jenks natural breaks optimization
given a vector of numeric values and the number of desired breaks, calculate the optimum breakpoints using jenks natural breaks optimization .
george f. jenks proposed jenks natural breaks in his 1977 article entitled ‘optimal data classification for choropleth maps’ .
, (or making maps with any software), this page explains about
natural breaks
for setting map ranges. jenks natural breaks is also called
fisher' s natural breaks
.
real statistics data analysis tool: the real statistics resource pack provides the jenks natural breaks data analysis tool to perform the optimization automatically .
the jenks natural breaks method was ported to c from code found in the classint r package .
i' m looking for an implementation of the jenks optimization algorithm for data classification .
jenks natural breaks optimization addresses the problem of how to split a range of numbers into contiguous classes so as to minimize the squared deviation within each class .
observation: the jenks natural breaks algorithm is commonly used to color maps as shown in figure 9 .
the jenks natural breaks optimization method allows you to break up data points into the best possible number of groupings, with the best possible contents of each group for choropleth mapping .
the jenks natural breaks classification method seeks to reduce the variance within classes and maximize the variance between classes. urltoken urltoken
fisher' s natural breaks uses dynamic programming to find the optimal solution and is deterministic .
you wish. here are stands displayed in 5 natural breaks classes, according to age .
different class break values between this map and the natural breaks map. each class here will
natural breaks is complicated because many steps are involved, but does not involve any higher math .
the first thing to remember about the natural breaks (jenks) classification is that it is an optimization method for choropleth maps. in short, it arranges each groupings so there is less variation in each class or shading .
equal intervals, quantile, natural breaks, pretty breaks – there’s a lot to choose from. but what is the difference between each of them ?
and so here’s a literate implementation of jenks natural breaks in javascript. if you can, please help to improve it by making the implementation more explanatory .
compute “natural” break values (jenks algorythm) on list / tuple / numpy. ndarray of integers / floats .
equal intervals, quantile, natural breaks, pretty breaks. each data classification methods generate different choropleth maps. we explain the advantages and disadvantages of using each type of choropleth map .
any pointers to any particular work that uses jenks natural break or even compares it with k - means would be great .
natural breaks classes are based on the nature of data. the range is set only when a surprise or big difference appears in data values. convergent data values put together under one class. different classes are properly distributed under this way. this method is applied to prepare this atlas, and it is calculated by the gis software according to the jenks natural breaks optimization (jenks). it depends on the adequate classification by the clearest values on the map .
don' t use multidimensional clustering algorithms for a one - dimensional problem. a single dimension is much more special than you naively think, because you can actually sort it, which makes things a lot easier. in fact, it is usually not even called clustering, but e. g. segmentation or natural breaks optimization. you might want to look at jenks natural ...
simple - statistics, my library for literate, beginner - friendly statistics, now has a literate implementation of jenks natural breaks. in the process of understanding the implementation, i found that the matrix system is a sort of dynamic programming that cleverly solves all breaks from 1 - k when you request k breaks. thus my implementation breaks it into two parts - creating the matrices and pulling a solution out of them .
to perform the analysis for example 1, press ctrl - m and select the multivariate analyses option from the main menu and then select jenks natural breaks from the dialog box that appears, as shown in figure 5 .
not the answer you' re looking for? browse other questions tagged python clustering optimization or ask your own question .
when the jenks natural breaks dialog box appears, fill in the various fields as shown in figure 5 and press the ok button. if you leave the number of iterations field blank then the algorithm will evaluate all possible partitions .
this page was updated 20 october 2002 to expand on the natural breaks description. thanks to daniel karnes of dartmouth college for pointing out the need for improvement .
the founder of the natural breaks data classification method was a cartographer by the name of george frederick jenks. he specialized in monitoring the eye movements of people when looking at a map. and the results for this map looked great too .
the jenks natural breaks algorithm, just like k - means, assigns data to one of k groups such that the within group distances are minimized. also just like k - means, one must select k prior to running the algorithm .
borden dent (cartography - - thematic map design, fifth edition, 1999, pages 147 - 149) relates that the natural breaks classes are found by an iterative search to minimize the sum of spreads of the classes. thus it is a one - dimensional example of the k - means clustering method. natural breaks, jenks' method, and k - means are all the same in this instance .
jenks (r1, , lab, iter) – performs jenks natural breaks optimization on the data in range r1 for k classes and outputs a k + 1 × 3 range, whose first k rows contain the left and right endpoints of each of the classes followed by the number of elements in the class. the last row contains the total squared deviation of the k classes, followed by the squared deviation for the data in r1 and gvf .
natural breaks (jenks) takes an iterative approach by comparing the sum of squared deviations between classes to the array mean. then, the algorithm uses a goodness of variance fit with 1 as a perfect fit and 0 as a poor fit .
features in each bin. here is the jenks classification for the image above .
what we don’t know is whether these four intervals is the partition of the full range that results in the smallest sum in cell t21. as we will see shortly, the optimum solution, using jenks natural breaks optimization, is shown in figure 3. as you can see, this is a better solution than the one shown in figure 2 (since 3107. 102 < 4496. 957) .
if you want round numbers in your ranges, then you should choose pretty breaks. all pretty breaks does is rounds each break - point up or down. so instead of having a break point as 599. 364 it will become 600, 000 with pretty breaks .
the quantile style provides quantile breaks; arguments to quantile may be passed through... .
the chart shows the number of acres in each unique age class. this method uses then jenks
computebayesfactors: compute bayes factors generatecontrolfile: generate control file for' bamm' getjenksbreaks: jenks natural breaks classification gettiprates: compute tip - specific macroevolutionary rates from' bammdata'... plotprior: plot the prior and posterior distribution of shifts setbammpriors: set bamm priors traitdependentbamm: strapp: structured rate permutations on phylogenies
create a “natural breaks” classification using two or more classes. look closely at the map. now reverse the order of the classes in the legend (use, for example). what changes? can you discover why ?
one problem is that your boss is not letting the data speak. perhaps there are three
natural
clusters. insisting on five will result in breaking apart two of the clusters. those break points will not be
natural .
i' m not sure what mathematical definition you are using for
natural breaks ,
but one interpretation is to identify places where there are
gaps
in the data. in other words, you want to cluster the data .
the hclust style uses hclust to generate the breaks using hierarchical clustering; the pars attribute returns the hclust object generated, and can be used to find other breaks using gethclustclassintervals; arguments to hclust may be passed through... .
natural breaks
finds the
best
way to split up the ranges. suppose we had 30 counties, 15 counties with 0 - 1 values, 10 counties with 16 - 18 values, and 5 counties with 24 - 29 values. obviously, the
best
ranges are 0 - 1, 16 - 18, 24 - 29. counties with similar values should have the same color .
natural breaks
is the only method that finds the
best
ranges .
given a desired number of classes, k, the natural breaks method partitions the data into k subsets that minimize the sum of the
spreads
within each subset. (
spread
is an informal term employed solely for this description. )
and this is not quibbling about style: the wikipedia article about jenks natural breaks describes an algorithm entirely unlike these ones. there’s no mention of dynamic - programming matrices, and there’s no hint in the implementations of something like the ‘sum of deviations between classes. ’ of course, there are no comments in any of the ports, so it’s hard to tell .
getclassjenks (nbclass): perform a jenks classification and return bounds into an array. alias: getjenks (nbclass )
as a result of making rounded numbers, pretty breaks will also be very picky for the number of classes you decide .
jenks is k - means in one dimension. the wikipedia article would be one of the references you seek. for work that uses the jenks method, just find any of several million maps created by arcview users over the last 20 years .
my boss has asked me to show the distribution of a variable (square footage of 17, 000 public library outlets / branches), breaking it into 5 categories using natural breaks. putting aside arguments of the
best
number of categories or the problems with categorizing a continuous variable... i' ve used proc rank for creating deciles, quantiles, etc. , but the only place i' ve seen that uses
natural breaks
( outside of distance sporting events) is chloropleth maps. i' ve googled for a solution. the closest i got, other than a bunch of articles on analyses using the jenks optimization in arcgis, was a post to the uga sas - listserv in 2000. i' ve seen a few possible links to using proc cluster, but that always seems to be when doing a multivariate analysis .
here 3107. 102 (cell l7) represents the total squared deviation for the partition found using the jenks natural breaks algorithm, 27504. 59 (cell m7) is the squared deviation of the input data, as calculated by = devsq (b3: i22) and 88. 7% is the gvf (cell n7), as calculated by the formula = 1 - l7 / m7 .
there’s a standard algorithm we use for jenks, with existing ports in in javascript, python, and in java and fortran .
here you have php urltoken this is implementation of jenks algorithm. david drake made these functions and integrated it with google api .
the range u10: u13 of figure 8 contains the array formula = jenks (b3: i22, 4, , 100000) .
when the data represent a random sample from a population consisting of two more more distinctly different subpopulations, and you know - - or can accurately guess - - how many different subpopulations there are, then natural breaks can do a good job of choosing classes which reflect the subpopulation groupings .
observation: in example 2, we were able to go through each possible partition and calculate its squared deviation. while in example 1, this is too time - consuming to do manually, we can use the real statistics jenks natural breaks data analysis tool, as described below, to do the calculations for us, for other problems the number of possible partitions is even too time - consuming for the computer to do .
jenks natural breaks works by optimizing the goodness of variance fit, a value from 0 to 1 where 0 = no fit and 1 = perfect fit. the key in selecting the number of classes is to find a balance between detecting differences and overfitting your data. to determine the optimum number of classes, i suggest you use a threshold gvf value you desire and use the number of classes that satisfies this value first .
it means the ranges where like areas are grouped together. obviously, we do not give a low rate area the same color as a high rate area. natural breaks minimizes the variation within each color, so the areas within each color are as close as possible in value to each other .
gvf (r1, r2) = gvf for the right endpoints (breaks) in range r2 based on the data in column range r1 .
i found this python implementation of the jenks natural breaks algorithm and i could make it run on my windows 7 machine. it is pretty fast and it finds the breaks in few time, considering the size of my geodata. before using this clustering algorithm for my data, i was using sklearn. clustering. kmeans (here) algorithm. the problem i had with kmeans, was finding the optimum k value parameter, but i
solved
it launching the algorithm for different k values and using sklearn. metrics. silhouette _ score (here) to find the best k .
the range k3: n7 of figure 7 contains the output of the array formula = jenks (b3: i22, , true). the range p10: s14 of figure 8 contains the output of the array formula = jenks (b3: i22, , true, 100000) .
i also read from another source that says jenks is usually used to spot gaps in ranged data. again i cannot find a reference to explain this .
try out jenks in simple - statistics. if the assortment of other algorithms included make you feel like you should create a new ‘micro library’, you can also just use the jenks implementation as a standalone. please don’t delete the comments: if you need compression, use uglify - js as a separate step .
widely used within gis packages, these are forms of variance - minimization classification. breaks are typically uneven, and are selected to separate values where large changes in value occur. may be significantly affected by the number of classes selected and tends to have unusual class boundaries. typically the method applied is due to jenks, as described in jenks and caspall (1971), which in turn follows fisher (1958). see figure 4‑22 for more details. unsuitable for map comparisons
the method requires an iterative process. that is, calculations must be repeated using different breaks in the dataset to determine which set of breaks has the smallest in - class variance. the process is started by dividing the ordered data into groups. initial group divisions can be arbitrary. there are four steps that must be repeated :
i am new to this topic. as far as i know both are data clustering methods. then my question is when is jenks prefered over k - means ?
let’s be clear: the algorithms we use for jenks are directly ported from an algorithm that accepted punch cards, documented in a paper composed and published on a typewriter .
the pretty style chooses a number of breaks not necessarily equal to n using pretty, but likely to be legible; arguments to pretty may be passed through... .
positioning of breaks may be pre - determined by the classification procedure (e. g. jenks natural breaks), but these are often manually adjustable for some of the schemes provided — this is particularly useful if specific values or intervals are preferred, such as whole numbers or some convenient values such as 1000s, or if comparisons are to be made across a series of maps. in some instances options are provided for dealing with zero - valued and / or missing data prior to classification, but if not provided the data should be inspected in advance and subsets selected prior to classification where necessary. some authors recommend that maps with large numbers of zero or missing data values should have such regions identified in a class of their own .
so: yes, this isn’t the first implementation of jenks in javascript. and it took me several times longer to do things this way than to just get the code working .
there is no such assurance. but natural breaks makes the most epidemiologically accurate map. making the map
pretty
is important, but that would involve such factors as right number of colors (not too few, not too many), right color palette (a standard colorbrewer palette provided by vitalnet), high resolution so the map does not look fuzzy, and nice legend .
however, jenks and k - means are different in how they minimize within group distances. jenks takes advantage of the fact that 1 - dimensional data is sortable which makes it a faster algorithm for 1 - dimensional data. k - means is more general in that it can handle data in any dimension; including dimensions greater than 1 where the data is not sortable .
’s thematic formatting facilities providing a wide range of options that reduce or remove the dependence on formal class boundaries. the jenks method, as implemented in a number of gis packages such as
the jenks style has been ported from jenks’ basic code, and has been checked for consistency with arcview, arcgis, and mapinfo (with some remaining differences); added here thanks to hisaji ono; note that the sense of interval closure is reversed from the other styles, and in this implementation has to be right - closed - use cutlabels = true downstream for clarity .
using silhouette with jenks should be comparable to using it with kmeans. it' s a heuristic and you shouldn' t trust it blindly. imho the best is to visualize your rrsults .
the sd style chooses breaks based on pretty of the centred and scaled variables, and may have a number of classes different from n; the returned par = includes the centre and scale values .
from jenks import jenks import numpy as np def goodness _ of _ variance _ fit (array, classes): # get the break points classes = jenks (array, classes) # do the actual classification classified = np. array ([ classify (i, classes) for i in array ]) # max value of zones maxz = max (classified) # nested list of zone indices zone _ indices = [ [ idx for idx, val in enumerate (classified) if zone + 1 = = val ] for zone in range (maxz) ] # sum of squared deviations from array mean sdam = np. sum (( array - array. mean ()) * * 2) # sorted polygon stats array _ sort = [ np. array ([ array [ index ] for index in zone ]) for zone in zone _ indices ] # sum of squared deviations of class means sdcm = sum ([ np. sum (( classified - classified. mean ()) * * 2) for classified in array _ sort ]) # goodness of variance fit gvf = (sdam - sdcm) / sdam return gvf def classify (value, breaks): for i in range (1, len (breaks) ): if value < breaks [ i ]: return i return len (breaks) - 1
due to numerical stability, the log likelihood is used, turning the product into a sum of logarithms, for optimization is the same, given that the log function is monotonicly increasing. by convention, loss functions are minimized with neural networks, this means we will negate it and call it the negative log - likelihood :
it depends on the number of areas, number of colors, and the cpu. remember that with 254 counties and 6 colors, there are 8, 301, 429, 675 possible range combinations. it might take the computer a little while to test so many combinations. there would be even more combinations (a lot more) with 7 colors. vitalnet natural breaks has been designed with special programming to run as fast as possible. it runs quickly, a second or two, with a modest number of ranges or if not that many areas. but it eventually slows down with a larger number of ranges and areas. so if using natural breaks, start with a low number of ranges, and only increase the number to a large number if needed, and then be prepared for a possible wait. if needed, you can minimize the window and do other tasks .
the standard deviation classification method shows how much an attribute value varies from the mean. particularly useful for economic and physical studies such as temperature, wind and other natural phenomena with planar distributions which depend on arithmetic averages to get analyses .
the bclust style uses bclust to generate the breaks using bagged clustering; it may be anchored using set. seed; the pars attribute returns the bclust object generated, and can be used to find other breaks using getbclustclassintervals; if bclust fails, a jittered input vector containing rtimes replications of var is tried — with few unique values in var, this can prove necessary; arguments to bclust may be passed through... .
in addition to selection of the scheme to use, the number of breaks or intervals and the positioning of these breaks are fundamental decisions. the number of breaks is often selected as an odd value: 5, 7 or 9. with an even number of classes there is no central class, and with a number of classes less than 4 or 5 the level of detail obtained may be too limited. with more than 9 classes gradations may be too fine to distinguish key differences between zones, but this will depend a great deal on the data and the purpose of the classification being selected. in a number of cases the gis package provides linked frequency diagrams with breaks identified and in some cases interactively adjustable. in other cases generation of frequency diagrams should be conducted in advance to help determine the ideal number of classes and type of classification to be used .
first is proc means using the groups / clusters picked by fastclus, next is the same using the 4 jenks categories out of arcmap. the min / max are the lower and upper level of each interval .
the fisher style uses the algorithm proposed by w. d. fisher (1958) and discussed by slocum et al. (2005) as the fisher - jenks algorithm; added here thanks to hisaji ono .
very interesting read. as someone who struggled through the implementation of jenks [ 1 ] a few years ago in php, i can attest to the fact that it was extremely difficult to find any programming of jenks that was understandable and clear. i was able to find some examples of code, but nothing could explain what was actually happening and, as you said, there were no comments or poor comments in those examples .
in this case, it' s a combination - the algorithm jenks arrived at is not just the math - implemented, but a clever solution that (afaik) has not been expressed in pure - math terms .
every implementation i could find of jenks is a port from fortran iv, with the same terrible variable names, no comments, and absolutely no evidence that the porting author knew absolutely anything about what the code was doing .
i read on this website that jenks is particularly suited for 1 - dimensional data, while k - means are for multi - dimensional. is this true? if so can you point me to some references about this ?
the fixed style permits a
classintervals
object to be specified with given breaks, set in the fixedbreaks argument; the length of fixedbreaks should be n + 1; this style can be used to insert rounded break values .
i thought jenk' s method of optimization was k - means in one dimension (see this comment by whuber). i believe there are multiple
jenk' s
methods though, so it could be some confusions around this. your second comment also seems to be confused, if you use k - means clusters in one dimension it doesn' t assign groups in non - contiguous rank order .
it’s a bit hard to see how round the numbers are (it’s grouping by 5’s) in this example because all the examples above also produce round numbers. but when you have large numbers like population estimates (see below), it will generate some very pretty breaks .
a common one is “k - means, ” which (given k) minimizes the total within - cluster variance. this method, which has been around a long time, was rediscovered by cartographers in 1963 and so goes by the name “jenks’ method” in that community .
it destroys the objective display of the data. of the few patterns the user can test, the
prettiest
pattern will almost certainly be selected, but that has nothing to do with the correct display of the data. if the pattern resulting from natural breaks is not
pretty
, it just means there is no strong geographic pattern to the data. artificially selecting ranges to produce a
pretty
or desired pattern can easily turn into
how to lie with maps
. a valid reason to manually set map ranges is to compare two maps, to use the same ranges as in an existing map .
step 4: individual values in each class are then systematically assigned to adjacent classes by adjusting the class boundaries to see if the tssd can be reduced. this is an iterative process, which ends when improvement in tssd falls below a threshold level, i. e. when the within class variance is as small as possible and between class variance is as large as possible. true optimization is not assured. the entire process can be optionally repeated from step 1 or 2 and tssd values compared
cartographica has added new classifications for displaying data. we have expanded the classifications from equal interval and jenks classifications to equal count, equal area, quartile distributions, quintile distributions, and classification that incorporates all unique values. the upgrades in classifications schemes gives users better capability for displaying data using chlorpleth maps .
i studied up on this and wrote it while working for a startup a few years ago, in 2009. i found that all of the available choroplethic mapping solutions available had inadequate splits of data when asked to create a map. unfortunately, very few implementations of jenks exist outside of professional cartography packages .
jenks (r1, k, lab, iter) – outputs a a k × 1 column range, whose first k− 1 rows contain the right endpoint of the first k− 1 classes (the endpoint of the k th category is the largest element in r1) and whose last element is the gvf .
interesting that you mention the
combination .
when i was able to find the aforementioned image that showed the jenks method, i knew i had to understand what it meant and what the symbols were. i found it very cool that the mathematics were simple and it was the process that made the method work so well .
jenks is an algorithm designed specifically for this case that tries to maximize the similarity of numbers in groups while maximizing the distance between the groups. it tends to look good and be easily understood, so it’s popular. it’s in arcgis, qgis, and lots of other tools for maps and statistics. it’s in university lectures and textbooks .
the kmeans style uses kmeans to generate the breaks; it may be anchored using set. seed; the pars attribute returns the kmeans object generated; if kmeans fails, a jittered input vector containing rtimes replications of var is tried — with few unique values in var, this can prove necessary; arguments to kmeans may be passed through... .
another thing to notice is that this concept is very natural to extend to multiple output dimensions. instead of using 1 - dimensional gaussian kernels, we can also use n - dimensional gaussian kernels, by using the squared norm between the $ \ mathbf { y } $ and $ \ mathbf { y } ’ $ output vectors. if we look at the uncorrelated multivariate gaussian distribution, our kernel function turns into :
this article isn’t available online, or in print. as far as i can tell, it has never been digitized. it’s ostensibly still owned by the university of kansas geography department, so the copy that bill morris was kind enough to scan and share i can’t safely post. even if the university doesn’t assert copyright, because professor jenks passed in 1996, it’ll be copyrighted in his name until 2072 .
note after looking into the jenks algorithm proposed in the answer, i found that the classint package in r apparently has a number of such algorithms. i post a note from its documentation to expand on the answer below. i have no idea how well these perform in practice, i post them just because of the variety of possibilities and because being in r makes them easy to try out for yourself .
my question: does anyone know whether sas has a procedure that does this? i could probably write up code to do the iterations, but if there is a ready - made proc, there' s no reason to do that. it also looks like the head / tail breaks approach is better (would be more appropriate for this data, since it has a long tail) and would be more straight - forward if i had to write something myself .
the breaks between the classes are set so that the total area of the features in each class is as close as possible to 1 / k times the total area of all features. (this of course makes sense only for features with areas—that is, for polygonal features .) the result usually is a map that has about the same amount of every symbol on it—a kind of visual balance. (thought question: could you describe, in detail, an algorithm for determining the equal area cutpoints? )
fantastic article. i like to think of this phenomenon with a positive spin though. . i can use the jenks algorithm without understanding anything about how it works, just plug it in and go. and with a few test cases i can even port it to a new language without really understanding it. i admire tom' s work in digging in and doing it right, but as a journeyman programmer i like that i can just use it without really understanding it. sometimes cargo cults work .
for each range combination, calculate
sum of squared deviations for class means
( sdcm _ all), and find the smallest one. sdcm _ all is similar to sdam, but uses class means and deviations. suppose we have four counties and two ranges. for [ 4 ] [ 5, 9, 10 ], sdcm _ all = (4 - 4) ^ 2 + (5 - 8) ^ 2 + (9 - 8) ^ 2 + (10 - 8) ^ 2 = 0 + 9 + 1 + 4 = 14. for [ 4, 5 ] [ 9, 10 ], sdcm _ all = (4 - 4. 5) ^ 2 + (5 - 4. 5) ^ 2 + (9 - 9. 5) ^ 2 + (10 - 9. 5) ^ 2 = 0. 25 + 0. 25 + 0. 25 + 0. 25 = 1. for [ 4, 5, 9 ] [ 10 ], sdcm _ all = (4 - 6) ^ 2 + (5 - 6) ^ 2 + (9 - 6) ^ 2 + (10 - 10) ^ 2 = 4 + 1 + 9 + 0 = 14. [ 4, 5 ] [ 9, 10 ] has the smallest sdcm _ all, so is
best ranges
, minimizes variation within classes. intuitively, it makes sense to use [ 4, 5 ] [ 9, 10 ], and the natural breaks algorithm automatically figures this out .
instead of using a non - parametric approach like histograms or kernel density estimation we can also use parametric distributions where we estimate the parameters to fit the assumed distribution. if the family of distributions fits the data well we can get very nice fits with relatively little amount of data, however, if the family does not fit the generating distribution properly you will get poor results. some standard distributions, for example gaussians are easy to fit by just looking at some statistics of your data. in case of the gaussian the mean $ \ hat { \ mu } = \ overline { x } = \ frac { \ sum _ { i = 1 } ^ nx _ i } { n } $ and $ \ hat { \ sigma } ^ 2 = \ frac { \ sum _ { i = 1 } ^ n (x _ i - \ overline { x }) ^ 2 } { n - 1 } $ will produce the most likely gaussian that generated your data. in more complicated distributions you will need to use some form of iterative optimization, for instance using gradient descent on the likelihood function to fit your parameters. below, i have illustrated a gaussian fit on our data generated by a default gaussian. notice that this animation uses smaller sample sizes .
the feature / symbol dichotomy: a gis divides the functions of representing data on a map into at least five conceptually different parts: the attributes are classified (classification) and associated with geometric figures (projection), the classifications are associated with graphic symbols, and those symbols determine how the geometric figures will appear (symbolization of the graphic shapes) .
emphasize substance over form: when working with complex software, like a gis, deal with the important things first and take care of the details as late in the process as possible .
look for the familiar within the strange terminology: many disciplines have, or could have, contributed to the development and evolution of gis. many techniques may already be familiar to you under different names. do not let the terminology daunt you !
maps are statistics: symbolizing features on a map to reflect underlying data is a statistical procedure. you can use statistical techniques to help determine appropriate symbolization methods .
you are in charge: always have a reason for your choice of data symbolization: do not let the computer choose .
the legend editor is a complex dialog providing access to almost all arcview’s capabilities to classify and modify the display of features. every feature has a geographic location. the view erects a graphic shape at each feature’s location in order to show it to a person. a graphic shape is a combination of a geometric shape and a symbol, which records the visual features of the shape, such as its colors, line styles, and so on. the shape is determined by the geographic data, but the means of displaying the shape are not. determining how the shape will display is the role of the legend editor. the legend editor can make this determination based on the values of one or more attributes of each feature. in this manner arcview can show data on a map .
the legend editor sets up a sequence of processes. the first is classification. this assigns features to groups according to the values of one or more of the feature attributes. the second is symbolization. this associates a symbol with each class. the figure sketches this process .
in addition, the legend editor controls how the classifications will be named or labeled in the view’s table of contents and provides capabilities to save and reload a legend .
gis software differs in how it implements the legend editor capabilities, but it must implement these capabilities at some level .
invoke the legend editor through a menu item (theme | edit l egend), a button, or most simply by double - clicking on a theme’s legend in the view .
the legend editor dialog is in a separate window that can be moved outside the arcview window. however, it will never disappear behind arcview. it can remain open while you do other arcview operations (it is “modeless ,
not a child window) .
the legend editor will act differently on different kinds of themes: feature themes, image themes, and grid themes .
the legend editor dialog has been long in need of improvement. typically, changes to the legend type, values field, or classification type will cause unwanted side - effects, often destroying any colors or symbols you have already specified. the trick lies in specifying this fundamental information first. only when you are sure you have chosen the right kind of legend, the correct attribute (s) to display, and the correct number of classifications should you bother with the details of specifying the symbols themselves .
this last rule applies to all complex software: deal with the important things first and take care of the details as late in the process as possible (or never, if you can get away with it) .
practice editing symbols. (you double - click on one to change it .) can you find a way to edit multiple symbols simultaneously ?
suppose two classifications overlap; for example, suppose one classification is 0 - 10 and the next is 5 - 20. how does arcview resolve this conflict; that is, how does it decide which classification to use for values within the overlapping region ?
figure out what this button does and how it is relevant to the previous question .
( optional) the “statistics” button is handy. experiment with it. note that there are two definitions of “standard deviation” (sd). one is the “root mean square” (of the deviations from the mean), or “population” standard deviation. the other has been adjusted to reduce its estimation bias; it is
n / (n - 1) ) times larger than the population sd when n values are involved. which sd does arcview compute ?
when is the “advanced” button enabled? optional (for now): determine what the advanced options do .
every map is to some extent a distortion of reality. through this distortion some maps reveal patterns and other maps lie [ see monmonier or tufte, for instance ]. one of the subtlest forms of lying with maps is associated with the method of classifying numerical attributes .
we will experiment with different classification methods. the material here is background. for more information see urltoken (section 6) .
for each classification type below, n numerical values are to be divided into k classifications (“classes”) according to the rules to be described. suppose the ordered numerical values are x
respectively. in every case the classifications consist of non - overlapping intervals of numbers .
the k classes will be constructed to contain as near to n / k values as practicable. for example, if n is 18 and k is 5, then n / k = 3. 6, so the classes will contain either 3 or 4 values each. the cut points will always coincide with one of the original data values .
some intervals may contain none of the original values (can you think of a simple example ?) .
this is almost another kind of equal interval classification. however, the interval length is set to be some multiple of the standard deviation of the data. multiples of 1 and 0. 5 are common; smaller multiples may be used with lots of data .
equal length and equal number legends are conceivable for polyline and multipoint themes, but arcview does not implement these. (the features of a
multipoint
theme consist of zero, one, or more points. )
any nonempty set of numbers can be centered by subtracting the mean of those numbers from each value. the resulting values are known as residuals. one measure of spread is the sum of squares of the residuals. it is a measure of how much the values vary about their means, and so is appropriate for identifying clusters of data .
there are many other types of classification, but (for data sets with no “ties”) all can be reduced to equal intervals after applying a preliminary transformation of the data, y = f (x). for example, dividing y = log (x) into equal intervals is equivalent to dividing the range of the x’s into intervals of equal ratios, such as 1 - 2, 2 - 4, 4 - 8, and so on. therefore classification methods for numerical data are usually determined by their effects on the resulting map rather than by some statistical method based purely on the data .
the classification method is an easy target for criticism when you make a map. therefore, always have a reason for selecting your method. it is not sufficient to say, “the computer chose it. ” who is in charge, you or the machine ?
experiment with the data provided in gtkav chapter 9 (counties. shp). apply all five classification methods and vary the numbers of classes. each team (computer) will select one of the several dozen numerical attributes to study .
create a histogram of the data, by hand or using software on your computer (such as excel or a statistics package). use this as a frame of reference for evaluating the maps you create. create at least two versions of this histogram by substantially varying the bin size (or, equivalently, by varying the number of histogram bars). using these histograms alone, predict your answers to the next three questions. then proceed to answer those questions through experimentation with arcview’s legend editor. check your answers against your predictions .
this page was last updated 11 march 2004. it was reformatted. the
summary of principles
section was added. minor editorial changes were made to clarify portions of the text .
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subtract the sdbc from the sdam (sdam - sdbc). this equals the sum of the squared deviations from the class means .
after inspecting each of the sdbc, a decision is made to move one unit from the class with the largest sdbc toward the class with the lowest sdbc .
new class deviations are then calculated, and the process is repeated until the sum of the within class deviations reaches a minimal value .
a is the set of values that have been ordered from 1 to n .
mean i. . j is the mean of the class bounded by i and j .
when re - writing this, i had chosen to use the google charts api for our maps so, you should be able to use this to simply output a map assuming you provide all the necessary parameters .
this was kind of based on another script that was in french (which i don' t speak very well, at all). the original script had many issues and bugs and wasn' t as flexible as i wanted it to be. so, i re - wrote it .
this one works well and is flexible in terms of the datasets. it has been tested pretty thoroughly and, as far as i can tell, is correct .
need a chance to go over it again and make improvements and / or more comments .
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example 1: define 4 classes for the data in figure 1 which achieves this objective .
suppose, for example, that we simply divide the 160 elements in figure 1 into four intervals each containing 40 elements, as shown in figure 2 .
here, for example, cell q18 contains the formula = small ($ b $ 3: $ i $ 22, 40 * (p18 - 1) + 1), cell r18 contains the formula = small ($ b $ 3: $ i $ 22, 40 * p18) and cell s18 contains the formula = countifs ($ b $ 3: $ i $ 22, ” > = ” & q18, $ b $ 3: $ i $ 22, ” < = ” & r18). note that cell r3 should contain the 120 th smallest element in figure 1, which is 80. 7, but since 80. 7 is also the 121 st smallest element, this would split 80. 7 between interval 3 and interval 4. this is why we are forced to make interval 3 slightly smaller and interval 4 slightly bigger .
the squared deviation of each of the four intervals is found in range t17: t20, with the sum in cell t21. note that cell t17 contains the array formula
example 2: find the optimal partition of the values 5, 8, 9, 12, 15 into 3 classes .
since this is a simpler problem than the one in example 1, we are able to evaluate all six possible solutions, as shown in figure 4 .
in the first possible solution, shown in range b3: d7, the three classes are 5, 8 and 9 - 15. the squared deviation for the third category (cell d6), for example, is calculated by the formula = devsq (d3: d5), with a total for all three classes of 18, using the formula = sum (b6: d6) .
we see from figure 4 that the optimal partition is the second one, with classes 5, 8 - 9 and 12 - 15, since the total squared deviation of 5 is less than that of the other partitions .
where d * is the minimum total squared deviation, as described above, and d is the squared deviation of the data set, which for example 2 is 58. 8 (cell aa3) as calculated by the formula = devsq (f3: h5). gvf takes a value between 0 and 1. the closer to 1 the better the fit .
for example, there are 8, 301, 429, 675 ways to partition 254 data elements in 6 classes. this number gets even larger the more data elements there are and the more classes we want to use. in such cases, we will need to use an iterative approach. in particular, we select a partition at random and see which class has the largest squared deviation and which class has the least squared deviation .
if, for example, we used 6 classes and class 3 had the largest squared deviation and class 5 had the smallest squared deviation, then we would move the last element in class 3 into class 4 (in the direction of class 5). if instead, class 3 had the largest squared deviation and class 1 had the smallest squared deviation, then we would move the first element in class 3 into class 2. we repeat this process again and again, continuing until the squared deviations of none of the classes changed or until a maximum number of iterations had been reached, at which point we would select the partition with the lowest total squared deviations. in order to avoid a local minimum, from time to time we could make a random perturbation.
the jenks optimization method , also called the jenks natural breaks classification method , is a data clustering method designed to determine the best arrangement of values into different classes . this is done by seeking to minimize each class ’s average deviation from the class mean , while maximizing each class ’s deviation from the means of the other groups . in other words , the method seeks to reduce the variance within classes and maximize the variance between classes . the jenks optimization method is directly related to otsu 's method and fisher 's discriminant analysis . 
the jenks optimization method, also called the jenks natural breaks classification method, is a data clustering method designed to determine the best arrangement of values into different classes. this is done by seeking to minimize each class ’s average deviation from the class mean, while maximizing each class ’s deviation from the means of the other groups. in other words, the method seeks to reduce the variance within classes and maximize the variance between classes. the jenks optimization method is directly related to otsu's method and fisher's discriminant analysis.
the jenks optimization method, also called the jenks natural breaks classification method, is a data clustering method designed to determine the best arrangement of values into different classes. this is done by seeking to minimize each class ’s average deviation from the class mean, while maximizing each class ’s deviation from the means of the other groups. in other words, the method seeks to reduce the variance within classes and maximize the variance between classes. the jenks optimization method is directly related to otsu's method and fisher's discriminant analysis.