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Followup to my earlier post on Hilbert curve timeseries plots

Monday, August 24th, 2009

Note that the Hilbert code is not mine.  Rather it is from a Haskell Golf Challenge post in a pastebin and appropriated because I wanted a quick Hilbert curve without having to think through a hazy minded Friday evening.  Can’t find the original post now, actually.  That’ll teach me to bookmark things that are important.  Here’s the other person’s code.  If anyone can help me claim it, I will post your name as author.  Again, apologies in advance for that :-(.

h :: [Bool] -> ([Bool], [Bool])
h = l

go :: Bool -> Bool -> ([Bool], [Bool]) -> ([Bool], [Bool])
go x y (xs,ys) = (x:xs, y:ys)

l, r :: [Bool] -> ([Bool], [Bool])
l (False:False:ns) = go False False $ right (r ns)
l (False:True:ns) = go False True $ l ns
l (True:False:ns) = go True True $ l ns
l (True:True:ns) = go True False $ left (r ns)
l _ = ([], [])

r (False:False:ns) = go False True $ left (l ns)
r (False:True:ns) = go False False $ r ns
r (True:False:ns) = go True False $ r ns
r (True:True:ns) = go True True $ right (l ns)
r _ = ([], [])

left, right :: ([Bool], [Bool]) -> ([Bool], [Bool])
left (False:xs, False:ys) = go False True $ left (xs,ys)
left (False:xs, True:ys) = go True True $ left (xs,ys)
left (True:xs, False:ys) = go False False $ left (xs,ys)
left (True:xs, True:ys) = go True False $ left (xs,ys)
left _ = ([], [])

right (False:xs, True:ys) = go False False $ right (xs,ys)
right (True:xs, True:ys) = go False True $ right (xs,ys)
right (False:xs, False:ys) = go True False $ right (xs,ys)
right (True:xs, False:ys) = go True True $ right (xs,ys)
right _ = ([], [])

-- Infrastructure for testing:
bits :: Int -> Int -> [Bool]
bits n k = go n k [] where
  go 0 k = id
  go n k = go (n-1) (k `div` 2) . (odd k:)

num :: [Bool] -> Double
num (False:xs) = num xs / 2
num (True:xs) = (num xs + 1) / 2
num [] = 0

hilbert :: Int -> Int -> (Double, Double)
hilbert n k = (\(x,y) -> (num x, num y)) (h (bits n k))

--

Here begins my own code.  To use the Data.Colour.blend function, I need to normalize all the values.  Here is that function, which could be made considerably more efficient with a minimax instead of independently calling minimum and maximum, but again, the point here is illustration of a technique, not the most beautiful code.

normalize :: [Double] -> [Double]
normalize values = [(val-minval) / (maxval-minval) | val <- values]
    where minval = minimum values
          maxval = maximum values

Following that, we have a function and its helper for creating a hilbert plot of the data.  Note the use of the constant 64.  The Hilbert code above keeps everything within a unit vector of the origin, so we scale out for the resolution.  The resolution should properly be ceiling . log2 of the number of items in the list, which could be calculated efficiently, but it would clutter the code.

vis :: Int -> [Double] -> BaseVisual
vis n values = concat $ zipWith vis'
                [(64*x,64*y) | (x,y) <- (map (hilbert n) [0..2^n-1])]
                (normalize values)

Finally here is the visualization of a single point whose x,y vector is now the Hilbert point for its position in the timeseries.  We blend between two colors, blue for 0 and orange for 1. This could just as easily be a more complicated colourmap, but this is the code that generated the colormap from the previous post.

vis' :: (Double,Double) -> Double -> BaseVisual
vis' (x,y) val = fillcolour (blend val c0 c1)
               . filled True
               . outlined False
               $ arc{ center = Point x y, radius=0.5 }
    where c0 = opaque blue
          c1 = opaque orange
--

And finally the main program.  All we do here is take the filename from the arguments, read in the lines as Double values, and map those values to colored hilbert points using our vis function.

main = do
  name <- (!!0) <$> getArgs
  k <- map read . tail . words <$> readFile name
  let visualization = vis degree k
      degree = (ceiling $ log (realToFrac . length $ k) / log 2)
  renderToSVG (name ++ ".svg") 128 128 visualization

A quick buster example

Thursday, July 9th, 2009

Some people have asked of late for examples of Buster.  What follows is the (annotated) code that I used to generate the FDL video of the particles swarming around the lambda in the last post:

module Main where

import qualified Data.Array.IO as A
import Graphics.Rendering.Hieroglyph
import Graphics.Rendering.Hieroglyph.OpenGL
import App.EventBus
import IsoFDL
import Random
import Data.Colour
import Data.Colour.Names
import System.Random
import Control.Concurrent

dim = 480
minSeedCoord = fromIntegral $ quot dim 2 - 80
maxSeedCoord = fromIntegral $ quot dim 2 + 80
minIdealDist = (-100)
maxIdealDist = 100
npts = 100
basestrength = 4 

randomRIOs :: Double -> Double -> IO [Double]
randomRIOs a b = do
    g <- getStdGen
    return $ randomRs (a, b) g

This is the only behaviour we define, adjustLayout, and it runs a single layout iteration over the data.  We take the old locales of points, compute an FDL iteration, take the new locales of points, and produce the event, which is “Visible” (this is important, as Hieroglyph looks for all items in the Visible group, assumes they are o the class Visual, and attempts to render them.  We also produce a re-rendering request, and return the list of rerender and our layout as a future to be used at the appropriate time by the buster framework.  One final thing to note about our event is that it should live for 10 iterations.  Often geometry is persistent, but this was used to give the “contrail” effect to the particles as they move, effectively layering slightly transparent geometry over the top of older slightly transparent geometry to create the effect.

The Hieroglyph code is also fairly straightforward.  We apply linewidth and line color combinators (using the colours in Russell O’Connor’s excellent Data.Colour library) to a list of paths which are slightly exaggerated (that’s the +/-(x1-x0) and +/-(y1-y0) operations on the Points in the path) vectors along the path that was most recently traveled.

adjustLayout :: [Attractor] -> (Arr (Int,Int) Double) -> (Arr Int Double) -> (Arr Int Double) -> Behaviour [EData HieroglyphGLRuntime]
adjustLayout poles dists xs ys b = do
    pts <- externalizeLayout xs ys
    layoutIteration basestrength poles xs ys dists
    pts' <- externalizeLayout xs ys
    event' <- produce "Visible" "" (show . head $ pts') (Iterations 10)
                . (:[])
                . EOther
                . Geometry
                . linewidth 1
                . strokecolour (dissolve 0.3 (opaque darkblue))
                . name "points"
                $ zipWith (\(x0,y0) (x1,y1) -> path{ begin=Point (x0-(x1-x0)) (y0-(y1-y0)), segments=[Line $ Point (x1+(x1-x0)) (y1+(y1-y0))] }) pts pts'
    rerender <- produce "Hieroglyph" "" "Rerender" once []
    future b . return $ [event',rerender]

The next function, behaviour would normally be pure, but because we have some data that is internal to the adjustLayout behaviour, it’s got a do.  One of the things that I will change in the near future about Hieroglyph and OpenGL is to make the events it can respond to more polymorphic.  Currently I require that to use Hieroglyph in OpenGL that the event data type is [EData HieroglyphGLRuntime].  That’s pretty inane if I do say so myself, and in the future, there will be a constraint more along the lines of (HieroglyphInteractiveEventData a) => a rather than a static type to be determined at compile time.  However, with the current state of things, this is our behaviour, and it’s still pretty straightforward.  renderBehaviour depends on the result of adjustLayout.  That’s it.  The <~< establishes the dependency relationship and buster takes care of the rest.  The “poles” are a series of attractors (defined in IsoFDL.hs), which are static fields attached to a fixed point in space, attracting or repelling depending on the constant given them.

behaviour = do
    randomDistances <- randomRIOs minIdealDist maxIdealDist >>= A.newListArray ((0,0),(npts-1,npts-1)) . take (npts^2) . drop (2*npts)
    randomXs <- randomRIOs minSeedCoord maxSeedCoord >>= A.newListArray (0,npts-1) . take npts
    randomYs <- randomRIOs minSeedCoord maxSeedCoord >>= A.newListArray (0,npts-1) . take npts . drop npts

    poles <- mapM (\(x,y) -> compileAttractor . Attractor x y . take npts $ repeat 150)
        [(100,(440-100)), (170,(440-160)), (220,(440-220)), (160,(440-280)),
         (100,(440-320)), (280,(440-280)), (350,(440-320)), (380,(440-300))]

    return $ renderBehaviour <~< adjustLayout poles randomDistances randomXs randomYs

main = behaviour >>= boilerplateOpenGLMain  [initializeBus "Testing Force Directed layouts" dim dim]

Finally, our main function is incredibly simple.  We simply bind the behaviour to the boilerplate buster OpenGL main.  The other argument to boilerplateOpenGLMain is a list of “widgets” to be bound to the bus at the beginning of the application.

Heiroglyph: Interactively graph a spreadsheet in 99 lines of code.

Thursday, February 5th, 2009

screenshot-sparklines-for-breast-cancer-wisconsin-conttab-1

99 lines of code, even with imports. The app is simple, but it builds upon the sparkline example of the other day. In this app, we import an entire spreadsheet of tabular data and create sparklines for each column of data where the data is floating point. The user can sort the data by a column by clicking on the left side of a sparkline. This doesn’t do a lot of error checking, and only works on spreadsheets where the columnar data type is homogenous, but that’s a large class of datasets, and it simplifies our code considerably. First, our boring imports:

> module Main where
>
> import Graphics.Rendering.Hieroglyph
> import Graphics.Rendering.Hieroglyph.Scaffolds.Interactive
> import qualified Data.Map as Map
> import Data.Map (Map)
> import qualified Data.Set as Set
> import Data.Set (Set)
> import qualified Data.ByteString.Lazy.Char8 as C
> import Data.List (foldl',transpose,mapAccumL,sort)
> import System.Environment (getArgs)
> import Data.Maybe (fromJust)
> import qualified Data.IntMap as IntMap

Now we setup a bit of color for our visualization: attribute sets for describing thin black lines (the sparklines), thicker black lines (for the text), and the background color.

> blackstroke = plain{ strokeRGBA=black, linewidth=0.5 }
> blacktext = plain{ strokeRGBA=black }
> whitefill = plain{ fillRGBA=white, filled=True, outlined=False }

Then we setup our datatype that we’re visualizing. Note that we’re using the interactive scaffolding, so this will be an interactive application.

> data MyData = MyData {
>     interactiveScaffolding :: Scaffolding
>   , spreadsheet :: Map String [Double]
>   , ncolumns :: Int
>   }

Next we setup the data so that it’s responsive to the user input.

> instance Interactive MyData where
>     getInteractiveScaffolding = interactiveScaffolding
>     setInteractiveScaffolding a b = a{ interactiveScaffolding = b }
>
>     interactionFold dat geom

Yes, the 80 is arbitrary. It seems to work well, and to some extent, this is a draft application and isn’t meant to be perfect. Here, if the user clicks on the left side of the application, we check to see which sparkline the user is over and sort that column in ascending order.

>         | xcoord < 80 && getMouseLeftButtonDown dat = dat { spreadsheet = sortSpreadsheet dat }

Then we ignore all other inputs.

>         | otherwise = dat
>                 where Point xcoord ycoord = getMousePosition dat

Next is our familiar sparkline function from my blog post the other day. As a review, we remap all the values in the line to be inside the height of a sparkline and we take those and create a path out of them.

> sparkline width height (Point startx starty) values = path{ begin=point0 , segments=map Line points , attribs=blackstroke }
>    where (point0:points) = zipWith Point xvals yvals
>          xvals = iterate (+(width/n)) startx
>          yvals = map (remap mn mx starty (starty+height)) values
>          (mx,mn,_,_,_,n) = stats values

To visualize a whole spreadsheet, we start at the origin and work our way down using Map.mapAccum to create a visualization that is the same structure as our spreadsheet. We could simply pull the columns out of the spreadsheet and return a list of primitives instead of a map from name to primitives, but this way, we could filter the output from this function based on the structure of the original data. To me, this is closer to the intent of visualization; it’s less about the drawing aspect and more about the data, and manipulations on visualizations are based on the data, not the geometry or the screen.

> visSpreadsheet width height (Point startx starty) mp = snd . Map.mapAccum stackSparklines starty $ mp
>     where stackSparklines starty' values = (starty'+height, sparkline width height (Point startx starty') values)

This function places the names of the columns to the left of the sparklines.

> visSpreadsheetNames height starty names = snd . mapAccumL stackNames starty $ names
>     where stackNames starty' name = (starty'+height, text{ str=name, bottomleft = Point 5 starty', attribs=blacktext })

This merely defines our background rectangle

> background w h = rectangle{ width = w , height = h , attribs=whitefill }

And finally, vis calls our vis functions and also sets up the occlusion order, putting the names and sparklines on top of the background. The height of each sparkline is the number of sparklines divided by the height of the window.

> vis dat = visSpreadsheetNames (sy/ncf) 10 names
>           `over` visSpreadsheet (sx-offset) (sy/ncf) (Point offset 0) sheet
>           `over` background sx sy
>     where sx = getSizeX dat
>           sy = getSizeY dat
>           nc = ncolumns dat
>           ncf = fromIntegral nc
>           sheet = spreadsheet dat
>           Point xcoord ycoord = getMousePosition dat
>           offset = 6 * (fromIntegral . maximum . map length $ names)
>           names = Map.keys sheet

This is a simple function to read a spreadsheet and filter out columns that aren’t numeric. Note that it only checks the first item of each column, so it really doesn’t do sufficient error checking, however it’s faster than loading the entire spreadsheet to check to make sure each is numeric. Yes, we could have used Parsec here instead of defining our own isNumeric, but I wanted to keep down the number of libraries I was importing for tutorial’s sake.

> readSpreadsheet name = do
>      sheet <- (transpose . map (C.split '\t') . C.lines) `fmap` C.readFile name
>      return $ foldl' go Map.empty sheet
>   where go m (x:xs) | isNumeric xs = Map.insert (C.unpack x) (map (read . C.unpack) xs) m
>                     | otherwise = m
>         isNumeric = C.all isNumeral . head
>         isNumeral = (flip Set.member) nums
>         nums = Set.fromList "0123456789."

Remap a range of values to a different range. This is used in the sparkline function, and since our origin is at the top left and values go down, the seeming reversal of the first two arguments of the function is correct for our purposes.

> remap mx mn mn' mx' a = (mx'-mn') * (a-mn) / (mx-mn) + mn'

Once again, you should recognize this stats function from the blog post from the other day.

> stats (x:xs) = finish . foldl' stats' (x,x,x,x*x,1) $ xs
>     where stats' (mx,mn,s,ss,n) x = ( max x mx , min x mn , s + x , ss + x*x , n+1 )
>
> finish (mx,mn,s,ss,n) = (mx,mn,av,va,stdev,n)
>    where av = s/n
>          va = ss/(n-1) - n*av*av/(n-1)
>          stdev = sqrt va

This sorts the spreadsheet based on which sparkline the mouse pointer is on using the mapsort algorithm below.

> sortSpreadsheet dat = mapsort sheet (names !! item)
>     where sy = getSizeY dat
>           nc = ncolumns dat
>           ncf = fromIntegral nc
>           item = round $ (ycoord / sy) * ncf
>           Point _ ycoord = getMousePosition dat
>           names = Map.keys sheet
>           sheet = spreadsheet dat

This function is a little more elegant than it looks. It’s not terribly useful to do things this way here, but this is a lazy version of sorting the entire map. The more straightforward way to sort the map uses transpose to turn columns into rows, but also makes the whole thing head strict on the first element retrieved from the map. This is only head strict per column.

> mapsort mp key = Map.map reorder mp
>     where sort' vs = sort . zip vs $ indices
>           ordering = map snd . sort' $ (mp Map.! key)
>           nitems = (length ordering :: Int)
>           indices = iterate (+1) (0::Int)
>           reorder vs = (let arr = IntMap.fromList . zip indices $ vs in map (arr IntMap.!) ordering)

Now finally our very simple main. Take the argument from the command line that is our spreadsheet and create the visualization from it. Note that this GUI is not motion sensitive, as it really doesn’t need to be.

> main = do
>     [fname] <- getArgs
>     sheet <- readSpreadsheet fname
>     let dat = MyData scaffold sheet (Map.size sheet)
>     simpleGui dat vis ("Sparklines for " ++ fname)

Hieroglyph Haddock docs

Wednesday, February 4th, 2009

Since Hackage is building with GHC 6.10 now and we’re still waiting on a final release of Gtk for 6.10, it’s not generating the Haddock docs for Hieroglyph.  I’m posting them here for the time being. That’s all!

Introducing Hieroglyph. Purely functional 2D visualization on top of Cairo.

Tuesday, February 3rd, 2009

As a followup to my tutorial last year on Beautiful Code, Compelling Evidence, the tutorial on doing infovis and visual analytics in Haskell, I set about trying to do a pure-functional wrapper to Cairo.  After a month or so of banging my and Conal’s head against it, I am ready to (with many MANY thanks to Conal Elliott for his help in getting me to think about this in a semantic way) release a first pass at purely functional 2D drawing, called Hieroglyph.

My original goal was to come up with something that simply produced no IO when creating a scene and was fast enough to create interactive visualizations.  What I was eventually directed toward was semantic design and trying to figure out what a 2D visualization means. Hieroglyph is a first cut at that — a compromise between strict semantic design and staying close enough to Cairo to not make things easy in Hieroglyph that create performance problems in Cairo.  Future versions of Hieroglyph or forks will be less based on Cairo and more based on semantic design, but for now, I’m satisfied with what I have.

So the first question is: What is a visualization?  My answer is that it’s a function from data to a visual.  A visual is an unstructured collection (bag) of primitives.  The data can be any data that is interesting to the user doing the visualizing.

To encapsulate the Visual, we have:

type BaseVisual = [Primitive] 

class Visual a where
  primitives :: a -> BaseVisual

In Graphics.Rendering.Hieroglyph.Visual, I define instances on each of the basic collection types in the Haskell standard library, Visual v =>  Data.Set.Set v, Data.IntMap.IntMap v, Data.Map.Map a v,  [v], and v itself.  This allows the programmer to be unconcerned with how the collection of visual primitives is structured, allowing in many cases a straightforward map from data to primitives with the same structure.  In addition, there are two relationships between Visuals that are important to describe.  One is occlusion.  To describe the occlusion characteristics of two visuals, I have defined the combinator “a `over` b” to describe a composed visual in which a occludes b.  The other is specificity.  There is a concept in visualization of level-of-detail, and this is encapsulated by the combinator “a `moreSpecific` b”, which declares that a is more specific than b.  Both over and moreSpecific describe partial orderings over primitives, which can be used for filtering, or used by the renderer to order and constrain drawing.

Now for the primitives.  A primitive is nothing more than a declaration of attributed geometry.  Our primitives in Hieroglyph are Arc, Path, Rectangle, Text, Union, Image, and Hidden.  I chose to implement primitives as “labeled” functions using the Haskell record modification syntax.  While you can declare a primitive using the type constructor, it is easier and more expedient to use the lower case version of each of the primitive constructors and modify them using the record modification syntax.  The Haddock documentation describes what each record field is for every primitive.

Over the next few days, I’ll be giving examples of Hieroglyph and talking a little bit more about my design process as I do.  In the meantime, enjoy Hieroglyph and let me know about any bugs!