Край­гинг-ин­тер­по­ля­ция

ВНИМАНИЕ, ЭТО НЕДОПИСАННАЯ СТАТЬЯ. Она мо­жет со­дер­жать ка­кие-угод­но не­точ­но­сти.

Край­гинг (kriging) был при­ду­ман в [та­ком-то] го­ду [та­ким-то] че­ло­ве­ком, от­сю­да и на­зва­ние ме­то­да. Ин­те­рес­ны­ми осо­бен­но­стя­ми дан­но­го ме­то­да яв­ля­ют­ся [осо­бен­но­сти]. До край­гин­га ис­поль­зо­ва­лись [та­кие-то] ме­то­ды. Край­гинг вы­год­но от­ли­чал­ся от них [тем-то], по­это­му стал очень по­пу­ля­рен вско­ре по­сле сво­е­го изоб­ре­те­ния.

Жир­ным шриф­том бу­ду обо­зна­чать точ­ки и век­то­ры, что­бы не пу­тать их с чис­ла­ми.

Опре­де­ле­ние ме­то­да

До­пу­стим, нам да­ны точ­ки \mathbf x_1, \mathbf x_2, ... , \mathbf  x_n в ев­кли­до­вом про­стран­стве \mathbb R^m лю­бой раз­мер­но­сти, и в этих точ­ках из­ве­ст­ны зна­че­ния не­ко­то­рой функ­ции: f_1\equiv\mathrm f(\mathbf x_1), f_2\equiv\mathrm f(\mathbf x_2), ... , f_n\equiv\mathrm f(\mathbf x_n), при­чём са­ма функ­ция \mathrm f(\cdot) нам не­из­ве­ст­на. Тре­бу­ет­ся по­стро­ить ин­тер­по­ля­ци­он­ную функ­цию \mathrm f^*(\cdot), яв­ля­ю­щу­ю­ся «хо­ро­шей» оцен­кой не­из­вест­ной функ­ции \mathrm f(\cdot): \mathrm f^*(\mathbf x) \approx \mathrm f(\mathbf x)\ \forall\mathbf x\in\mathbb R^m.

Как и дру­гие клас­си­че­ские ме­то­ды ин­тер­по­ля­ции, край­гинг ос­но­ван на вы­чис­ле­нии для каж­дой нуж­ной нам точ­ки \mathbf x ве­сов \lambda_1(\mathbf x), \lambda_2(\mathbf x), ... , \lambda_n(\mathbf x), и взя­тии ли­ней­ной ком­би­на­ции зна­че­ний в из­ве­ст­ных точ­ках с эти­ми ве­са­ми:

\mathrm f^*(\mathbf x) \equiv \sum_{i=1}^n\lambda_i(\mathbf x)\cdot f_i.
(1)

Край­гинг — ли­ней­ный ме­тод. Это озна­ча­ет, что для вы­чис­ле­ния ко­эф­фи­ци­ен­тов \lambda_1(\mathbf x), ... , \lambda_n(\mathbf x) зна­че­ния f_1, ... , f_n не ис­поль­зу­ют­ся; ис­поль­зу­ет­ся по­ло­же­ние то­чек \mathbf x, \mathbf x_1, ... , \mathbf x_n, и мо­дель слу­чай­но­го про­цес­са (ва­рио­грам­ма, о ней поз­же). Од­на­ко, ес­ли для по­строе­ния ва­рио­грам­мы вы поль­зу­е­тесь ве­ли­чи­на­ми f_1, ... , f_n, то ре­зуль­тат ин­тер­по­ля­ции бу­дет не­ли­ней­но за­ви­сеть от этих ве­ли­чин.

В ме­то­де край­гин­га пред­по­ла­га­ет­ся, что функ­ция \mathrm f(\cdot) — это не­ко­то­рый слу­чай­ный про­цесс. Со­от­вет­ствен­но, f_i=\mathrm f(\mathbf x_i) — слу­чай­ные ве­ли­чи­ны. То­гда их ли­ней­ная ком­би­на­ция \mathrm f^*(\mathbf x) то­же яв­ля­ет­ся слу­чай­ной ве­ли­чи­ной. Ко­эф­фи­ци­ен­ты \lambda_i(\mathbf x) вы­чис­ля­ют­ся так, что­бы ма­те­ма­ти­че­ское ожи­да­ние ре­зуль­ти­рую­щей ве­ли­чи­ны \mathrm{f}^*(\mathbf x) бы­ло рав­но ма­те­ма­ти­че­ско­му ожи­да­нию зна­че­ния слу­чай­но­го про­цес­са в этой точ­ке, а дис­пер­сия раз­но­сти \mathrm{f}^*(\mathbf x)-\mathrm f(\mathbf x) бы­ла ми­ни­маль­ной:

\mathrm M(\mathrm{f}^*(\mathbf x)) =  \mathrm M(\mathrm f(\mathbf x)),\quad \mathrm D(\mathrm{f}^*(\mathbf x)-\mathrm f(\mathbf x))\to\min.
(2)

Та­кой под­ход был на­зван оп­ти­маль­ным ли­ней­ным пред­ска­за­ни­ем.

Для вы­пол­не­ния пред­ска­за­ния ме­тод край­гин­га дол­жен иметь не­ко­то­рые зна­ния о слу­чай­ном про­цес­се \mathrm f(\cdot) (об­ла­дать мо­де­лью слу­чай­но­го про­цес­са). В ка­че­стве мо­де­ли ис­поль­зу­ет­ся функ­ция \mathrm d(\mathbf a,\mathbf b), ха­рак­те­ри­зую­щая за­ви­си­мость ожи­да­е­мо­го раз­ли­чия меж­ду зна­че­ни­я­ми про­цес­са в не­ко­то­рых точ­ках \mathbf a и \mathbf b от рас­по­ло­же­ния этих то­чек. Ме­рой ожи­да­е­мо­го раз­ли­чия яв­ля­ет­ся услов­ная дис­пер­сия раз­но­сти зна­че­ний:

\mathrm d(\mathbf a,\mathbf b) \equiv \left.\mathrm D(\mathrm f(\mathbf a)-\mathrm f(\mathbf b))\right|_{\mathbf a,\mathbf b}.
(3)

Функ­ция \mathrm d(\cdot,\cdot) на­зы­ва­ет­ся ва­рио­грам­мой слу­чай­но­го про­цес­са. По­ка мы её не зна­ем (и вряд ли узна­ем, так как функ­ция \mathrm f(\cdot) нам не­из­ве­ст­на и, во­об­ще го­во­ря, мо­жет не яв­лять­ся слу­чай­ным про­цес­сом).

Пред­по­ло­жим, что ва­рио­грам­ма \mathrm d(\cdot,\cdot) из­ве­ст­на. То­гда зна­че­ния ве­сов \lambda_i(\mathbf x) опре­де­ля­ют­ся сле­дую­щим об­ра­зом:

\begin{pmatrix}\lambda_1(\mathbf x) \\ \vdots \\ \lambda_n(\mathbf x) \\ \mu(\mathbf x) \end{pmatrix}=
A^{-1}
\begin{pmatrix}\mathrm d( \mathbf x_1, \mathbf x) \\ \vdots \\ \mathrm d( \mathbf x_n, \mathbf x) \\ 1\end{pmatrix},
(4)

где

A = \begin{pmatrix}\mathrm d( \mathbf x_1, \mathbf x_1) & \cdots & \mathrm d( \mathbf x_1, \mathbf x_n) &1 \\
\vdots & \ddots & \vdots  & \vdots \\
\mathrm d( \mathbf x_n, \mathbf x_1) & \cdots & \mathrm d( \mathbf x_n, \mathbf x_n) & 1 \\
1 &\cdots& 1 & 0 
\end{pmatrix}.
(5)

В фор­му­ле (5) име­ет­ся в ви­ду тео­ре­ти­че­ская ва­рио­грам­ма, ко­то­рая по опре­де­ле­нию рав­на ну­лю при сов­па­даю­щих точ­ках-ар­гу­мен­тах. Это озна­ча­ет, что в мат­ри­це A на диа­го­на­ли все эле­мен­ты долж­ны быть рав­ны ну­лю, да­же ес­ли ва­ша экс­пе­ри­мен­таль­ная ва­рио­грам­ма, ко­то­рую вы под­ста­ви­те в эту мат­ри­цу, не рав­на ну­лю при сов­па­даю­щих точ­ках.

По­лу­чен­ные ве­са мож­но под­ста­вить в фор­му­лу (1) и по­лу­чить тре­бу­е­мое зна­че­ние \mathrm f^*(\mathbf x). Ве­ли­чи­на \mu вспо­мо­га­тель­ная (это мно­жи­тель Лагран­жа); она мо­жет при­го­дить­ся для по­сле­дую­щей оцен­ки по­греш­но­сти ме­то­да.

Важ­но, что ес­ли нуж­но вы­чис­лить зна­че­ния ин­тер­по­ля­ци­он­ной функ­ции во мно­гих точ­ках, то мат­ри­цу A до­ста­точ­но раз­ло­жить один раз, так как она не за­ви­сит от те­ку­щей точ­ки \mathbf x. Об­ра­ти­те вни­ма­ние, что при прак­ти­че­ской реа­ли­за­ции мат­ри­цу нуж­но имен­но раз­ло­жить, а не об­ра­тить (ра­зум­ные лю­ди ни­ко­гда не об­ра­ща­ют мат­ри­цы). Мож­но, на­при­мер, ис­поль­зо­вать LU-раз­ло­же­ние. С раз­ло­жен­ной мат­ри­цей си­сте­ма урав­не­ний (4) бу­дет ре­шать­ся для каж­дой вы­чис­ля­е­мой точ­ки со ско­ро­стью умно­же­ния мат­ри­цы на век­тор (то есть вы не те­ря­е­те в ско­ро­сти по срав­не­нию с од­но­крат­ным об­ра­ще­ни­ем мат­ри­цы), од­на­ко точ­ность ре­ше­ния бу­дет вы­ше, чем при об­ра­ще­нии мат­ри­цы.

Край­гинг поз­во­ля­ет оце­нить по­греш­ность са­мо­го се­бя. По­греш­ность вы­чис­ля­ет­ся в ви­де дис­пер­сии от­кло­не­ния вы­чис­лен­но­го зна­че­ния от ис­тин­но­го не­из­вест­но­го:

\mathrm D(\mathrm{f}^*(\mathbf x)-\mathrm f(\mathbf x)) = \begin{pmatrix}\lambda_1(\mathbf x) \\ \vdots \\ \lambda_n(\mathbf x) \\ \mu(\mathbf x) \end{pmatrix}^T
\cdot
\begin{pmatrix}\mathrm d( \mathbf x_1, \mathbf x) \\ \vdots \\ \mathrm d( \mathbf x_n, \mathbf x) \\ 1\end{pmatrix}.
(6)

Ана­ло­гич­но мы по­лу­ча­ем за­ни­жен­ное зна­че­ние вы­бо­роч­ной дис­пер­сии, ес­ли оце­ни­ва­ем её как сред­нее зна­че­ние квад­ра­тов от­кло­не­ний имею­щих­ся реа­ли­за­ций слу­чай­ной ве­ли­чи­ны от их сред­не­го зна­че­ния. При­чи­на это­го в том, что сред­нее ариф­ме­ти­че­ское реа­ли­за­ций име­ет тен­ден­цию от­кло­нять­ся от ма­те­ма­ти­че­ско­го ожи­да­ния, и это от­кло­не­ние та­ко­во, что­бы ми­ни­ми­зи­ро­вать вы­чис­ля­е­мую впо­след­ствии дис­пер­сию. В свя­зи с этим обыч­но вво­дят по­прав­ку: при вы­чис­ле­нии вы­бо­роч­ной дис­пер­сии сум­му квад­ра­тов от­кло­не­ний де­лят не на ко­ли­че­ство ис­ход­ных реа­ли­за­ций, а на ко­ли­че­ство реа­ли­за­ций без еди­ни­цы.

Ре­зю­ми­ро­вать это мож­но сле­дую­щим об­ра­зом: оцен­ка ка­че­ства ра­бо­ты мо­де­ли все­гда за­вы­ше­на на тех дан­ных, по ко­то­рым под­би­ра­лись па­ра­мет­ры этой мо­де­ли.

Сле­ду­ет иметь в ви­ду, что ес­ли са­ми ис­ход­ные точ­ки бы­ли ис­поль­зо­ва­ны для оцен­ки ва­рио­грам­мы \mathrm d(\cdot,\cdot), то зна­че­ние дис­пер­сии (6) яв­ля­ет­ся за­ни­жен­ным. Чем мень­ше то­чек бы­ло ис­поль­зо­ва­но для по­строе­ния ва­рио­грам­мы, тем бо­лее за­ни­же­на оцен­ка (6).

Слу­чай ста­ци­о­нар­но­го изо­троп­но­го про­цес­са

Рас­ска­жу про са­мую ча­сто ис­поль­зу­е­мую мо­дель, в ко­то­рой пред­по­ла­га­ет­ся, что слу­чай­ный про­цесс \mathrm f(\cdot) яв­ля­ет­ся ста­ци­о­нар­ным и изо­троп­ным. Ста­ци­о­нар­ность про­цес­са озна­ча­ет, что его ма­те­ма­ти­че­ское ожи­да­ние не за­ви­сит от точ­ки про­стран­ства, а кор­ре­ля­ция зна­че­ний в двух про­из­воль­ных точ­ках \mathbf a и \mathbf b за­ви­сит лишь от вза­имно­го рас­по­ло­же­ния этих то­чек (от раз­но­сти меж­ду ни­ми). Изо­троп­ность озна­ча­ет, что кор­ре­ля­ция за­ви­сит лишь от рас­стоя­ния меж­ду точ­ка­ми.

Ва­рио­грам­ма ста­ци­о­нар­но­го и изо­троп­но­го слу­чай­но­го про­цес­са яв­ля­ет­ся функ­ци­ей од­ной пе­ре­мен­ной — рас­стоя­ния меж­ду точ­ка­ми:

\mathrm d(\mathbf a,\mathbf b) = \mathrm v(\rho) \equiv \left.\mathrm D(\mathrm f(\mathbf a)-\mathrm f(\mathbf b))\right|_{\left|\mathbf a-\mathbf b\right|=\rho}.
(7)

Пе­рей­дём к са­мо­му ин­те­рес­но­му — оцен­ке ва­рио­грам­мы \mathrm v(\rho). Её мож­но при­мер­но по­стро­ить на ос­но­ве ана­ли­за имею­щих­ся ис­ход­ных дан­ных. Из ста­ци­о­нар­но­сти про­цес­са сле­ду­ет, что раз­ность \mathrm f(\mathbf a)-\mathrm f(\mathbf b) име­ет ну­ле­вое ма­те­ма­ти­че­ское ожи­да­ние. Дис­пер­сия слу­чай­ной ве­ли­чи­ны, имею­щей ну­ле­вое мат. ожи­да­ние — это ма­те­ма­ти­че­ское ожи­да­ние квад­ра­та этой слу­чай­ной ве­ли­чи­ны:

\mathrm v(\rho) = \left.\mathrm M\left(\left(\mathrm f(\mathbf a)-\mathrm f(\mathbf b)\right)^2\right)\right|_{\left|\mathbf a-\mathbf b\right|=\rho}.
(8)

Ма­те­ма­ти­че­ское ожи­да­ние, как мы зна­ем, мож­но оце­нить сред­ним ариф­ме­ти­че­ским имею­щих­ся реа­ли­за­ций слу­чай­ной ве­ли­чи­ны. Но у нас си­ту­а­ция не­мно­го слож­нее: дис­пер­сия за­ви­сит от рас­стоя­ния меж­ду точ­ка­ми. По­это­му в ка­че­стве оцен­ки мат. ожи­да­ния у нас бу­дет вы­сту­пать не чис­ло, а не­ко­то­рая кри­вая, ко­то­рая пред­став­ля­ет со­бой за­ви­си­мость сред­не­го ариф­ме­ти­че­ско­го от рас­стоя­ния.

Посмот­рим, ка­кие реа­ли­за­ции слу­чай­ной ве­ли­чи­ны \left(\mathrm f(\mathbf a)-\mathrm f(\mathbf b)\right)^2 у нас име­ют­ся. В ка­че­стве то­чек \mathbf a и \mathbf b мож­но взять все­воз­мож­ные па­ры \mathbf x_i, \mathbf x_j, 1\leqslant i < j\leqslant n имею­щих­ся у нас то­чек с из­ве­ст­ны­ми зна­че­ни­я­ми.

Итак, мы пе­ре­би­ра­ем все па­ры то­чек \mathbf x_i, \mathbf x_j, для каж­дой па­ры вы­чис­ля­ем рас­стоя­ние \rho=\left|\mathbf x_i-\mathbf x_j\right| и квад­рат раз­но­сти имею­щих­ся зна­че­ний в этих точ­ках: v=\left(f_i-f_j\right)^2. От­кла­ды­ва­ем на гра­фи­ке точ­ку (\rho,v), и пе­ре­хо­дим к сле­дую­щей па­ре имею­щих­ся то­чек. В ито­ге по­лу­чим что-ни­будь ти­па это­го:

Точ­ки для по­строе­ния ва­рио­грам­мы

Ри­су­нок 1. Точ­ки для по­строе­ния ва­рио­грам­мы

Те­перь стро­им по этим точ­кам плав­ную кри­вую (есть мно­го ме­то­дов сде­лать это):

Экс­пе­ри­мен­таль­ная ва­рио­грам­ма

Ри­су­нок 2. Экс­пе­ри­мен­таль­ная ва­рио­грам­ма

Эту кри­вую и ис­поль­зу­ем в ка­че­стве функ­ции \mathrm d(\mathbf a,\mathbf b)=\mathrm v(\left|\mathbf a-\mathbf b\right|) в фор­му­лах (4)–(5).

Кста­ти, ме­тод поз­во­ля­ет учесть по­греш­ность ис­ход­ных дан­ных: \mathrm v(0) есть ни что иное, как удво­ен­ная дис­пер­сия из­ме­ря­е­мой ве­ли­чи­ны. Ес­ли тре­бу­ет­ся, что­бы по­верх­ность про­шла точ­но по имею­щим­ся точ­кам, то \mathrm v(0) долж­но быть рав­но ну­лю (кри­вая на ри­сун­ке 2 долж­на вы­хо­дить из на­ча­ла ко­ор­ди­нат). Ес­ли же ис­ход­ные дан­ные не­точ­ны, и име­ют дис­пер­сию d, то долж­но быть \mathrm v(0)=2d.

Ещё од­но за­ме­ча­ние. Глад­кость ре­зуль­ти­рую­щей ин­тер­по­ля­ци­он­ной функ­ции \mathrm{f}^*(\cdot) сов­па­да­ет с глад­ко­стью функ­ции \mathrm d(\cdot,\cdot). Ес­ли нам нуж­но, что­бы \mathrm{f}^*(\cdot) бы­ла не­пре­рыв­но диф­фе­рен­ци­ру­е­ма (не име­ла «из­ло­мов»), то \mathrm d(\cdot,\cdot) долж­на быть то­же не­пре­рыв­но диф­фе­рен­ци­ру­е­мой. Не­труд­но ви­деть, что ес­ли \mathrm v(\rho) не­пре­рыв­но диф­фе­рен­ци­ру­е­ма, и \mathrm v'(0)=0, то функ­ция \mathrm d(\mathbf a,\mathbf b)=\mathrm v(\left|\mathbf a - \mathbf b\right|) бу­дет то­же не­пре­рыв­но диф­фе­рен­ци­ру­е­мой. В свя­зи с этим ва­рио­грам­му ча­сто ищут в сле­дую­щем ви­де:

\mathrm v(\rho) = \alpha + \beta \cdot \left( 1 - e^{-\gamma\cdot\rho^2} \right),
(9)

где ко­эф­фи­ци­ен­ты \alpha\beta и \gamma вы­чис­ля­ют­ся ме­то­дом наи­мень­ших квад­ра­тов по имею­щим­ся точ­кам ва­рио­грам­мы. Ес­ли v(0)=0, то \alpha=0, и ис­кать нуж­но толь­ко \beta и \gamma.

При­мер

[Здесь бу­дет при­мер для не­сколь­ких то­чек]

Про­грамм­ная реа­ли­за­ция

[Здесь бу­дет бу­дет при­мер про­грам­мы с опи­са­ни­ем]

Ма­те­ри­а­лы по те­ме

[Здесь бу­дут ссыл­ки]

54 отзыва на запись «Край­гинг-ин­тер­по­ля­ция»

Спа­си­бо боль­шое!!! Очень ин­фор­ма­тив­ная и до­ход­чи­вая ста­тья. Луч­шее, что по­ка я ви­дел. Бла­го­да­рю за ско­рый от­вет на .Хо­те­лось бы уви­деть про­дол­же­ние ее в ви­де при­ме­ров. Еще раз бла­го­да­рю Вас!!!!
Да, на са­мом де­ле на рус­ском язы­ке про кри­гин­га боль­ше чем здесь я еще не встре­чал… но хо­чет­ся про­дол­же­ния…
От­лич­ная ста­тья, ООО4ень по­нят­но. Про­грамм­ку бы уви­деть!!!
До­пу­стим на экс­пе­ри­мен­таль­ной ва­рио­грам­ме по­лу­дис­пер­сия с ро­стом ша­га сна­ча­ла уве­ли­чи­ва­ет­ся до ка­ко­го-то зна­че­ния, а да­лее пе­ре­ста­ет рас­ти, на­чи­на­ет рез­ко или по­сте­пен­но умень­шать­ся.
Ес­ли на­при­мер, мак­си­му­ма в рас­пре­де­ле­нии свой­ства по­лу­дис­пер­сия до­сти­га­ет при ла­ге 4 м, зна­чит ли это что име­ет­ся не­од­но­род­ность с ша­гом 4 м, или шаг дру­гой?
Ес­ли в ва­рио­грам­ме име­ют­ся «ямы», то это очень не­при­ят­ная си­ту­а­ция. Обыч­но при ап­прок­си­ма­ции та­ких ва­рио­грамм от ям из­бав­ля­ют­ся. Шаг не­од­но­род­но­сти, по-ви­ди­мо­му, ра­вен рас­стоя­нию от ну­ля до ми­ни­му­ма пер­вой ямы (не до мак­си­му­ма).
Посмот­ри­те опи­са­ние Geostatistical Analyst из ArcGis — очень по­дроб­но и на­гляд­но все опи­са­но. А по по­во­ду кри­гин­га есть боль­шая кни­га Ж. Ма­те­рон., Вве­де­ние в гео­ста­ти­сти­ку, 1969 г.
A нет же­ла­ния по­ме­стить ста­тью в Wikipedia?
Обид­но, что на Ан­глий­ском так мно­го ин­фор­ма­ции, а на рус­ском нет ни­че­го…
http://en.wikipedia.org/wiki/Kriging
От­лич­ный об­зор, ла­ко­нич­но и как раз то, что нуж­но.
Дан­ный ме­тод ма­те­ма­ти­че­ски был де­таль­но раз­ра­бо­тан Жор­жем Ма­те­ро­ном в 1961-62 гг, он и дал ему это на­зва­ние в честь юж­но­аф­ри­кан­ско­го гео­ло­га Да­ни­э­ля Край­га — на­сколь­ко я по­нял из крат­кой ан­глий­ской справ­ки, тот при­ме­нял ка­кие-то при­бли­жен­ные по­прав­ки для той же за­да­чи.
Здрав­ствуй­те.
Про­чи­тал. За­шел в ан­глий­скую ви­ки. Ни­че­го се­бе… мда…
По­че­му же я рань­ше не слы­шал об этом?
По­вто­ре­нье — мать уче­нья. При­гла­шаю сю­да: сайт http://www.machinelearning.ru
Раз­дел — «Ре­грес­си­он­ный ана­лиз».
Ста­тья на «М» — «Мно­го­мер­ная ин­тер­по­ля­ция и ап­прок­си­ма­ция на ос­но­ве тео­рии слу­чай­ных функ­ций». Фак­ти­че­ски те­перь уже пе­реот­кры­тие, хо­тя… Есть спе­циа­ли­сты по край­нин­гу? На­сколь­ко он успел про­дви­нуть­ся впе­ред за по­след­нее вре­мя по срав­не­нию с тем что в en.winipedia? У ме­ня в ста­тье в до­пол­не­ние есть раз­бор мно­го­мер­ной ап­прок­си­ма­ции и вы­вод уни­вер­саль­ной ва­рио­грам­мы (так ока­зы­ва­ет­ся это на­зы­ва­ет­ся…), на­сколь­ко это но­во? Мо­гу раз­вить тео­рию даль­ше, ин­тер­по­ля­цию/ап­прок­си­ма­цию с уче­том из­ве­ст­ных про­из­вод­ных лю­бо­го по­ряд­ка в лю­бых точ­ках где из­вест­но.
Юрий, у вас очень хо­ро­шая ста­тья. Ре­ко­мен­дую так­же посмот­реть про­грам­му MLDemos, она со­дер­жит ви­зуа­ли­за­цию по­пу­ляр­ных ал­го­рит­мов ма­шин­но­го обу­че­ния. Там есть кар­тин­ки, по­хо­жие на ва­ши. Мно­го­мер­ная ап­прок­си­ма­ция и уни­вер­саль­ная ва­рио­грам­ма — не но­во, а вот про учёт из­ве­ст­ных про­из­вод­ных лю­бо­го по­ряд­ка я не слы­шал. Хо­тя, для уве­рен­но­сти на­до бу­дет посмот­реть ка­кой-ни­будь учеб­ник по край­гин­гу. К со­жа­ле­нию, мно­гие об­ла­сти на­у­ки и тех­но­ло­гии на­прочь от­сут­ству­ют в на­шей (рос­сий­ской) об­ра­зо­ва­тель­ной про­грам­ме. По­лу­ча­ет­ся так, что этих раз­де­лов для нас как бы не су­ще­ству­ет. Вот и про­ис­хо­дят пе­реот­кры­тия. При­чём ста­тьи с пе­реот­кры­ти­я­ми успеш­но про­хо­дят про­це­ду­ру ре­цен­зи­ро­ва­ния, что свя­за­но с тем, что са­ми ре­цен­зен­ты учи­лись в Со­вет­ском со­ю­зе, а по­сле его рас­па­да не по­тру­ди­лись «за­но­во по­лу­чить об­ра­зо­ва­ние» по ино­стран­ным учеб­ни­кам. Я сам с удив­ле­ни­ем узнал о край­гин­ге па­ру лет на­зад, ко­гда пи­сал ста­тью по ин­тер­по­ля­ции (до­га­ды­ва­юсь, что мой под­ход так­же дав­но из­ве­с­тен). P.S. Я про­чёл ва­шу ста­тью, и уви­дел в ней эле­мент, с ко­то­рым рань­ше не встре­чал­ся при изу­че­нии край­гин­га. Ес­ли я пра­виль­но по­нял, вы мо­же­те од­но­знач­но опре­де­лить ав­то­кор­ре­ля­ци­он­ную функ­цию, вве­дя (в до­пол­не­ние к ста­ци­о­нар­но­сти и изо­троп­но­сти) пред­по­ло­же­ние о рав­ной ве­ро­ят­но­сти «раз­но­мас­штаб­ных» реа­ли­за­ций слу­чай­но­го про­цес­са. К со­жа­ле­нию, это пред­по­ло­же­ние не вы­пол­ня­ет­ся для боль­шин­ства за­дач ин­тер­по­ля­ции (осо­бен­но в гео­ста­ти­сти­ке, где край­гинг наи­бо­лее рас­про­стра­нён), имен­но по­это­му ва­рио­грам­ма вы­хо­дит на кон­стан­ту на­чи­ная с ка­ко­го-то рас­стоя­ния, и по­это­му вы­бор мо­де­ли для ва­рио­грам­мы обыч­но оста­ёт­ся за поль­зо­ва­те­лем ме­то­да. Но в об­ра­бот­ке изоб­ра­же­ний (чем я за­ни­ма­юсь) это пред­по­ло­же­ние ино­гда мож­но при­нять с вы­со­кой точ­но­стью, по­это­му оно ме­ня за­ин­те­ре­со­ва­ло.
Мо­жет, я че­го-то не по­нял… Но по­сле соб­ствен­но­руч­ной реа­ли­за­ции кри­гин­га об­на­ру­жил, что в ка­че­стве ко­эф­фи­ци­ен­тов мат­ри­цы на­до брать не ва­рио­грам­му, а ав­то­ко­ва­ри­а­ци­он­ную функ­цию (в ну­ле — еди­ни­ца, на бес­ко­неч­но­сти стре­мит­ся к ну­лю). Я брал гаус­со­ву функ­цию При­чем её рас­тя­ну­тость по го­ри­зон­та­ли (ко­эф­фи­ци­ент а) про­из­воль­ная — в за­ви­си­мо­сти от это­го бу­дет ме­нять­ся вид ре­зуль­ти­рую­щей по­верх­но­сти, а имен­но — на­сколь­ко ост­рые бу­дут пи­ки в опор­ных точ­ках.
По­про­буй взять t и n мож­но устре­мить в бес­ко­неч­ность. Толь­ко на­до кор­рект­но об­ра­бо­тать в окрест­но­стях ну­ля.
Про­чи­тал в книж­ке у Дю­б­ру­ла.
В ка­че­стве ко­эф­фи­ци­эн­тов мож­но ис­поль­зо­вать как тренд из ва­рио­грам­мы, так и ав­то­кор­ре­ля­цию.
Про­ве­рил. дей­стви­тель­но, мож­но. ра­бо­та­ет.
Толь­ко то­гда по­лу­ча­ет­ся вы слег­ка вво­ди­те лю­дей в за­блуж­де­ние та­ким при­сталь­ным вни­ма­ни­ем к ва­рио­грам­ме и что тренд на­до по­лу­чать не­пре­мен­но из неё.
Спа­си­бо, ста­тья про­сто спа­са­ет, я по­чти от­ча­ял­ся ис­кать что-ли­бо адек­ват­ное на эту те­му.
Дійс­но, про­сто і зро­зуміло. Дя­кую ав­то­ру. За­про­гра­му­ва­ла за до­по­мо­гою статті крігінг. Пра­цює.
нор­маль­но. в ра­бо­те по­мог­ло. про раз­ло­же­ние мат­риц знал, но ни­ко­гда не ис­поль­зо­вал… те­перь ис­прав­люсь
хел­ло фо­рум­ча­не iproc.ru
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