2021-11-07 20:53:18 +00:00
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< title > pyerrors API documentation< / title >
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< style > / * ! * B o o t s t r a p R e b o o t v 5 . 0 . 0 ( h t t p s : / / g e t b o o t s t r a p . c o m / ) * C o p y r i g h t 2 0 1 1 - 2 0 2 1 T h e B o o t s t r a p A u t h o r s * C o p y r i g h t 2 0 1 1 - 2 0 2 1 T w i t t e r , I n c . * L i c e n s e d u n d e r M I T ( h t t p s : / / g i t h u b . c o m / t w b s / b o o t s t r a p / b l o b / m a i n / L I C E N S E ) * F o r k e d f r o m N o r m a l i z e . c s s , l i c e n s e d M I T ( h t t p s : / / g i t h u b . c o m / n e c o l a s / n o r m a l i z e . c s s / b l o b / m a s t e r / L I C E N S E . m d ) * / * , : : a f t e r , : : b e f o r e { b o x - s i z i n g : b o r d e r - b o x } @ m e d i a ( p r e f e r s - r e d u c e d - m o t i o n : n o - p r e f e r e n c e ) { : r o o t { s c r o l l - b e h a v i o r : s m o o t h } } b o d y { m a r g i n : 0 ; f o n t - f a m i l y : s y s t e m - u i , - a p p l e - s y s t e m , " S e g o e U I " , R o b o t o , " H e l v e t i c a N e u e " , A r i a l , " N o t o S a n s " , " L i b e r a t i o n S a n s " , s a n s - s e r i f , " A p p l e C o l o r E m o j i " , " S e g o e U I E m o j i " , " S e g o e U I S y m b o l " , " N o t o C o l o r E m o j i " ; f o n t - s i z e : 1 r e m ; f o n t - w e i g h t : 4 0 0 ; l i n e - h e i g h t : 1 . 5 ; c o l o r : # 2 1 2 5 2 9 ; b a c k g r o u n d - c o l o r : # f f f ; - w e b k i t - t e x t - s i z e - a d j u s t : 1 0 0 % ; - w e b k i t - t a p - h i g h l i g h t - c o l o r : t r a n s p a r e n t } h r { m a r g i n : 1 r e m 0 ; c o l o r : i n h e r i t ; b a c k g r o u n d - c o l o r : c u r r e n t C o l o r ; b o r d e r : 0 ; o p a c i t y : . 2 5 } h r : n o t ( [ s i z e ] ) { h e i g h t : 1 p x } h 1 , h 2 , h 3 , h 4 , h 5 , h 6 { m a r g i n - t o p : 0 ; m a r g i n - b o t t o m : . 5 r e m ; f o n t - w e i g h t : 5 0 0 ; l i n e - h e i g h t : 1 . 2 } h 1 { f o n t - s i z e : c a l c ( 1 . 3 7 5 r e m + 1 . 5 v w ) } @ m e d i a ( m i n - w i d t h : 1 2 0 0 p x ) { h 1 { f o n t - s i z e : 2 . 5 r e m } } h 2 { f o n t - s i z e : c a l c ( 1 . 3 2 5 r e m + . 9 v w ) } @ m e d i a ( m i n - w i d t h : 1 2 0 0 p x ) { h 2 { f o n t - s i z e : 2 r e m } } h 3 { f o n t - s i z e : c a l c ( 1 . 3 r e m + . 6 v w ) } @ m e d i a ( m i n - w i d t h : 1 2 0 0 p x ) { h 3 { f o n t - s i z e : 1 . 7 5 r e m } } h 4 { f o n t - s i z e : c a l c ( 1 . 2 7 5 r e m + . 3 v w ) } @ m e d i a ( m i n - w i d t h : 1 2 0 0 p x ) { h 4 { f o n t - s i z e : 1 . 5 r e m } } h 5 { f o n t - s i z e : 1 . 2 5 r e m } h 6 { f o n t - s i z e : 1 r e m } p { m a r g i n - t o p : 0 ; m a r g i n - b o t t o m : 1 r e m } a b b r [ d a t a - b s - o r i g i n a l - t i t l e ] , a b b r [ t i t l e ] { - w e b k i t - t e x t - d e c o r a t i o n : u n d e r l i n e d o t t e d ; t e x t - d e c o r a t i o n : u n d e r l i n e d o t t e d ; c u r s o r : h e l p ; - w e b k i t - t e x t - d e c o r a t i o n - s k i p - i n k : n o n e ; t e x t - d e c o r a t i o n - s k i p - i n k : n o n e } a d d r e s s { m a r g i n - b o t t o m : 1 r e m ; f o n t - s t y l e : n o r m a l ; l i n e - h e i g h t : i n h e r i t } o l , u l { p a d d i n g - l e f t : 2 r e m } d l , o l , u l { m a r g i n - t o p : 0 ; m a r g i n - b o t t o m : 1 r e m } o l o l , o l u l , u l o l , u l u l { m a r g i n - b o t t o m : 0 } d t { f o n t - w e i g h t : 7 0 0 } d d { m a r g i n - b o t t o m : . 5 r e m ; m a r g i n - l e f t : 0 } b l o c k q u o t e { m a r g i n : 0 0 1 r e m } b , s t r o n g { f o n t - w e i g h t : b o l d e r } s m a l l { f o n t - s i z e : . 8 7 5 e m } m a r k { p a d d i n g : . 2 e m ; b a c k g r o u n d - c o l o r : # f c f 8 e 3 } s u b , s u p { p o s i t i o n : r e l a t i v e ; f o n t - s i z e : . 7 5 e m ; l i n e - h e i g h t : 0 ; v e r t i c a l - a l i g n : b a s e l i n e } s u b { b o t t o m : - . 2 5 e m } s u p { t o p : - . 5 e m } a { c o l o r : # 0 d 6 e f d ; t e x t - d e c o r a t i o n : u n d e r l i n e } a : h o v e r { c o l o r : # 0 a 5 8 c a } a : n o t ( [ h r e f ] ) : n o t ( [ c l a s s ] ) , a : n o t ( [ h r e f ] ) : n o t ( [ c l a s s ] ) : h o v e r { c o l o r : i n h e r i t ; t e x t - d e c o r a t i o n : n o n e } c o d e , k b d , p r e , s a m p { f o n t - f a m i l y : S F M o n o - R e g u l a r , M e n l o , M o n a c o , C o n s o l a s , " L i b e r a t i o n M o n o " , " C o u r i e r N e w " , m o n o s p a c e ; f o n t - s i z e : 1 e m ; d i r e c t i o n : l t r ; u n i c o d e - b i d i : b i d i - o v e r r i d e } p r e { d i s p l a y : b l o c k ; m a r g i n - t o p : 0 ; m a r g i n - b o t t o m : 1 r e m ; o v e r f l o w : a u t o ; f o n t - s i z e : . 8 7 5 e m } p r e c o d e { f o n t - s i z e : i n h e r i t ; c o l o r : i n h e r i t ; w o r d - b r e a k : n o r m a l } c o d e { f o n t - s i z e : . 8 7 5 e m ; c o l o r : # d 6 3 3 8 4 ; w o r d - w r a p : b r e a k - w o r d } a > c o d e { c o l o r : i n h e r i t } k b d { p a d d i n g : . 2 r e m . 4 r e m ; f o n t - s i z e : . 8 7 5 e m ; c o l o r : # f f f ; b a c k g r o u n d - c o l o r : # 2 1 2 5 2 9 ; b o r d e r - r a d i u s : . 2 r e m } k b d k b d { p a d d i n g : 0 ; f o n t - s i z e : 1 e m ; f o n t - w e i g h t : 7 0 0 } f i g u r e { m a r g i n : 0 0 1 r e m } i m g , s v g { v e r t i c a l - a l i g n : m i d d l e } t a b l e { c a p t i o n - s i d e : b o t t o m ; b o r d e r - c o l l a p s e : c o l l a p s e } c a p t i o n { p a d d i n g - t o p : . 5 r e m ; p a d d i n g - b o t t o m : . 5 r e m ; c o l o r : # 6 c 7 5 7 d ; t e x t - a l i g n : l e f t } t h { t e x t - a l i g n : i n h e r i t ; t e x t - a l i g n : - w e b k i t - m a t c h - p a r e n t } t b o d y , t d , t f o o t , t h , t h e a d , t r { b o r d e r - c o l o r : i n h e r i t ; b o r d e r - s t y l e : s o l i d ; b o r d e r - w i d t h : 0 } l a b e l { d i s p l a y : i n l i n e - b l o c k } b u t t o n { b o r d e r - r a d i u s : 0 } b u t t o n : f o c u s : n o t ( : f o c u s - v i s i b l e ) { o u t l i n e : 0 } b u t t o n , i n p u t , o p t g r o u p , s e l e c t , t e x t a r e a { m a r g i n : 0 ; f o n t - f a m i l y : i n h e r i t ; f o n t - s i z e : i n h e r i t ; l i n e - h e i g h t : i n h e r i t } b u t t o n , s e l e c t { t e x t - t r a n s f o r m : n o n e } [ r o l e = b u t t o n ] { c u r s o r : p o i n t e r } s e l e c t { w o r d - w r a p : n o r m a l } s e l e c t : d i s a b l e d { o p a c i t y : 1 } [ l i s t ] : : - w e b k i t - c a l e n d a r - p i c k e r - i n d i c a t o r { d i s p l a y : n o n e } [ t y p e = b u t t o n ] , [ t y p e = r e s e t ] , [ t y p e = s u b m i t ] , b u t t o n { - w e b k i t - a p p e a r a n c e : b u t t o n } [ t y p e = b u t t o n ] : n o t ( : d i s a b l e d ) , [ t y p e = r e s e t ] : n o t ( : d i s a b l e d ) , [ t y p e = s u b m i t ] : n o t ( : d i s a b l e d ) , b u t t o n : n o t ( : d i s a b l e d ) { c u r s o r : p o i n t e r } : : - m o z - f o c u s - i n n e r { p a d d i n g : 0 ; b o r d e r - s t y l e : n o n e } t e x t a r e a { r e s i z e : v e r t i c a l } f i e l d s e t { m i n - w i d t h : 0 ; p a d d i n g : 0 ; m a r g i n : 0 ; b o r d e r : 0 } l e g e n d { f l o a t : l e f t ; w i d t h : 1 0 0 % ; p a d d i n g : 0 ; m a r g i n - b o t t o m : . 5 r e m ; f o n t - s i z e : c a l c ( 1 . 2 7 5 r e m + . 3 v w ) ; l i n e - h e i g h t : i n h e r i t } @ m e d i a ( m i n - w i d t h : 1 2 0 0 p x ) { l e g e n d { f o n t - s i z e : 1 . 5 r e m } } l e g e n d + * { c l e a r : l e f t } : : - w e b k i t - d a t e t i m e - e d i t - d a y - f i e l d , : : - w e b k i t - d a t e t i m e - e d i t - f i e l d s - w r a p p e r , : : - w e b k i t - d a t e t i m e - e d i t - h o u r - f i e l d , : : - w e b k i t - d a t e t i m e - e d i t - m i n u t e , : : - w e b k i t - d a t e t i m e - e d i t - m o n t h - f i e l d , :
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< style > / * ! p d o c * / : r o o t { - - p d o c - b a c k g r o u n d : # f f f ; } . p d o c { - - t e x t : # 2 1 2 5 2 9 ; - - m u t e d : # 6 c 7 5 7 d ; - - l i n k : # 3 6 6 0 a 5 ; - - l i n k - h o v e r : # 1 6 5 9 c 5 ; - - c o d e : # f 7 f 7 f 7 ; - - a c t i v e : # f f f 5 9 8 ; - - a c c e n t : # e e e ; - - a c c e n t 2 : # c 1 c 1 c 1 ; - - n a v - h o v e r : r g b a ( 2 5 5 , 2 5 5 , 2 5 5 , 0 . 5 ) ; - - n a m e : # 0 0 6 6 B B ; - - d e f : # 0 0 8 8 0 0 ; - - a n n o t a t i o n : # 0 0 7 0 2 0 ; } b o d y { b a c k g r o u n d - c o l o r : v a r ( - - p d o c - b a c k g r o u n d ) ; } h t m l , b o d y { w i d t h : 1 0 0 % ; h e i g h t : 1 0 0 % ; } @ m e d i a ( m a x - w i d t h : 7 6 9 p x ) { # n a v t o g g l e { c u r s o r : p o i n t e r ; p o s i t i o n : a b s o l u t e ; w i d t h : 5 0 p x ; h e i g h t : 4 0 p x ; t o p : 1 r e m ; r i g h t : 1 r e m ; b o r d e r - c o l o r : v a r ( - - t e x t ) ; c o l o r : v a r ( - - t e x t ) ; d i s p l a y : f l e x ; o p a c i t y : 0 . 8 ; } # n a v t o g g l e : h o v e r { o p a c i t y : 1 ; } # t o g g l e s t a t e + d i v { d i s p l a y : n o n e ; } # t o g g l e s t a t e : c h e c k e d + d i v { d i s p l a y : i n h e r i t ; } m a i n , h e a d e r { p a d d i n g : 2 r e m 3 v w ; } . g i t - b u t t o n { d i s p l a y : n o n e ! i m p o r t a n t ; } n a v i n p u t [ t y p e = " s e a r c h " ] : v a l i d ~ * { d i s p l a y : n o n e ! i m p o r t a n t ; } } @ m e d i a ( m i n - w i d t h : 7 7 0 p x ) { : r o o t { - - s i d e b a r - w i d t h : c l a m p ( 1 2 . 5 r e m , 2 8 v w , 2 2 r e m ) ; } n a v { p o s i t i o n : f i x e d ; o v e r f l o w : a u t o ; h e i g h t : 1 0 0 v h ; w i d t h : v a r ( - - s i d e b a r - w i d t h ) ; } m a i n , h e a d e r { p a d d i n g : 3 r e m 2 r e m 3 r e m c a l c ( v a r ( - - s i d e b a r - w i d t h ) + 3 r e m ) ; w i d t h : c a l c ( 5 4 r e m + v a r ( - - s i d e b a r - w i d t h ) ) ; m a x - w i d t h : 1 0 0 % ; } # n a v t o g g l e { d i s p l a y : n o n e ; } } # t o g g l e s t a t e { d i s p l a y : n o n e ; } n a v . p d o c { - - p a d : 1 . 7 5 r e m ; - - i n d e n t : 1 . 5 r e m ; b a c k g r o u n d - c o l o r : v a r ( - - a c c e n t ) ; b o r d e r - r i g h t : 1 p x s o l i d v a r ( - - a c c e n t 2 ) ; b o x - s h a d o w : 0 0 2 0 p x r g b a ( 5 0 , 5 0 , 5 0 , . 2 ) i n s e t ; p a d d i n g : 0 0 0 v a r ( - - p a d ) ; o v e r f l o w - w r a p : a n y w h e r e ; s c r o l l b a r - w i d t h : t h i n ; s c r o l l b a r - c o l o r : v a r ( - - a c c e n t 2 ) t r a n s p a r e n t } n a v . p d o c : : - w e b k i t - s c r o l l b a r { w i d t h : . 4 r e m ; } n a v . p d o c : : - w e b k i t - s c r o l l b a r - t h u m b { b a c k g r o u n d - c o l o r : v a r ( - - a c c e n t 2 ) ; } n a v . p d o c > d i v { p a d d i n g : v a r ( - - p a d ) 0 ; } n a v . p d o c . m o d u l e - l i s t - b u t t o n { d i s p l a y : i n l i n e - f l e x ; a l i g n - i t e m s : c e n t e r ; c o l o r : v a r ( - - t e x t ) ; b o r d e r - c o l o r : v a r ( - - m u t e d ) ; m a r g i n - b o t t o m : 1 r e m ; } n a v . p d o c . m o d u l e - l i s t - b u t t o n : h o v e r { b o r d e r - c o l o r : v a r ( - - t e x t ) ; } n a v . p d o c i n p u t [ t y p e = s e a r c h ] { d i s p l a y : b l o c k ; o u t l i n e - o f f s e t : 0 ; w i d t h : c a l c ( 1 0 0 % - v a r ( - - p a d ) ) ; } n a v . p d o c . l o g o { m a x - w i d t h : c a l c ( 1 0 0 % - v a r ( - - p a d ) ) ; m a x - h e i g h t : 3 5 v h ; d i s p l a y : b l o c k ; m a r g i n : 0 a u t o 1 r e m ; t r a n s f o r m : t r a n s l a t e ( c a l c ( - . 5 * v a r ( - - p a d ) ) , 0 ) ; } n a v . p d o c u l { l i s t - s t y l e : n o n e ; p a d d i n g - l e f t : 0 ; } n a v . p d o c l i { d i s p l a y : b l o c k ; m a r g i n : 0 ; p a d d i n g : . 2 r e m 0 . 2 r e m v a r ( - - i n d e n t ) ; t r a n s i t i o n : a l l 1 0 0 m s ; } n a v . p d o c > d i v > u l > l i { p a d d i n g - l e f t : 0 ; } n a v . p d o c l i : h o v e r { b a c k g r o u n d - c o l o r : v a r ( - - n a v - h o v e r ) ; } n a v . p d o c a , n a v . p d o c a : h o v e r { c o l o r : v a r ( - - t e x t ) ; } n a v . p d o c a { d i s p l a y : b l o c k ; } n a v . p d o c > h 2 : f i r s t - o f - t y p e { m a r g i n - t o p : 1 . 5 r e m ; } n a v . p d o c . c l a s s : b e f o r e { c o n t e n t : " c l a s s " ; c o l o r : v a r ( - - m u t e d ) ; } n a v . p d o c . f u n c t i o n : a f t e r { c o n t e n t : " ( ) " ; c o l o r : v a r ( - - m u t e d ) ; } n a v . p d o c f o o t e r : b e f o r e { c o n t e n t : " " ; d i s p l a y : b l o c k ; w i d t h : c a l c ( 1 0 0 % - v a r ( - - p a d ) ) ; b o r d e r - t o p : s o l i d v a r ( - - a c c e n t 2 ) 1 p x ; m a r g i n - t o p : 1 . 5 r e m ; p a d d i n g - t o p : . 5 r e m ; } n a v . p d o c f o o t e r { f o n t - s i z e : s m a l l ; } h t m l , m a i n { s c r o l l - b e h a v i o r : s m o o t h ; } . p d o c { c o l o r : v a r ( - - t e x t ) ; b o x - s i z i n g : b o r d e r - b o x ; l i n e - h e i g h t : 1 . 5 ; b a c k g r o u n d : n o n e ; } . p d o c . p d o c - b u t t o n { d i s p l a y : i n l i n e - b l o c k ; b o r d e r : s o l i d b l a c k 1 p x ; b o r d e r - r a d i u s : 2 p x ; f o n t - s i z e : . 7 5 r e m ; p a d d i n g : c a l c ( 0 . 5 e m - 1 p x ) 1 e m ; t r a n s i t i o n : 1 0 0 m s a l l ; } . p d o c . v i s u a l l y - h i d d e n { p o s i t i o n : a b s o l u t e ! i m p o r t a n t ; w i d t h : 1 p x ! i m p o r t a n t ; h e i g h t : 1 p x ! i m p o r t a n t ; p a d d i n g : 0 ! i m p o r t a n t ; m a r g i n : - 1 p x ! i m p o r t a n t ; o v e r f l o w : h i d d e n ! i m p o r t a n t ; c l i p : r e c t ( 0 , 0 , 0 , 0 ) ! i m p o r t a n t ; w h i t e - s p a c e : n o w r a p ! i m p o r t a n t ; b o r d e r : 0 ! i m p o r t a n t ; } . p d o c h 1 , . p d o c h 2 , . p d o c h 3 { f o n t - w e i g h t : 3 0 0 ; m a r g i n : . 3 e m 0 ; p a d d i n g : . 2 e m 0 ; } . p d o c a { t e x t - d e c o r a t i o n : n o n e ; c o l o r : v a r ( - - l i n k ) ; } . p d o c a : h o v e r { c o l o r : v a r ( - - l i n k - h o v e r ) ; } . p d o c b l o c k q u o t e { m a r g i n - l e f t : 2 r e m ; } . p d o c p r e { b a c k g r o u n d - c o l o r : v a r ( - - c o d e ) ; b o r d e r - t o p : 1 p x s o l i d v a r ( - - a c c e n t 2 ) ; b o r d e r - b o t t o m : 1 p x s o l i d v a r ( - - a c c e n t 2 ) ; m a r g i n - b o t t o m : 1 e m ; p a d d i n g : . 5 r e m 0 . 5 r e m . 5 r e m ; o v e r f l o w - x : a u t o ; } . p d o c c o d e { c o l o r : v a r ( - - t e x t ) ; p a d d i n g : . 2 e m . 4 e m ; m a r g i n : 0 ; f o n t - s i z e : 8 5 % ; b a c k g r o u n d - c o l o r : v a r ( - - c o d e ) ; b o r d e r - r a d i u s : 6 p x ; } . p d o c a > c o d e { c o l o r : i n h e r i t ; } . p d o c p r e > c o d e { d i s p l a y : i n l i n e - b l o c k ; f o n t - s i z e : i n h e r i t ; b a c k g r o u n d : n o n e ; b o r d e r : n o n e ; p a d d i n g : 0 ; } . p d o c . m o d u l e n a m e { m a r g i n - t o p : 0 ; f o n t - w e i g h t : b o l d ; } . p d o c . m o d u l e n a m e a { c o l o r : v a r ( - - l i n k ) ; t r a n s i t i o n : 1 0 0 m s a l l ; } . p d o c . g i t - b u t t o n { f l o a t : r i g h t ; b o r d e r : s o l i d v a r ( - - l i n k ) 1 p x ; } . p d o c . g i t - b u t t o n : h o v e r { b a c k g r o u n d - c o l o r : v a r ( - - l i n k ) ; c o l o r : v a r ( - - p d o c - b a c k g r o u n d ) ; } . p d o c d e t a i l s { - - s h i f t : - 4 0 p x ; t e x t - a l i g n : r i g h t ; m a r g i n - t o p : v a r ( - - s h i f t ) ; m a r g i n - b o t t o m : c a l c ( 0 p x - v a r ( - - s h i f t ) ) ; c l e a r :
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Module Index
< / a >
< input type = "search" placeholder = "Search..." role = "searchbox" aria-label = "search"
pattern=".+" required>
< h2 > Contents< / h2 >
< ul >
< li > < a href = "#what-is-pyerrors" > What is pyerrors?< / a >
< ul >
2021-11-16 11:46:28 +00:00
< li > < a href = "#basic-example" > Basic example< / a > < / li >
2021-11-07 20:53:18 +00:00
< / ul > < / li >
< li > < a href = "#the-obs-class" > The < code > Obs< / code > class< / a >
< ul >
2021-11-09 11:51:27 +00:00
< li > < a href = "#error-propagation" > Error propagation< / a > < / li >
< li > < a href = "#error-estimation" > Error estimation< / a >
< ul >
< li > < a href = "#exponential-tails" > Exponential tails< / a > < / li >
< / ul > < / li >
2021-11-15 10:12:16 +00:00
< li > < a href = "#multiple-ensemblesreplica" > Multiple ensembles/replica< / a >
< ul >
< li > < a href = "#error-estimation-for-multiple-ensembles" > Error estimation for multiple ensembles< / a > < / li >
< / ul > < / li >
2021-11-07 20:53:18 +00:00
< li > < a href = "#irregular-monte-carlo-chains" > Irregular Monte Carlo chains< / a > < / li >
< / ul > < / li >
2021-11-07 21:09:48 +00:00
< li > < a href = "#correlators" > Correlators< / a > < / li >
2021-11-07 20:53:18 +00:00
< li > < a href = "#complex-observables" > Complex observables< / a > < / li >
2021-11-09 11:51:27 +00:00
< li > < a href = "#optimization-fits-roots" > Optimization / fits / roots< / a > < / li >
2021-11-07 20:53:18 +00:00
< li > < a href = "#matrix-operations" > Matrix operations< / a > < / li >
2021-11-15 14:30:41 +00:00
< li > < a href = "#export-data" > Export data< / a >
< ul >
< li > < a href = "#jackknife-samples" > Jackknife samples< / a > < / li >
< / ul > < / li >
2021-11-07 20:53:18 +00:00
< li > < a href = "#input" > Input< / a > < / li >
< / ul >
< h2 > Submodules< / h2 >
< ul >
< li > < a href = "pyerrors/correlators.html" > pyerrors.correlators< / a > < / li >
2021-12-01 14:49:57 +00:00
< li > < a href = "pyerrors/covobs.html" > pyerrors.covobs< / a > < / li >
2021-11-07 20:53:18 +00:00
< li > < a href = "pyerrors/dirac.html" > pyerrors.dirac< / a > < / li >
< li > < a href = "pyerrors/fits.html" > pyerrors.fits< / a > < / li >
< li > < a href = "pyerrors/input.html" > pyerrors.input< / a > < / li >
< li > < a href = "pyerrors/linalg.html" > pyerrors.linalg< / a > < / li >
< li > < a href = "pyerrors/misc.html" > pyerrors.misc< / a > < / li >
< li > < a href = "pyerrors/mpm.html" > pyerrors.mpm< / a > < / li >
< li > < a href = "pyerrors/obs.html" > pyerrors.obs< / a > < / li >
< li > < a href = "pyerrors/roots.html" > pyerrors.roots< / a > < / li >
< li > < a href = "pyerrors/version.html" > pyerrors.version< / a > < / li >
< / ul >
< a class = "attribution" title = "pdoc: Python API documentation generator" href = "https://pdoc.dev" >
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pyerrors < / h1 >
< div class = "docstring" > < h1 id = "what-is-pyerrors" > What is pyerrors?< / h1 >
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< p > < code > < a href = "" > pyerrors< / a > < / code > is a python package for error computation and propagation of Markov chain Monte Carlo data.
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It is based on the gamma method < a href = "https://arxiv.org/abs/hep-lat/0306017" > arXiv:hep-lat/0306017< / a > . Some of its features are:< / p >
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< ul >
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< li > automatic differentiation for exact liner error propagation as suggested in < a href = "https://arxiv.org/abs/1809.01289" > arXiv:1809.01289< / a > (partly based on the < a href = "https://github.com/HIPS/autograd" > autograd< / a > package).< / li >
< li > treatment of slow modes in the simulation as suggested in < a href = "https://arxiv.org/abs/1009.5228" > arXiv:1009.5228< / a > .< / li >
< li > coherent error propagation for data from different Markov chains.< / li >
< li > non-linear fits with x- and y-errors and exact linear error propagation based on automatic differentiation as introduced in < a href = "https://arxiv.org/abs/1809.01289" > arXiv:1809.01289< / a > .< / li >
< li > real and complex matrix operations and their error propagation based on automatic differentiation (Matrix inverse, Cholesky decomposition, calculation of eigenvalues and eigenvectors, singular value decomposition...).< / li >
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< / ul >
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< p > There exist similar publicly available implementations of gamma method error analysis suites in < a href = "https://gitlab.ift.uam-csic.es/alberto/aderrors" > Fortran< / a > , < a href = "https://gitlab.ift.uam-csic.es/alberto/aderrors.jl" > Julia< / a > and < a href = "https://github.com/mbruno46/pyobs" > Python< / a > .< / p >
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< h2 id = "basic-example" > Basic example< / h2 >
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< div class = "codehilite" > < pre > < span > < / span > < code > < span class = "kn" > import< / span > < span class = "nn" > numpy< / span > < span class = "k" > as< / span > < span class = "nn" > np< / span >
< span class = "kn" > import< / span > < span class = "nn" > pyerrors< / span > < span class = "k" > as< / span > < span class = "nn" > pe< / span >
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< span class = "n" > my_obs< / span > < span class = "o" > =< / span > < span class = "n" > pe< / span > < span class = "o" > .< / span > < span class = "n" > Obs< / span > < span class = "p" > ([< / span > < span class = "n" > samples< / span > < span class = "p" > ],< / span > < span class = "p" > [< / span > < span class = "s1" > ' ensemble_name' < / span > < span class = "p" > ])< / span > < span class = "c1" > # Initialize an Obs object< / span >
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< span class = "n" > my_new_obs< / span > < span class = "o" > =< / span > < span class = "mi" > 2< / span > < span class = "o" > *< / span > < span class = "n" > np< / span > < span class = "o" > .< / span > < span class = "n" > log< / span > < span class = "p" > (< / span > < span class = "n" > my_obs< / span > < span class = "p" > )< / span > < span class = "o" > /< / span > < span class = "n" > my_obs< / span > < span class = "o" > **< / span > < span class = "mi" > 2< / span > < span class = "c1" > # Construct derived Obs object< / span >
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< span class = "n" > my_new_obs< / span > < span class = "o" > .< / span > < span class = "n" > gamma_method< / span > < span class = "p" > ()< / span > < span class = "c1" > # Estimate the statistical error< / span >
< span class = "nb" > print< / span > < span class = "p" > (< / span > < span class = "n" > my_new_obs< / span > < span class = "p" > )< / span > < span class = "c1" > # Print the result to stdout< / span >
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< span class = "o" > > < / span > < span class = "mf" > 0.31498< / span > < span class = "p" > (< / span > < span class = "mi" > 72< / span > < span class = "p" > )< / span >
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< / code > < / pre > < / div >
< h1 id = "the-obs-class" > The < code > Obs< / code > class< / h1 >
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< p > < code > < a href = "" > pyerrors< / a > < / code > introduces a new datatype, < code > Obs< / code > , which simplifies error propagation and estimation for auto- and cross-correlated data.
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An < code > Obs< / code > object can be initialized with two arguments, the first is a list containing the samples for an Observable from a Monte Carlo chain.
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The samples can either be provided as python list or as numpy array.
The second argument is a list containing the names of the respective Monte Carlo chains as strings. These strings uniquely identify a Monte Carlo chain/ensemble.< / p >
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< div class = "codehilite" > < pre > < span > < / span > < code > < span class = "kn" > import< / span > < span class = "nn" > pyerrors< / span > < span class = "k" > as< / span > < span class = "nn" > pe< / span >
< span class = "n" > my_obs< / span > < span class = "o" > =< / span > < span class = "n" > pe< / span > < span class = "o" > .< / span > < span class = "n" > Obs< / span > < span class = "p" > ([< / span > < span class = "n" > samples< / span > < span class = "p" > ],< / span > < span class = "p" > [< / span > < span class = "s1" > ' ensemble_name' < / span > < span class = "p" > ])< / span >
< / code > < / pre > < / div >
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< h2 id = "error-propagation" > Error propagation< / h2 >
< p > When performing mathematical operations on < code > Obs< / code > objects the correct error propagation is intrinsically taken care using a first order Taylor expansion
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$$\delta_f^i=\sum_\alpha \bar{f}_\alpha \delta_\alpha^i\,,\quad \delta_\alpha^i=a_\alpha^i-\bar{a}_\alpha\,,$$
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as introduced in < a href = "https://arxiv.org/abs/hep-lat/0306017" > arXiv:hep-lat/0306017< / a > .
The required derivatives $\bar{f}_\alpha$ are evaluated up to machine precision via automatic differentiation as suggested in < a href = "https://arxiv.org/abs/1809.01289" > arXiv:1809.01289< / a > .< / p >
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< p > The < code > Obs< / code > class is designed such that mathematical numpy functions can be used on < code > Obs< / code > just as for regular floats.< / p >
< div class = "codehilite" > < pre > < span > < / span > < code > < span class = "kn" > import< / span > < span class = "nn" > numpy< / span > < span class = "k" > as< / span > < span class = "nn" > np< / span >
< span class = "kn" > import< / span > < span class = "nn" > pyerrors< / span > < span class = "k" > as< / span > < span class = "nn" > pe< / span >
< span class = "n" > my_obs1< / span > < span class = "o" > =< / span > < span class = "n" > pe< / span > < span class = "o" > .< / span > < span class = "n" > Obs< / span > < span class = "p" > ([< / span > < span class = "n" > samples1< / span > < span class = "p" > ],< / span > < span class = "p" > [< / span > < span class = "s1" > ' ensemble_name' < / span > < span class = "p" > ])< / span >
< span class = "n" > my_obs2< / span > < span class = "o" > =< / span > < span class = "n" > pe< / span > < span class = "o" > .< / span > < span class = "n" > Obs< / span > < span class = "p" > ([< / span > < span class = "n" > samples2< / span > < span class = "p" > ],< / span > < span class = "p" > [< / span > < span class = "s1" > ' ensemble_name' < / span > < span class = "p" > ])< / span >
< span class = "n" > my_sum< / span > < span class = "o" > =< / span > < span class = "n" > my_obs1< / span > < span class = "o" > +< / span > < span class = "n" > my_obs2< / span >
< span class = "n" > my_m_eff< / span > < span class = "o" > =< / span > < span class = "n" > np< / span > < span class = "o" > .< / span > < span class = "n" > log< / span > < span class = "p" > (< / span > < span class = "n" > my_obs1< / span > < span class = "o" > /< / span > < span class = "n" > my_obs2< / span > < span class = "p" > )< / span >
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< span class = "n" > iamzero< / span > < span class = "o" > =< / span > < span class = "n" > my_m_eff< / span > < span class = "o" > -< / span > < span class = "n" > my_m_eff< / span >
< span class = "c1" > # Check that value and fluctuations are zero within machine precision< / span >
< span class = "nb" > print< / span > < span class = "p" > (< / span > < span class = "n" > iamzero< / span > < span class = "o" > ==< / span > < span class = "mf" > 0.0< / span > < span class = "p" > )< / span >
< span class = "o" > > < / span > < span class = "kc" > True< / span >
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< / code > < / pre > < / div >
< h2 id = "error-estimation" > Error estimation< / h2 >
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< p > The error estimation within < code > < a href = "" > pyerrors< / a > < / code > is based on the gamma method introduced in < a href = "https://arxiv.org/abs/hep-lat/0306017" > arXiv:hep-lat/0306017< / a > .
After having arrived at the derived quantity of interest the < code > gamma_method< / code > can be called as detailed in the following example.< / p >
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< div class = "codehilite" > < pre > < span > < / span > < code > < span class = "n" > my_sum< / span > < span class = "o" > .< / span > < span class = "n" > gamma_method< / span > < span class = "p" > ()< / span >
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< span class = "nb" > print< / span > < span class = "p" > (< / span > < span class = "n" > my_sum< / span > < span class = "p" > )< / span >
< span class = "o" > > < / span > < span class = "mf" > 1.70< / span > < span class = "p" > (< / span > < span class = "mi" > 57< / span > < span class = "p" > )< / span >
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< span class = "n" > my_sum< / span > < span class = "o" > .< / span > < span class = "n" > details< / span > < span class = "p" > ()< / span >
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< span class = "o" > > < / span > < span class = "n" > Result< / span > < span class = "mf" > 1.70000000e+00< / span > < span class = "o" > +/-< / span > < span class = "mf" > 5.72046658e-01< / span > < span class = "o" > +/-< / span > < span class = "mf" > 7.56746598e-02< / span > < span class = "p" > (< / span > < span class = "mf" > 33.650< / span > < span class = "o" > %< / span > < span class = "p" > )< / span >
< span class = "o" > > < / span > < span class = "n" > t_int< / span > < span class = "mf" > 2.71422900e+00< / span > < span class = "o" > +/-< / span > < span class = "mf" > 6.40320983e-01< / span > < span class = "n" > S< / span > < span class = "o" > =< / span > < span class = "mf" > 2.00< / span >
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< span class = "o" > > < / span > < span class = "mi" > 1000< / span > < span class = "n" > samples< / span > < span class = "ow" > in< / span > < span class = "mi" > 1< / span > < span class = "n" > ensemble< / span > < span class = "p" > :< / span >
< span class = "o" > > < / span > < span class = "err" > ·< / span > < span class = "n" > Ensemble< / span > < span class = "s1" > ' ensemble_name' < / span > < span class = "p" > :< / span > < span class = "mi" > 1000< / span > < span class = "n" > configurations< / span > < span class = "p" > (< / span > < span class = "kn" > from< / span > < span class = "mi" > 1< / span > < span class = "n" > to< / span > < span class = "mi" > 1000< / span > < span class = "p" > )< / span >
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< / code > < / pre > < / div >
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< p > We use the following definition of the integrated autocorrelation time established in < a href = "https://link.springer.com/article/10.1007/BF01022990" > Madras & Sokal 1988< / a >
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$$\tau_\mathrm{int}=\frac{1}{2}+\sum_{t=1}^{W}\rho(t)\geq \frac{1}{2}\,.$$
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The window $W$ is determined via the automatic windowing procedure described in < a href = "https://arxiv.org/abs/hep-lat/0306017" > arXiv:hep-lat/0306017< / a > .
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The standard value for the parameter $S$ of this automatic windowing procedure is $S=2$. Other values for $S$ can be passed to the < code > gamma_method< / code > as parameter.< / p >
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< div class = "codehilite" > < pre > < span > < / span > < code > < span class = "n" > my_sum< / span > < span class = "o" > .< / span > < span class = "n" > gamma_method< / span > < span class = "p" > (< / span > < span class = "n" > S< / span > < span class = "o" > =< / span > < span class = "mf" > 3.0< / span > < span class = "p" > )< / span >
< span class = "n" > my_sum< / span > < span class = "o" > .< / span > < span class = "n" > details< / span > < span class = "p" > ()< / span >
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< span class = "o" > > < / span > < span class = "n" > Result< / span > < span class = "mf" > 1.70000000e+00< / span > < span class = "o" > +/-< / span > < span class = "mf" > 6.30675201e-01< / span > < span class = "o" > +/-< / span > < span class = "mf" > 1.04585650e-01< / span > < span class = "p" > (< / span > < span class = "mf" > 37.099< / span > < span class = "o" > %< / span > < span class = "p" > )< / span >
< span class = "o" > > < / span > < span class = "n" > t_int< / span > < span class = "mf" > 3.29909703e+00< / span > < span class = "o" > +/-< / span > < span class = "mf" > 9.77310102e-01< / span > < span class = "n" > S< / span > < span class = "o" > =< / span > < span class = "mf" > 3.00< / span >
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< span class = "o" > > < / span > < span class = "mi" > 1000< / span > < span class = "n" > samples< / span > < span class = "ow" > in< / span > < span class = "mi" > 1< / span > < span class = "n" > ensemble< / span > < span class = "p" > :< / span >
< span class = "o" > > < / span > < span class = "err" > ·< / span > < span class = "n" > Ensemble< / span > < span class = "s1" > ' ensemble_name' < / span > < span class = "p" > :< / span > < span class = "mi" > 1000< / span > < span class = "n" > configurations< / span > < span class = "p" > (< / span > < span class = "kn" > from< / span > < span class = "mi" > 1< / span > < span class = "n" > to< / span > < span class = "mi" > 1000< / span > < span class = "p" > )< / span >
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< / code > < / pre > < / div >
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< p > The integrated autocorrelation time $\tau_\mathrm{int}$ and the autocorrelation function $\rho(W)$ can be monitored via the methods < code > < a href = "pyerrors/obs.html#Obs.plot_tauint" > pyerrors.obs.Obs.plot_tauint< / a > < / code > and < code > < a href = "pyerrors/obs.html#Obs.plot_tauint" > pyerrors.obs.Obs.plot_tauint< / a > < / code > .< / p >
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< p > If the parameter $S$ is set to zero it is assumed that dataset does not exhibit any autocorrelation and the windowsize is chosen to be zero.
In this case the error estimate is identical to the sample standard error.< / p >
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< h3 id = "exponential-tails" > Exponential tails< / h3 >
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< p > Slow modes in the Monte Carlo history can be accounted for by attaching an exponential tail to the autocorrelation function $\rho$ as suggested in < a href = "https://arxiv.org/abs/1009.5228" > arXiv:1009.5228< / a > . The longest autocorrelation time in the history, $\tau_\mathrm{exp}$, can be passed to the < code > gamma_method< / code > as parameter. In this case the automatic windowing procedure is vacated and the parameter $S$ does not affect the error estimate.< / p >
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< div class = "codehilite" > < pre > < span > < / span > < code > < span class = "n" > my_sum< / span > < span class = "o" > .< / span > < span class = "n" > gamma_method< / span > < span class = "p" > (< / span > < span class = "n" > tau_exp< / span > < span class = "o" > =< / span > < span class = "mf" > 7.2< / span > < span class = "p" > )< / span >
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< span class = "n" > my_sum< / span > < span class = "o" > .< / span > < span class = "n" > details< / span > < span class = "p" > ()< / span >
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< span class = "o" > > < / span > < span class = "n" > Result< / span > < span class = "mf" > 1.70000000e+00< / span > < span class = "o" > +/-< / span > < span class = "mf" > 6.28097762e-01< / span > < span class = "o" > +/-< / span > < span class = "mf" > 5.79077524e-02< / span > < span class = "p" > (< / span > < span class = "mf" > 36.947< / span > < span class = "o" > %< / span > < span class = "p" > )< / span >
< span class = "o" > > < / span > < span class = "n" > t_int< / span > < span class = "mf" > 3.27218667e+00< / span > < span class = "o" > +/-< / span > < span class = "mf" > 7.99583654e-01< / span > < span class = "n" > tau_exp< / span > < span class = "o" > =< / span > < span class = "mf" > 7.20< / span > < span class = "p" > ,< / span > < span class = "n" > N_sigma< / span > < span class = "o" > =< / span > < span class = "mi" > 1< / span >
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< span class = "o" > > < / span > < span class = "mi" > 1000< / span > < span class = "n" > samples< / span > < span class = "ow" > in< / span > < span class = "mi" > 1< / span > < span class = "n" > ensemble< / span > < span class = "p" > :< / span >
< span class = "o" > > < / span > < span class = "err" > ·< / span > < span class = "n" > Ensemble< / span > < span class = "s1" > ' ensemble_name' < / span > < span class = "p" > :< / span > < span class = "mi" > 1000< / span > < span class = "n" > configurations< / span > < span class = "p" > (< / span > < span class = "kn" > from< / span > < span class = "mi" > 1< / span > < span class = "n" > to< / span > < span class = "mi" > 1000< / span > < span class = "p" > )< / span >
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< / code > < / pre > < / div >
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< p > For the full API see < code > < a href = "pyerrors/obs.html#Obs.gamma_method" > pyerrors.obs.Obs.gamma_method< / a > < / code > .< / p >
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< h2 id = "multiple-ensemblesreplica" > Multiple ensembles/replica< / h2 >
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< p > Error propagation for multiple ensembles (Markov chains with different simulation parameters) is handled automatically. Ensembles are uniquely identified by their < code > name< / code > .< / p >
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< div class = "codehilite" > < pre > < span > < / span > < code > < span class = "n" > obs1< / span > < span class = "o" > =< / span > < span class = "n" > pe< / span > < span class = "o" > .< / span > < span class = "n" > Obs< / span > < span class = "p" > ([< / span > < span class = "n" > samples1< / span > < span class = "p" > ],< / span > < span class = "p" > [< / span > < span class = "s1" > ' ensemble1' < / span > < span class = "p" > ])< / span >
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< span class = "n" > obs2< / span > < span class = "o" > =< / span > < span class = "n" > pe< / span > < span class = "o" > .< / span > < span class = "n" > Obs< / span > < span class = "p" > ([< / span > < span class = "n" > samples2< / span > < span class = "p" > ],< / span > < span class = "p" > [< / span > < span class = "s1" > ' ensemble2' < / span > < span class = "p" > ])< / span >
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< span class = "n" > my_sum< / span > < span class = "o" > =< / span > < span class = "n" > obs1< / span > < span class = "o" > +< / span > < span class = "n" > obs2< / span >
< span class = "n" > my_sum< / span > < span class = "o" > .< / span > < span class = "n" > details< / span > < span class = "p" > ()< / span >
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< span class = "o" > > < / span > < span class = "n" > Result< / span > < span class = "mf" > 2.00697958e+00< / span >
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< span class = "o" > > < / span > < span class = "mi" > 1500< / span > < span class = "n" > samples< / span > < span class = "ow" > in< / span > < span class = "mi" > 2< / span > < span class = "n" > ensembles< / span > < span class = "p" > :< / span >
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< span class = "o" > > < / span > < span class = "err" > ·< / span > < span class = "n" > Ensemble< / span > < span class = "s1" > ' ensemble1' < / span > < span class = "p" > :< / span > < span class = "mi" > 1000< / span > < span class = "n" > configurations< / span > < span class = "p" > (< / span > < span class = "kn" > from< / span > < span class = "mi" > 1< / span > < span class = "n" > to< / span > < span class = "mi" > 1000< / span > < span class = "p" > )< / span >
< span class = "o" > > < / span > < span class = "err" > ·< / span > < span class = "n" > Ensemble< / span > < span class = "s1" > ' ensemble2' < / span > < span class = "p" > :< / span > < span class = "mi" > 500< / span > < span class = "n" > configurations< / span > < span class = "p" > (< / span > < span class = "kn" > from< / span > < span class = "mi" > 1< / span > < span class = "n" > to< / span > < span class = "mi" > 500< / span > < span class = "p" > )< / span >
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< / code > < / pre > < / div >
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< p > < code > < a href = "" > pyerrors< / a > < / code > identifies multiple replica (independent Markov chains with identical simulation parameters) by the vertical bar < code > |< / code > in the name of the data set.< / p >
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< div class = "codehilite" > < pre > < span > < / span > < code > < span class = "n" > obs1< / span > < span class = "o" > =< / span > < span class = "n" > pe< / span > < span class = "o" > .< / span > < span class = "n" > Obs< / span > < span class = "p" > ([< / span > < span class = "n" > samples1< / span > < span class = "p" > ],< / span > < span class = "p" > [< / span > < span class = "s1" > ' ensemble1|r01' < / span > < span class = "p" > ])< / span >
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< span class = "n" > obs2< / span > < span class = "o" > =< / span > < span class = "n" > pe< / span > < span class = "o" > .< / span > < span class = "n" > Obs< / span > < span class = "p" > ([< / span > < span class = "n" > samples2< / span > < span class = "p" > ],< / span > < span class = "p" > [< / span > < span class = "s1" > ' ensemble1|r02' < / span > < span class = "p" > ])< / span >
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< span class = "o" > > < / span > < span class = "n" > my_sum< / span > < span class = "o" > =< / span > < span class = "n" > obs1< / span > < span class = "o" > +< / span > < span class = "n" > obs2< / span >
< span class = "o" > > < / span > < span class = "n" > my_sum< / span > < span class = "o" > .< / span > < span class = "n" > details< / span > < span class = "p" > ()< / span >
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< span class = "o" > > < / span > < span class = "n" > Result< / span > < span class = "mf" > 2.00697958e+00< / span >
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< span class = "o" > > < / span > < span class = "mi" > 1500< / span > < span class = "n" > samples< / span > < span class = "ow" > in< / span > < span class = "mi" > 1< / span > < span class = "n" > ensemble< / span > < span class = "p" > :< / span >
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< span class = "o" > > < / span > < span class = "err" > ·< / span > < span class = "n" > Ensemble< / span > < span class = "s1" > ' ensemble1' < / span >
< span class = "o" > > < / span > < span class = "err" > ·< / span > < span class = "n" > Replicum< / span > < span class = "s1" > ' r01' < / span > < span class = "p" > :< / span > < span class = "mi" > 1000< / span > < span class = "n" > configurations< / span > < span class = "p" > (< / span > < span class = "kn" > from< / span > < span class = "mi" > 1< / span > < span class = "n" > to< / span > < span class = "mi" > 1000< / span > < span class = "p" > )< / span >
< span class = "o" > > < / span > < span class = "err" > ·< / span > < span class = "n" > Replicum< / span > < span class = "s1" > ' r02' < / span > < span class = "p" > :< / span > < span class = "mi" > 500< / span > < span class = "n" > configurations< / span > < span class = "p" > (< / span > < span class = "kn" > from< / span > < span class = "mi" > 1< / span > < span class = "n" > to< / span > < span class = "mi" > 500< / span > < span class = "p" > )< / span >
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< / code > < / pre > < / div >
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< h3 id = "error-estimation-for-multiple-ensembles" > Error estimation for multiple ensembles< / h3 >
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< p > In order to keep track of different error analysis parameters for different ensembles one can make use of global dictionaries as detailed in the following example.< / p >
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< div class = "codehilite" > < pre > < span > < / span > < code > < span class = "n" > pe< / span > < span class = "o" > .< / span > < span class = "n" > Obs< / span > < span class = "o" > .< / span > < span class = "n" > S_dict< / span > < span class = "p" > [< / span > < span class = "s1" > ' ensemble1' < / span > < span class = "p" > ]< / span > < span class = "o" > =< / span > < span class = "mf" > 2.5< / span >
< span class = "n" > pe< / span > < span class = "o" > .< / span > < span class = "n" > Obs< / span > < span class = "o" > .< / span > < span class = "n" > tau_exp_dict< / span > < span class = "p" > [< / span > < span class = "s1" > ' ensemble2' < / span > < span class = "p" > ]< / span > < span class = "o" > =< / span > < span class = "mf" > 8.0< / span >
< span class = "n" > pe< / span > < span class = "o" > .< / span > < span class = "n" > Obs< / span > < span class = "o" > .< / span > < span class = "n" > tau_exp_dict< / span > < span class = "p" > [< / span > < span class = "s1" > ' ensemble3' < / span > < span class = "p" > ]< / span > < span class = "o" > =< / span > < span class = "mf" > 2.0< / span >
< / code > < / pre > < / div >
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< p > In case the < code > gamma_method< / code > is called without any parameters it will use the values specified in the dictionaries for the respective ensembles.
Passing arguments to the < code > gamma_method< / code > still dominates over the dictionaries.< / p >
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< h2 id = "irregular-monte-carlo-chains" > Irregular Monte Carlo chains< / h2 >
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< p > Irregular Monte Carlo chains can be initialized with the parameter < code > idl< / code > .< / p >
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< div class = "codehilite" > < pre > < span > < / span > < code > < span class = "c1" > # Observable defined on configurations 20 to 519< / span >
< span class = "n" > obs1< / span > < span class = "o" > =< / span > < span class = "n" > pe< / span > < span class = "o" > .< / span > < span class = "n" > Obs< / span > < span class = "p" > ([< / span > < span class = "n" > samples1< / span > < span class = "p" > ],< / span > < span class = "p" > [< / span > < span class = "s1" > ' ensemble1' < / span > < span class = "p" > ],< / span > < span class = "n" > idl< / span > < span class = "o" > =< / span > < span class = "p" > [< / span > < span class = "nb" > range< / span > < span class = "p" > (< / span > < span class = "mi" > 20< / span > < span class = "p" > ,< / span > < span class = "mi" > 520< / span > < span class = "p" > )])< / span >
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< span class = "n" > obs1< / span > < span class = "o" > .< / span > < span class = "n" > details< / span > < span class = "p" > ()< / span >
< span class = "o" > > < / span > < span class = "n" > Result< / span > < span class = "mf" > 9.98319881e-01< / span >
< span class = "o" > > < / span > < span class = "mi" > 500< / span > < span class = "n" > samples< / span > < span class = "ow" > in< / span > < span class = "mi" > 1< / span > < span class = "n" > ensemble< / span > < span class = "p" > :< / span >
< span class = "o" > > < / span > < span class = "err" > ·< / span > < span class = "n" > Ensemble< / span > < span class = "s1" > ' ensemble1' < / span > < span class = "p" > :< / span > < span class = "mi" > 500< / span > < span class = "n" > configurations< / span > < span class = "p" > (< / span > < span class = "kn" > from< / span > < span class = "mi" > 20< / span > < span class = "n" > to< / span > < span class = "mi" > 519< / span > < span class = "p" > )< / span >
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< span class = "c1" > # Observable defined on every second configuration between 5 and 1003< / span >
< span class = "n" > obs2< / span > < span class = "o" > =< / span > < span class = "n" > pe< / span > < span class = "o" > .< / span > < span class = "n" > Obs< / span > < span class = "p" > ([< / span > < span class = "n" > samples2< / span > < span class = "p" > ],< / span > < span class = "p" > [< / span > < span class = "s1" > ' ensemble1' < / span > < span class = "p" > ],< / span > < span class = "n" > idl< / span > < span class = "o" > =< / span > < span class = "p" > [< / span > < span class = "nb" > range< / span > < span class = "p" > (< / span > < span class = "mi" > 5< / span > < span class = "p" > ,< / span > < span class = "mi" > 1005< / span > < span class = "p" > ,< / span > < span class = "mi" > 2< / span > < span class = "p" > )])< / span >
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< span class = "n" > obs2< / span > < span class = "o" > .< / span > < span class = "n" > details< / span > < span class = "p" > ()< / span >
< span class = "o" > > < / span > < span class = "n" > Result< / span > < span class = "mf" > 9.99100712e-01< / span >
< span class = "o" > > < / span > < span class = "mi" > 500< / span > < span class = "n" > samples< / span > < span class = "ow" > in< / span > < span class = "mi" > 1< / span > < span class = "n" > ensemble< / span > < span class = "p" > :< / span >
< span class = "o" > > < / span > < span class = "err" > ·< / span > < span class = "n" > Ensemble< / span > < span class = "s1" > ' ensemble1' < / span > < span class = "p" > :< / span > < span class = "mi" > 500< / span > < span class = "n" > configurations< / span > < span class = "p" > (< / span > < span class = "kn" > from< / span > < span class = "mi" > 5< / span > < span class = "n" > to< / span > < span class = "mi" > 1003< / span > < span class = "ow" > in< / span > < span class = "n" > steps< / span > < span class = "n" > of< / span > < span class = "mi" > 2< / span > < span class = "p" > )< / span >
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< span class = "c1" > # Observable defined on configurations 2, 9, 28, 29 and 501< / span >
< span class = "n" > obs3< / span > < span class = "o" > =< / span > < span class = "n" > pe< / span > < span class = "o" > .< / span > < span class = "n" > Obs< / span > < span class = "p" > ([< / span > < span class = "n" > samples3< / span > < span class = "p" > ],< / span > < span class = "p" > [< / span > < span class = "s1" > ' ensemble1' < / span > < span class = "p" > ],< / span > < span class = "n" > idl< / span > < span class = "o" > =< / span > < span class = "p" > [[< / span > < span class = "mi" > 2< / span > < span class = "p" > ,< / span > < span class = "mi" > 9< / span > < span class = "p" > ,< / span > < span class = "mi" > 28< / span > < span class = "p" > ,< / span > < span class = "mi" > 29< / span > < span class = "p" > ,< / span > < span class = "mi" > 501< / span > < span class = "p" > ]])< / span >
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< span class = "n" > obs3< / span > < span class = "o" > .< / span > < span class = "n" > details< / span > < span class = "p" > ()< / span >
< span class = "o" > > < / span > < span class = "n" > Result< / span > < span class = "mf" > 1.01718064e+00< / span >
< span class = "o" > > < / span > < span class = "mi" > 5< / span > < span class = "n" > samples< / span > < span class = "ow" > in< / span > < span class = "mi" > 1< / span > < span class = "n" > ensemble< / span > < span class = "p" > :< / span >
< span class = "o" > > < / span > < span class = "err" > ·< / span > < span class = "n" > Ensemble< / span > < span class = "s1" > ' ensemble1' < / span > < span class = "p" > :< / span > < span class = "mi" > 5< / span > < span class = "n" > configurations< / span > < span class = "p" > (< / span > < span class = "n" > irregular< / span > < span class = "nb" > range< / span > < span class = "p" > )< / span >
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< / code > < / pre > < / div >
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< p > < strong > Warning:< / strong > Irregular Monte Carlo chains can result in odd patterns in the autocorrelation functions.
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Make sure to check the autocorrelation time with e.g. < code > < a href = "pyerrors/obs.html#Obs.plot_rho" > pyerrors.obs.Obs.plot_rho< / a > < / code > or < code > < a href = "pyerrors/obs.html#Obs.plot_tauint" > pyerrors.obs.Obs.plot_tauint< / a > < / code > .< / p >
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< p > For the full API see < code > < a href = "pyerrors/obs.html#Obs" > pyerrors.obs.Obs< / a > < / code > .< / p >
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< h1 id = "correlators" > Correlators< / h1 >
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< p > For the full API see < code > < a href = "pyerrors/correlators.html#Corr" > pyerrors.correlators.Corr< / a > < / code > .< / p >
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< h1 id = "complex-observables" > Complex observables< / h1 >
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< p > < code > < a href = "" > pyerrors< / a > < / code > can handle complex valued observables via the class < code > < a href = "pyerrors/obs.html#CObs" > pyerrors.obs.CObs< / a > < / code > .
< code > CObs< / code > are initialized with a real and an imaginary part which both can be < code > Obs< / code > valued.< / p >
< div class = "codehilite" > < pre > < span > < / span > < code > < span class = "n" > my_real_part< / span > < span class = "o" > =< / span > < span class = "n" > pe< / span > < span class = "o" > .< / span > < span class = "n" > Obs< / span > < span class = "p" > ([< / span > < span class = "n" > samples1< / span > < span class = "p" > ],< / span > < span class = "p" > [< / span > < span class = "s1" > ' ensemble1' < / span > < span class = "p" > ])< / span >
< span class = "n" > my_imag_part< / span > < span class = "o" > =< / span > < span class = "n" > pe< / span > < span class = "o" > .< / span > < span class = "n" > Obs< / span > < span class = "p" > ([< / span > < span class = "n" > samples2< / span > < span class = "p" > ],< / span > < span class = "p" > [< / span > < span class = "s1" > ' ensemble1' < / span > < span class = "p" > ])< / span >
< span class = "n" > my_cobs< / span > < span class = "o" > =< / span > < span class = "n" > pe< / span > < span class = "o" > .< / span > < span class = "n" > CObs< / span > < span class = "p" > (< / span > < span class = "n" > my_real_part< / span > < span class = "p" > ,< / span > < span class = "n" > my_imag_part< / span > < span class = "p" > )< / span >
< span class = "n" > my_cobs< / span > < span class = "o" > .< / span > < span class = "n" > gamma_method< / span > < span class = "p" > ()< / span >
< span class = "nb" > print< / span > < span class = "p" > (< / span > < span class = "n" > my_cobs< / span > < span class = "p" > )< / span >
< span class = "o" > > < / span > < span class = "p" > (< / span > < span class = "mf" > 0.9959< / span > < span class = "p" > (< / span > < span class = "mi" > 91< / span > < span class = "p" > )< / span > < span class = "o" > +< / span > < span class = "mf" > 0.659< / span > < span class = "p" > (< / span > < span class = "mi" > 28< / span > < span class = "p" > )< / span > < span class = "n" > j< / span > < span class = "p" > )< / span >
< / code > < / pre > < / div >
< p > Elementary mathematical operations are overloaded and samples are properly propagated as for the < code > Obs< / code > class.< / p >
< div class = "codehilite" > < pre > < span > < / span > < code > < span class = "n" > my_derived_cobs< / span > < span class = "o" > =< / span > < span class = "p" > (< / span > < span class = "n" > my_cobs< / span > < span class = "o" > +< / span > < span class = "n" > my_cobs< / span > < span class = "o" > .< / span > < span class = "n" > conjugate< / span > < span class = "p" > ())< / span > < span class = "o" > /< / span > < span class = "n" > np< / span > < span class = "o" > .< / span > < span class = "n" > abs< / span > < span class = "p" > (< / span > < span class = "n" > my_cobs< / span > < span class = "p" > )< / span >
< span class = "n" > my_derived_cobs< / span > < span class = "o" > .< / span > < span class = "n" > gamma_method< / span > < span class = "p" > ()< / span >
< span class = "nb" > print< / span > < span class = "p" > (< / span > < span class = "n" > my_derived_cobs< / span > < span class = "p" > )< / span >
< span class = "o" > > < / span > < span class = "p" > (< / span > < span class = "mf" > 1.668< / span > < span class = "p" > (< / span > < span class = "mi" > 23< / span > < span class = "p" > )< / span > < span class = "o" > +< / span > < span class = "mf" > 0.0< / span > < span class = "n" > j< / span > < span class = "p" > )< / span >
< / code > < / pre > < / div >
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< h1 id = "optimization-fits-roots" > Optimization / fits / roots< / h1 >
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< p > < code > < a href = "pyerrors/fits.html" > pyerrors.fits< / a > < / code >
< code > < a href = "pyerrors/roots.html" > pyerrors.roots< / a > < / code > < / p >
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< h1 id = "matrix-operations" > Matrix operations< / h1 >
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< p > < code > < a href = "pyerrors/linalg.html" > pyerrors.linalg< / a > < / code > < / p >
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< h1 id = "export-data" > Export data< / h1 >
< p > The preferred exported file format within < code > < a href = "" > pyerrors< / a > < / code > is< / p >
< h2 id = "jackknife-samples" > Jackknife samples< / h2 >
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< p > For comparison with other analysis workflows < code > < a href = "" > pyerrors< / a > < / code > can generate jackknife samples from an < code > Obs< / code > object or import jackknife samples into an < code > Obs< / code > object.
See < code > < a href = "pyerrors/obs.html#Obs.export_jackknife" > pyerrors.obs.Obs.export_jackknife< / a > < / code > and < code > < a href = "pyerrors/obs.html#import_jackknife" > pyerrors.obs.import_jackknife< / a > < / code > for details.< / p >
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< h1 id = "input" > Input< / h1 >
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< p > < code > < a href = "pyerrors/input.html" > pyerrors.input< / a > < / code > < / p >
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< / div >
< details >
< summary > View Source< / summary >
< div class = "codehilite" > < pre > < span > < / span > < span class = "sa" > r< / span > < span class = "sd" > ' ' ' < / span >
< span class = "sd" > # What is pyerrors?< / span >
< span class = "sd" > `pyerrors` is a python package for error computation and propagation of Markov chain Monte Carlo data.< / span >
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< span class = "sd" > It is based on the gamma method [arXiv:hep-lat/0306017](https://arxiv.org/abs/hep-lat/0306017). Some of its features are:< / span >
< span class = "sd" > - automatic differentiation for exact liner error propagation as suggested in [arXiv:1809.01289](https://arxiv.org/abs/1809.01289) (partly based on the [autograd](https://github.com/HIPS/autograd) package).< / span >
< span class = "sd" > - treatment of slow modes in the simulation as suggested in [arXiv:1009.5228](https://arxiv.org/abs/1009.5228).< / span >
< span class = "sd" > - coherent error propagation for data from different Markov chains.< / span >
< span class = "sd" > - non-linear fits with x- and y-errors and exact linear error propagation based on automatic differentiation as introduced in [arXiv:1809.01289](https://arxiv.org/abs/1809.01289).< / span >
< span class = "sd" > - real and complex matrix operations and their error propagation based on automatic differentiation (Matrix inverse, Cholesky decomposition, calculation of eigenvalues and eigenvectors, singular value decomposition...).< / span >
< span class = "sd" > There exist similar publicly available implementations of gamma method error analysis suites in [Fortran](https://gitlab.ift.uam-csic.es/alberto/aderrors), [Julia](https://gitlab.ift.uam-csic.es/alberto/aderrors.jl) and [Python](https://github.com/mbruno46/pyobs).< / span >
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< span class = "sd" > ## Basic example< / span >
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< span class = "sd" > ```python< / span >
< span class = "sd" > import numpy as np< / span >
< span class = "sd" > import pyerrors as pe< / span >
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< span class = "sd" > my_obs = pe.Obs([samples], [' ensemble_name' ]) # Initialize an Obs object< / span >
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< span class = "sd" > my_new_obs = 2 * np.log(my_obs) / my_obs ** 2 # Construct derived Obs object< / span >
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< span class = "sd" > my_new_obs.gamma_method() # Estimate the statistical error< / span >
< span class = "sd" > print(my_new_obs) # Print the result to stdout< / span >
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< span class = "sd" > > 0.31498(72)< / span >
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< span class = "sd" > ```< / span >
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< span class = "sd" > # The `Obs` class< / span >
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< span class = "sd" > `pyerrors` introduces a new datatype, `Obs`, which simplifies error propagation and estimation for auto- and cross-correlated data.< / span >
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< span class = "sd" > An `Obs` object can be initialized with two arguments, the first is a list containing the samples for an Observable from a Monte Carlo chain.< / span >
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< span class = "sd" > The samples can either be provided as python list or as numpy array.< / span >
< span class = "sd" > The second argument is a list containing the names of the respective Monte Carlo chains as strings. These strings uniquely identify a Monte Carlo chain/ensemble.< / span >
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< span class = "sd" > ```python< / span >
< span class = "sd" > import pyerrors as pe< / span >
< span class = "sd" > my_obs = pe.Obs([samples], [' ensemble_name' ])< / span >
< span class = "sd" > ```< / span >
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< span class = "sd" > ## Error propagation< / span >
< span class = "sd" > When performing mathematical operations on `Obs` objects the correct error propagation is intrinsically taken care using a first order Taylor expansion< / span >
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< span class = "sd" > $$\delta_f^i=\sum_\alpha \bar{f}_\alpha \delta_\alpha^i\,,\quad \delta_\alpha^i=a_\alpha^i-\bar{a}_\alpha\,,$$< / span >
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< span class = "sd" > as introduced in [arXiv:hep-lat/0306017](https://arxiv.org/abs/hep-lat/0306017).< / span >
< span class = "sd" > The required derivatives $\bar{f}_\alpha$ are evaluated up to machine precision via automatic differentiation as suggested in [arXiv:1809.01289](https://arxiv.org/abs/1809.01289).< / span >
< span class = "sd" > The `Obs` class is designed such that mathematical numpy functions can be used on `Obs` just as for regular floats.< / span >
< span class = "sd" > ```python< / span >
< span class = "sd" > import numpy as np< / span >
< span class = "sd" > import pyerrors as pe< / span >
< span class = "sd" > my_obs1 = pe.Obs([samples1], [' ensemble_name' ])< / span >
< span class = "sd" > my_obs2 = pe.Obs([samples2], [' ensemble_name' ])< / span >
< span class = "sd" > my_sum = my_obs1 + my_obs2< / span >
< span class = "sd" > my_m_eff = np.log(my_obs1 / my_obs2)< / span >
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< span class = "sd" > iamzero = my_m_eff - my_m_eff< / span >
< span class = "sd" > # Check that value and fluctuations are zero within machine precision< / span >
< span class = "sd" > print(iamzero == 0.0)< / span >
< span class = "sd" > > True< / span >
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< span class = "sd" > ```< / span >
< span class = "sd" > ## Error estimation< / span >
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< span class = "sd" > The error estimation within `pyerrors` is based on the gamma method introduced in [arXiv:hep-lat/0306017](https://arxiv.org/abs/hep-lat/0306017).< / span >
< span class = "sd" > After having arrived at the derived quantity of interest the `gamma_method` can be called as detailed in the following example.< / span >
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< span class = "sd" > ```python< / span >
< span class = "sd" > my_sum.gamma_method()< / span >
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< span class = "sd" > print(my_sum)< / span >
< span class = "sd" > > 1.70(57)< / span >
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< span class = "sd" > my_sum.details()< / span >
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< span class = "sd" > > Result 1.70000000e+00 +/- 5.72046658e-01 +/- 7.56746598e-02 (33.650%)< / span >
< span class = "sd" > > t_int 2.71422900e+00 +/- 6.40320983e-01 S = 2.00< / span >
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< span class = "sd" > > 1000 samples in 1 ensemble:< / span >
< span class = "sd" > > · Ensemble ' ensemble_name' : 1000 configurations (from 1 to 1000)< / span >
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< span class = "sd" > ```< / span >
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< span class = "sd" > We use the following definition of the integrated autocorrelation time established in [Madras & Sokal 1988](https://link.springer.com/article/10.1007/BF01022990)< / span >
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< span class = "sd" > $$\tau_\mathrm{int}=\frac{1}{2}+\sum_{t=1}^{W}\rho(t)\geq \frac{1}{2}\,.$$< / span >
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< span class = "sd" > The window $W$ is determined via the automatic windowing procedure described in [arXiv:hep-lat/0306017](https://arxiv.org/abs/hep-lat/0306017).< / span >
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< span class = "sd" > The standard value for the parameter $S$ of this automatic windowing procedure is $S=2$. Other values for $S$ can be passed to the `gamma_method` as parameter.< / span >
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< span class = "sd" > ```python< / span >
< span class = "sd" > my_sum.gamma_method(S=3.0)< / span >
< span class = "sd" > my_sum.details()< / span >
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< span class = "sd" > > Result 1.70000000e+00 +/- 6.30675201e-01 +/- 1.04585650e-01 (37.099%)< / span >
< span class = "sd" > > t_int 3.29909703e+00 +/- 9.77310102e-01 S = 3.00< / span >
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< span class = "sd" > > 1000 samples in 1 ensemble:< / span >
< span class = "sd" > > · Ensemble ' ensemble_name' : 1000 configurations (from 1 to 1000)< / span >
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< span class = "sd" > ```< / span >
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< span class = "sd" > The integrated autocorrelation time $\tau_\mathrm{int}$ and the autocorrelation function $\rho(W)$ can be monitored via the methods `pyerrors.obs.Obs.plot_tauint` and `pyerrors.obs.Obs.plot_tauint`.< / span >
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< span class = "sd" > If the parameter $S$ is set to zero it is assumed that dataset does not exhibit any autocorrelation and the windowsize is chosen to be zero.< / span >
< span class = "sd" > In this case the error estimate is identical to the sample standard error.< / span >
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< span class = "sd" > ### Exponential tails< / span >
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< span class = "sd" > Slow modes in the Monte Carlo history can be accounted for by attaching an exponential tail to the autocorrelation function $\rho$ as suggested in [arXiv:1009.5228](https://arxiv.org/abs/1009.5228). The longest autocorrelation time in the history, $\tau_\mathrm{exp}$, can be passed to the `gamma_method` as parameter. In this case the automatic windowing procedure is vacated and the parameter $S$ does not affect the error estimate.< / span >
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< span class = "sd" > ```python< / span >
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< span class = "sd" > my_sum.gamma_method(tau_exp=7.2)< / span >
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< span class = "sd" > my_sum.details()< / span >
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< span class = "sd" > > Result 1.70000000e+00 +/- 6.28097762e-01 +/- 5.79077524e-02 (36.947%)< / span >
< span class = "sd" > > t_int 3.27218667e+00 +/- 7.99583654e-01 tau_exp = 7.20, N_sigma = 1< / span >
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< span class = "sd" > > 1000 samples in 1 ensemble:< / span >
< span class = "sd" > > · Ensemble ' ensemble_name' : 1000 configurations (from 1 to 1000)< / span >
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< span class = "sd" > ```< / span >
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< span class = "sd" > For the full API see `pyerrors.obs.Obs.gamma_method`.< / span >
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< span class = "sd" > ## Multiple ensembles/replica< / span >
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< span class = "sd" > Error propagation for multiple ensembles (Markov chains with different simulation parameters) is handled automatically. Ensembles are uniquely identified by their `name`.< / span >
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< span class = "sd" > ```python< / span >
< span class = "sd" > obs1 = pe.Obs([samples1], [' ensemble1' ])< / span >
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< span class = "sd" > obs2 = pe.Obs([samples2], [' ensemble2' ])< / span >
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< span class = "sd" > my_sum = obs1 + obs2< / span >
< span class = "sd" > my_sum.details()< / span >
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< span class = "sd" > > Result 2.00697958e+00< / span >
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< span class = "sd" > > 1500 samples in 2 ensembles:< / span >
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< span class = "sd" > > · Ensemble ' ensemble1' : 1000 configurations (from 1 to 1000)< / span >
< span class = "sd" > > · Ensemble ' ensemble2' : 500 configurations (from 1 to 500)< / span >
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< span class = "sd" > ```< / span >
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< span class = "sd" > `pyerrors` identifies multiple replica (independent Markov chains with identical simulation parameters) by the vertical bar `|` in the name of the data set.< / span >
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< span class = "sd" > ```python< / span >
< span class = "sd" > obs1 = pe.Obs([samples1], [' ensemble1|r01' ])< / span >
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< span class = "sd" > obs2 = pe.Obs([samples2], [' ensemble1|r02' ])< / span >
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< span class = "sd" > > my_sum = obs1 + obs2< / span >
< span class = "sd" > > my_sum.details()< / span >
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< span class = "sd" > > Result 2.00697958e+00< / span >
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< span class = "sd" > > 1500 samples in 1 ensemble:< / span >
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< span class = "sd" > > · Ensemble ' ensemble1' < / span >
< span class = "sd" > > · Replicum ' r01' : 1000 configurations (from 1 to 1000)< / span >
< span class = "sd" > > · Replicum ' r02' : 500 configurations (from 1 to 500)< / span >
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< span class = "sd" > ```< / span >
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< span class = "sd" > ### Error estimation for multiple ensembles< / span >
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< span class = "sd" > In order to keep track of different error analysis parameters for different ensembles one can make use of global dictionaries as detailed in the following example.< / span >
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< span class = "sd" > ```python< / span >
< span class = "sd" > pe.Obs.S_dict[' ensemble1' ] = 2.5< / span >
< span class = "sd" > pe.Obs.tau_exp_dict[' ensemble2' ] = 8.0< / span >
< span class = "sd" > pe.Obs.tau_exp_dict[' ensemble3' ] = 2.0< / span >
< span class = "sd" > ```< / span >
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< span class = "sd" > In case the `gamma_method` is called without any parameters it will use the values specified in the dictionaries for the respective ensembles.< / span >
< span class = "sd" > Passing arguments to the `gamma_method` still dominates over the dictionaries.< / span >
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< span class = "sd" > ## Irregular Monte Carlo chains< / span >
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< span class = "sd" > Irregular Monte Carlo chains can be initialized with the parameter `idl`.< / span >
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< span class = "sd" > ```python< / span >
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< span class = "sd" > # Observable defined on configurations 20 to 519< / span >
< span class = "sd" > obs1 = pe.Obs([samples1], [' ensemble1' ], idl=[range(20, 520)])< / span >
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< span class = "sd" > obs1.details()< / span >
< span class = "sd" > > Result 9.98319881e-01< / span >
< span class = "sd" > > 500 samples in 1 ensemble:< / span >
< span class = "sd" > > · Ensemble ' ensemble1' : 500 configurations (from 20 to 519)< / span >
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< span class = "sd" > # Observable defined on every second configuration between 5 and 1003< / span >
< span class = "sd" > obs2 = pe.Obs([samples2], [' ensemble1' ], idl=[range(5, 1005, 2)])< / span >
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< span class = "sd" > obs2.details()< / span >
< span class = "sd" > > Result 9.99100712e-01< / span >
< span class = "sd" > > 500 samples in 1 ensemble:< / span >
< span class = "sd" > > · Ensemble ' ensemble1' : 500 configurations (from 5 to 1003 in steps of 2)< / span >
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< span class = "sd" > # Observable defined on configurations 2, 9, 28, 29 and 501< / span >
< span class = "sd" > obs3 = pe.Obs([samples3], [' ensemble1' ], idl=[[2, 9, 28, 29, 501]])< / span >
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< span class = "sd" > obs3.details()< / span >
< span class = "sd" > > Result 1.01718064e+00< / span >
< span class = "sd" > > 5 samples in 1 ensemble:< / span >
< span class = "sd" > > · Ensemble ' ensemble1' : 5 configurations (irregular range)< / span >
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< span class = "sd" > ```< / span >
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< span class = "sd" > **Warning:** Irregular Monte Carlo chains can result in odd patterns in the autocorrelation functions.< / span >
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< span class = "sd" > Make sure to check the autocorrelation time with e.g. `pyerrors.obs.Obs.plot_rho` or `pyerrors.obs.Obs.plot_tauint`.< / span >
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< span class = "sd" > For the full API see `pyerrors.obs.Obs`.< / span >
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< span class = "sd" > # Correlators< / span >
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< span class = "sd" > For the full API see `pyerrors.correlators.Corr`.< / span >
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< span class = "sd" > # Complex observables< / span >
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< span class = "sd" > `pyerrors` can handle complex valued observables via the class `pyerrors.obs.CObs`.< / span >
< span class = "sd" > `CObs` are initialized with a real and an imaginary part which both can be `Obs` valued.< / span >
< span class = "sd" > ```python< / span >
< span class = "sd" > my_real_part = pe.Obs([samples1], [' ensemble1' ])< / span >
< span class = "sd" > my_imag_part = pe.Obs([samples2], [' ensemble1' ])< / span >
< span class = "sd" > my_cobs = pe.CObs(my_real_part, my_imag_part)< / span >
< span class = "sd" > my_cobs.gamma_method()< / span >
< span class = "sd" > print(my_cobs)< / span >
< span class = "sd" > > (0.9959(91)+0.659(28)j)< / span >
< span class = "sd" > ```< / span >
< span class = "sd" > Elementary mathematical operations are overloaded and samples are properly propagated as for the `Obs` class.< / span >
< span class = "sd" > ```python< / span >
< span class = "sd" > my_derived_cobs = (my_cobs + my_cobs.conjugate()) / np.abs(my_cobs)< / span >
< span class = "sd" > my_derived_cobs.gamma_method()< / span >
< span class = "sd" > print(my_derived_cobs)< / span >
< span class = "sd" > > (1.668(23)+0.0j)< / span >
< span class = "sd" > ```< / span >
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< span class = "sd" > # Optimization / fits / roots< / span >
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< span class = "sd" > `pyerrors.fits`< / span >
< span class = "sd" > `pyerrors.roots`< / span >
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< span class = "sd" > # Matrix operations< / span >
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< span class = "sd" > `pyerrors.linalg`< / span >
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< span class = "sd" > # Export data< / span >
< span class = "sd" > The preferred exported file format within `pyerrors` is< / span >
< span class = "sd" > ## Jackknife samples< / span >
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< span class = "sd" > For comparison with other analysis workflows `pyerrors` can generate jackknife samples from an `Obs` object or import jackknife samples into an `Obs` object.< / span >
< span class = "sd" > See `pyerrors.obs.Obs.export_jackknife` and `pyerrors.obs.import_jackknife` for details.< / span >
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< span class = "sd" > # Input< / span >
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< span class = "sd" > `pyerrors.input`< / span >
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< span class = "sd" > ' ' ' < / span >
< span class = "kn" > from< / span > < span class = "nn" > .obs< / span > < span class = "kn" > import< / span > < span class = "o" > *< / span >
< span class = "kn" > from< / span > < span class = "nn" > .correlators< / span > < span class = "kn" > import< / span > < span class = "o" > *< / span >
< span class = "kn" > from< / span > < span class = "nn" > .fits< / span > < span class = "kn" > import< / span > < span class = "o" > *< / span >
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< span class = "kn" > from< / span > < span class = "nn" > .misc< / span > < span class = "kn" > import< / span > < span class = "o" > *< / span >
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< span class = "kn" > from< / span > < span class = "nn" > .< / span > < span class = "kn" > import< / span > < span class = "n" > dirac< / span >
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< span class = "kn" > from< / span > < span class = "nn" > .< / span > < span class = "kn" > import< / span > < span class = "nb" > input< / span >
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< span class = "kn" > from< / span > < span class = "nn" > .< / span > < span class = "kn" > import< / span > < span class = "n" > linalg< / span >
< span class = "kn" > from< / span > < span class = "nn" > .< / span > < span class = "kn" > import< / span > < span class = "n" > mpm< / span >
< span class = "kn" > from< / span > < span class = "nn" > .< / span > < span class = "kn" > import< / span > < span class = "n" > roots< / span >
< span class = "kn" > from< / span > < span class = "nn" > .version< / span > < span class = "kn" > import< / span > < span class = "n" > __version__< / span >
< / pre > < / div >
< / details >
< / section >
< / main >
< script >
function escapeHTML(html) {
return document.createElement('div').appendChild(document.createTextNode(html)).parentNode.innerHTML;
}
const originalContent = document.querySelector("main.pdoc");
let currentContent = originalContent;
function setContent(innerHTML) {
let elem;
if (innerHTML) {
elem = document.createElement("main");
elem.classList.add("pdoc");
elem.innerHTML = innerHTML;
} else {
elem = originalContent;
}
if (currentContent !== elem) {
currentContent.replaceWith(elem);
currentContent = elem;
}
}
function getSearchTerm() {
return (new URL(window.location)).searchParams.get("search");
}
const searchBox = document.querySelector(".pdoc input[type=search]");
searchBox.addEventListener("input", function () {
let url = new URL(window.location);
if (searchBox.value.trim()) {
url.hash = "";
url.searchParams.set("search", searchBox.value);
} else {
url.searchParams.delete("search");
}
history.replaceState("", "", url.toString());
onInput();
});
window.addEventListener("popstate", onInput);
let search, searchErr;
async function initialize() {
try {
search = await new Promise((resolve, reject) => {
const script = document.createElement("script");
script.type = "text/javascript";
script.async = true;
script.onload = () => resolve(window.pdocSearch);
script.onerror = (e) => reject(e);
script.src = "search.js";
document.getElementsByTagName("head")[0].appendChild(script);
});
} catch (e) {
console.error("Cannot fetch pdoc search index");
searchErr = "Cannot fetch search index.";
}
onInput();
document.querySelector("nav.pdoc").addEventListener("click", e => {
if (e.target.hash) {
searchBox.value = "";
searchBox.dispatchEvent(new Event("input"));
}
});
}
function onInput() {
setContent((() => {
const term = getSearchTerm();
if (!term) {
return null
}
if (searchErr) {
return `< h3 > Error: ${searchErr}< / h3 > `
}
if (!search) {
return "< h3 > Searching...< / h3 > "
}
window.scrollTo({top: 0, left: 0, behavior: 'auto'});
const results = search(term);
let html;
if (results.length === 0) {
html = `No search results for '${escapeHTML(term)}'.`
} else {
html = `< h4 > ${results.length} search result${results.length > 1 ? "s" : ""} for '${escapeHTML(term)}'.< / h4 > `;
}
for (let result of results.slice(0, 10)) {
let doc = result.doc;
let url = `${doc.modulename.replaceAll(".", "/")}.html`;
if (doc.qualname) {
url += `#${doc.qualname}`;
}
let heading;
switch (result.doc.type) {
case "function":
heading = `< span class = "def" > ${doc.funcdef}< / span > < span class = "name" > ${doc.fullname}< / span > < span class = "signature" > (${doc.parameters.join(", ")})< / span > `;
break;
case "class":
heading = `< span class = "def" > class< / span > < span class = "name" > ${doc.fullname}< / span > `;
break;
default:
heading = `< span class = "name" > ${doc.fullname}< / span > `;
break;
}
html += `
< section class = "search-result" >
< a href = "${url}" class = "attr ${doc.type}" > ${heading}< / a >
< div class = "docstring" > ${doc.doc}< / div >
< / section >
`;
}
return html;
})());
}
if (getSearchTerm()) {
initialize();
searchBox.value = getSearchTerm();
onInput();
} else {
searchBox.addEventListener("focus", initialize, {once: true});
}
searchBox.addEventListener("keydown", e => {
if (["ArrowDown", "ArrowUp", "Enter"].includes(e.key)) {
let focused = currentContent.querySelector(".search-result.focused");
if (!focused) {
currentContent.querySelector(".search-result").classList.add("focused");
} else if (
e.key === "ArrowDown"
& & focused.nextElementSibling
& & focused.nextElementSibling.classList.contains("search-result")
) {
focused.classList.remove("focused");
focused.nextElementSibling.classList.add("focused");
focused.nextElementSibling.scrollIntoView({
behavior: "smooth",
block: "nearest",
inline: "nearest"
});
} else if (
e.key === "ArrowUp"
& & focused.previousElementSibling
& & focused.previousElementSibling.classList.contains("search-result")
) {
focused.classList.remove("focused");
focused.previousElementSibling.classList.add("focused");
focused.previousElementSibling.scrollIntoView({
behavior: "smooth",
block: "nearest",
inline: "nearest"
});
} else if (
e.key === "Enter"
) {
focused.querySelector("a").click();
}
}
});
< / script > < / body >
< / html >
