8651:
8638:
8633:
8622:
8616:
8611:
8605:
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8547:
8532:
8528:
8517:
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8513:
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8478:
8475:
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8447:
8445:
8442:
8441:
8417:
8414:
8413:
8390:
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8381:
8380:
8350:
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8341:
8340:
8315:
8311:
8306:
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8283:
8275:
8269:
8265:
8260:
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8254:
8219:
8215:
8210:
8206:
8200:
8196:
8194:
8190:
8178:
8174:
8164:
8159:
8150:
8146:
8144:
8141:
8140:
8113:
8109:
8097:
8092:
8086:
8083:
8082:
8057:
8056:
8051:
8045:
8044:
8038:
8034:
8032:
8026:
8022:
8015:
8014:
8012:
8009:
8008:
7988:
7984:
7982:
7979:
7978:
7961:
7957:
7955:
7952:
7951:
7924:
7920:
7908:
7903:
7897:
7888:
7884:
7883:
7879:
7870:
7866:
7862:
7857:
7848:
7844:
7835:
7831:
7829:
7826:
7825:
7792:
7788:
7782:
7778:
7766:
7762:
7756:
7752:
7744:
7741:
7740:
7711:
7707:
7698:
7694:
7688:
7684:
7672:
7668:
7662:
7658:
7640:
7636:
7634:
7631:
7630:
7590:
7586:
7580:
7576:
7564:
7560:
7552:
7549:
7548:
7534:
7501:
7497:
7495:
7492:
7491:
7471:
7467:
7465:
7462:
7461:
7434:
7430:
7428:
7425:
7424:
7385:
7381:
7369:
7364:
7353:
7347:
7342:
7336:
7324:
7319:
7318:
7300:
7296:
7290:
7286:
7275:
7274:
7268:
7263:
7257:
7240:
7237:
7236:
7230:
7205:
7200:
7179:
7176:
7175:
7169:
7141:
7136:
7135:
7114:
7110:
7104:
7100:
7094:
7083:
7068:
7067:
7061:
7056:
7050:
7033:
7030:
7029:
7009:
7004:
6988:
6984:
6970:
6968:
6965:
6964:
6959:(PSD) of an AR(
6939:
6910:
6878:
6873:
6854:
6850:
6844:
6840:
6834:
6823:
6810:
6806:
6804:
6801:
6800:
6779:
6753:
6724:
6720:
6713:
6706:
6702:
6693:
6688:
6675:
6670:
6665:
6663:
6654:
6650:
6645:
6639:
6635:
6626:
6622:
6620:
6617:
6616:
6594:
6590:
6583:
6577:
6573:
6571:
6562:
6558:
6553:
6547:
6543:
6534:
6530:
6528:
6525:
6524:
6505:
6501:
6489:
6485:
6483:
6480:
6479:
6458:
6454:
6448:
6444:
6435:
6431:
6425:
6421:
6412:
6408:
6406:
6403:
6402:
6381:
6377:
6371:
6367:
6358:
6354:
6348:
6344:
6335:
6331:
6329:
6326:
6325:
6296:
6292:
6283:
6279:
6274:
6268:
6264:
6255:
6251:
6249:
6246:
6245:
6226:
6222:
6216:
6212:
6203:
6199:
6197:
6194:
6193:
6142:
6138:
6132:
6121:
6100:
6097:
6096:
6068:
6065:
6064:
6034:
6029:
6023:
6020:
6019:
5969:
5965:
5960:
5957:
5956:
5933:
5928:
5912:
5908:
5902:
5898:
5892:
5881:
5868:
5864:
5862:
5859:
5858:
5798:
5794:
5789:
5786:
5785:
5764:
5763:
5757:
5753:
5750:
5749:
5743:
5742:
5736:
5732:
5729:
5728:
5722:
5718:
5715:
5714:
5708:
5704:
5697:
5696:
5689:
5688:
5683:
5671:
5667:
5665:
5653:
5649:
5647:
5635:
5631:
5628:
5627:
5622:
5617:
5612:
5606:
5605:
5600:
5594:
5590:
5588:
5582:
5578:
5576:
5570:
5566:
5563:
5562:
5557:
5548:
5544:
5542:
5536:
5532:
5530:
5524:
5520:
5517:
5516:
5511:
5502:
5498:
5496:
5487:
5483:
5481:
5475:
5471:
5464:
5463:
5453:
5452:
5446:
5442:
5439:
5438:
5432:
5431:
5425:
5421:
5418:
5417:
5411:
5407:
5404:
5403:
5397:
5393:
5386:
5385:
5383:
5380:
5379:
5369:
5362:
5332:
5328:
5326:
5323:
5322:
5305:
5301:
5299:
5296:
5295:
5293:
5274:
5270:
5268:
5265:
5264:
5257:
5247:
5218:
5214:
5208:
5203:
5184:
5180:
5174:
5170:
5164:
5153:
5140:
5136:
5134:
5131:
5130:
5116:
5086:
5082:
5080:
5077:
5076:
5052:
5048:
5033:
5029:
5023:
5019:
5013:
5002:
4989:
4985:
4983:
4980:
4979:
4958:
4950:
4944:
4918:
4914:
4907:
4897:
4893:
4886:
4884:
4878:
4874:
4862:
4858:
4853:
4841:
4837:
4826:
4823:
4822:
4803:
4799:
4793:
4789:
4775:
4771:
4764:
4760:
4745:
4741:
4736:
4724:
4720:
4709:
4706:
4705:
4674:
4670:
4658:
4654:
4653:
4649:
4643:
4632:
4613:
4609:
4591:
4587:
4581:
4577:
4562:
4558:
4556:
4553:
4552:
4529:
4525:
4495:
4491:
4473:
4469:
4467:
4464:
4463:
4443:
4440:
4439:
4419:
4415:
4410:
4407:
4406:
4375:
4372:
4371:
4348:
4340:
4338:
4335:
4334:
4311:
4307:
4295:
4291:
4258:
4254:
4239:
4235:
4233:
4230:
4229:
4210:
4202:
4199:
4198:
4191:
4166:
4163:
4162:
4145:
4141:
4129:
4125:
4123:
4120:
4119:
4088:
4084:
4072:
4068:
4066:
4063:
4062:
4039:
4035:
4033:
4030:
4029:
4010:
4007:
4006:
3989:
3985:
3983:
3980:
3979:
3978:indicates that
3958:
3954:
3952:
3949:
3948:
3927:
3923:
3921:
3918:
3917:
3900:
3896:
3894:
3891:
3890:
3873:
3869:
3867:
3864:
3863:
3834:
3830:
3824:
3820:
3814:
3803:
3790:
3786:
3784:
3781:
3780:
3760:
3756:
3754:
3751:
3750:
3717:
3713:
3707:
3703:
3691:
3680:
3661:
3657:
3651:
3647:
3638:
3634:
3632:
3629:
3628:
3598:
3594:
3592:
3589:
3588:
3565:
3561:
3546:
3542:
3537:
3534:
3533:
3516:
3512:
3510:
3507:
3506:
3487:
3484:
3483:
3464:
3453:
3450:
3449:
3423:
3419:
3410:
3406:
3399:
3394:
3384:
3380:
3373:
3367:
3362:
3356:
3340:
3323:
3320:
3319:
3311:which yields a
3291:
3283:
3282:
3278:
3267:
3263:
3256:
3250:
3245:
3239:
3222:
3219:
3218:
3198:
3194:
3192:
3189:
3188:
3172:
3169:
3168:
3143:
3140:
3139:
3122:
3118:
3116:
3113:
3112:
3058:
3054:
3047:
3041:
3036:
3030:
3026:
3010:
2992:
2988:
2982:
2978:
2972:
2958:
2938:
2921:
2918:
2917:
2875:
2872:
2871:
2843:
2835:
2834:
2830:
2819:
2815:
2808:
2802:
2797:
2791:
2782:
2778:
2766:
2762:
2750:
2746:
2728:
2724:
2722:
2719:
2718:
2685:
2680:
2658:
2654:
2645:
2644:
2638:
2634:
2622:
2618:
2609:
2608:
2606:
2603:
2602:
2582:
2578:
2576:
2573:
2572:
2555:
2551:
2549:
2546:
2545:
2519:
2515:
2508:
2502:
2497:
2491:
2482:
2478:
2466:
2461:
2436:
2432:
2423:
2422:
2420:
2417:
2416:
2384:
2381:
2380:
2343:
2340:
2339:
2316:
2312:
2285:
2281:
2257:
2253:
2242:
2239:
2238:
2222:
2219:
2218:
2194:
2190:
2179:
2176:
2175:
2153:
2145:
2143:
2140:
2139:
2122:
2118:
2116:
2113:
2112:
2090:
2087:
2086:
2064:
2056:
2054:
2051:
2050:
2029:
2025:
2023:
2020:
2019:
2002:
1997:
1991:
1988:
1987:
1970:
1966:
1964:
1961:
1960:
1942:
1938:
1923:
1919:
1907:
1903:
1901:
1898:
1897:
1894:
1872:
1868:
1866:
1863:
1862:
1845:
1841:
1839:
1836:
1835:
1818:
1814:
1812:
1809:
1808:
1791:
1787:
1785:
1782:
1781:
1775:low pass filter
1758:
1755:
1754:
1738:
1735:
1734:
1718:
1715:
1714:
1699:
1658:
1654:
1652:
1649:
1648:
1623:
1620:
1619:
1591:
1587:
1581:
1577:
1571:
1560:
1533:
1530:
1529:
1509:
1505:
1503:
1500:
1499:
1475:
1467:
1463:
1458:
1448:
1444:
1438:
1427:
1406:
1403:
1402:
1388:
1366:
1362:
1360:
1357:
1356:
1339:
1335:
1333:
1330:
1329:
1301:
1297:
1281:
1276:
1267:
1263:
1261:
1258:
1257:
1233:
1229:
1220:
1216:
1202:
1199:
1198:
1193:
1158:
1154:
1152:
1149:
1148:
1131:
1127:
1121:
1116:
1110:
1107:
1106:
1089:
1085:
1083:
1080:
1079:
1078:, this affects
1062:
1058:
1056:
1053:
1052:
1035:
1031:
1029:
1026:
1025:
1008:
1004:
998:
994:
992:
989:
988:
971:
967:
965:
962:
961:
960:, this affects
944:
940:
938:
935:
934:
917:
913:
911:
908:
907:
890:
886:
884:
881:
880:
879:by the amount
863:
859:
857:
854:
853:
832:
828:
826:
823:
822:
805:
801:
786:
782:
776:
772:
763:
759:
757:
754:
753:
750:
723:
717:
713:
708:
706:
703:
702:
685:
681:
679:
676:
675:
653:
649:
643:
639:
633:
622:
594:
591:
590:
564:
558:
554:
549:
547:
544:
543:
509:
505:
496:
492:
478:
475:
474:
447:
443:
434:
430:
424:
420:
414:
410:
404:
393:
380:
376:
374:
371:
370:
339:
335:
333:
330:
329:
308:
304:
289:
285:
283:
280:
279:
259:
255:
240:
236:
230:
226:
220:
209:
196:
192:
190:
187:
186:
147:
144:
143:
140:
86:
75:
69:
66:
56:Please help to
55:
39:
35:
28:
23:
22:
15:
12:
11:
5:
12085:
12075:
12074:
12069:
12052:
12051:
12049:
12048:
12043:
12041:List of topics
12037:
12034:
12033:
12031:
12030:
12025:
12020:
12015:
12010:
12005:
12000:
11998:Renewal theory
11995:
11990:
11985:
11980:
11975:
11970:
11965:
11963:Ergodic theory
11960:
11955:
11953:Control theory
11950:
11944:
11942:
11938:
11937:
11935:
11934:
11933:
11932:
11927:
11917:
11912:
11907:
11902:
11897:
11896:
11895:
11885:
11883:Snell envelope
11880:
11875:
11870:
11865:
11860:
11855:
11850:
11845:
11840:
11835:
11830:
11825:
11820:
11815:
11810:
11805:
11800:
11795:
11790:
11785:
11780:
11775:
11770:
11765:
11760:
11755:
11749:
11747:
11743:
11742:
11740:
11739:
11734:
11729:
11724:
11719:
11713:
11711:
11705:
11704:
11702:
11701:
11682:BorelâCantelli
11671:
11666:
11661:
11656:
11651:
11646:
11641:
11636:
11631:
11626:
11620:
11618:
11617:Limit theorems
11614:
11613:
11611:
11610:
11605:
11600:
11595:
11590:
11585:
11580:
11575:
11570:
11565:
11560:
11555:
11550:
11545:
11540:
11534:
11532:
11528:
11527:
11525:
11524:
11519:
11514:
11509:
11504:
11499:
11493:
11491:
11485:
11484:
11482:
11481:
11476:
11471:
11466:
11460:
11458:
11452:
11451:
11449:
11448:
11443:
11438:
11433:
11428:
11423:
11418:
11413:
11408:
11403:
11398:
11393:
11388:
11383:
11378:
11373:
11368:
11363:
11358:
11352:
11350:
11344:
11343:
11341:
11340:
11335:
11330:
11325:
11320:
11315:
11309:
11307:
11301:
11300:
11298:
11297:
11292:
11287:
11286:
11285:
11280:
11270:
11265:
11260:
11255:
11254:
11253:
11248:
11238:
11236:Hopfield model
11233:
11228:
11223:
11217:
11215:
11211:
11210:
11208:
11207:
11202:
11197:
11192:
11187:
11182:
11181:
11180:
11175:
11170:
11165:
11155:
11153:Markov process
11150:
11145:
11140:
11134:
11132:
11128:
11127:
11125:
11124:
11122:Wiener sausage
11119:
11117:Wiener process
11114:
11109:
11104:
11099:
11097:Stable process
11094:
11089:
11087:Semimartingale
11084:
11079:
11078:
11077:
11072:
11062:
11057:
11052:
11047:
11042:
11037:
11032:
11030:Jump diffusion
11027:
11022:
11017:
11012:
11007:
11005:Hawkes process
11002:
10997:
10992:
10987:
10985:Feller process
10982:
10977:
10972:
10967:
10962:
10957:
10952:
10950:Cauchy process
10947:
10946:
10945:
10940:
10935:
10930:
10925:
10915:
10914:
10913:
10903:
10901:Bessel process
10898:
10892:
10890:
10884:
10883:
10881:
10880:
10879:
10878:
10873:
10868:
10863:
10853:
10848:
10843:
10838:
10833:
10828:
10823:
10817:
10815:
10809:
10808:
10801:
10800:
10793:
10786:
10778:
10769:
10768:
10766:
10765:
10764:
10763:
10758:
10745:
10744:
10743:
10738:
10724:
10721:
10720:
10718:
10717:
10712:
10707:
10702:
10697:
10692:
10687:
10682:
10677:
10672:
10667:
10662:
10657:
10652:
10647:
10641:
10639:
10635:
10634:
10632:
10631:
10626:
10621:
10616:
10611:
10606:
10601:
10596:
10591:
10585:
10583:
10579:
10578:
10576:
10575:
10573:Ilya Sutskever
10570:
10565:
10560:
10555:
10550:
10545:
10540:
10538:Demis Hassabis
10535:
10530:
10528:Ian Goodfellow
10525:
10520:
10514:
10512:
10508:
10507:
10504:
10503:
10501:
10500:
10495:
10494:
10493:
10483:
10478:
10473:
10468:
10463:
10458:
10453:
10447:
10445:
10441:
10440:
10438:
10437:
10432:
10427:
10422:
10417:
10412:
10407:
10402:
10397:
10392:
10387:
10382:
10377:
10372:
10367:
10362:
10357:
10356:
10355:
10345:
10340:
10335:
10330:
10325:
10319:
10317:
10313:
10312:
10310:
10309:
10304:
10303:
10302:
10297:
10287:
10286:
10285:
10280:
10275:
10265:
10260:
10255:
10250:
10245:
10240:
10235:
10230:
10225:
10219:
10217:
10210:
10206:
10205:
10203:
10202:
10197:
10192:
10187:
10182:
10177:
10172:
10166:
10164:
10160:
10159:
10157:
10156:
10151:
10146:
10141:
10136:
10130:
10128:
10124:
10123:
10121:
10120:
10119:
10118:
10111:Language model
10108:
10103:
10098:
10097:
10096:
10086:
10085:
10084:
10073:
10071:
10067:
10066:
10064:
10063:
10061:Autoregression
10058:
10053:
10052:
10051:
10041:
10039:Regularization
10036:
10035:
10034:
10029:
10024:
10014:
10009:
10004:
10002:Loss functions
9999:
9994:
9989:
9984:
9979:
9978:
9977:
9967:
9962:
9961:
9960:
9949:
9947:
9943:
9942:
9940:
9939:
9937:Inductive bias
9934:
9929:
9924:
9919:
9914:
9909:
9904:
9899:
9891:
9889:
9883:
9882:
9877:
9876:
9869:
9862:
9854:
9848:
9847:
9832:
9831:by Paul Bourke
9824:
9823:External links
9821:
9820:
9819:
9810:
9793:
9787:
9768:
9765:
9762:
9761:
9732:
9706:
9677:
9648:
9619:
9590:
9568:
9549:
9499:
9486:
9483:on 2012-10-21.
9453:
9440:
9412:Eshel, Gidon.
9404:
9397:
9368:
9361:
9343:
9316:
9289:
9265:
9259:978-1441978646
9258:
9240:
9214:
9194:
9179:
9158:
9157:
9155:
9152:
9151:
9150:
9145:
9140:
9135:
9130:
9125:
9120:
9115:
9108:
9105:
9077:
9073:
9051:until, after
9033:
9016:
9012:
8989:
8979:
8978:
8964:
8960:
8956:
8951:
8948:
8945:
8941:
8935:
8931:
8925:
8920:
8917:
8914:
8910:
8906:
8901:
8897:
8881:
8875:
8873:applies here.
8854:
8851:
8850:
8849:
8843:
8837:
8827:
8809:
8803:
8792:
8776:
8773:
8772:
8771:
8757:
8754:
8751:
8748:
8745:
8742:
8739:
8734:
8730:
8726:
8723:
8720:
8717:
8714:
8711:
8708:
8705:
8702:
8699:
8694:
8690:
8686:
8683:
8680:
8675:
8671:
8667:
8664:
8659:
8654:
8650:
8646:
8641:
8636:
8632:
8628:
8625:
8619:
8614:
8610:
8604:
8601:
8598:
8595:
8592:
8565:
8559:
8555:
8550:
8546:
8543:
8540:
8535:
8531:
8527:
8524:
8521:
8509:
8508:
8496:
8492:
8488:
8485:
8482:
8462:
8459:
8454:
8450:
8438:
8427:
8424:
8421:
8401:
8398:
8393:
8389:
8361:
8358:
8353:
8349:
8337:
8336:
8325:
8318:
8314:
8310:
8305:
8301:
8295:
8291:
8286:
8282:
8278:
8272:
8268:
8263:
8248:
8247:
8236:
8232:
8222:
8218:
8214:
8209:
8203:
8199:
8193:
8189:
8184:
8181:
8177:
8170:
8167:
8163:
8158:
8153:
8149:
8137:
8136:
8124:
8121:
8116:
8112:
8108:
8105:
8100:
8095:
8091:
8075:
8074:
8061:
8055:
8052:
8050:
8047:
8046:
8041:
8037:
8033:
8029:
8025:
8021:
8020:
8018:
7991:
7987:
7964:
7960:
7949:
7948:
7935:
7927:
7923:
7919:
7916:
7911:
7906:
7902:
7896:
7891:
7887:
7882:
7873:
7869:
7865:
7861:
7856:
7851:
7847:
7843:
7838:
7834:
7821:which yields:
7819:
7818:
7806:
7803:
7798:
7795:
7791:
7785:
7781:
7777:
7772:
7769:
7765:
7759:
7755:
7751:
7748:
7734:
7733:
7722:
7717:
7714:
7710:
7704:
7701:
7697:
7691:
7687:
7683:
7678:
7675:
7671:
7665:
7661:
7657:
7654:
7651:
7648:
7643:
7639:
7616:
7615:
7604:
7601:
7598:
7593:
7589:
7583:
7579:
7575:
7572:
7567:
7563:
7559:
7556:
7533:
7530:
7529:
7528:
7512:
7509:
7504:
7500:
7488:
7474:
7470:
7445:
7442:
7437:
7433:
7420:
7419:
7405:
7402:
7399:
7396:
7393:
7388:
7384:
7380:
7377:
7372:
7367:
7363:
7359:
7356:
7350:
7345:
7341:
7335:
7327:
7322:
7315:
7312:
7309:
7306:
7303:
7299:
7293:
7289:
7285:
7282:
7278:
7271:
7266:
7262:
7256:
7253:
7250:
7247:
7244:
7229:
7226:
7225:
7224:
7213:
7208:
7203:
7199:
7195:
7192:
7189:
7186:
7183:
7168:
7165:
7164:
7163:
7152:
7144:
7139:
7132:
7129:
7126:
7123:
7120:
7117:
7113:
7107:
7103:
7097:
7092:
7089:
7086:
7082:
7078:
7075:
7071:
7064:
7059:
7055:
7049:
7046:
7043:
7040:
7037:
7012:
7007:
7003:
6999:
6996:
6991:
6987:
6983:
6979:
6976:
6973:
6938:
6935:
6922:
6921:
6906:
6900:
6899:
6898:
6887:
6881:
6876:
6872:
6868:
6863:
6860:
6857:
6853:
6847:
6843:
6837:
6832:
6829:
6826:
6822:
6818:
6813:
6809:
6795:
6794:
6790:
6775:
6765:
6752:
6749:
6748:
6747:
6746:
6745:
6744:
6743:
6727:
6723:
6719:
6716:
6709:
6705:
6701:
6696:
6691:
6687:
6683:
6678:
6673:
6669:
6662:
6657:
6653:
6648:
6642:
6638:
6634:
6629:
6625:
6613:
6597:
6593:
6589:
6586:
6580:
6576:
6570:
6565:
6561:
6556:
6550:
6546:
6542:
6537:
6533:
6521:
6508:
6504:
6500:
6495:
6492:
6488:
6478:Remember that
6475:
6474:
6461:
6457:
6451:
6447:
6443:
6438:
6434:
6428:
6424:
6420:
6415:
6411:
6400:
6387:
6384:
6380:
6374:
6370:
6366:
6361:
6357:
6351:
6347:
6343:
6338:
6334:
6314:
6313:
6312:
6299:
6295:
6291:
6286:
6282:
6277:
6271:
6267:
6263:
6258:
6254:
6242:
6229:
6225:
6219:
6215:
6211:
6206:
6202:
6177:
6176:
6165:
6162:
6159:
6156:
6153:
6150:
6145:
6141:
6135:
6130:
6127:
6124:
6120:
6116:
6113:
6110:
6107:
6104:
6081:
6078:
6075:
6072:
6042:
6037:
6032:
6028:
6007:
6004:
6001:
5998:
5995:
5992:
5989:
5986:
5983:
5980:
5977:
5972:
5968:
5964:
5953:
5952:
5941:
5936:
5931:
5927:
5923:
5918:
5915:
5911:
5905:
5901:
5895:
5890:
5887:
5884:
5880:
5876:
5871:
5867:
5839:
5836:
5833:
5830:
5827:
5824:
5821:
5818:
5815:
5812:
5809:
5806:
5801:
5797:
5793:
5782:
5781:
5768:
5760:
5756:
5752:
5751:
5748:
5745:
5744:
5739:
5735:
5731:
5730:
5725:
5721:
5717:
5716:
5711:
5707:
5703:
5702:
5700:
5693:
5687:
5684:
5680:
5677:
5674:
5670:
5666:
5662:
5659:
5656:
5652:
5648:
5644:
5641:
5638:
5634:
5630:
5629:
5626:
5623:
5621:
5618:
5616:
5613:
5611:
5608:
5607:
5604:
5601:
5597:
5593:
5589:
5585:
5581:
5577:
5573:
5569:
5565:
5564:
5561:
5558:
5554:
5551:
5547:
5543:
5539:
5535:
5531:
5527:
5523:
5519:
5518:
5515:
5512:
5508:
5505:
5501:
5497:
5493:
5490:
5486:
5482:
5478:
5474:
5470:
5469:
5467:
5462:
5457:
5449:
5445:
5441:
5440:
5437:
5434:
5433:
5428:
5424:
5420:
5419:
5414:
5410:
5406:
5405:
5400:
5396:
5392:
5391:
5389:
5341:
5338:
5335:
5331:
5308:
5304:
5291:
5277:
5273:
5244:
5243:
5232:
5227:
5224:
5221:
5217:
5211:
5206:
5202:
5198:
5193:
5190:
5187:
5183:
5177:
5173:
5167:
5162:
5159:
5156:
5152:
5148:
5143:
5139:
5124:Gilbert Walker
5115:
5112:
5089:
5085:
5073:
5072:
5060:
5055:
5051:
5047:
5042:
5039:
5036:
5032:
5026:
5022:
5016:
5011:
5008:
5005:
5001:
4997:
4992:
4988:
4957:
4954:
4946:Main article:
4943:
4940:
4939:
4938:
4921:
4917:
4913:
4910:
4903:
4900:
4896:
4892:
4889:
4881:
4877:
4873:
4870:
4865:
4861:
4856:
4850:
4847:
4844:
4840:
4836:
4833:
4830:
4820:
4806:
4802:
4796:
4792:
4788:
4784:
4778:
4774:
4770:
4767:
4763:
4759:
4756:
4753:
4748:
4744:
4739:
4733:
4730:
4727:
4723:
4719:
4716:
4713:
4689:
4683:
4680:
4677:
4673:
4667:
4664:
4661:
4657:
4652:
4646:
4641:
4638:
4635:
4631:
4627:
4624:
4621:
4616:
4612:
4608:
4605:
4602:
4599:
4594:
4590:
4584:
4580:
4576:
4571:
4568:
4565:
4561:
4538:
4535:
4532:
4528:
4524:
4521:
4518:
4515:
4512:
4509:
4506:
4503:
4498:
4494:
4490:
4487:
4482:
4479:
4476:
4472:
4460:
4459:
4447:
4427:
4422:
4418:
4414:
4394:
4391:
4388:
4385:
4382:
4379:
4358:
4355:
4351:
4347:
4343:
4320:
4317:
4314:
4310:
4306:
4303:
4298:
4294:
4290:
4287:
4284:
4281:
4278:
4275:
4272:
4269:
4266:
4261:
4257:
4253:
4248:
4245:
4242:
4238:
4225:, is given by
4213:
4209:
4206:
4190:
4187:
4170:
4148:
4144:
4140:
4137:
4132:
4128:
4097:
4094:
4091:
4087:
4083:
4080:
4075:
4071:
4061:, the process
4050:
4047:
4042:
4038:
4014:
3992:
3988:
3961:
3957:
3930:
3926:
3903:
3899:
3876:
3872:
3860:
3859:
3848:
3843:
3840:
3837:
3833:
3827:
3823:
3817:
3812:
3809:
3806:
3802:
3798:
3793:
3789:
3763:
3759:
3743:
3742:
3731:
3726:
3723:
3720:
3716:
3710:
3706:
3700:
3697:
3694:
3689:
3686:
3683:
3679:
3675:
3670:
3667:
3664:
3660:
3654:
3650:
3646:
3641:
3637:
3607:
3604:
3601:
3597:
3574:
3571:
3568:
3564:
3560:
3555:
3552:
3549:
3545:
3541:
3519:
3515:
3491:
3471:
3467:
3463:
3460:
3457:
3446:
3445:
3431:
3426:
3422:
3418:
3413:
3409:
3405:
3402:
3398:
3387:
3383:
3379:
3376:
3370:
3365:
3361:
3351:
3348:
3344:
3339:
3336:
3333:
3330:
3327:
3309:
3308:
3294:
3290:
3286:
3281:
3270:
3266:
3262:
3259:
3253:
3248:
3244:
3238:
3235:
3232:
3229:
3226:
3201:
3197:
3176:
3156:
3153:
3150:
3147:
3125:
3121:
3109:
3108:
3097:
3093:
3087:
3084:
3081:
3078:
3075:
3072:
3069:
3066:
3061:
3057:
3053:
3050:
3044:
3039:
3035:
3029:
3021:
3018:
3014:
3009:
3004:
3001:
2998:
2995:
2991:
2985:
2981:
2975:
2970:
2967:
2964:
2961:
2957:
2949:
2946:
2942:
2937:
2934:
2931:
2928:
2925:
2891:
2888:
2885:
2882:
2879:
2864:
2863:
2852:
2846:
2842:
2838:
2833:
2822:
2818:
2814:
2811:
2805:
2800:
2796:
2790:
2785:
2781:
2777:
2774:
2769:
2765:
2759:
2756:
2753:
2749:
2745:
2742:
2739:
2736:
2731:
2727:
2712:autocovariance
2705:
2704:
2693:
2688:
2683:
2679:
2675:
2672:
2667:
2664:
2661:
2657:
2653:
2641:
2637:
2633:
2630:
2625:
2621:
2617:
2585:
2581:
2558:
2554:
2542:
2541:
2530:
2522:
2518:
2514:
2511:
2505:
2500:
2496:
2490:
2485:
2481:
2477:
2474:
2469:
2464:
2460:
2456:
2453:
2450:
2447:
2444:
2439:
2435:
2431:
2406:
2405:
2394:
2391:
2388:
2365:
2362:
2359:
2356:
2353:
2350:
2347:
2327:
2324:
2319:
2315:
2311:
2308:
2305:
2302:
2299:
2294:
2291:
2288:
2284:
2280:
2277:
2274:
2271:
2268:
2265:
2260:
2256:
2252:
2249:
2246:
2226:
2202:
2197:
2193:
2189:
2186:
2183:
2163:
2160:
2156:
2152:
2148:
2125:
2121:
2100:
2097:
2094:
2074:
2071:
2067:
2063:
2059:
2032:
2028:
2005:
2000:
1996:
1973:
1969:
1945:
1941:
1937:
1932:
1929:
1926:
1922:
1918:
1915:
1910:
1906:
1893:
1890:
1875:
1871:
1848:
1844:
1821:
1817:
1794:
1790:
1762:
1742:
1722:
1698:
1691:
1690:
1689:
1686:
1661:
1657:
1636:
1633:
1630:
1627:
1608:
1607:
1594:
1590:
1584:
1580:
1574:
1569:
1566:
1563:
1559:
1555:
1552:
1549:
1546:
1543:
1540:
1537:
1512:
1508:
1496:
1495:
1484:
1478:
1474:
1470:
1466:
1461:
1457:
1451:
1447:
1441:
1436:
1433:
1430:
1426:
1422:
1419:
1416:
1413:
1410:
1387:
1384:
1369:
1365:
1342:
1338:
1322:
1321:
1310:
1304:
1300:
1293:
1290:
1287:
1284:
1280:
1275:
1270:
1266:
1251:
1250:
1236:
1232:
1228:
1223:
1219:
1215:
1212:
1209:
1206:
1189:
1161:
1157:
1134:
1130:
1124:
1119:
1115:
1105:by the amount
1092:
1088:
1065:
1061:
1038:
1034:
1011:
1007:
1001:
997:
987:by the amount
974:
970:
947:
943:
920:
916:
893:
889:
866:
862:
835:
831:
808:
804:
800:
795:
792:
789:
785:
779:
775:
771:
766:
762:
749:
746:
733:
730:
726:
720:
716:
711:
688:
684:
656:
652:
646:
642:
636:
631:
628:
625:
621:
617:
614:
610:
607:
604:
601:
598:
574:
571:
567:
561:
557:
552:
526:
525:
512:
508:
504:
499:
495:
491:
488:
485:
482:
464:
463:
450:
446:
442:
437:
433:
427:
423:
417:
413:
407:
402:
399:
396:
392:
388:
383:
379:
342:
338:
311:
307:
303:
300:
297:
292:
288:
276:
275:
262:
258:
254:
249:
246:
243:
239:
233:
229:
223:
218:
215:
212:
208:
204:
199:
195:
163:
160:
157:
154:
151:
139:
136:
94:autoregressive
88:
87:
42:
40:
33:
26:
9:
6:
4:
3:
2:
12084:
12073:
12070:
12068:
12065:
12064:
12062:
12047:
12044:
12042:
12039:
12038:
12035:
12029:
12026:
12024:
12021:
12019:
12016:
12014:
12011:
12009:
12006:
12004:
12001:
11999:
11996:
11994:
11991:
11989:
11986:
11984:
11981:
11979:
11976:
11974:
11971:
11969:
11966:
11964:
11961:
11959:
11956:
11954:
11951:
11949:
11946:
11945:
11943:
11939:
11931:
11928:
11926:
11923:
11922:
11921:
11918:
11916:
11913:
11911:
11908:
11906:
11903:
11901:
11900:Stopping time
11898:
11894:
11891:
11890:
11889:
11886:
11884:
11881:
11879:
11876:
11874:
11871:
11869:
11866:
11864:
11861:
11859:
11856:
11854:
11851:
11849:
11846:
11844:
11841:
11839:
11836:
11834:
11831:
11829:
11826:
11824:
11821:
11819:
11816:
11814:
11811:
11809:
11806:
11804:
11801:
11799:
11796:
11794:
11791:
11789:
11786:
11784:
11781:
11779:
11776:
11774:
11771:
11769:
11766:
11764:
11761:
11759:
11756:
11754:
11751:
11750:
11748:
11744:
11738:
11735:
11733:
11730:
11728:
11725:
11723:
11720:
11718:
11715:
11714:
11712:
11710:
11706:
11699:
11695:
11691:
11690:HewittâSavage
11687:
11683:
11679:
11675:
11674:Zeroâone laws
11672:
11670:
11667:
11665:
11662:
11660:
11657:
11655:
11652:
11650:
11647:
11645:
11642:
11640:
11637:
11635:
11632:
11630:
11627:
11625:
11622:
11621:
11619:
11615:
11609:
11606:
11604:
11601:
11599:
11596:
11594:
11591:
11589:
11586:
11584:
11581:
11579:
11576:
11574:
11571:
11569:
11566:
11564:
11561:
11559:
11556:
11554:
11551:
11549:
11546:
11544:
11541:
11539:
11536:
11535:
11533:
11529:
11523:
11520:
11518:
11515:
11513:
11510:
11508:
11505:
11503:
11500:
11498:
11495:
11494:
11492:
11490:
11486:
11480:
11477:
11475:
11472:
11470:
11467:
11465:
11462:
11461:
11459:
11457:
11453:
11447:
11444:
11442:
11439:
11437:
11434:
11432:
11429:
11427:
11424:
11422:
11419:
11417:
11414:
11412:
11409:
11407:
11404:
11402:
11399:
11397:
11394:
11392:
11389:
11387:
11384:
11382:
11379:
11377:
11374:
11372:
11371:BlackâScholes
11369:
11367:
11364:
11362:
11359:
11357:
11354:
11353:
11351:
11349:
11345:
11339:
11336:
11334:
11331:
11329:
11326:
11324:
11321:
11319:
11316:
11314:
11311:
11310:
11308:
11306:
11302:
11296:
11293:
11291:
11288:
11284:
11281:
11279:
11276:
11275:
11274:
11273:Point process
11271:
11269:
11266:
11264:
11261:
11259:
11256:
11252:
11249:
11247:
11244:
11243:
11242:
11239:
11237:
11234:
11232:
11231:Gibbs measure
11229:
11227:
11224:
11222:
11219:
11218:
11216:
11212:
11206:
11203:
11201:
11198:
11196:
11193:
11191:
11188:
11186:
11183:
11179:
11176:
11174:
11171:
11169:
11166:
11164:
11161:
11160:
11159:
11156:
11154:
11151:
11149:
11146:
11144:
11141:
11139:
11136:
11135:
11133:
11129:
11123:
11120:
11118:
11115:
11113:
11110:
11108:
11105:
11103:
11100:
11098:
11095:
11093:
11090:
11088:
11085:
11083:
11080:
11076:
11073:
11071:
11068:
11067:
11066:
11063:
11061:
11058:
11056:
11053:
11051:
11048:
11046:
11043:
11041:
11038:
11036:
11033:
11031:
11028:
11026:
11023:
11021:
11020:ItĂ´ diffusion
11018:
11016:
11013:
11011:
11008:
11006:
11003:
11001:
10998:
10996:
10995:Gamma process
10993:
10991:
10988:
10986:
10983:
10981:
10978:
10976:
10973:
10971:
10968:
10966:
10963:
10961:
10958:
10956:
10953:
10951:
10948:
10944:
10941:
10939:
10936:
10934:
10931:
10929:
10926:
10924:
10921:
10920:
10919:
10916:
10912:
10909:
10908:
10907:
10904:
10902:
10899:
10897:
10894:
10893:
10891:
10889:
10885:
10877:
10874:
10872:
10869:
10867:
10866:Self-avoiding
10864:
10862:
10859:
10858:
10857:
10854:
10852:
10851:Moran process
10849:
10847:
10844:
10842:
10839:
10837:
10834:
10832:
10829:
10827:
10824:
10822:
10819:
10818:
10816:
10814:
10813:Discrete time
10810:
10806:
10799:
10794:
10792:
10787:
10785:
10780:
10779:
10776:
10762:
10759:
10757:
10754:
10753:
10746:
10742:
10739:
10737:
10734:
10733:
10730:
10726:
10725:
10722:
10716:
10713:
10711:
10708:
10706:
10703:
10701:
10698:
10696:
10693:
10691:
10688:
10686:
10683:
10681:
10678:
10676:
10673:
10671:
10668:
10666:
10663:
10661:
10658:
10656:
10653:
10651:
10648:
10646:
10643:
10642:
10640:
10638:Architectures
10636:
10630:
10627:
10625:
10622:
10620:
10617:
10615:
10612:
10610:
10607:
10605:
10602:
10600:
10597:
10595:
10592:
10590:
10587:
10586:
10584:
10582:Organizations
10580:
10574:
10571:
10569:
10566:
10564:
10561:
10559:
10556:
10554:
10551:
10549:
10546:
10544:
10541:
10539:
10536:
10534:
10531:
10529:
10526:
10524:
10521:
10519:
10518:Yoshua Bengio
10516:
10515:
10513:
10509:
10499:
10498:Robot control
10496:
10492:
10489:
10488:
10487:
10484:
10482:
10479:
10477:
10474:
10472:
10469:
10467:
10464:
10462:
10459:
10457:
10454:
10452:
10449:
10448:
10446:
10442:
10436:
10433:
10431:
10428:
10426:
10423:
10421:
10418:
10416:
10415:Chinchilla AI
10413:
10411:
10408:
10406:
10403:
10401:
10398:
10396:
10393:
10391:
10388:
10386:
10383:
10381:
10378:
10376:
10373:
10371:
10368:
10366:
10363:
10361:
10358:
10354:
10351:
10350:
10349:
10346:
10344:
10341:
10339:
10336:
10334:
10331:
10329:
10326:
10324:
10321:
10320:
10318:
10314:
10308:
10305:
10301:
10298:
10296:
10293:
10292:
10291:
10288:
10284:
10281:
10279:
10276:
10274:
10271:
10270:
10269:
10266:
10264:
10261:
10259:
10256:
10254:
10251:
10249:
10246:
10244:
10241:
10239:
10236:
10234:
10231:
10229:
10226:
10224:
10221:
10220:
10218:
10214:
10211:
10207:
10201:
10198:
10196:
10193:
10191:
10188:
10186:
10183:
10181:
10178:
10176:
10173:
10171:
10168:
10167:
10165:
10161:
10155:
10152:
10150:
10147:
10145:
10142:
10140:
10137:
10135:
10132:
10131:
10129:
10125:
10117:
10114:
10113:
10112:
10109:
10107:
10104:
10102:
10099:
10095:
10094:Deep learning
10092:
10091:
10090:
10087:
10083:
10080:
10079:
10078:
10075:
10074:
10072:
10068:
10062:
10059:
10057:
10054:
10050:
10047:
10046:
10045:
10042:
10040:
10037:
10033:
10030:
10028:
10025:
10023:
10020:
10019:
10018:
10015:
10013:
10010:
10008:
10005:
10003:
10000:
9998:
9995:
9993:
9990:
9988:
9985:
9983:
9982:Hallucination
9980:
9976:
9973:
9972:
9971:
9968:
9966:
9963:
9959:
9956:
9955:
9954:
9951:
9950:
9948:
9944:
9938:
9935:
9933:
9930:
9928:
9925:
9923:
9920:
9918:
9915:
9913:
9910:
9908:
9905:
9903:
9900:
9898:
9897:
9893:
9892:
9890:
9888:
9884:
9875:
9870:
9868:
9863:
9861:
9856:
9855:
9852:
9846:
9842:
9838:
9833:
9830:
9827:
9826:
9816:
9811:
9807:
9803:
9799:
9794:
9790:
9788:9780521343398
9784:
9779:
9778:
9771:
9770:
9750:
9746:
9742:
9736:
9720:
9716:
9710:
9695:
9691:
9690:pub.ist.ac.at
9687:
9681:
9666:
9662:
9658:
9652:
9637:
9633:
9629:
9623:
9608:
9604:
9600:
9594:
9581:
9580:
9572:
9565:
9561:
9558:
9553:
9538:
9534:
9530:
9526:
9522:
9518:
9514:
9510:
9503:
9496:
9490:
9479:
9475:
9471:
9464:
9457:
9450:
9444:
9426:
9422:
9415:
9408:
9400:
9398:0-521-01230-9
9394:
9390:
9386:
9382:
9375:
9373:
9364:
9358:
9354:
9347:
9340:
9338:
9333:
9329:
9326:
9320:
9313:
9311:
9306:
9302:
9299:
9293:
9286:
9282:
9278:
9275:
9269:
9261:
9255:
9251:
9244:
9229:
9225:
9221:
9217:
9215:0-387-98950-1
9211:
9207:
9206:
9198:
9190:
9186:
9182:
9180:0-13-060774-6
9176:
9172:
9171:
9163:
9159:
9149:
9146:
9144:
9141:
9139:
9136:
9134:
9131:
9129:
9126:
9124:
9121:
9119:
9116:
9114:
9111:
9110:
9104:
9102:
9098:
9094:
9075:
9071:
9060:
9058:
9054:
9049:
9045:
9041:
9036:
9032:
9014:
9010:
9001:
8997:
8992:
8988:
8984:
8962:
8958:
8954:
8949:
8946:
8943:
8939:
8933:
8929:
8923:
8918:
8915:
8912:
8908:
8904:
8899:
8895:
8887:
8886:
8885:
8879:
8874:
8872:
8868:
8864:
8860:
8847:
8844:
8841:
8838:
8835:
8831:
8828:
8825:
8821:
8817:
8813:
8810:
8807:
8804:
8801:
8797:
8793:
8790:
8786:
8782:
8779:
8778:
8752:
8749:
8746:
8740:
8737:
8732:
8728:
8724:
8721:
8715:
8712:
8709:
8703:
8700:
8692:
8688:
8684:
8681:
8673:
8669:
8665:
8662:
8657:
8652:
8648:
8644:
8639:
8634:
8630:
8626:
8623:
8617:
8612:
8608:
8602:
8596:
8590:
8583:
8582:
8581:
8578:
8557:
8553:
8544:
8541:
8538:
8533:
8529:
8525:
8522:
8519:
8494:
8490:
8486:
8483:
8480:
8460:
8457:
8452:
8448:
8439:
8425:
8422:
8419:
8399:
8396:
8391:
8387:
8378:
8377:
8376:
8373:
8359:
8356:
8351:
8347:
8323:
8316:
8312:
8308:
8303:
8293:
8289:
8280:
8270:
8266:
8253:
8252:
8251:
8234:
8230:
8220:
8216:
8212:
8207:
8201:
8197:
8191:
8187:
8182:
8179:
8175:
8168:
8165:
8161:
8156:
8151:
8147:
8139:
8138:
8122:
8119:
8114:
8110:
8106:
8103:
8098:
8093:
8089:
8080:
8079:
8078:
8059:
8053:
8048:
8039:
8035:
8027:
8023:
8016:
8007:
8006:
8005:
7989:
7985:
7962:
7958:
7933:
7925:
7921:
7917:
7914:
7909:
7904:
7900:
7894:
7889:
7885:
7880:
7871:
7867:
7863:
7859:
7854:
7849:
7845:
7841:
7836:
7832:
7824:
7823:
7822:
7804:
7801:
7796:
7793:
7789:
7783:
7779:
7775:
7770:
7767:
7763:
7757:
7753:
7749:
7746:
7739:
7738:
7737:
7720:
7715:
7712:
7702:
7699:
7695:
7689:
7685:
7681:
7676:
7673:
7669:
7663:
7659:
7655:
7652:
7646:
7641:
7637:
7629:
7628:
7627:
7625:
7621:
7602:
7599:
7596:
7591:
7587:
7581:
7577:
7573:
7570:
7565:
7561:
7557:
7554:
7547:
7546:
7545:
7543:
7539:
7526:
7510:
7507:
7502:
7498:
7489:
7472:
7468:
7459:
7443:
7440:
7435:
7431:
7422:
7421:
7403:
7400:
7397:
7394:
7391:
7386:
7382:
7378:
7375:
7370:
7365:
7361:
7357:
7354:
7348:
7343:
7339:
7333:
7325:
7313:
7310:
7307:
7304:
7301:
7297:
7291:
7287:
7283:
7280:
7269:
7264:
7260:
7254:
7248:
7242:
7235:
7234:
7233:
7211:
7206:
7201:
7197:
7193:
7187:
7181:
7174:
7173:
7172:
7150:
7142:
7130:
7127:
7124:
7121:
7118:
7115:
7111:
7105:
7101:
7095:
7090:
7087:
7084:
7080:
7076:
7073:
7062:
7057:
7053:
7047:
7041:
7035:
7028:
7027:
7026:
7010:
7005:
7001:
6997:
6989:
6985:
6962:
6958:
6950:
6943:
6934:
6931:
6927:
6919:
6914:
6909:
6905:
6901:
6885:
6879:
6874:
6870:
6866:
6861:
6858:
6855:
6851:
6845:
6841:
6835:
6830:
6827:
6824:
6820:
6816:
6811:
6807:
6799:
6798:
6797:
6796:
6791:
6787:
6783:
6778:
6774:
6770:
6766:
6762:
6761:
6760:
6758:
6725:
6721:
6717:
6714:
6707:
6703:
6699:
6694:
6689:
6685:
6681:
6676:
6671:
6667:
6660:
6655:
6651:
6646:
6640:
6636:
6632:
6627:
6623:
6614:
6595:
6591:
6587:
6584:
6578:
6574:
6568:
6563:
6559:
6554:
6548:
6544:
6540:
6535:
6531:
6522:
6506:
6502:
6498:
6493:
6490:
6486:
6477:
6476:
6459:
6455:
6449:
6445:
6441:
6436:
6432:
6426:
6422:
6418:
6413:
6409:
6401:
6385:
6382:
6378:
6372:
6368:
6364:
6359:
6355:
6349:
6345:
6341:
6336:
6332:
6324:
6323:
6321:
6320:
6318:
6315:
6297:
6293:
6289:
6284:
6280:
6275:
6269:
6265:
6261:
6256:
6252:
6243:
6227:
6223:
6217:
6213:
6209:
6204:
6200:
6192:
6191:
6189:
6186:
6185:
6184:
6182:
6160:
6157:
6154:
6148:
6143:
6139:
6133:
6128:
6125:
6122:
6118:
6114:
6108:
6102:
6095:
6094:
6093:
6076:
6070:
6062:
6058:
6053:
6040:
6035:
6030:
6026:
6002:
5999:
5996:
5993:
5990:
5987:
5984:
5981:
5978:
5975:
5970:
5966:
5955:which, once
5939:
5934:
5929:
5925:
5921:
5916:
5913:
5909:
5903:
5899:
5893:
5888:
5885:
5882:
5878:
5874:
5869:
5865:
5857:
5856:
5855:
5853:
5837:
5831:
5828:
5825:
5822:
5819:
5816:
5813:
5810:
5807:
5804:
5799:
5795:
5766:
5758:
5754:
5746:
5737:
5733:
5723:
5719:
5709:
5705:
5698:
5691:
5685:
5678:
5675:
5672:
5668:
5660:
5657:
5654:
5650:
5642:
5639:
5636:
5632:
5624:
5619:
5614:
5609:
5602:
5595:
5591:
5583:
5579:
5571:
5567:
5559:
5552:
5549:
5545:
5537:
5533:
5525:
5521:
5513:
5506:
5503:
5499:
5491:
5488:
5484:
5476:
5472:
5465:
5460:
5455:
5447:
5443:
5435:
5426:
5422:
5412:
5408:
5398:
5394:
5387:
5378:
5377:
5376:
5372:
5365:
5359:
5357:
5339:
5336:
5333:
5329:
5306:
5302:
5275:
5271:
5260:
5254:
5250:
5230:
5225:
5222:
5219:
5215:
5209:
5204:
5200:
5196:
5191:
5188:
5185:
5181:
5175:
5171:
5165:
5160:
5157:
5154:
5150:
5146:
5141:
5137:
5129:
5128:
5127:
5125:
5121:
5111:
5109:
5105:
5087:
5083:
5058:
5053:
5049:
5045:
5040:
5037:
5034:
5030:
5024:
5020:
5014:
5009:
5006:
5003:
4999:
4995:
4990:
4986:
4978:
4977:
4976:
4974:
4969:
4967:
4964:procedure or
4963:
4953:
4949:
4919:
4915:
4911:
4908:
4901:
4898:
4894:
4890:
4887:
4879:
4875:
4871:
4863:
4859:
4848:
4845:
4842:
4838:
4831:
4828:
4821:
4804:
4800:
4794:
4790:
4786:
4782:
4776:
4772:
4768:
4765:
4761:
4757:
4754:
4746:
4742:
4731:
4728:
4725:
4721:
4714:
4704:
4703:
4702:
4687:
4681:
4678:
4675:
4671:
4665:
4662:
4659:
4655:
4650:
4644:
4639:
4636:
4633:
4625:
4622:
4614:
4610:
4606:
4603:
4597:
4592:
4588:
4582:
4578:
4574:
4569:
4566:
4563:
4559:
4536:
4533:
4530:
4526:
4522:
4519:
4513:
4510:
4507:
4501:
4496:
4492:
4488:
4485:
4480:
4477:
4474:
4470:
4445:
4420:
4416:
4389:
4383:
4380:
4377:
4356:
4353:
4345:
4318:
4315:
4312:
4308:
4304:
4296:
4292:
4288:
4285:
4276:
4273:
4270:
4264:
4259:
4255:
4251:
4246:
4243:
4240:
4236:
4228:
4227:
4226:
4207:
4204:
4196:
4186:
4184:
4168:
4146:
4142:
4138:
4135:
4130:
4126:
4117:
4113:
4095:
4092:
4089:
4085:
4081:
4078:
4073:
4069:
4048:
4045:
4040:
4036:
4026:
4012:
3990:
3986:
3977:
3959:
3955:
3946:
3928:
3924:
3901:
3897:
3874:
3870:
3846:
3841:
3838:
3835:
3831:
3825:
3821:
3810:
3807:
3804:
3800:
3796:
3791:
3787:
3779:
3778:
3777:
3761:
3757:
3748:
3729:
3724:
3721:
3718:
3714:
3708:
3704:
3698:
3695:
3692:
3687:
3684:
3681:
3677:
3673:
3668:
3665:
3662:
3658:
3652:
3648:
3644:
3639:
3635:
3627:
3626:
3625:
3624:times yields
3623:
3605:
3602:
3599:
3595:
3572:
3569:
3566:
3562:
3558:
3553:
3550:
3547:
3543:
3539:
3517:
3513:
3503:
3489:
3469:
3465:
3461:
3458:
3455:
3424:
3420:
3416:
3411:
3407:
3400:
3396:
3385:
3381:
3377:
3374:
3368:
3363:
3359:
3349:
3346:
3342:
3337:
3331:
3318:
3317:
3316:
3314:
3288:
3279:
3268:
3264:
3260:
3257:
3251:
3246:
3242:
3236:
3230:
3224:
3217:
3216:
3215:
3199:
3195:
3174:
3154:
3151:
3148:
3123:
3119:
3095:
3091:
3082:
3076:
3073:
3070:
3067:
3064:
3059:
3055:
3051:
3048:
3042:
3037:
3033:
3027:
3019:
3016:
3012:
3007:
3002:
2999:
2996:
2993:
2989:
2983:
2979:
2965:
2962:
2959:
2955:
2947:
2944:
2940:
2935:
2929:
2916:
2915:
2914:
2912:
2908:
2903:
2889:
2886:
2883:
2880:
2877:
2869:
2868:time constant
2850:
2840:
2831:
2820:
2816:
2812:
2809:
2803:
2798:
2794:
2788:
2783:
2779:
2775:
2767:
2763:
2757:
2754:
2751:
2747:
2740:
2734:
2729:
2725:
2717:
2716:
2715:
2713:
2708:
2691:
2686:
2681:
2677:
2673:
2665:
2662:
2659:
2655:
2639:
2635:
2631:
2623:
2619:
2601:
2600:
2599:
2583:
2579:
2556:
2552:
2528:
2520:
2516:
2512:
2509:
2503:
2498:
2494:
2488:
2483:
2479:
2475:
2467:
2462:
2458:
2451:
2445:
2437:
2433:
2415:
2414:
2413:
2411:
2392:
2389:
2386:
2379:
2378:
2377:
2363:
2360:
2357:
2354:
2351:
2348:
2345:
2325:
2317:
2313:
2306:
2300:
2292:
2289:
2286:
2282:
2275:
2269:
2266:
2258:
2254:
2247:
2224:
2216:
2195:
2191:
2184:
2161:
2158:
2150:
2123:
2119:
2098:
2095:
2092:
2072:
2069:
2061:
2048:
2030:
2026:
2003:
1998:
1994:
1971:
1967:
1943:
1939:
1935:
1930:
1927:
1924:
1920:
1916:
1913:
1908:
1904:
1889:
1873:
1869:
1846:
1842:
1819:
1815:
1792:
1788:
1778:
1776:
1760:
1740:
1720:
1711:
1703:
1696:
1693:Graphs of AR(
1687:
1684:
1683:
1682:
1680:
1675:
1659:
1655:
1631:
1625:
1617:
1613:
1592:
1588:
1582:
1578:
1572:
1567:
1564:
1561:
1557:
1553:
1550:
1547:
1541:
1535:
1528:
1527:
1526:
1510:
1506:
1482:
1472:
1464:
1459:
1455:
1449:
1445:
1439:
1434:
1431:
1428:
1424:
1420:
1414:
1408:
1401:
1400:
1399:
1397:
1393:
1383:
1367:
1363:
1340:
1336:
1327:
1308:
1302:
1298:
1288:
1282:
1278:
1273:
1268:
1264:
1256:
1255:
1254:
1234:
1230:
1226:
1221:
1217:
1210:
1204:
1197:
1196:
1195:
1192:
1188:
1184:
1179:
1177:
1159:
1155:
1132:
1128:
1122:
1117:
1113:
1090:
1086:
1063:
1059:
1036:
1032:
1009:
1005:
999:
995:
972:
968:
945:
941:
918:
914:
891:
887:
864:
860:
851:
833:
829:
806:
802:
798:
793:
790:
787:
783:
777:
773:
769:
764:
760:
745:
731:
728:
718:
714:
701:must satisfy
686:
682:
673:
654:
650:
644:
640:
634:
629:
626:
623:
619:
615:
612:
608:
602:
588:
572:
569:
559:
555:
541:
536:
534:
531:
510:
506:
502:
497:
493:
486:
480:
473:
472:
471:
469:
448:
444:
440:
435:
431:
425:
421:
415:
411:
405:
400:
397:
394:
390:
386:
381:
377:
369:
368:
367:
365:
362:
358:
340:
336:
327:
309:
305:
301:
298:
295:
290:
286:
260:
256:
252:
247:
244:
241:
237:
231:
227:
221:
216:
213:
210:
206:
202:
197:
193:
185:
184:
183:
181:
177:
158:
152:
149:
142:The notation
135:
133:
129:
126:
124:
120:
116:
112:
108:
103:
99:
95:
84:
81:
73:
63:
59:
53:
52:
46:
41:
32:
31:
19:
11958:Econometrics
11920:Wiener space
11808:ItĂ´ integral
11709:Inequalities
11598:Self-similar
11568:GaussâMarkov
11558:Exchangeable
11538:CĂ dlĂ g paths
11474:Risk process
11426:LIBOR market
11322:
11295:Random graph
11290:Random field
11102:Superprocess
11040:LĂŠvy process
11035:Jump process
11010:Hunt process
10846:Markov chain
10604:Hugging Face
10568:David Silver
10216:Audioâvisual
10070:Applications
10060:
10049:Augmentation
9894:
9814:
9797:
9776:
9753:. Retrieved
9744:
9735:
9725:September 4,
9723:. Retrieved
9709:
9698:. Retrieved
9689:
9680:
9669:. Retrieved
9660:
9651:
9640:. Retrieved
9631:
9622:
9611:. Retrieved
9602:
9593:
9583:, retrieved
9578:
9571:
9552:
9541:. Retrieved
9516:
9512:
9502:
9494:
9489:
9478:the original
9473:
9469:
9456:
9448:
9443:
9432:. Retrieved
9420:
9407:
9380:
9352:
9346:
9335:
9319:
9308:
9292:
9284:
9268:
9249:
9243:
9232:. Retrieved
9204:
9197:
9169:
9162:
9100:
9096:
9061:
9056:
9052:
9047:
9043:
9039:
9034:
9030:
8999:
8995:
8990:
8986:
8982:
8980:
8883:
8877:
8866:
8862:
8856:
8839:
8833:
8824:multivariate
8819:
8799:
8798:includes an
8795:
8788:
8784:
8579:
8510:
8374:
8338:
8249:
8076:
7950:
7820:
7735:
7617:
7542:lag operator
7535:
7231:
7170:
6960:
6954:
6929:
6923:
6912:
6907:
6903:
6781:
6776:
6772:
6756:
6754:
6316:
6187:
6183:) processes
6180:
6178:
6063:+1 elements
6060:
6054:
5954:
5851:
5783:
5370:
5363:
5360:
5258:
5252:
5248:
5245:
5117:
5107:
5103:
5074:
4972:
4970:
4959:
4951:
4461:
4192:
4115:
4027:
3861:
3746:
3744:
3621:
3504:
3447:
3310:
3110:
2904:
2865:
2714:is given by
2709:
2706:
2543:
2407:
2214:
1895:
1779:
1712:
1708:
1694:
1678:
1676:
1611:
1609:
1497:
1395:
1389:
1323:
1252:
1190:
1186:
1182:
1180:
1051:in terms of
933:in terms of
849:
848:at say time
751:
586:
537:
527:
465:
363:
325:
277:
179:
175:
141:
130:
127:
101:
97:
93:
91:
76:
67:
48:
12003:Ruin theory
11941:Disciplines
11813:ItĂ´'s lemma
11588:Predictable
11263:Percolation
11246:Potts model
11241:Ising model
11205:White noise
11163:Differences
11025:ItĂ´ process
10965:Cox process
10861:Loop-erased
10856:Random walk
10752:Categories
10700:Autoencoder
10655:Transformer
10523:Alex Graves
10471:OpenAI Five
10375:IBM Watsonx
9997:Convolution
9975:Overfitting
9519:(6): 1289.
8820:TSA toolbox
5256:, yielding
4116:exponential
2174:, the mean
1697:) processes
852:=1 affects
672:unit circle
357:white noise
121:(ARMA) and
62:introducing
12061:Categories
12013:Statistics
11793:Filtration
11694:Kolmogorov
11678:Blumenthal
11603:Stationary
11543:Continuous
11531:Properties
11416:HullâWhite
11158:Martingale
11045:Local time
10933:Fractional
10911:pure birth
10741:Technology
10594:EleutherAI
10553:Fei-Fei Li
10548:Yann LeCun
10461:Q-learning
10444:Decisional
10370:IBM Watson
10278:Midjourney
10170:TensorFlow
10017:Activation
9970:Regression
9965:Clustering
9845:Mark Thoma
9767:References
9755:2021-04-29
9700:2012-04-03
9671:2022-02-16
9642:2022-02-16
9613:2022-02-16
9585:2023-08-20
9543:2019-12-11
9434:2019-01-27
9234:2022-09-03
7525:blue noise
7232:For AR(1)
5106:= 1, ...,
4110:will be a
2376:and hence
1176:stationary
470:, we have
326:parameters
138:Definition
107:stochastic
70:March 2011
45:references
11925:Classical
10938:Geometric
10928:Excursion
10624:MIT CSAIL
10589:Anthropic
10558:Andrew Ng
10456:AlphaZero
10300:VideoPoet
10263:AlphaFold
10200:MindSpore
10154:SpiNNaker
10149:Memristor
10056:Diffusion
10032:Rectifier
10012:Batchnorm
9992:Attention
9987:Adversary
9339:of London
9312:of London
9133:Resonance
9072:ε
9011:ε
8959:ε
8947:−
8930:φ
8909:∑
8840:bayesloop
8791:function.
8750:π
8741:
8729:φ
8722:−
8713:π
8704:
8689:φ
8685:−
8670:φ
8663:−
8649:φ
8631:φ
8609:σ
8554:φ
8545:−
8539:≤
8530:φ
8526:≤
8520:−
8449:φ
8388:φ
8348:φ
8313:φ
8309:−
8217:φ
8213:−
8198:φ
8188:
8180:−
8169:π
8152:∗
8111:φ
8090:φ
8036:φ
8024:φ
7922:φ
7901:φ
7895:±
7886:φ
7868:φ
7794:−
7780:φ
7776:−
7768:−
7754:φ
7750:−
7713:−
7700:−
7686:φ
7682:−
7674:−
7660:φ
7656:−
7578:φ
7574:−
7562:φ
7558:−
7499:φ
7469:φ
7458:red noise
7432:φ
7401:π
7395:
7383:φ
7376:−
7362:φ
7340:σ
7308:π
7302:−
7288:φ
7284:−
7261:σ
7198:σ
7125:π
7116:−
7102:φ
7081:∑
7077:−
7054:σ
7002:σ
6880:∗
6871:ε
6842:φ
6821:∑
6789:estimate.
6722:φ
6718:−
6704:φ
6686:φ
6682:−
6668:φ
6652:γ
6637:γ
6624:ρ
6592:φ
6588:−
6575:φ
6560:γ
6545:γ
6532:ρ
6503:γ
6491:−
6487:γ
6456:γ
6446:φ
6433:γ
6423:φ
6410:γ
6383:−
6379:γ
6369:φ
6356:γ
6346:φ
6333:γ
6294:φ
6281:γ
6266:γ
6253:ρ
6224:γ
6214:φ
6201:γ
6161:τ
6158:−
6149:ρ
6140:φ
6119:∑
6109:τ
6103:ρ
6077:τ
6071:ρ
6031:ε
6027:σ
5997:…
5967:φ
5930:ε
5926:σ
5914:−
5910:γ
5900:φ
5879:∑
5866:γ
5826:…
5796:φ
5755:φ
5747:⋮
5734:φ
5720:φ
5706:φ
5686:⋯
5676:−
5669:γ
5658:−
5651:γ
5640:−
5633:γ
5625:⋱
5620:⋮
5615:⋮
5610:⋮
5603:⋯
5592:γ
5580:γ
5568:γ
5560:⋯
5550:−
5546:γ
5534:γ
5522:γ
5514:⋯
5504:−
5500:γ
5489:−
5485:γ
5473:γ
5444:γ
5436:⋮
5423:γ
5409:γ
5395:γ
5330:δ
5307:ε
5303:σ
5272:γ
5216:δ
5205:ε
5201:σ
5189:−
5182:γ
5172:φ
5151:∑
5138:γ
5120:Udny Yule
5084:φ
5050:ε
5038:−
5021:φ
5000:∑
4916:θ
4912:−
4895:θ
4891:−
4876:σ
4832:
4801:θ
4773:θ
4769:−
4758:μ
4715:
4672:ϵ
4663:−
4656:θ
4630:Σ
4623:μ
4611:θ
4607:−
4579:θ
4527:ε
4520:μ
4514:θ
4511:−
4489:θ
4446:σ
4417:ϵ
4378:μ
4346:θ
4309:ε
4289:−
4286:μ
4277:θ
4274:−
4208:∈
4205:θ
4143:φ
4093:−
4082:φ
4037:ε
4013:φ
3925:ε
3898:φ
3839:−
3832:ε
3822:φ
3816:∞
3801:∑
3758:φ
3722:−
3715:ε
3705:φ
3696:−
3678:∑
3666:−
3649:φ
3603:−
3570:−
3563:ε
3551:−
3540:φ
3490:τ
3470:τ
3456:γ
3421:ω
3408:γ
3401:π
3397:γ
3382:φ
3378:−
3364:ε
3360:σ
3350:π
3332:ω
3326:Φ
3280:φ
3265:φ
3261:−
3247:ε
3243:σ
3237:≈
3175:τ
3146:Δ
3083:ω
3077:
3071:φ
3065:−
3056:φ
3038:ε
3034:σ
3020:π
3000:ω
2994:−
2974:∞
2969:∞
2966:−
2956:∑
2948:π
2930:ω
2924:Φ
2890:φ
2887:−
2878:τ
2832:φ
2817:φ
2813:−
2799:ε
2795:σ
2780:μ
2776:−
2741:
2682:ε
2678:σ
2663:−
2636:φ
2580:ε
2557:ε
2553:σ
2517:φ
2513:−
2499:ε
2495:σ
2480:μ
2476:−
2452:
2387:μ
2355:μ
2352:φ
2346:μ
2314:ε
2307:
2290:−
2276:
2270:φ
2248:
2225:μ
2185:
2151:φ
2093:φ
2062:φ
2027:φ
1999:ε
1995:σ
1968:ε
1940:ε
1928:−
1917:φ
1870:φ
1843:φ
1816:φ
1789:φ
1761:φ
1741:φ
1721:φ
1656:φ
1632:⋅
1626:ϕ
1579:φ
1558:∑
1554:−
1536:ϕ
1473:τ
1465:−
1425:∑
1415:τ
1409:ρ
1394:of an AR(
1364:ε
1337:ε
1324:When the
1299:ε
1283:ϕ
1231:ε
1205:ϕ
1156:ε
1129:ε
1114:φ
1006:ε
996:φ
888:ε
830:ε
803:ε
791:−
774:φ
641:φ
620:∑
616:−
597:Φ
570:≥
556:φ
507:ε
481:ϕ
445:ε
412:φ
391:∑
337:ε
306:φ
299:…
287:φ
257:ε
245:−
228:φ
207:∑
178:. The AR(
12046:Category
11930:Abstract
11464:BĂźhlmann
11070:Compound
10732:Portals
10491:Auto-GPT
10323:Word2vec
10127:Hardware
10044:Datasets
9946:Concepts
9749:Archived
9719:Archived
9694:Archived
9665:Archived
9636:Archived
9607:Archived
9560:Archived
9537:Archived
9425:Archived
9328:Archived
9301:Archived
9277:Archived
9228:Archived
9224:42392178
9189:28888762
9107:See also
9095:for the
8998:1, ...,
7624:Z domain
6937:Spectrum
5251:=â0,ââŚ,
4333:, where
4161:whereby
2410:variance
1618:, where
324:are the
18:AR model
11553:Ergodic
11441:VaĹĄĂÄek
11283:Poisson
10943:Meander
10614:Meta AI
10451:AlphaGo
10435:PanGu-ÎŁ
10405:ChatGPT
10380:Granite
10328:Seq2seq
10307:Whisper
10228:WaveNet
10223:AlexNet
10195:Flux.jl
10175:PyTorch
10027:Sigmoid
10022:Softmax
9887:General
9841:YouTube
9802:Bibcode
9521:Bibcode
6780:on the
5854:= 0 is
5354:is the
4971:The AR(
1614:is the
58:improve
11893:Tanaka
11578:Mixing
11573:Markov
11446:Wilkie
11411:HoâLee
11406:Heston
11178:Super-
10923:Bridge
10871:Biased
10629:Huawei
10609:OpenAI
10511:People
10481:MuZero
10343:Gemini
10338:Claude
10273:DALL-E
10185:Theano
9785:
9566:(in R)
9395:
9359:
9256:
9222:
9212:
9187:
9177:
8846:Python
8818:: the
8816:Octave
8812:Matlab
8806:MATLAB
8800:sarima
8783:, the
6244:Hence
5373:>â0
5246:where
5102:where
3448:where
2544:where
1959:where
1610:where
1498:where
278:where
47:, but
11746:Tools
11522:M/M/c
11517:M/M/1
11512:M/G/1
11502:Fluid
11168:Local
10695:Mamba
10466:SARSA
10430:LLaMA
10425:BLOOM
10410:GPT-J
10400:GPT-4
10395:GPT-3
10390:GPT-2
10385:GPT-1
10348:LaMDA
10180:Keras
9481:(PDF)
9466:(PDF)
9428:(PDF)
9417:(PDF)
9283:, in
9154:Notes
8836:lags.
8830:PyMC3
8796:astsa
8785:stats
8440:When
8379:When
8081:When
7532:AR(2)
7460:. As
7228:AR(1)
7167:AR(0)
3947:then
3943:is a
2870:) of
102:model
11698:LĂŠvy
11497:Bulk
11381:Chen
11173:Sub-
11131:Both
10619:Mila
10420:PaLM
10353:Bard
10333:BERT
10316:Text
10295:Sora
9783:ISBN
9727:2018
9393:ISBN
9357:ISBN
9254:ISBN
9220:OCLC
9210:ISBN
9185:OCLC
9175:ISBN
9044:next
8994:for
8857:The
8814:and
8458:<
8397:>
8357:<
8120:<
7977:and
7626:by:
7544:as:
7508:<
7441:>
6955:The
5122:and
4354:<
4028:For
3745:For
3587:for
2905:The
2710:The
2408:The
2338:that
2159:<
2070:<
1807:and
1390:The
729:>
530:pole
11278:Cox
10360:NMT
10243:OCR
10238:HWR
10190:JAX
10144:VPU
10139:TPU
10134:IPU
9958:SGD
9843:by
9839:on
9529:doi
9385:doi
8991:t-i
8738:cos
8701:cos
8176:cos
7490:If
7423:If
7392:cos
7025:is
6319:=2
6190:=1
5366:=â0
5261:+â1
4829:Var
4819:and
4185:).
3074:cos
2647:var
2611:var
2425:var
2412:is
2049:if
366:as
355:is
12063::
11696:,
11692:,
11688:,
11684:,
11680:,
9747:.
9743:.
9692:.
9688:.
9663:.
9659:.
9634:.
9630:.
9605:.
9601:.
9535:.
9527:.
9517:51
9515:.
9511:.
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9468:.
9423:.
9419:.
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9226:.
9218:.
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8789:ar
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1777:.
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98:AR
11700:)
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10790:t
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9873:e
9866:t
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9808:.
9804::
9791:.
9758:.
9703:.
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9645:.
9616:.
9546:.
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9401:.
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9365:.
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9191:.
9101:n
9097:n
9076:t
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9040:t
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8987:X
8983:t
8963:t
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8950:i
8944:t
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8934:i
8924:p
8919:1
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8913:i
8905:=
8900:t
8896:X
8878:n
8867:k
8863:k
8834:p
8781:R
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8747:4
8744:(
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8725:2
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8423:=
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8317:2
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7864:2
7860:1
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7837:1
7833:z
7817:,
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7797:2
7790:z
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7764:z
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7716:1
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7677:1
7670:z
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7650:(
7647:=
7642:z
7638:H
7603:,
7600:0
7597:=
7592:2
7588:B
7582:2
7571:B
7566:1
7555:1
7511:0
7503:1
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7436:1
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7321:|
7314:f
7311:i
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7252:)
7249:f
7246:(
7243:S
7212:.
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7191:)
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7045:)
7042:f
7039:(
7036:S
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6998:=
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6990:t
6986:Z
6982:(
6978:r
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6972:V
6961:p
6930:p
6920:.
6913:p
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6886:.
6875:t
6867:+
6862:i
6859:+
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6836:p
6831:1
6828:=
6825:i
6817:=
6812:t
6808:X
6782:p
6777:t
6773:X
6757:p
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2489:=
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2304:E
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2031:1
2004:2
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1914:=
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1874:2
1847:1
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1660:k
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1583:k
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1551:1
1548:=
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1542:B
1539:(
1511:k
1507:y
1483:,
1477:|
1469:|
1460:k
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1446:a
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1412:(
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1309:.
1303:t
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1274:=
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861:X
850:t
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807:t
799:+
794:1
788:t
784:X
778:1
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761:X
732:1
725:|
719:i
715:z
710:|
687:i
683:z
655:i
651:z
645:i
635:p
630:1
627:=
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613:1
606:)
603:z
600:(
587:p
573:1
566:|
560:1
551:|
511:t
503:=
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494:X
490:]
487:B
484:[
449:t
441:+
436:t
432:X
426:i
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416:i
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401:1
398:=
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387:=
382:t
378:X
364:B
341:t
310:p
302:,
296:,
291:1
261:t
253:+
248:i
242:t
238:X
232:i
222:p
217:1
214:=
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203:=
198:t
194:X
180:p
176:p
162:)
159:p
156:(
153:R
150:A
96:(
83:)
77:(
72:)
68:(
54:.
20:)
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