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Classification Trees and Regression Trees

On this page…

What Are Classification Trees and Regression Trees?

Example: Creating a Classification Tree

Example: Creating a Regression Tree

Viewing a Tree

How the Fit Methods Create Trees

redicting Responses With Classification Trees and Regression Trees

!mpro"ing Classification Trees and Regression Trees

Alternati"e: classregtree

What Are Classification Trees and Regression Trees?

Classification trees and regression trees predict responses to data# To predict a response$follow the decisions in the tree from the root %&eginning' node down to a leaf node# The leafnode contains the response# Classification trees gi"e responses that are nominal$ s(ch as'true' or 'false' # Regression trees gi"e n(meric responses#

)tatistics Tool&ox trees are &inar*# Each step in a prediction in"ol"es chec+ing the "al(e ofone predictor %"aria&le'# For example$ here is a simple classification tree:

This tree predicts classifications &ased on two predictors$ x1 and x2 # To predict$ start at thetop node$ represented &* a triangle %,'# The first decision is whether x1 is smaller than 0.5 # !fso$ follow the left &ranch$ and see that the tree classifies the data as t*pe 0#

!f$ howe"er$ x1 exceeds 0.5 $ then follow the right &ranch to the lower-right triangle node#Here the tree as+s if x2 is smaller than 0.5 # !f so$ then follow the left &ranch to see that thetree classifies the data as t*pe 0# !f not$ then follow the right &ranch to see that the that thetree classifies the data as t*pe 1#

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To learn how to prepare *o(r data for classification or regression (sing decision trees$ see)teps in )(per"ised .earning %Machine .earning' #

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Example: Creating a Classification Tree

To create a classification tree for the ionosphere data:

load ionosphere % contains X and Y variablesctree = ClassificationTree.fit X!Y"

ctree =

ClassificationTree# $redictor a&es# 1x() cell* Cate+orical$redictors# ,-

esponse a&e# 'Y' Class a&es# 'b' '+'* /coreTransfor&# 'none' bservations# (51

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Example: Creating a Regression Tree

To create a regression tree for the cars&all data &ased on the orsepo er and 3ei+ht "ectors for data$ and 4$ "ector for response:

load cars&all % contains orsepo er! 3ei+ht! 4$X = , orsepo er 3ei+ht-6rtree = e+ressionTree.fit X!4$ "

rtree =

e+ressionTree# $redictor a&es# 'x1' 'x2'* Cate+orical$redictors# ,- esponse a&e# 'Y' esponseTransfor&# 'none' bservations# 7)

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Viewing a Tree

There are two wa*s to "iew a tree:

• vie tree" ret(rns a text description of the tree#• vie tree!'&ode'!'+raph'" ret(rns a graphic description of the tree#

Example: Creating a Classification Tree has the following two "iews:

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load fisheririsctree = ClassificationTree.fit &eas!species"6vie ctree"

8ecision tree for classification1 if x(92.)5 then node 2 elseif x(:=2.)5 then node ( else setosa2 class = setosa( if x)91.;5 then node ) elseif x):=1.;5 then node 5 else versicolor) if x(9).75 then node < elseif x(:=).75 then node ; else versicolor5 class = vir+inica< if x)91.<5 then node elseif x):=1.<5 then node 7 else versicolor; class = vir+inica

class = versicolor7 class = vir+inica

vie ctree!'&ode'!'+raph'"

)imilarl*$ Example: Creating a Regression Tree has the following two "iews:

load cars&all % contains orsepo er! 3ei+ht! 4$X = , orsepo er 3ei+ht-6rtree = e+ressionTree.fit X!4$ !'4in$arent'!(0"6vie rtree"

8ecision tree for re+ression1 if x29(0 5.5 then node 2 elseif x2:=(0 5.5 then node ( else 2(.;1 12 if x19 7 then node ) elseif x1:= 7 then node 5 else 2 .;7(1( if x19115 then node < elseif x1:=115 then node ; else 15.5)1;) if x2921<2 then node elseif x2:=21<2 then node 7 else (0.7(;55 fit = 2).0 2< fit = 17.<25; fit = 1).(;5

fit = ((.(05<7 fit = 27

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vie rtree!'&ode'!'+raph'"

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How the it !ethods Create Trees

The ClassificationTree.fit and e+ressionTree.fit methods perform the followingsteps to create decision trees:

0# )tart with all inp(t data$ and examine all possi&le &inar* splits on e"er* predictor#1# )elect a split with &est optimi2ation criterion#

• !f the split leads to a child node ha"ing too few o&ser"ations %less than the4in>eaf parameter'$ select a split with the &est optimi2ation criterion s(&3ectto the 4in>eaf constraint#

4# !mpose the split#5# Repeat rec(rsi"el* for the two child nodes#

The explanation re6(ires two more items: description of the optimi2ation criterion$ andstopping r(le#

"topping r#le: )top splitting when an* of the following hold:

• The node is pure #o For classification$ a node is p(re if it contains onl* o&ser"ations of one class#

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o For regression$ a node is p(re if the mean s6(ared error %M)E' for theo&ser"ed response in this node drops &elow the M)E for the o&ser"edresponse in the entire data m(ltiplied &* the tolerance on 6(adratic error pernode %?etoler parameter'#

•

There are fewer than 4in$arent o&ser"ations in this node#• An* split imposed on this node wo(ld prod(ce children with fewer than 4in>eaf

o&ser"ations#

Optimi$ation criterion:

• Regression: mean-s6(ared error %M)E'# Choose a split to minimi2e the M)E of predictions compared to the training data#

• Classification: 7ne of three meas(res$ depending on the setting of the/plitCriterion name-"al(e pair:

o '+di' %8ini9s di"ersit* index$ the defa(lt'

o 't oin+'

o 'deviance'

For details$ see ClassificationTree efinitions #

For a contin(o(s predictor$ a tree can split halfwa* &etween an* two ad3acent (ni6(e "al(esfo(nd for this predictor# For a categorical predictor with L le"els$ a classification tree needs toconsider 1 L ;0 ;0 splits# To o&tain this form(la$ o&ser"e that *o( can assign L distinct "al(es to

the left and right nodes in 1 L wa*s# Two o(t of these 1 L config(rations wo(ld lea"e either leftor right node empt*$ and therefore sho(ld &e discarded# <ow di"ide &* 1 &eca(se left andright can &e swapped# A classification tree can th(s process onl* categorical predictors with amoderate n(m&er of le"els# A regression tree emplo*s a comp(tational shortc(t: it sorts thele"els &* the o&ser"ed mean response$ and considers onl* the L ;0 splits &etween the sortedle"els#

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%redicting Responses With Classification Trees and Regression Trees

After creating a tree$ *o( can easil* predict responses for new data# )(ppose Xne is new datathat has the same n(m&er of col(mns as the original data X# To predict the classification orregression &ased on the tree and the new data$ enter

Yne = predict tree!Xne "6

For each row of data in Xne $predict r(ns thro(gh the decisions in tree and gi"es theres(lting prediction in the corresponding element of Yne # For more information forclassification$ see the classification predict reference page= for regression$ see the regressionpredict reference page#

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For example$ to find the predicted classification of a point at the mean of the ionosphere data:

load ionosphere % contains X and Y variablesctree = ClassificationTree.fit X!Y"6Yne = predict ctree!&ean X""

Yne ='+'

To find the predicted 4$ of a point at the mean of the cars&all data:

load cars&all % contains orsepo er! 3ei+ht! 4$X = , orsepo er 3ei+ht-6rtree = e+ressionTree.fit X!4$ "6Yne = predict rtree!&ean X""

Yne = 2 .;7(1

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&mpro'ing Classification Trees and Regression Trees

>o( can t(ne trees &* setting name-"al(e pairs in ClassificationTree.fit ande+ressionTree.fit # The remainder of this section descri&es how to determine the 6(alit*

of a tree$ how to decide which name-"al(e pairs to set$ and how to control the si2e of a tree:

• Examining Res(&stit(tion Error• Cross Validation

• Control epth or .eafiness

• r(ning

Examining Res#(stit#tion Error

Resubstitution error is the difference &etween the response training data and the predictionsthe tree ma+es of the response &ased on the inp(t training data# !f the res(&stit(tion error ishigh$ *o( cannot expect the predictions of the tree to &e good# Howe"er$ ha"ing lowres(&stit(tion error does not g(arantee good predictions for new data# Res(&stit(tion error isoften an o"erl* optimistic estimate of the predicti"e error on new data#

Example: Res#(stit#tion Error of a Classification Tree) Examine the res(&stit(tion errorof a defa(lt classification tree for the Fisher iris data:

load fisheririsctree = ClassificationTree.fit &eas!species"6resuberror = resub>oss ctree"

resuberror = 0.0200

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The tree classifies nearl* all the Fisher iris data correctl*#

Cross Validation

To get a &etter sense of the predicti"e acc(rac* of *o(r tree for new data$ cross "alidate the

tree# /* defa(lt$ cross "alidation splits the training data into 0@ parts at random# !t trains 0@new trees$ each one on nine parts of the data# !t then examines the predicti"e acc(rac* of eachnew tree on the data not incl(ded in training that tree# This method gi"es a good estimate ofthe predicti"e acc(rac* of the res(lting tree$ since it tests the new trees on new data#

Example: Cross Validating a Regression Tree) Examine the res(&stit(tion and cross-"alidation acc(rac* of a regression tree for predicting mileage &ased on the cars&all data:

load cars&allX = ,@cceleration 8isplace&ent orsepo er 3ei+ht-6rtree = e+ressionTree.fit X!4$ "6

resuberror = resub>oss rtree"resuberror = ).;1

The res(&stit(tion loss for a regression tree is the mean-s6(ared error# The res(lting "al(eindicates that a t*pical predicti"e error for the tree is a&o(t the s6(are root of 5# $ or a &it o"er1#

<ow calc(late the error &* cross "alidating the tree:

cvrtree = crossval rtree"6cvloss = Afold>oss cvrtree"

cvloss = 2(.) 0

The cross-"alidated loss is almost 1B$ meaning a t*pical predicti"e error for the tree on newdata is a&o(t B# This demonstrates that cross-"alidated loss is (s(all* higher than simpleres(&stit(tion loss#

Control *epth or +,eafiness+

When *o( grow a decision tree$ consider its simplicit* and predicti"e power# A deep treewith man* lea"es is (s(all* highl* acc(rate on the training data# Howe"er$ the tree is notg(aranteed to show a compara&le acc(rac* on an independent test set# A leaf* tree tends too"ertrain$ and its test acc(rac* is often far less than its training %res(&stit(tion' acc(rac*# !ncontrast$ a shallow tree does not attain high training acc(rac*# /(t a shallow tree can &e morero&(st its training acc(rac* co(ld &e close to that of a representati"e test set# Also$ ashallow tree is eas* to interpret#

!f *o( do not ha"e eno(gh data for training and test$ estimate tree acc(rac* &* cross"alidation#

For an alternati"e method of controlling the tree depth$ see r(ning #

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Example: "electing Appropriate Tree *epth) This example shows how to control thedepth of a decision tree$ and how to choose an appropriate depth#

1. .oad the ionosphere data:

load ionosphere2. 8enerate minim(m leaf occ(pancies for classification trees from 10 to 100 $ spaced

exponentiall* apart:

leafs = lo+space 1!2!10"63. Create cross "alidated classification trees for the ionosphere data with minim(m leaf

occ(pancies from leafs :). = nu&el leafs"65. err = Beros !1"6<. for n=1#;. t = ClassificationTree.fit X!Y!'crossval'!'on'!...

. '&inleaf'!leafs n""67. err n" = Afold>oss t"610. end11. plot leafs!err"612. xlabel '4in >eaf /iBe'"6

label 'crossDvalidated error'"6

The &est leaf si2e is &etween a&o(t 20 and 50 o&ser"ations per leaf#

13. Compare the near-optimal tree with at least )0 o&ser"ations per leaf with the defa(lttree$ which (ses 10 o&ser"ations per parent node and 1 o&ser"ation per leaf#

1). 8efaultTree = ClassificationTree.fit X!Y"6vie 8efaultTree!'&ode'!'+raph'"

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pti&alTree = ClassificationTree.fit X!Y!'&inleaf'!)0"6vie pti&alTree!'&ode'!'+raph'"

resub pt = resub>oss pti&alTree"6loss pt = Afold>oss crossval pti&alTree""6resub8efault = resub>oss 8efaultTree"6loss8efault = Afold>oss crossval 8efaultTree""6resub pt!resub8efault!loss pt!loss8efault

resub pt = 0.0 (

resub8efault = 0.011)

loss pt = 0.105)

loss8efault = 0.102<

The near-optimal tree is m(ch smaller and gi"es a m(ch higher res(&stit(tion error#>et it gi"es similar acc(rac* for cross-"alidated data#

%r#ning

r(ning optimi2es tree depth %leafiness' is &* merging lea"es on the same tree &ranch#Control epth or .eafiness descri&es one method for selecting the optimal depth for a tree#

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Dnli+e in that section$ *o( do not need to grow a new tree for e"er* node si2e# !nstead$ growa deep tree$ and pr(ne it to the le"el *o( choose#

r(ne a tree at the command line (sing the prune method %classification' or prune method%regression'# Alternati"el*$ pr(ne a tree interacti"el* with the tree "iewer:

vie tree!'&ode'!'+raph'"

To pr(ne a tree$ the tree m(st contain a pr(ning se6(ence# /* defa(lt$ &othClassificationTree.fit and e+ressionTree.fit calc(late a pr(ning se6(ence for a treed(ring constr(ction# !f *o( constr(ct a tree with the '$rune' name-"al(e pair set to 'off' $ orif *o( pr(ne a tree to a smaller le"el$ the tree does not contain the f(ll pr(ning se6(ence#8enerate the f(ll pr(ning se6(ence with the prune method %classification' or prune method%regression'#

Example: %r#ning a Classification Tree) This example creates a classification tree for theionosphere data$ and pr(nes it to a good le"el#

1. .oad the ionosphere data:

load ionosphere1# Constr(ct a defa(lt classification tree for the data:

tree = ClassificationTree.fit X!Y"64# View the tree in the interacti"e "iewer:

vie tree!'&ode'!'+raph'"

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The pr(ned tree is the same as the near-optimal tree in Example: )electingAppropriate Tree epth #

10. )et 'treesiBe' to 'se' %defa(lt' to find the maximal pr(ning le"el for which the treeerror does not exceed the error from the &est le"el pl(s one standard de"iation:

11. ,E!E!E!bestlevel- = cv>oss tree!'subtrees'!'all'"12.1(. bestlevel =

<

!n this case the le"el is the same for either setting of 'treesiBe' #

05# r(ne the tree to (se it for other p(rposes:15. tree = prune tree!'>evel'!<"6

vie tree!'&ode'!'+raph'"

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Alternati'e: classregtree

The ClassificationTree and e+ressionTree classes are new in MAT.A/ R1@00a#re"io(sl*$ *o( represented &oth classification trees and regression trees with a

classre+tree o&3ect# The new classes pro"ide all the f(nctionalit* of the classre+tree class$ and are more con"enient when (sed in con3(nction with Ensem&le Methods #

/efore the classre+tree class$ there were treefit $treedisp $treeval $treeprune $ andtreetest f(nctions# )tatistics Tool&ox software maintains these onl* for &ac+wardcompati&ilit*#

Example: Creating Classification Trees -sing classregtree

This example (ses Fisher9s iris data in fisheriris.&at to create a classification tree for predicting species (sing meas(rements of sepal length$ sepal width$ petal length$ and petalwidth as predictors# Here$ the predictors are contin(o(s and the response is categorical#

1. .oad the data and (se the classre+tree constr(ctor of the classre+tree class tocreate the classification tree:

2. load fisheriris(.

). t = classre+tree &eas!species!...5. 'na&es'! '/>' '/3' '$>' '$3'*"<. t =;. 8ecision tree for classification

. 1 if $>92.)5 then node 2 elseif $>:=2.)5 then node ( else setosa7. 2 class = setosa10. ( if $391.;5 then node ) elseif $3:=1.;5 then node 5 else

versicolor11. ) if $>9).75 then node < elseif $>:=).75 then node ; else

versicolor12. 5 class = vir+inica1(. < if $391.<5 then node elseif $3:=1.<5 then node 7 else

versicolor

1). ; class = vir+inica15. class = versicolor7 class = vir+inica

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t is a classre+tree o&3ect and can &e operated on with an* class method#

16. Dse the t pe method of the classre+tree class to show the t*pe of the tree:1;. treet pe = t pe t"1 . treet pe =17. classification

classre+tree creates a classification tree &eca(se species is a cell arra* of strings$and the response is ass(med to &e categorical#

20. To "iew the tree$ (se the vie method of the classre+tree class:

vie t"

The tree predicts the response "al(es at the circ(lar leaf nodes &ased on a series of6(estions a&o(t the iris at the triang(lar &ranching nodes# A true answer to an*6(estion follows the &ranch to the left# A false follows the &ranch to the right#

21. The tree does not (se sepal meas(rements for predicting species# These can go(nmeas(red in new data$ and *o( can enter them as a "al(es for predictions# Forexample$ to (se the tree to predict the species of an iris with petal length ). and petalwidth 1.< $ t*pe:

22. predicted = t , a a ). 1.<-"2(. predicted =

'versicolor'

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The o&3ect allows for f(nctional e"al(ation$ of the form t X" # This is a shorthand wa*of calling the eval method of the classre+tree class# The predicted species is theleft leaf node at the &ottom of the tree in the pre"io(s "iew#

24. >o( can (se a "ariet* of methods of the classre+tree class$ s(ch as cutvar andcutt pe to get more information a&o(t the split at node that ma+es the finaldistinction &etween versicolor and vir+inica :

25. var< = cutvar t!<" % 3hat variable deter&ines the splitF2<. var< =2;. '$3'2 .27. t pe< = cutt pe t!<" % 3hat t pe of split is itF(0. t pe< =(1. 'continuous'32. Classification trees fit the original %training' data well$ &(t can do a poor 3o& of

classif*ing new "al(es# .ower &ranches$ especiall*$ can &e strongl* affected &*o(tliers# A simpler tree often a"oids o"erfitting# >o( can (se the prune method of theclassre+tree class to find the next largest tree from an optimal pr(ning se6(ence:

((. pruned = prune t!'level'!1"(). pruned =(5. 8ecision tree for classification(<. 1 if $>92.)5 then node 2 elseif $>:=2.)5 then node ( else setosa(;. 2 class = setosa( . ( if $391.;5 then node ) elseif $3:=1.;5 then node 5 else

versicolor(7. ) if $>9).75 then node < elseif $>:=).75 then node ; else

versicolor)0. 5 class = vir+inica)1. < class = versicolor)2. ; class = vir+inica)(.

vie pruned"

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To find the &est classification tree$ emplo*ing the techni6(es of res(&stit(tion and cross"alidation$ (se the test method of the classre+tree class#

Example: Creating Regression Trees -sing classregtree

This example (ses the data on cars in cars&all.&at to create a regression tree for predictingmileage (sing meas(rements of weight and the n(m&er of c*linders as predictors# Here$ one

predictor %weight' is contin(o(s and the other %c*linders' is categorical# The response%mileage' is contin(o(s#

1. .oad the data and (se the classre+tree constr(ctor of the classre+tree class tocreate the regression tree:

2. load cars&all(.). t = classre+tree ,3ei+ht! C linders-!4$ !...5. 'cat'!2!'split&in'!20!...<. 'na&es'! '3'!'C'*";.

. t =7.10. 8ecision tree for re+ression11. 1 if 39(0 5.5 then node 2 elseif 3:=(0 5.5 then node ( else

2(.;1 112. 2 if 392(;1 then node ) elseif 3:=2(;1 then node 5 else 2 .;7(11(. ( if C= then node < elseif C in ) <* then node ; else 15.5)1;1). ) if 3921<2 then node elseif 3:=21<2 then node 7 else (2.0;)115. 5 if C=< then node 10 elseif C=) then node 11 else 25.7(551<. < if 39)( 1 then node 12 elseif 3:=)( 1 then node 1( else 1).27<(1;. ; fit = 17.2;;

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1 . fit = ((.(05<17. 7 fit = 27.<11120. 10 fit = 2(.2521. 11 if 392 2;.5 then node 1) elseif 3:=2 2;.5 then node 15 else

2;.21)(22. 12 if 39(5((.5 then node 1< elseif 3:=(5((.5 then node 1; else

1). <7<2(. 1( fit = 112). 1) fit = 2;.<( 725. 15 fit = 2).<<<;2<. 1< fit = 1<.<

1; fit = 1).( 7

t is a classre+tree o&3ect and can &e operated on with an* of the methods of theclass#

27. Dse the t pe method of the classre+tree class to show the t*pe of the tree:2 . treet pe = t pe t"

27. treet pe =(0. re+ression

classre+tree creates a regression tree &eca(se 4$ is a n(merical "ector$ and theresponse is ass(med to &e contin(o(s#

31. To "iew the tree$ (se the vie method of the classre+tree class:

vie t"

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&ntrod#ction to !A.OVA

The anal*sis of "ariance techni6(e in Example: 7ne-Wa* A<7VA ta+es a set of gro(peddata and determine whether the mean of a "aria&le differs significantl* among gro(ps# 7ftenthere are m(ltiple response "aria&les$ and *o( are interested in determining whether the entireset of means is different from one gro(p to the next# There is a m(lti"ariate "ersion ofanal*sis of "ariance that can address the pro&lem#

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A.OVA with !#ltiple Responses

The cars&all data set has meas(rements on a "ariet* of car models from the *ears 0G @$0G $ and 0G 1# )(ppose *o( are interested in whether the characteristics of the cars ha"echanged o"er time#

First$ load the data#

load cars&allhos

a&e /iBe H tes Class @cceleration 100x1 00 double arra C linders 100x1 00 double arra 8isplace&ent 100x1 00 double arra orsepo er 100x1 00 double arra 4$ 100x1 00 double arra 4odel 100x(< ;200 char arra 4odelIYear 100x1 00 double arra

ri+in 100x; 1)00 char arra 3ei+ht 100x1 00 double arra

Fo(r of these "aria&les % @cceleration $8isplace&ent $ orsepo er $ and 4$ ' arecontin(o(s meas(rements on indi"id(al car models# The "aria&le 4odelIYear indicates the*ear in which the car was made# >o( can create a gro(ped plot matrix of these "aria&les(sing the +plot&atrix f(nction#

x = ,4$ orsepo er 8isplace&ent 3ei+ht-6+plot&atrix x!,-!4odelIYear!,-!'Jxo'"

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%When the second arg(ment of +plot&atrix is empt*$ the f(nction graphs the col(mns ofthe x arg(ment against each other$ and places histograms along the diagonals# The empt*fo(rth arg(ment prod(ces a graph with the defa(lt colors# The fifth arg(ment controls thes*m&ols (sed to disting(ish &etween gro(ps#'

!t appears the cars do differ from *ear to *ear# The (pper right plot$ for example$ is a graph of4$ "ers(s 3ei+ht # The 0G 1 cars appear to ha"e higher mileage than the older cars$ and the*

appear to weigh less on a"erage# /(t as a gro(p$ are the three *ears significantl* differentfrom one another? The &anova1 f(nction can answer that 6(estion#

,d!p!stats- = &anova1 x!4odelIYear"d = 2p = 1.0eD00< K 0 0.11)1stats =

3# ,)x) double- H# ,)x) double-

T# ,)x) double- df3# 70 dfH# 2 dfT# 72 la&bda# ,2x1 double- chis?# ,2x1 double- chis?df# ,2x1 double- ei+enval# ,)x1 double- ei+envec# ,)x) double- canon# ,100x) double- &dist# ,100x1 double- +&dist# ,(x( double-

The &anova1 f(nction prod(ces three o(tp(ts:

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• The first o(tp(t$ d$ is an estimate of the dimension of the gro(p means# !f the meanswere all the same$ the dimension wo(ld &e @$ indicating that the means are at the same

point# !f the means differed &(t fell along a line$ the dimension wo(ld &e 0# !n theexample the dimension is 1$ indicating that the gro(p means fall in a plane &(t notalong a line# This is the largest possi&le dimension for the means of three gro(ps#

• The second o(tp(t$ p$ is a "ector of p-"al(es for a se6(ence of tests# The first p "al(etests whether the dimension is @$ the next whether the dimension is 0$ and so on# !nthis case &oth p-"al(es are small# That9s wh* the estimated dimension is 1#

• The third o(tp(t$ stats $ is a str(ct(re containing se"eral fields$ descri&ed in thefollowing section#

The ields of the stats "tr#ct#re

The 3$H$ and T fields are matrix analogs to the within$ &etween$ and total s(ms of s6(ares inordinar* one-wa* anal*sis of "ariance# The next three fields are the degrees of freedom forthese matrices# Fields la&bda $chis? $ and chis?df are the ingredients of the test for thedimensionalit* of the gro(p means# %The p-"al(es for these tests are the first o(tp(t arg(mentof &anova1 #'

The next three fields are (sed to do a canonical anal*sis# Recall that in principal componentsanal*sis % rincipal Component Anal*sis % CA' ' *o( loo+ for the com&ination of the original"aria&les that has the largest possi&le "ariation# !n m(lti"ariate anal*sis of "ariance$ *o(instead loo+ for the linear com&ination of the original "aria&les that has the largest separation

&etween gro(ps# !t is the single "aria&le that wo(ld gi"e the most significant res(lt in a(ni"ariate one-wa* anal*sis of "ariance# Ha"ing fo(nd that com&ination$ *o( next loo+ for

the com&ination with the second highest separation$ and so on#

The ei+envec field is a matrix that defines the coefficients of the linear com&inations of theoriginal "aria&les# The ei+enval field is a "ector meas(ring the ratio of the &etween-gro(p"ariance to the within-gro(p "ariance for the corresponding linear com&ination# The canon field is a matrix of the canonical "aria&le "al(es# Each col(mn is a linear com&ination of themean-centered original "aria&les$ (sing coefficients from the ei+envec matrix#

A gro(ped scatter plot of the first two canonical "aria&les shows more separation &etweengro(ps then a gro(ped scatter plot of an* pair of original "aria&les# !n this example it showsthree clo(ds of points$ o"erlapping &(t with distinct centers# 7ne point in the &ottom right sitsapart from the others# /* (sing the +na&e f(nction$ *o( can see that this is the 1@th point#

c1 = stats.canon #!1"6c2 = stats.canon #!2"6+scatter c2!c1!4odelIYear!,-!'oxs'"+na&e

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atten(ation in the high fre6(enc* components# !f discontin(ities exist in the << inter"alseries$ either &eca(se of the presence of a&normal &eats or &eca(se of gaps or extreme noisein the original EC8 recording$ traditional approaches re6(ire either discarding the data org(esswor+ to estimate the locations of missing normal &eats# To eliminate the need for e"enl*sampled data re6(ired &* Fo(rier or maxim(m entrop* methods$ fre6(enc* domain spectra

can &e calc(lated (sing the .om& periodogram for (ne"enl* sampled data I $ J %the method(sed in this tool+it'#

Altho(gh the long term %15-ho(r' statistics of ) A<<$ ) <<! L and D.F power can &ecalc(lated for shorter data lengths$ the* will &ecome increasingl* (nrelia&le# For short-termdata %less than 0B min(tes in length'$ onl* the time domain meas(res of AV<<$ ) <<$rM)) and p<<B@ and the fre6(enc* domain meas(res of total power$ V.F power$ HF

power and .F HF ratio sho(ld &e (sed#

A n(m&er of the HRV meas(res are highl* correlated with each other# These incl(de ) <<$) A<<$ total power and D.F power= ) <<! L$ V.F power and .F power= and rM)) $

p<<B@ and HF power# The .F HF ratio does not correlate strongl* with an* other HRVmeas(res I5J#

Heart rate "aria&ilit* %HRV' has &een widel* applied in &asic and clinical research st(dies# !tsclinical application is "er* limited at present$ howe"er# These limitations are d(e to lac+ ofstandardi2ation of methodolog* and application to different non-compara&le s(&sets ofs(&3ects$ as well as to the confo(nding effects of age$ gender$ dr(gs$ health stat(s$ andchrono&iologic "ariations$ among others# F(rthermore$ o(tliers d(e to ectop* and artifact canha"e ma3or effects on comp(ted HRV "al(es# !n elderl* s(&3ects$ especiall*$ a sp(rio(sl*high "al(e of certain meas(res ma* &e d(e to the effects of erratic s(pra"entric(lar rh*thmIGJ d(e to s(&tle atrial ectop*$ wandering atrial pacema+er$ or sin(s node cond(ctiona&normalities# Additional information on heart rate d*namics and anal*sis techni6(es$incl(ding non-linear and complexit* &ased meas(res$ can &e fo(nd in the HRV 1@@ co(rsenotes and elsewhere on h*sio<et %see for example: etrended Fl(ct(ation Anal*sis $M(ltiscale Entrop* Anal*sis $ and !nformation-/ased )imilarit* $ among others'#

h*sio<et9s HRV Tool+it$ a"aila&le here$ is a rigoro(sl* "alidated pac+age of open so(rcesoftware for HRV anal*sis$ incl(ding "is(ali2ation of << inter"al time series$ a(tomatedo(tlier remo"al$ and calc(lation of the &asic time- and fre6(enc*-domain HRV statisticswidel* (sed in the literat(re$ incl(ding all of those listed in the ta&les &elow#

)e"eral other high-6(alit*$ freel* a"aila&le HRV tool+its ma* also &e of interest toresearchers= lin+s to them are pro"ided at the end of this page#

Ta(le 1: Commonl2 #sed time3domain meas#res

AVNN * Average of all NN intervals

SDNN * Standard deviation of all NN intervals

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SDANNStandard deviation of the averages of NN intervals in all 5 !in"teseg!ents of a 24 ho"r re#ording

SDNN$D%

&ean of the standard deviations of NN intervals in all 5 !in"teseg!ents of a 24 ho"r re#ording

r&SSD * S'"are root of the !ean of the s'"ares of differen#es (et)eenad a#ent NN intervals

+NN50 * ,er#entage of differen#es (et)een ad a#ent NN intervals that aregreater than 50 !s- a !e!(er of the larger +NN fa!il/ 6

N )hort-term HRV statistics

Ta(le 4: Commonl2 #sed fre5#enc23domain meas#res

, * otal s+e#tral +o)er of all NN intervals "+ to 0.04

8 : otal s+e#tral +o)er of all NN intervals "+ to 0.003

V : * otal s+e#tral +o)er of all NN intervals (et)een 0.003 and 0.04

: * otal s+e#tral +o)er of all NN intervals (et)een 0.04 and 0.15 .

: * otal s+e#tral +o)er of all NN intervals (et)een 0.15 and 0.4

:; : * atio of lo) to high fre'"en#/ +o)er

N )hort-term HRV statistics %V.F O spectral power &etween @ and @#@5 H2#'

"elected References:

I0J Wolf MM$ Varigos 8A$ H(nt $ )loman P8# )in(s arrh*thmia in ac(te m*ocardialinfarction# Med P A(st 0G =1:B1-B4#

I1J Qleiger RE$ Miller P $ /igger PT$ Moss AP$ and the M(lticenter ost-!nfarction Research8ro(p# ecreased heart rate "aria&ilit* and its association with increased mortalit* after ac(te

m*ocardial infarction# Am P Cardiol 0G =BG:1B -1 1#

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I4J Tas+ Force of the E(ropean )ociet* of Cardiolog* and the <orth American )ociet* ofacing and Electroph*siolog*# Heart rate "aria&ilit*: )tandards of meas(rement$

ph*siological interpretation$ and clinical (se# Circ(lation 0GG = G4:0@54-0@ B#

I5J Miet(s PE# Time domain meas(res: from "ariance to p<<x #

http: ph*sionet#org e"ents hr"-1@@ miet(s-0#pdf

IBJ arati 8$ Mancia 8$ i Rien2o M$ Castiglioni $ Ta*lor PA$ )t(dinger # oint-Co(nterpoint: Cardio"asc(lar "aria&ilit* is is not an index of a(tonomic control ofcirc(lation# P Appl h*siol 1@@ = 0@0: - 1#

I J Miet(s PE$ eng C-Q$ Henr* !$ 8oldsmith R.$ 8old&erger A.# The p<<x-files:Reexamining a widel*-(sed heart rate "aria&ilit* meas(re# Heart 1@@1= :4 -4 @#

I J ress WH$ Te(+ols+* )A$ Vetterling WT$ Flanner* / # <(merical Recipes in C: The Artof )cientific Comp(ting$ 1nd ed# Cam&ridge Dni"# ress$ 0GG1$ pp# B B-B 5#

I J Mood* 8/# )pectral anal*sis of heart rate witho(t resampling # Comp(ters in Cardiolog*0GG4= 0B- 0 #

IGJ )tein Q$ >ane2 $ omitro"ich $ 8ottdiener P$ Cha"es $ Qronmal R$ Ra(tahar3( #Heart rate "aria&ilit* is confo(nded &* the presence of erratic sin(s rh*thm# Comp(ters inCardiolog* 1@@1= G- 1#

&&) O(taining the HRV Tool0it

The HRV Tool0it: Contents

The HRV Tool+it consists of a Lsa+es file$ a 4aAefile $ and the following programs:

pltIrrs a shell s#ri+t for +lotting ;NN interval series

+etIhrv a shell s#ri+t for #al#"lating ti!e and fre'"en#/ do!ain V statisti#s

Scripts above use the programs below, and others from the WFDB and plt packages

rrlist.c < #ode for e tra#ting an interval list fro! an annotation file

filt.c < #ode for filtering ;NN intervals

filtnn.c < #ode for filtering ;NN intervals

statnn.c < #ode for #al#"lating ti!e do!ain statisti#s

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p r.c< #ode for #al#"lating +o)er in "+ to 10 fre'"en#/ (ands fro! a +o)ers+e#tr"!

seconds.c < #ode for #onverting hh=!!=ss to se#onds

hours.c < #ode for #onverting se#onds to hh=!!=ss

pltIrrs and +etIhrv are the main scripts (sed in calc(lating HRV as ill(strated &elow#These two scripts call the "ario(s C programs to accomplish their calc(lations# The C

programs can &e (sed separatel* and their (sage and "ario(s options can &e fo(nd in Dsages $or &* r(nning an* of these programs with the Dh option#

*ownloading and &nstalling the HRV Tool0it!nstalling the HRV tool+it is eas*:

1. n indo)s> install the free </g)in soft)are first- see o"r t"torial for details. </g)inin#l"des the +cc < #o!+iler and all other "tilities needed to ("ild and r"n the#o!+onents of the V ool?it on indo)s. Start a </g)in @ter!inal e!"lator

)indo) and "se it for all of the re!aining ste+s.2. $nstall the free :DB and +lt soft)are +a#?ages.

3. he V tool?it is availa(le as a tar(all of so"r#es @for all +latfor!s or as tar(alls of+re("ilt (inaries for CN8; in" @ 6 > &a# S % @ 6 >Solaris @S,A < > or

indo)s;</g)in . Do)nload the version of /o"r #hoi#e.

4. 8n+a#? the V tool?it tar(all /o" do)nloaded. @See the ,h/sioNet :AE forinfor!ation on "n+a#?ing tar(alls .

5. $f /o" do)nloaded the so"r#es> enter the so"r#e dire#tor/ @ M.src and #o!+ileand install the tool?it (/ t/+ing=

<. &aAe install

$f /o" do)nloaded the (inaries> !ove the #ontents of the M dire#tor/ into so!edire#tor/ in /o"r $@T @or add the M dire#tor/ to /o"r $@T . he (inaries re'"ire

the sa!e additional +a#?ages as the so"r#e distri("tion @ 3N8H>plt > and @onindo)s </g)in .

&&&) -sing the HRV tool0it

-ser interface

he V tool?it does not in#l"de a gra+hi#al "ser interfa#e. $ts #o!+onents are #o!!andline tools that !"st (e r"n fro! a ter!inal )indo) @"nder &S indo)s> a </g)in )indo)or (/ a shell s#ri+t. @Fven pltIrrs !"st (e started fro! the #o!!and line or a s#ri+t>altho"gh it +rod"#es gra+hi#al o"t+"t.

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&np#t data format

/oth pltIrrs %for plotting the RR << inter"al series' and +etIhrv %for calc(lating the HRVstatistics' can ta+e as inp(t either a h*sio/an+-compati&le &eat annotation file or a text filecontaining an RR inter"al list# RR inter"al lists can &e in an* of fo(r formats:

• 3 #ol"!ns @ > > A• 2 #ol"!ns @ > A

• 2 #ol"!ns @ >

• 1 #ol"!n @

)here is the ti!e of o##"rren#e of the (eginning of the interval> is the d"ration ofthe interval> and A is a (eat la(el. Nor!al sin"s (eats are la(eled .

Altho(gh T is ass(med to &e expressed in seconds &* defa(lt$ the DO option of pltIrrs and+etIhrv %see &elow' can &e (sed if the RR inter"al list contains T "al(es expressed as cloc+time %hh:mm:ss#xxx'$ ho(rs %hh#xxxxxxx' or min(tes %mm#xxxxx'# )imilarl*$ altho(gh RR isass(med to &e expressed in seconds$ inter"als in milliseconds can &e inp(t &* (sing the -m9option# %)ee details &elow or in Dsages '#

/eat annotation files are a"aila&le for most of the h*sio/an+ records that incl(de EC8s# Forinformation a&o(t record and annotation con"entions see the h*sio<et FA # !f *o( wish tost(d* a recording for which no &eat annotation file or RR inter"al listing is a"aila&le$ *o(ma* &e a&le to create an annotation file (sing software from the WF / software pac+age#Additional information on RR inter"als$ heart rate$ and HRV can &e fo(nd at theRR HR HRV Howto #

&np#t #sed in this t#torial

The examples &elow read the ec+ annotations from record chf0( of the /! MC Congesti"eHeart Fail(re ata&ase # To reprod(ce the res(lts shown &elow$ it is not necessar* todownload this annotation file$ &eca(se applications that (se the WF / li&rar* %incl(dingthose in the HRV Tool+it that read annotation files' can locate and read the cop* from the

h*sio<et we& ser"er if no local cop* exists# !n order to do so s(ccessf(ll*$ it is necessar* tospecif* the path to the record from the h*sio/an+ archi"e director* within the record name$as shown in these examples %(se chfdbPchf0( rather than simpl* chf0( '# Examples thatill(strate inp(t from RR inter"al lists ass(me that the inp(t file is named chf0(.rr $ and thatit is in the c(rrent %local' director*#

plt_rrs : plotting the RR6.. inter'al time series

<< inter"al o(tliers d(e to missed or false &eat detections can serio(sl* corr(pt HRVstatistics# Most fre6(enc* domain meas(res are especiall* s(scepti&le to o(tliers$ partic(larl*.F and HF power which can &e in error &* greater than 0@@@S# Most time domain meas(resare less affected &* o(tliers$ &(t still can gi"e errors in excess of 0@@S# AV<<$ p<<B@ andD.F power are least affected$ generall* with errors less than 0@S %see Time domain

Meas(res: From Variance to p<<x '#

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To red(ce the errors d(e to o(tliers$ the first step in HRV anal*sis is to "is(all* examine theRR << inter"al data and determine the necessit* of o(tlier filtering# This can &e done (singpltIrrs #

pltIrrs can &e (sed with either an annotation file or an RR inter"al listing %see &elow' and

has the following options:

pltIrrs ,options- D rrfile Q record annotator ,start ,end-- $lot intervals or interval heart rates options # ,D$ 2Q)Q Q1<Q2)Q(2- # plot 2! )! ! 1<! 2) or (2 hours per pa+e default# scale pa+e len+th to data len+th" ,D rrfile- # interval file # ti&e sec"! interval ,D - # plot intervals instead of intervals ,D - # plot P interval heart rate ,DN Rfilt h inR- # filter intervals! plot filtered data ,Df Rfilt h inR- # filter intervals ,Dp- # plot points ,DO cQhQ&- # input ti&e for&at# hh##&&#ss! hours! &inutes default#seconds" ,D&- # intervals in &sec ,D R &in &axR- # axis li&its ,Do- # output postscriptB/ defa"lt> pltIrrs )ill o+en a plt )indo) and dis+la/ the +lot on the ter!inal s#reen. oo"t+"t the +lot as a +osts#ri+t file "se the Do o+tion. o +rint the +lot to a +osts#ri+t +rinter"se

pltIrrs Do other arguments Q lpr

Dsing an annotation file$ li+e this:

pltIrrs chfdbPchf0( ec+ )ill +lot the entire interval series for the #ongestive heart fail"re re#ord #hfd(;#hf03 "singthe ec+ annotator. B/ defa"lt> pltIrrs s#ales the +lot so that the entire interval series is+lotted on one +age> )ith a !a i!"! of lines +er +age. o +lot a s+e#ified n"!(er ofho"rs +er +age> "se the D$ o+tion. :or e a!+le> D$ ) )ill +rint 4 ho"rs +er +age @15 !in"tes+er line > D$ 1< )ill +rint 16 ho"rs +er +age @2 ho"rs +er line > et#.

To plot selected portions of the data$ specif* a start time and optionall* an end time# Forexample$

pltIrrs D$ chfdbPchf0( ec+ 00#00#00 01#00#00+lots the first ho"r of the interval se'"en#e=

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With the Dp option$ as in

pltIrrs D$ Dp chfdbPchf0( ec+ 00#00#00 01#00#00pltIrrs +lots the interval se'"en#e as individ"al +oints=

With the D option$ as in

pltIrrs D$ D chfdbPchf0( ec+ 00#00#00 01#00#00pltIrrs +lots NN intervals onl/> o!itting intervals that are not (o"nded (/ nor!al sin"s(eats at (oth ends=

The # = ( 2( # )20( = 0.710 ,( 0 nonD - in the title indicates the totaln(m&er of << inter"als$ the total n(m&er of RR inter"als$ the fraction of RR inter"als that are

<< inter"als and the n(m&er of non-<< inter"als#

To filter o(tliers$ (se the DN or Df options$ as in

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pltIrrs D$ D DN R0.2 20 Dx 0.) 2.0R chfdbPchf0( ec+ 00#00#0001#00#00

his +ro#ed"re )ill +lot the first ho"r of the filtered NN intervals se'"en#e. ere the G DN 0.220 Dx 0.) 2.0 G s+e#ifies the filtering +ara!eters as follo)s. :irst> an/ intervals less than0.4 se# or greater than 2.0 se# are e #l"ded. Ne t> "sing a )indo) of 41 intervals @20

intervals on either side of the #entral +oint > the average over the )indo) is #al#"latede #l"ding the #entral interval. $f the #entral interval lies o"tside 20H @0.2 of the )indo)average this interval is flagged as an o"tlier and e #l"ded. hen the )indo) is advan#ed tothe ne t interval. hese +ara!eters #an (e ad "sted as a++ro+riate for different data sets.

Dsing DN allows pltIrrs to plot the excl(ded inter"als as small filled circles:

The Df option s(ppresses o(tp(t of excl(ded inter"als:

The RNilt # # = (;<2 # ( 2( # )20( = 7 ) # 0.710 = 0. 75 ,<1Niltered! ( 0 nonD -R in the title of the two plots a&o"e gi"es the total n(m&er of <<inter"als remaining after filtering$ the total n(m&er of << inter"als &efore filtering$ the totaln(m&er of RR inter"als$ the fraction of << inter"als remaining after filtering$ the fraction ofRR inter"als that are << inter"als$ the fraction of the total n(m&er of RR inter"als that are

<< inter"als remaining after filtering$ and the n(m&er of << inter"als filtered o(t togetherwith the n(m&er of non-<< inter"als#

Filtering the << inter"al data ma* not &e necessar* if there are no extreme o(tliers#

To plot the RR inter"al series of an RR inter"al file containing three col(mns of data : time in

seconds$ inter"al in seconds and &eat la&el$ (se the D option followed &* the name of the

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file$ rather than specif*ing the record and annotator names# For example$ if the file chf0(.rr contains:

1.(5< 0.752 2.(12 0.75< (.252 0.7)0 ).212 0.7<0 5.15< 0.7)) <.112 0.75< ;.0) 0.7(< ;.77< 0.7)

.7<0 0.7<) 7.700 0.7)0 ...

"se pltIrrs D chf0(.rr )ith an/ of the a(ove o+tions to +lot its #ontents.

The a&o"e RR inter"al list was generated (sing the command

rrlist ec+ chfdbPchf0( Ds :chf0(.rr@Noti#e that the re#ord and annotator arg"!ents a++ear in reverse order in rrlist #o!!ands. he vario"s other o+tions availa(le for rrlist are given in 8sages .

!f an RR inter"al listing contains onl* time in seconds and inter"al in seconds$ e#g#$

1.(5< 0.7522.(12 0.75<(.252 0.7)0

).212 0.7<05.15< 0.7))<.112 0.75<;.0) 0.7(<;.77< 0.7)

.7<0 0.7<)7.700 0.7)0...

then a #o!!and s"#h as pltIrrs D chf0(.rr +lots the interval se'"en#e #orre#tl/>("t )ill not (e a(le to +lot the NN interval se'"en#e "sing the D o+tion> sin#e the (eatla(els are !issing.

To plot an RR inter"al listing containing onl* RR inter"als in seconds and &eat la&els$ e#g#$

0.752 0.75< 0.7)0 0.7<0 0.7)) 0.75< 0.7(< 0.7) 0.7<) 0.7)0 .

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.

.the #o!!and pltIrrs D chf0(.rr #al#"lates the ti!e of ea#h interval fro! the interval se'"en#e> !a?ing the ass"!+tion that the intervals are #onse#"tive.

Finall*$ if the RR inter"al listing contains onl* RR inter"als in seconds$ e#g#$

0.7520.75<0.7)00.7<00.7))0.75<0.7(<0.7)0.7<)0.7)0.

.

.then pltIrrs D chf0(.rr #al#"lates the ti!e of the ti!e of the interval fro! the interval se'"en#e> ("t as in a +revio"s e a!+le )ill not (e a(le to +lot the NN intervalse'"en#e "sing the D o+tion.

!f the RR inter"als in an* of the a&o"e formats are listed in milliseconds rather than seconds$(se pltIrrs with the D& option#

get_hrv : calc#lating the HRV statistics

To calc(late the time and fre6(enc* domain HRV statistics (se +etIhrv $ where the optionsare:

+etIhrv ,options- D rrfile Q record annotator ,start ,end--et M statistics #

SC # P @M /8 /8@ /8 O 8X 4//8 $ # T T$3 L>N M>N>N N >NP N

options # ,D rrfile- # interval file # ti&e sec"! interval ,Df Rfilt h inR- # filter outliers ,Dp Rnndiff ...R- # nn difference for pnn default# 50 &sec" ,D$ Rlo1 hi1 lo2 hi2 lo( hi( lo) hi)R- # po er bands

default # 0 0.00(( 0.00(( 0.0) 0.0) 0.15 0.15 0.)" ,Ds- # short ter& stats of SC # P @M /8 4//8 $ # T T$3 M>N >N N >NP N ,DO cQhQ&- # input ti&e for&at# hh##&&#ss! hours! &inutes

default# seconds" ,D&- # intervals in &sec

,D4- # output statistics in &sec rather than sec ,D>- # output statistics on one line

,D/- # plot M results on screen

plottin+ options # ,DN Rfilt h inR- # filter outliers! plot filtered data ,D R &in &axR- # ti&e series Daxis li&its RD DR for selfD

scalin+" ,DX &axfre?- # fft &axi&u& fre?uenc default # 0.) B"

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,DY fft&ax- # fft &axi&u& RDR for selfDscalin+" ,Do- # output plot in postscript

+etIhrv "ses statnn to #al#"late the ti!e do!ain statisti#s> and lo&b @fro! the :DBsoft)are +a#?age and p r to #al#"late the fre'"en#/ do!ain statisti#s. NN; is thefra#tion of total intervals that are #lassified as nor!al to nor!al @NN intervals and

in#l"ded in the #al#"lation of V statisti#s. his ratio #an (e "sed as a !eas"re of datarelia(ilit/. :or e a!+le> if the NN; ratio is less than 0. > fe)er than 0H of the intervals are #lassified as NN intervals> and the res"lts )ill (e so!e)hat "nrelia(le.

The command

+etIhrv Df R0.2 20 Dx 0.) 2.0R Dp R20 50R chfdbPchf0( ec+sho"ld give the follo)ing o"t+"t=

chfdbPchf0( # P = 0.7)2 77 @M = 0. 7220< /8 = 0.05);12 /8@ = 0.0)<<;;( /8 O8X = 0.02)112) r4//8 = 0.01;;<7) p 20 = 0.0< <7 p 50 = 0.0252( 7 T T $3 = 0.00(( 2 L>N $3 = 0.002;0) 7 M>N $3 = 0.000(701 >N $3 = 0.00012)10) N $3 = 0.0001<702; >NP N = 0.;()22<$n this #ase AVNN> SDNN> SDANN> SDNN$D% are r&SSD are given in se#onds> +NN val"esas ratios and , val"es in se#onds s'"ared. Note that the e a#t val"es o(tained ondifferent +latfor!s !a/ var/ (/ a++ro i!atel/ 10 6 d"e to ro"nd off errors.

/* defa(lt$ +etIhrv o(tp(ts the HRV statistics in seconds# To o(tp(t res(lts in milliseconds(se the D4 option$ as in

+etIhrv D4 Df R0.2 20 Dx 0.) 2.0R Dp R20 50R chfdbPchf0( ec+his sho"ld give the follo)ing o"t+"t=

chfdbPchf0( # P = 0.7)2 77 @M = 72.20< /8 = 5).;12 /8@ = )<.<;;( /8 O8X = 2).112) r4//8 = 1;.;<7) p 20 = <. <7 p 50 = 2.52( 7 T T $3 = (( .<5 L>N $3 = 2;05.17 M>N $3 = (70.25 >N $3 = 12).15< N $3 = 1<7.05; >NP N = 0.;())0(

$n this #ase AVNN> SDNN> SDANN> SDNN$D% are r&SSD are given in !illise#onds> +NNval"es as +er#entages and , val"es in !illise#onds s'"ared.

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inter"als of the << inter"al time series: the non-normal &eats as small open circles ando(tliers as small filled circles# For example

+etIhrv D/ Ds D4 DN R0.2 20 Dx 0.) 2.0R Dp R20 50R chf0( ec+ 0#00#001#00#00

)ill generate the follo)ing +lot=

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