Simon Funk: Netflix provided a database of 100M ratings (1 to 5) of 17K movies by 500K users. as a triplet of numbers: (User,Movie,Rating).
The challenge: For (User,Movie,?) not in the database, predict how the given User would rate the given Movie.
Think of the data as a big sparsely filled matrix, with userIDs across the top and movieIDs down the side (or vice versa then transpose everything),
and each cell contains an observed rating (1-5) for that movie (row) by that user (column), or is blank meaning you don't know.
This matrix would have 8.5B entries, but you are only given values for 1/85th of those 8.5B cells (or 100M of them). The rest are all blank.
Netflix posed a "quiz" of a bunch of question marks plopped into previously blank slots, and your job is to fill in best-guess ratings in their place.
Squared error (se) measures accuracy (Your guess = 1.5, actual = 2, you get docked for (2-1.5)^2 or .25. They use root mean squared error (rmse),
but rmse and mse monotonically related.) There is a date for ratings and question marks (so a cell can potentially have >=1 rating in it.
Any movie can be described in terms of some aspects or attributes such as overall quality, action(y/n?), comedy(y/n?), stars, producer, etc.
Every user's preferences can be roughly described in terms of whether they tend to rate quality/action/comedy/star/producer/etc. high or low.
If true, then ratings ought to be explainable by a lot less than 8.5 billion numbers (e.g., a single number specifying how much action a particular
movie has may help explain why a few million action-buffs like that movie.)
SVD assumes rating(u,m) is sum of preferences about the various aspects. E.g., take 40 aspects - a movie, m, is described by 40 values, m(f),
saying how much that movie exemplifies that aspect, and a user is described by 40 values, u(f), saying how much they prefer each aspect.
A rating(u,m) = u(f) dot m(f) (40*(17K+500K) values =~20M << 8.5B.) or:
ratingsMatrix[user][movie] = sum (userFeature[f][user] * movieFeature[f][movie]) for f from 1 to 40 (or 1 to F in general)
R = UT o I
ru,i = u o i = f=1..F ru,f * rf,i
The original matrix has been decomposed to 2 oblong matrices:
17Kx40 movie aspect matrix, 500Kx40 user preference matrix.
T
SVD
is
a
trick for finding the 2 smaller matrices which minimize the
R i1 i
iTestSizeI U f1 f fF
I i1
i
iTestSizeI
resulting
approx error--specifically the mean squared error.
f1
u1
u1
So, if we take the rank=40 SVD of the 8.5B matrix, we have the best
o f
rf,i
(least error) approx we can within the limits of our user-movie.
:
fF
rating model. I.e., the SVD has found "best" generalizations.
ru,i
= u
ru,f
u
Take the derivative of the approx error and follow it. This has the bonus
.
:
that we can ignore the unknown error on the 8.4B empty slots.
Take the derivative of the equations for the error--just the given values,
not the empties--with respect to the parameters:
uTestSizeU
uTestSizeU
userValue[user]
+= lrate*err*movieValue[movie];
movieValue[movie] += lrate*err*userValue[user];
With Horizontal data, the code is evaluated for each rating. So, to train for one sample: real *userValue= userFeature[featureBeingTrained];
real *movieValue= movieFeature[featureBeingTrained]; real lrate = 0.001;
More correctly: ru,f = uv = userValue[user] += err * movieValue[movie];
rf,i = movieValue[movie] += err * uv;
finds the most prominent feature remaining (most reduces error).
When it's good, shift it onto done features, start a new one (cache residuals of the 100M. "What does that mean for us???).
This Gradient descent has no local minima, which means it doesn't really matter how it's initialized.
u+ = lrate ( u,i * iT -  * u ) where u,i = ru,i - r^u,i and r^u,i = actual rating
Refinements: Prior to starting SVD, Note: AvgRating(movie), AvgOffset(UserRating, MovieAvgRating), for every user. I.e.:
static inline
real predictRating_Baseline(int movie, int user)
{return averageRating[movie] + averageOffset[user];}
So, that's the return value of predictRating before the first SVD feature even starts training.
You'd think avg rating for a movie would just be... its average rating! Alas, Occam's razor was a little rusty that day.
If m only appears once with r(m,u)=1 say, AvgRating(m)=1? Probably not! View r(m,u)=1 as a draw from a true prob dist who's avg you want...
View that true average itself as a draw from a prob dist of averages--the histogram of average movie ratings. Assume both distributions Gaussian,
then the best-guess mean should be lin combo of observed mean and apriori mean, with a blending ratio equal to the ratio of variances.
If Ra and Va are the mean and variance (squared standard deviation) of all of the movies' average ratings (which defines your prior expectation for
a new movie's average rating before you've observed any actual ratings) and Vb is the average variance of individual movie ratings
(which tells you how indicative each new observation is of the true mean--e.g,. if the average variance is low, then ratings tend to be near
the movie's true mean, whereas if the avg variance is high, ratings tend to be more random and less indicative) then:
BogusMean = sum(ObservedRatings)/count(ObservedRatings) K = Vb/Va
BetterMean = [GlobalAverage*K + sum(ObservedRatings)] / [K + count(ObservedRatings)]
The point here is simply that any time you're averaging a small number of examples, the true average is most likely nearer the apriori average
than the sparsely observed average. Note if the number of observed ratings for a particular movie is zero, the BetterMean (best guess)
above defaults to the global average movie rating as one would expect.
Moving on: 20M free params is a lot for a 100M TrainSet. Seems neat to just ignore all blanks, but we have expectations about them.
As-is, this modified SVD algorithm tends to make a mess of sparsely observed movies or users. If you have a user who has only rated 1 movie,
say American Beauty=2 while the avg is 4.5, and further that their offset is only -1, we'd, prior to SVD, expect them to rate it 3.5.
So the error given to the SVD is -1.5 (the true rating is 1.5 less than we expect).
m(Action) is training up to measure the amount of Action, say, .01 for American Beauty (ust slightly more than avg). SVD optimize predictions,
which it can do by eventually setting our user's preference for Action to a huge -150. I.e., the alg naively looks at the only example it has
of this user's preferences and in the context of only the one feature it knows about so far (Action), determines that our user so hates action
movies that even the tiniest bit of action in American Beauty makes it suck a lot more than it otherwise might. This is not a problem for
users we have lots of observations for because those random apparent correlations average out and the true trends dominate.
We need to account for priors. As with the average movie ratings, blend our sparse observations in with some sort of prior, but it's a little less clear
how to do that with this incremental algorithm. But if you look at where the incremental algorithm theoretically converges, you get:
userValue[user] = [sum residual[user,movie]*movieValue[movie]] / [sum (movieValue[movie]^2)]
The numerator there will fall in a roughly zero-mean Gaussian distribution when charted over all users, which through various gyrations:
userValue[user] = [sum residual[user,movie]*movieValue[movie]] / [sum (movieValue[movie]^2 + K)]
userValue[user] += lrate * (err * movieValue[movie] - K * userValue[user]);
movieValue[movie] += lrate * (err * userValue[user] - K * movieValue[movie]);
And finally back to:
This is equivalent to penalizing the magnitude of the features. To cut over fitting, allowing use of more features.
Moving on: Linear models are limiting. We've bastardized the whole matrix analogy so much that we aren't really restricted to linear models:
We can add non-linear outputs such that instead of predicting with: sum (userFeature[f][user] * movieFeature[f][movie]) for f from 1 to 40.
We can use: sum G(userFeature[f][user] * movieFeature[f][movie]) for f from 1 to 40.
Two choices for G proved useful. 1. clip the prediction to 1-5 after each component is added. E.g., each feature is limited to only swaying rating
within the valid range, and any excess beyond that is lost rather than carried over. So, if the first feature suggests +10 on a scale of 1-5,
and the second feature suggests -1, then instead of getting a 5 for the final clipped score, it gets a 4 because the score was clipped after
each stage. The intuitive rationale here is that we tend to reserve the top of our scale for the perfect movie, and the bottom for one with
no redeeming qualities whatsoever, and so there's a sort of measuring back from the edges that we do with each aspect independently.
More pragmatically, since the target range has a known limit, clipping is guaranteed to improve our perf, and having trained a stage with
clipping on, use it with clipping on. I did not really play with this extensively enough to determine there wasn't a better strategy.
A second choice for G is to introduce some functional non-linearity such as a sigmoid. I.e., G(x) = sigmoid(x). Even if G is fixed, this requires
modifying the learning rule slightly to include the slope of G, but that's straightforward. The next question is how to adapt G to the data. I
tried a couple of options, including an adaptive sigmoid, but the most general and the one that worked the best was to simply fit a
piecewise linear approximation to the true output/output curve. That is, if you plot the true output of a given stage vs the average target
output, the linear model assumes this is a nice 45 degree line. But in truth, for the first feature for instance, you end up with a kink around
the origin such that the impact of negative values is greater than the impact of positive ones. That is, for two groups of users with
opposite preferences, each side tends to penalize more strongly than the other side rewards for the same quality. Or put another way,
below-average quality (subjective) hurts more than above-average quality helps. There is also a bit of a sigmoid to the natural data
beyond just what is accounted for by the clipping. The linear model can't account for these, so it just finds a middle compromise; but
even at this compromise, the inherent non-linearity shows through in an actual-output vs. average-target-output plot, and if G is then
simply set to fit this, the model can further adapt with this new performance edge, which leads to potentially more beneficial nonlinearity and so on... This introduces new free parameters and encourages over fitting especially for the later features which tend to
represent small groups. We found it beneficial to use this non-linearity only for the first twenty or so features and to disable it after that.
Moving on: Despite the regularization term in the final incremental law above, over fitting remains a problem. Plotting the progress over time,
the probe rmse eventually turns upward and starts getting worse (even though the training error is still inching down). We found that
simply choosing a fixed number of training epochs appropriate to the learning rate and regularization constant resulted in the best overall
performance. I think for the numbers mentioned above it was about 120 epochs per feature, at which point the feature was considered
done and we moved on to the next before it started over fitting. Note that now it does matter how you initialize the vectors: Since we're
stopping the path before it gets to the (common) end, where we started will affect where we are at that point. I wonder if a better
regularization couldn't eliminate overfitting altogether, something like Dirichlet priors in an EM approach--but I tried that and a few
others and none worked as well as the above.
Here is the probe and training rmse for the first few features with and w/o regularization term "decay" enabled.
Same thing, just the probe set rmse, further along where you can see the regularized version pulling ahead:
This time showing probe rmse (vertical) against train rmse (horizontal). Note how the regularized version has better probe performance relative to
the training performance:
Anyway, that's about it. I've tried a few other ideas over the last couple of weeks, including a couple of ways of using the date information, and
while many of them have worked well up front, none held their advantage long enough to actually improve the final result.
If you notice any obvious errors or have reasonably quick suggestions for better notation or whatnot to make this explanation more clear, let me
know. And of course, I'd love to hear what y'all are doing and how well it's working, whether it's improvements to the above or something
completely different. Whatever you're willing to share,
//=======================================================
// SVD Sample Code (C) 2007 Timely Development (www.timelydevelopment.com)
// STANDARD DISCLAIMER:
// - THIS CODE AND INFORMATION IS PROVIDED "AS IS" WITHOUT WARRANTY
// - OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING BUT NOT
// - LIMITED TO THE IMPLIED WARRANTIES OF MERCHANTABILITY AND/OR
// - FITNESS FOR A PARTICULAR PURPOSE.
//====================================================
#define WIN32_LEAN_AND_MEAN
#include <windows.h>
#include <stdio.h>
#include <math.h>
#include <tchar.h>
#include <map>
using namespace std;
//================================================
// Constants and Type Declarations
//================================================
#define TRAINING_PATH
L"C:
etflix raining_set*.txt"
#define TRAINING_FILE
L"C:
etflix raining_set\%s"
#define FEATURE_FILE
L"C:
etflixfeatures.txt"
#define TEST_PATH
L"C:
etflix\%s"
#define PREDICTION_FILE L"C:
etflixprediction.txt"
#define
#define
#define
#define
#define
#define
MAX_RATINGS
MAX_CUSTOMERS
MAX_MOVIES
MAX_FEATURES
MIN_EPOCHS
MAX_EPOCHS
00480508
480190
17771
64
120
200
#define
#define
#define
#define
MIN_IMPROVEMENT
INIT
LRATE
K
0.0001
0.1
0.001
0.015
//Ratings in entire training set (+1)
//Custs in entire training set (+1)
//Movies in entire training set (+1)
//Number of features to use
//Min number of epochs per feature
// Max epochs per feature
//Min improve req cont current feature
// Initialization value for features
// Learning rate parameter
// Reg param to min over-fitting
typedef unsigned char BYTE;
typedef map<int, int> IdMap;
typedef IdMap::iterator IdItr;
struct Movie
{
int
int
double
double
};
RatingCount;
RatingSum;
RatingAvg;
PseudoAvg; //Wtd avg to deal with small movie counts
struct Customer
{
int
int
int
};
struct Data
{
int
short
BYTE
float
};
CustomerId;
RatingCount;
RatingSum;
CustId;
MovieId;
Rating;
Cache;
class Engine
{
private:
int
m_nRatingCount;
// Current number of loaded ratings
Data
m_aRatings[MAX_RATINGS];
//Array of ratings data
Movie
m_aMovies[MAX_MOVIES];
//Array of movie metrics
Customer m_aCustomers[MAX_CUSTOMERS];
//Array of customer metrics
float
m_aMovieFeatures[MAX_FEATURES][MAX_MOVIES]; //Array of feat by mov
float
m_aCustFeatures[MAX_FEATURES][MAX_CUSTOMERS];//Array feas by cust
IdMap
m_mCustIds;
// Map for one time translation of ids to compact array index
inline double PredictRating(short movieId, int custId, int feature, float cache, bool bTrailing=true);
inline double PredictRating(short movieId, int custId);
bool
bool
bool
ReadNumber(wchar_t* pwzBufferIn, int nLength, int &nPosition, wchar_t* pwzBufferOut);
ParseInt(wchar_t* pwzBuffer, int nLength, int &nPosition, int& nValue);
ParseFloat(wchar_t*pwzBuffer, int nLength,int &nPosition,float& fValue);
public:
Engine(void);
~Engine(void) { };
void
void
void
void
void
CalcMetrics();
CalcFeatures();
LoadHistory();
ProcessTest(wchar_t* pwzFile);
ProcessFile(wchar_t* pwzFile);
};
//=============
// Program Main
int _tmain(int argc, _TCHAR* argv[])
{
Engine* engine = new Engine();
engine->LoadHistory();
engine->CalcMetrics();
engine->CalcFeatures();
engine->ProcessTest(L"qualifying.txt");
wprintf(L"
Done
");
getchar();
return 0;
}
//=====================================
// Engine Class
// Initialization
Engine::Engine(void)
{
m_nRatingCount = 0;
for (int f=0; f<MAX_FEATURES; f++)
{
for (int i=0; i<MAX_MOVIES; i++) m_aMovieFeatures[f][i] = (float)INIT;
for (int i=0; i<MAX_CUSTOMERS; i++) m_aCustFeatures[f][i] = (float)INIT;
}
}
//-----------------------------------------// Calculations - This Paragraph contains all of the relevant code
// CalcMetrics
// - Loop through the history and pre-calculate metrics used in the training
// - Also re-number the customer id's to fit in a fixed array
void Engine::CalcMetrics()
{
int i, cid;
IdItr itr;
wprintf(L"
Calculating intermediate metrics
");
// Process each row in the training set
for (i=0; i<m_nRatingCount; i++)
{
Data* rating = m_aRatings + i;
// Increment movie stats
m_aMovies[rating->MovieId].RatingCount++;
m_aMovies[rating->MovieId].RatingSum += rating->Rating;
// Add customers (using a map to re-number id's to array indexes)
itr = m_mCustIds.find(rating->CustId);
if (itr == m_mCustIds.end())
{
cid = 1 + (int)m_mCustIds.size();
// Swap sparse id for compact one
rating->CustId = cid;
m_aCustomers[cid].RatingCount++;
m_aCustomers[cid].RatingSum += rating->Rating;
}
// Do a follow-up loop to calc movie averages
for (i=0; i<MAX_MOVIES; i++)
{
Movie* movie = m_aMovies+i;
movie->RatingAvg = movie->RatingSum / (1.0 * movie->RatingCount);
movie->PseudoAvg = (3.23 * 25 + movie->RatingSum) / (25.0 + movie->RatingCount);
}
}
// CalcFeatures - Iteratively train each feature on entire data set
//
- Once sufficient progress has been made, move on
void Engine::CalcFeatures()
{
int f, e, i, custId, cnt = 0;
Data* rating;
double err, p, sq, rmse_last, rmse = 2.0;
short movieId;
float cf, mf;
for (f=0; f<MAX_FEATURES; f++)
{
wprintf(L" --- Calculating feature: %d --- ", f);
// Keep looping until you have passed a minimum number
// of epochs or have stopped making significant progress
for (e=0; (e < MIN_EPOCHS) || (rmse <= rmse_last - MIN_IMPROVEMENT); e++)
{
cnt++;
sq = 0;
rmse_last = rmse;
for (i=0; i<m_nRatingCount; i++)
{
rating = m_aRatings + i;
movieId = rating->MovieId;
custId = rating->CustId;
// Predict rating and calc error
p = PredictRating(movieId, custId, f, rating->Cache, true);
err = (1.0 * rating->Rating - p);
sq += err*err;
// Reserve new id and add lookup
m_mCustIds[rating->CustId] = cid;
// Cache off old feature values
cf = m_aCustFeatures[f][custId];
mf = m_aMovieFeatures[f][movieId];
// Store off old sparse id for later
m_aCustomers[cid].CustomerId = rating->CustId;
// Init vars to zero
m_aCustomers[cid].RatingCount = 0;
m_aCustomers[cid].RatingSum = 0;
}
else
{
cid = itr->second;
}
// Cross-train the features
m_aCustFeatures[f][custId] += (float)(LRATE * (err * mf - K * cf));
m_aMovieFeatures[f][movieId] += (float)(LRATE * (err * cf - K * mf));
rmse = sqrt(sq/m_nRatingCount);
wprintf(L"
<set x='%d' y='%f' />
}
",cnt,rmse);
// Cache off old predictions
for (i=0; i<m_nRatingCount; i++)
{
rating = m_aRatings + i;
rating->Cache = (float)PredictRating(rating->MovieId, rating->CustId, f, rating->Cache, false);
}
}
}
// PredictRating - During training there is no need to loop through all of the features
// - Use a cache for the leading features and do a quick calculation for the trailing
// - The trailing can be optionally removed when calculating a new cache value
double Engine::PredictRating(short movieId, int custId, int
feature, float cache, bool bTrailing)
{
// Get cached value for old features or default to an average
double sum = (cache > 0) ? cache : 1; //m_aMovies[movieId].PseudoAvg;
//Add contribution of current feature
sum += m_aMovieFeatures[feature][movieId] *
m_aCustFeatures[feature][custId];
if (sum > 5) sum = 5;
if (sum < 1) sum = 1;
// Add up trailing defaults values
if (bTrailing)
{
sum += (MAX_FEATURES-feature-1) * (INIT * INIT);
if (sum > 5) sum = 5;
if (sum < 1) sum = 1;
}
return sum;
}
// PredictRating - This version is used for calculating the final results
// - It loops through the entire list of finished features
double Engine::PredictRating(short movieId, int custId)
{
double sum = 1; //m_aMovies[movieId].PseudoAvg;
for (int f=0; f<MAX_FEATURES; f++)
{
sum += m_aMovieFeatures[f][movieId] *
m_aCustFeatures[f][custId];
if (sum > 5) sum = 5;
if (sum < 1) sum = 1;
}
return sum;
}
// Data Loading / Saving
// LoadHistory
// - Loop through all of the files in the training directory
void Engine::LoadHistory()
{
WIN32_FIND_DATA FindFileData;
HANDLE hFind;
bool bContinue = true;
// Loop through all of the files in the training directory
hFind = FindFirstFile(TRAINING_PATH, &FindFileData);
if (hFind == INVALID_HANDLE_VALUE) return;
while (bContinue)
{
this->ProcessFile(FindFileData.cFileName);
bContinue = (FindNextFile(hFind, &FindFileData) != 0);
//if (++count > 999) break; // TEST: Uncomment to only test with the first X movies
}
FindClose(hFind);
}
// ProcessFile Load a history: <MovieId>:<CustomerId>,<Rating> <CustomerId>,<Rating>...
void Engine::ProcessFile(wchar_t* pwzFile)
{
FILE *stream;
wchar_t pwzBuffer[1000];
wsprintf(pwzBuffer,TRAINING_FILE,pwzFile);
int custId, movieId, rating, pos = 0;
wprintf(L"Processing file: %s ", pwzBuffer);
if (_wfopen_s(&stream, pwzBuffer, L"r") != 0) return;
// First line is the movie id
fgetws(pwzBuffer, 1000, stream);
ParseInt(pwzBuffer, (int)wcslen(pwzBuffer), pos, movieId);
m_aMovies[movieId].RatingCount = 0;
m_aMovies[movieId].RatingSum = 0;
// Get all remaining rows
fgetws(pwzBuffer, 1000, stream);
while ( !feof( stream ) )
{
pos = 0;
ParseInt(pwzBuffer, (int)wcslen(pwzBuffer), pos, custId);
ParseInt(pwzBuffer, (int)wcslen(pwzBuffer), pos, rating);
m_aRatings[m_nRatingCount].MovieId = (short)movieId;
m_aRatings[m_nRatingCount].CustId = custId;
m_aRatings[m_nRatingCount].Rating = (BYTE)rating;
m_aRatings[m_nRatingCount].Cache = 0;
m_nRatingCount++;
fgetws(pwzBuffer, 1000, stream);
}
// Cleanup
fclose( stream );
}
// ProcessTest - Load a sample set in the following format
// <Movie1Id>: <CustomerId> <CustomerId> ... <Movie2Id>: <CustomerId>
// - And write results: <Movie1Id>: <Rating> <Raing>
...
void Engine::ProcessTest(wchar_t* pwzFile)
{
FILE *streamIn, *streamOut;
wchar_t pwzBuffer[1000];
int custId, movieId, pos = 0;
double rating;
bool bMovieRow;
wsprintf(pwzBuffer, TEST_PATH, pwzFile);
Processing test: %s ", pwzBuffer);
if (_wfopen_s(&streamIn, pwzBuffer, L"r") != 0) return;
if (_wfopen_s(&streamOut, PREDICTION_FILE, L"w") != 0) return;
fgetws(pwzBuffer, 1000, streamIn);
while ( !feof( streamIn ) )
{
bMovieRow = false;
for (int i=0; i<(int)wcslen(pwzBuffer); i++)
{
bMovieRow |= (pwzBuffer[i] == 58);
}
pos = 0;
if (bMovieRow)
{
ParseInt(pwzBuffer, (int)wcslen(pwzBuffer), pos, movieId);
// Find start of number
while (start < nLength)
{
wc = pwzBufferIn[start];
if ((wc >= 48 && wc <= 57) || (wc == 45)) break;
start++;
}
// Copy each character into the output buffer
nPosition = start;
while(nPosition<nLength&&((wc>=48&&wc<=57)||wc==69 || wc==101 || wc==45 || wc==46))
{
pwzBufferOut[count++] = wc;
wc = pwzBufferIn[++nPosition];
}
// Null terminate and return
pwzBufferOut[count] = 0;
return (count > 0);
// Write same row to results
fputws(pwzBuffer,streamOut);
}
else
{
ParseInt(pwzBuffer, (int)wcslen(pwzBuffer), pos, custId);
custId = m_mCustIds[custId];
rating = PredictRating(movieId, custId);
// Write predicted value
swprintf(pwzBuffer,1000,L"%5.3f ",rating);
fputws(pwzBuffer,streamOut);
}
//wprintf(L"Got Line: %d %d %d ", movieId, custId, rating);
fgetws(pwzBuffer, 1000, streamIn);
}
// Cleanup
fclose( streamIn );
fclose( streamOut );
}
bool Engine::ParseFloat(wchar_t* pwzBuffer, int nLength, int &nPosition, float& fValue)
{
wchar_t pwzNumber[20];
bool bResult = ReadNumber(pwzBuffer, nLength, nPosition, pwzNumber);
fValue = (bResult) ? (float)_wtof(pwzNumber) : 0;
return false;
}
bool Engine::ParseInt(wchar_t* pwzBuffer, int nLength, int &nPosition, int& nValue)
{
wchar_t pwzNumber[20];
bool bResult = ReadNumber(pwzBuffer, nLength, nPosition, pwzNumber);
nValue = (bResult) ? _wtoi(pwzNumber) : 0;
return bResult;
}
}
//---------------------------------------// Helper Functions
//---------------------------------------bool Engine::ReadNumber(wchar_t* pwzBufferIn, int nLength, int &nPosition, wchar_t* pwzBufferOut)
{
int count = 0;
int start = nPosition;
wchar_t wc = 0;
Maximizing theVariance
Xod=Fd(X)=DPPd(X)
dn
j
(Xj2 - Xj2)dj2 +(2
+ j=1..n<k=1..n(XjXk - XjXk)djdk )
o
A
j=1..n
dT
d1 ... dn
o
i
XiXj-XiX,j
:
subject to i=1..ndi2=1
d = V(d)
d1
:
V
dn
We can separate out the diagonal
or not:
V(d)=jajjdj2 + jkajkdjdk
V(d) =
ijaijdidj
d1≡(V(d0));
 d0, one can hill-climb it to locally
maximize the variance, V, as follows: d2≡(V(d1)):... where
ora d
V(d)= 2a11d1 +j1
1j j
2a22d2 +j2a2jdj
:
2anndn +jnanjdj
Ubhaya Theorem1:
Nod
and XiXj = Mean Mi1 Mj1 classes) and are instantaneous. Once we have
.
the matrix A, we can hill-climb to obtain a d
FAUST Classifier MVDI
:
that maximizes the variance of the dot product
(Maximized Variance
MiC MjC
projections of the class means.
Definite Indefinite:
- "
=
=x
M1
For Dot Product Gap based Classification, we can start
M2
with X = the table of the C Training Set Class Means,
:
where Mk≡MeanVectorOfClassk .
MC
Then Xi = Mean(X)i and
These computations are O(C) (C=number of
V(d)≡VarianceXod=(Xod)2 - (Xod)2
2
= 1
( x d )2 - ( j=1..nXj dj )
N i=1..N j=1..n i,j j
= N1  (j xi,jdj)(k xi,kdk) - ( Xj dj)( Xk dk)
i
j
k
2 2 2
2 2
1
=  jxi,j dj +Nj<k xi,jxi,kdjdk - jXj dj +2j<k XjXkdjdk
N i
=  Xj2 dj2 +2j<kXjXkdjdk
x1od
x2od
d1
x1
x2
:
xN
Given any table, X(X1, ..., Xn), and any
unit vector, d, in n-space, let
V(d)≡Gradient(V)=2Aod
2a11 2a12 ... 2a1n
2a21 2a22 ... 2a2n
:
'
2an1
... 2ann
d1
:
di
:
dn
 k{1,...,n} s.t. d=ek will hill-climb V to its globally max.
Theorem2 (working on it):
Let d=ek s.t. akk is a maximal diagonal element of A,
d=ek will hill-climb V to its globally maximum.
How do we use this theory?
For Dot Product gap based Clustering, we can hill-climb akk
below to a d that gives us the global maximum variance.
Heuristically, higher variance means more prominent gaps.
Build a Decision tree. 1. Find d that maximizes variance of dot product
projections of class means each round. 2. Apply DI each round
FAUST technology relies on:
1. a distance dominating functional, F.
2. Use of gaps in range(F) to separate.
For Unsupervised (Clustering) Hierarchical Divisive? Piecewise
Linear? other? Perf Anal (which approach is best for which type of table?)
For Supervised (Classification), Decision Tree? Nearest Nbr?
Piecewise Linear? Perf Anal (which is best for training set?)
White papers: Terabyte Head Wall. The Only Good Data is Data in Motion
Multilevel pTrees: k=0,1 suffices! A PTreeSet is defined by specifying a table, an
array of stride_lengths (usually equi-length so just that one length is specified) and
a stride_predicate (T\F condition on a stride (stride=bag [or array?] of bits):
So the metadata of PTreeSet(T,sl,sp) specifies T, sl and sp.
A “raw”
PTreeSet has sl=1 and the identity predicate (sl and sp not used).
A “cooked” PTreeSet (AKA Level-1 PTreeSet) for a table with sl1
(main purpose: provide compact summary information on the table.)
Let PTS(T) be a raw PTreeSet, then it, plus PTS(T,64,p), ..., PTS(T,64^k,p) form a
tree of vertical summarizations of T.
Note that P(T, 64*64, p) is different from P(P(T,64,p), 64, p), but both make sense
since P(t, 64, p) is a table and P(P(T, 64, p), 64, p) is just a cooked pTree on it.
FAUST MVDI
on IRIS 15 records from each Class for Testing (Virg39 was removed as an outlier.)
s-Mean 50.49 34.74 14.74 2.43
e-Mean 63.50 30.00 44.00 13.50
i-Mean 61.00 31.50 55.50 21.50
(-1, 16.5=avg{23,10})s sCt=50
(16.5, 38)e eCt=24
i-Mean 62.8 29.2 46.1 14.5
e-Mean 59
26.9 49.6 18.4
(-1,8)e Ct=21
Definite_____
s
-1
10
e
23
48
i
38
70
Indefinite
s_ei
se_i
23
38
10
48
indef[38, 48]se_i seCt=26 iCt=13
Definite
i
-1
e
10
8
17
indef[8,10]e_i eCt=5 iCt=4
empty
(48.128)i iCt=39 d=(.33, -.1, .86, .38)
Indefinite
i_e
8
10
empty
(10,128)i Ct=9
d=(-.55, -.33, .51, .57)
In this case, since the indefinite interval is so narrow, we absorb it into the two definite intervals; resulting in decision tree:
Setosa
Versicolor
38  xod0  48
d0=(.33, -.1, .86,.38)
Virginica
d1=(-.55, -.33, .51, .57)
Versicolor
Virginica
FAUST MVDI SatLog 413train 4atr 6cls 127test
Gradient Hill Climb of Variance(d)
d1
d2
d3
d4
Vd)
0.00
0.00
1.00
0.00
282
mn2 49 40
0.13
0.38
0.64
0.65
700
mn5 58 58
0.20
0.51
0.62
0.57
742
mn7 69 77
0.26
0.62
0.57
0.47
781
mn4 78 91
0.30
0.70
0.53
0.38
810
mn1 67 103
0.34
0.76
0.48
0.30
830
0.36
0.79
0.44
0.23
841
mn3 89 107
0.37
0.81
0.40
0.18
847
0.38
0.83
0.38
0.15
850
0.39
0.84
0.36
0.12
852
0.39
0.84
0.35
0.10
853
Gradient Hill Climb of Var(d)on t25
d1
d2
d3
d4
Vd)
0.00
0.00
0.00
1.00
1137
-0.11 -0.22
0.54
0.81
1747
MNod Ct ClMn ClMx ClMx+1
mn2 45 33 115 124 150 54 102 177
178
mn5 55 52 72 59 69 33
45
88
89
Gradient Hill Climb of Var(d)on t257
0.00
0.00
1.00
0.00
496
-0.15 -0.29
0.56
0.76
1595
Same using class means or training subset.
115
76
81
96
114
112
119
64
64
74
94
88
Fomn Ct
106 108
108
61
131 154
152
60
167
27
178 155
min
91
92
104
127
118
157
max max+1
155 156
145 146
160 161
178 179
189 190
206 207
Using class means: FoMN
Ct
mn4 83 101 104 82
113
8
mn3 85 103 108 85
117
79
mn1 69 106 115 94
133
12
Using full data: (much better!)
mn4 83 101 104 82
59
8
mn3 85 103 108 85
62
79
mn1 69 106 115 94
81
12
min
110
105
123
max max+1
121
122
128
129
148
149
56
52
73
65
74
95
66
75
96
d=(0.39 0.89 0.35 0.10 )
F[a,b) 0
Class
2
d=(-.11 -.22 .54
92 104
2
2
5
5
.81)
7
F[a,b) 89 102
Class
5 2
d=(-.15 -.29 .56 .76)
F[a,b) 47
Class
7
5
65 81 101
5
2
2
118
2
5
7
1
127
2
5
7
1
4
146
2
156
157
161
179
7
1
4
7
1
4
7
1
4
3
1
4
3
1
3
190
3
d=(-.81, .17, .45, .33)
F[a,b) 21 35 41 59
Class
3
1
d=(-.66, .19, .47, .56)
F[a,b) 52 56 66 73 75
Class 3 3 3 3
4
1 1
cl=4
Gradient Hill Climb of Var(d)on t75
cl=7
0.00
0.00
1.00
0.00
12
d=(-.01, -.19, .7, .69)
Cl=7
0.04 -0.09
0.83
0.55
20
F[a,b) 57 61 69 87
-0.01 -0.19
0.70
0.69
21
Class
5
Gradient Hill Climb of Var(d)on t13
0.00
0.00
1.00
0.00
29
7
-0.83
0.17
0.42
0.34
166
On the 127 sample SatLog TestSet: 4 errors or 96.8% accuracy.
0.00
0.00
1.00
0.00
25
-0.66
0.14
0.65
0.36
81
speed? With horizontal data, DTI is applied one unclassified sample at a time (per execution thread).
-0.81
0.17
0.45
0.33
88
Gradient Hill Climb of Var(d)on t143 With this pTree Decision Tree, we take the entire TestSet (a PTreeSet), create the various dot product
0.00
0.00
1.00
0.00
19
-0.66
0.19
0.47
0.56
95
SPTS (one for each inode), create ut SPTS Masks. These masks mask the results for the entire TestSet.
0.00
0.00
1.00
0.00
27
-0.17
0.35
0.75
0.53
54
-0.32
0.36
0.65
0.58
57
-0.41
0.34
0.62
0.58
58
For WINE:
min max+1
Gradient Hill Climb of Var t156161
8.40 10.33 27.00
9.63 28.65
9.9
53.4
0.00
0.00
1.00
0.00
5
7.56 11.19 32.61 10.38 34.32
7.7 111.8
-0.23 -0.28
0.89
0.28
19
8.57 12.84 30.55 11.65 32.72
8.7 108.4
-0.02 -0.06
0.12
0.99
157
8.91 13.64 34.93 11.97 37.16
13.1
92.2
Awful results!
0.02 -0.02
0.02
1.00
159
0.00
0.00
1.00
0.00
1
-0.46 -0.53
0.57
0.43
2
Inconclusive both ways so predict
purality=4(17) (3ct=3 tct=6
Gradient Hill Climb of Var t146156
0.00
0.00
1.00
0.00
0
0.03 -0.08
0.81 -0.58
1
0.00
0.00
1.00
0.00
13
0.02
0.20
0.92
0.34
16
0.02
0.25
0.86
0.45
17
Inconclusive both ways so predict
purality=4(17) (7ct=15
Gradient2ct=2
Hill Climb of Var t127
0.00
0.00
1.00
0.00
41
-0.01 -0.01
0.70
0.71
90
-0.04 -0.04
0.65
0.75
91
0.00
0.00
1.00
0.00
35
-0.32 -0.14
0.59
0.73
105
Inconclusive predict purality=7(62
4(15) 1(5) 2(8) 5(7)
FAUST MVDI
d0= -0.34 -0.16 0.81 -0.45
Concrete
xod0<320
For Concrete
min max+1
335.3 657.1
120.5 611.6
321.1 633.5
Test
******
******
******
3.0
3.0
28.0
57.0
361.0
92.0
0
11
0
0
2
0
92
999
*****
*****
*****
d3
547.9 860.9
617.1 957.3
762.5 867.7
*******
*******
.
*******
d2
544.2
515.7
591.0
******
******
******
train
0 l
12 m
0 h
0 l
1 m
0 h
0
321
651.5
661.1
847.4
l
m
h
l
m
h
d1= .85 -.03 .52 -.02
Class=m (test:1/1)
xod2<28
Class= l or m
d3= .81 .04 .58 .01
xod3<969
Cl=l *test 6/9)
Class=l (test:1/1)
xod2>=92
d2= .85 -.00 .53 .05
Class=m (test:2/2)
xod2>=662
xod3>=868
Cl=h (test:11/12)
Cl=m (test:1/1)
xod3<544
d4 = .79 .14 .60 .03
Cl=m *test 0/0)
xod4>=681
xod4<640
Cl=l (test:0/3)
Cl=l *test 2/2)
4 l
0 m
0 h
0 l
0 m
0 h
0
617
0
0
40
1
0
11
662
999
xod0>=634
7 test errors / 30 = 77%
l
m
h
l
m
h
.97 .17 -.02
13.3
19.3
16.4
23.5
12.2
15.2
0
13.2
.15
0
0
25
19.3
d0
0 l
0 m
5 h
23.5
e
r
r
s
:
0
xod<13.2
/
5
Class=h
e )
r
r
s
:
0
/
5
Class=h )
e
r
r
s
:
0
/
1
)
Class=l
e
r
8 test errors / 32 = 75%
xod>=19.3
xod<13.2
Class=h
Seeds
.97 .19
.08
13.4
19.6
16.9
19.9
13.5
16.0
0 13.45
.16 d1
0
0 l
4
3 m
0
0 h
18.6
99
Class=m errs0/1)
xod>=18.6
0.97
0.19
14.4
19.6
16.8
18.8
13.5
15.8
0 14.366
.00 .00 1.00
1.0
8.0
4.0
5.0
2.0
9.0
0
2
0.06
0.15
0
0 l
0
0 m
11
1 h
17.816 99
.00
6
0
0
2
4 l
0 m
0 h
99
Class=m errs0/4)
Class=m errs0/0)
Class=m errs8/12)
FAUST Classifier
0. Cut in middle of the means: D≡mRmV d=D/|D| PR=Pxod<a PV=Pxoda a= (mR+(mV-mR)/2)od = (mR+mV)/2od
1. Cut in the middle of:VectorOfMedians (VOM), not the means. Use stdev ratio not middle for even better cut placement?
2. Cut in the middle of {Max{Rod}, Min{Vod}. (assuming mRodmVod) If no gap, move cut to minimize Rerrors + Verrors.
3. Hill-climb d to maximize gap or to minimize training set errors or (simplest) to minimize dis(max{r od},min{vod}) .
4. Replace mr, mv with the avg of the margin points?
5. PR=Px d<CutR
o
PV=Px d>CutV
o
Min{Vod}Max{Rod} CutR=CutV=avg{minVod,minRod}, else CutR≡Min{Vod}, Cut≡Max{Rod}
y PR or yPV , Definite classifications; else re-do on Indefinite region, PCutRxodCutV until actual  gap (AND with
certain stop cond? E.g., "On nth round, use definite only (cut at midpt(mR,mV)."
Another way to view FAUST DI is
that it is a Decision Tree Method.
With each non-empty indefinite set,
descend down the tree to a new level
For each definite set, terminate the
descent and make the classification.
Each round, it may be advisable to go
through an outlier removal process on
each class before setting Min{Vod} and
Max{Rod} (E.g., Iteratively check if F1(Min{Vod}) consists of V-outliers).
dim 2
vomR
vomV
MaxRod
MnVod
r
r
v v
mR
r v v vv
r r
v
mV
r
v v
r
v
v
dim 1
FAUST DI
K-class training set, TK, and a given d (e.g., from D≡MeanTKMedTK):
Let mi≡meanCi s.t. dom1dom2 ...domK Mni≡Min{doCi} Mxi≡Max{doCi} Mn>i≡Minj>i{Mnj} Mx<i≡Maxj<i{Mxj}
Definitei = ( Mx<i, Mn>i )
Indefinitei,i+1 = [ Mn>i, Mx<i+1 ]
Then recurse on each Indefinite.
For IRIS 15 records were extracted from each Class for Testing. The rest are the Training Set, TK. D=MEAN sMEANe
s-Mean 50.49 34.74 14.74 2.43
e-Mean 63.50 30.00 44.00 13.50
i-Mean 61.00 31.50 55.50 21.50
F < 18
 setosa
18 < F < 37
 versicolor
37  F  48
 IndefiniteSet2
48 < F  virginica
Definite_____
Indefinite__
s
-1
25
e
10
37
se 25
10
i
48
128
ei 37
48
(35 seto)
1ST ROUND D=MeansMeane
(15 vers)
(20 vers, 10 virg)
(25 virg)
F<7
 versicolor (17 vers. 0 virg)
7  F  10
 IndefSet3 ( 3 vers, 5 virg)
10 < F  virginica ( 0 vers, 5 virg)
empty
IndefSet2 ROUND D=MeaneMeani
F<3
 versicolor ( 2 vers. 0 virg)
IndefSet3 ROUND D=MeaneMeani
3F 7
 IndefSet4 ( 2 vers, 1 virg)
Here we will assign 0  F  7 versicolor
7 < F  virginica ( 0 vers, 3 virg)
7<F
virginica
Test:
F < 15
 setosa
(15 seto)
1ST ROUND D=MeansMeane
15 < F < 15
 versicolor
( 0 vers, 0 virg)
15  F  41
 IndefiniteSet2 (15 vers, 1 virg)
41 < F  virginica
(
14 virg)
F < 20
 versicolor (15 vers. 0 virg) IndefSet2 ROUND D=MeaneMeani
20 < F  virginica ( 0 vers, 1 virg)
100% accuracy.
Option-1: The sequence of D's is: Mean(Classk)Mean(Classk+1) k=1... (and Mean could be replaced by VOM or?)
Option-2: The sequence of D's is: Mean(Classk)Mean(h=k+1..nClassh) k=1... (and Mean could be replaced by VOM or?)
Option-3: D seq: Mean(Classk)Mean(h not used yetClassh) where k is the Class with max count in subcluster (VoM instead?)
Option-2: D seq.: Mean(Classk)Mean(h=k+1..nClassh) (VOM?) where k is Class with max count in subcluster.
Option-4: D seq.: always pick the means pair which are furthest separated from each other.
Option-5: D Start with Median-to-Mean of IndefiniteSet, then means pair corresp to max separation of F(mean i), F(meanj)
Option-6: D Always use Median-to-Mean of IndefiniteSet, IS. (initially, IS=X)
FAUST DI sequential
For SEEDS 15 records were extracted from each Class for Testing.
Option-4, means pair most separated in X.
m1 14.4
m2 18.6
m3 11.8
5.6
6.2
5.0
2.7
3.7
4.7
5.1
6.0
5.0
4.4 d(m1,m2)
3.4 d(m1,m3)
7.0 d(m2,m3)
DEFINITE
2 -inf
0
1 106
0
3 106 inf
INDEFINITE
12
23
0
0
106
106
0  F  106,
so totally non-productive!
Option-6: D Median-to-Mean of IndefSet (initially IS=X)
m1
m2
m3
On
14.4
18.6
11.8
whole
5.6
6.2
5.0
TR
2.7
3.7
4.7
5.1
6.0
5.0
37.3 meanF1
71.2 meanF2
`2.0 meanF3
5.1
3.7
5.0
30 avF1
m3 13.0 5.0
On Indef-1
4.0
5.0
m1 13.2
5.2
4.0
5.0
m3 13.0 5.0
On Indef-11
4.0
5.0
m1 13.0
5.2
3.6
5.0
m3 13.0 5.0
On Indef-111
4.0
5.0
m1 13.0
3.6
5.0
m3 13.0 5.0 4.0
On Indef-1111
5.0
m1 13.0
5.2
DEFINITE
Cl=1
def3[ -inf 21) 0
def1[ 28 49) 22
def2[ 58 inf) 0
Cl=1
DEFINITE
def3[ -inf 0 ) 0
27 avF3 def1[ 37 inf ) 1
Cl=1
DEFINITE
def3[ -inf 0 ) 0
6 avF3 def1[ 13 inf ) 1
9 avF1
Cl=1
DEFINITE
def3[ -inf 9 ) 0
9 avF3 def1[ 19 inf ) 0
13 avF1
Cl=1
DEFINITE
def3[ -inf 9 ) 0
9 avF3 def1[ 19 inf ) 0
13 avF1
2
0
0
30
2
3
0
0
0
0
Cls1
outlier(F=
2
3
54)
0
0
0
0
Cls1
outlier
2
3
(F=29)
0
0
0
1
Cls3
outlier
2
3
(F=0)
0
0
0
0
done! declare Class=1
3
32
0
0
INDEFINITE
ind1[ 21
ind2[ 49
INDEFINITE
in11[
0
37 )
INDEFINITE
in11[
0
13 )
INDEFINITE
in111[
9
19 )
INDEFINITE
in1111[ 9
19 )
28 )
58 )
Cl=1
2
3
6
0
3
Cl=1
2
3
5
0
3
Cl=1
2
3
5
0
2
Cl=1
2
3
5
0
2
FAUST DI sequential
For SEEDS 15 records were extracted from each Class for Testing.
Option-6: D Median-to-Mean of X
m1
m2
m3
On
14.4
18.6
11.8
whole
5.6
6.2
5.0
TR
2.7
3.7
4.7
5.1
6.0
5.0
37.3 meanF1
71.2 meanF2
`2.0 meanF3
DEFINITE
Cl=1
def3[ -inf 21) 0
def1[ 28 49) 22
def2[ 58 inf) 0
2
0
0
30
3
32
0
0
INDEFINITE
ind31[ 21
ind12[ 49
28 )
58 )
D Mean(loF)-to-Mean(hiF) of IndefSet31
m1 13.0
m3 13.0
.
5.1
5.0
3.7
4.0
5.0
5.0
Cl=1
30 avF1
DEFINITE
1
27 avF3 def1[-inf 18 )
0
def3[ 55 inf )
2
0
0
3
0
0
2
0
0
3
1
0
INDEFINITE
Cl=1
2
3
4 avF1
DEFINITE
0
0
0
2 avF3 def1[ -inf 0 )
0
0
def3[
5 inf ) 1
The rest, Class=1
INDEFINITE
INDEFINITE
in1313[
18
55 )
Cl=1
2
3
.
6
0
3
Cl=1
2
3
5
0
.2
Cl=1
2
3
.
4
0
2
Cl=1
2
3
0
0
D Mean(loF)-to-Mean(hiF) of IndefSet1313
m1 12.8
m3 13.0
.
5.2
5.0
3.2
4.0
5.0
5.0
Cl=1
18 avF1
DEFINITE
10 avF3 def3[ -inf 10 ) 0
def1[ 20 inf ) 1
in313131[ 10
20 ) .
D Mean(loF)-to-Mean(hiF) of IndefSet313131 (d repeats after this so=C1
m1 13.0
m3 13.0
5.2
5.0
3.6
3.5
5.0
5.0
C1=
[ 0
5 )
D Mean(loF)-to-Mean(hiF) of IndefSet12
m1 16.2
m2 16.6
6.0
6.0
1.8
4.6
Cl=1
5.2 5.8 avF1
DEFINITE
6.0 6.2 avF2 def1[ -inf 2 ) 5
def2[ 15 inf ) 0
[-inf, 21)class=3
2
0
5
3
0
0
INDEFINITE
in1212[
[28, 49)class=2
[21,28)ind31 d=(-.9, -.1, .14, -.1)
[-inf,18)def
2
15 ) .
0
[58.inf) class=3 d=(.,9, -,1, -.2, -.2)
[49, 58)ind12 d=(0, .31, -.9, 0)
[49, 58)ind23
FAUST CLUSTERING
V(d)≡VarDPPd(X)= (Xod)2 - (Xod)2
Xod=Fd(X)=DPPd(X)
x1od
d1
2
x1
Xj dj )2
= 1
(
j=1..n xi,jdj) - ( 
x2od
x
j=1..n
N i=1..N
2
=
:
dn
xNod
= N1  (j xi,jdj)(k xi,kdk) - ( Xj dj)( Xk dk) xN
i
j
k
2 2 2
2 2
1
=  jxi,j dj +Nj<k xi,jxi,kdjdk - jXj dj +2j<k XjXkdjdk
N i
Use DPPd(x), but which unit vector, d*, provides the best gap(s)?
1. DPPd exhaustively searches a grid of d's for the best gap provider.
2. Use some heuristic to choose a good d?
GV: Gradient-optimized Variance
MM: Use the d that maximizes |MedianF(X)-Mean(F(X))|.
=  Xj2 dj2 +2j<kXjXkdjdk
j
=
We have Avg as a function of d. Median? (Can you do it?)
HMM: Use a heuristic for MedianF(X): F(VectorOfMedians)=VOMod
MVM: Use D=MEAN(X)VOM(X), d=D/|D|
j=1..n
dT
d1 ... dn
d1≡(V(d0)) d2≡(V(d1)) til F(dk)
do=ek s.t. akk is max or d0k=akk
V(d)= 2a11d1 +j1a1jdj
2a22d2 +j2a2jdj
:
2a11 2a12 ... 2a1n
2a21 2a22 ... 2a2n
:
'
2an1
... 2ann
2anndn +jnanjdj
median
std
variance
Avg
consecutive
differences
avgCD
maxCD
||mean-VOM|
0
1
2
3
4
5
6
7
8
9
10
3.16
10.0
5.00
0
0
0
0
0
0
0
0
0
0
10
2.87
8.3
0.91
0
5
5
5
5
5
5
5
5
5
10
2.13
4.5
5.00
0
0
2
2
4
4
6
6
8
8
10
3.20
10.2
4.55
0
0
0
3
3
3
6
6
6
9
10
3.35
11.2
4.18
0
0
0
0
6
6
6
6
9
9
10
3.82
14.6
4.73
0
0
0
0
0
9
9
9
9
9
10
4.57
20.9
5.00
0
0
0
0
0
0
10
10
10
10
10
4.98
24.8
4.55
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
10
5
0
0
0
0
0
0
0
0
5
0
2
0
2
0
2
0
2
0
2
0
0
3
0
0
3
0
0
3
1
0
0
0
6
0
0
0
3
0
1
0
0
0
0
9
0
0
0
0
1
0
0
0
0
0
10
0
0
0
0
1.00
1.00
0.00
1.00
10.00
0.91
1.00
5.00
0.00
1.00
2.00
0.55
1.00
3.00
1.18
1.00
6.00
1.27
1.00
9.00
4.00
1.00
10.00
4.55
d1
:
di
:
dn
o
d = VarDPPdX≡V subject to i=1..ndi2=1
d1
:
V
i
GRADIENT(V) = 2A o d
(Xj2 - Xj2)dj2 +(2
+ j=1..n<k=1..n(XjXk - XjXk)djdk )
VX
o
- "
XiXj-XiX,j
dn
:
V(d)=jajjdj2 + jkajkdjdk
V(d) =
ijaijdidj
Maximize variance - is it wise?
MEDIAN picks out last 2 sequences which have best gaps (discounting outlier
gaps at the extremes) and it discards 1,3,4 which are not so good.
Finding good unit vector, d, for Dot Prod functional, DPP. to maximize gaps
Maximize wrt d, |Mean(DPPd(X)) - Median(DPPd(X)| sub to i di2=1
Mean(DPPdX)=(1/N)
j=1..n xi,jdj = (1/Ni xi,j ) dj = j=1..n Xjdj
j
i=1..N
Compute Median(DPPd(X)? Want to use only pTree processing.
Want a formula in d and numbers only (like the one above for the
mean (involves only the vector d and the numbers X1 ,..., Xn )
FAUST Clustering, simple example: Gd(x)=xod Fd(x)=Gd(x)-MinG on a dataset of 15 image points
X
x1 x2
1 1
3 1
2 2
3 3
5 2
9 3
15 1
14 2
15 3
13 4
10 9
1110
9 11
1111
7 8
The 15 Value_Arrays (one for each q=z1,z2,z3,...)
The 15 Count_Arrays
z1
z1z2
z2
z3
z4z3
z5
z6
z7
z8z4
z9
za
zbz5
zc
zd
ze
zfz6
z7
z8
z9
0
1
2
5
6 10 11 12 14
0 2
0 5
0 6
0 10
0 11
0 12
0 14
1
0 1
0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 1 0
0 1
0 0 5
0 0 10
0 1 12
0 0
0
0 0 2
0 0 6
0 1 11
0 0 14
0
0 0 0 0 1 0 0 0 0
0 1 0 0 0 0 0 0 0
0 3
1 6
0 10
0 11
0 12
0 14
0 0
0 1
1 0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0
0 1
0 1 3
0 0 6
0 0 11
0 0 14
0
0 1 2
0 0 5
0 0 10
0 0 12
0
0 0 1 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0
0 1
0 2
0 3
1 7
0 8
0 9
0 10
0 0
Level0, stride=z1 PointSet (as a pTree mask)
0
0
0
1
1
1
2
2
2
3
3
3
4
4
4
6
6
6
2
3
4
5
7 11 12 13
zb
0
1
2
3
4
6
8 10 11 12
zc
0
1
2
3
5
6
7
8
0
1
2
3
7
8
9 10
ze
0
1
2
3
5
7
9 11 12 13
5
6
7
1
1
1
1
2
1
z2
2
2
4
1
1
1
1
2
1
z3
1
5
2
1
1
1
1
2
1
z4
2
4
2
2
1
1
2
1
z5
2
2
3
1
1
1
1
1
z6
2
1
1
1
1
3
3
3
z7
1
4
1
3
1
1
1
2
1
z8
1
2
3
1
3
1
1
2
1
z9
2
1
1
2
1
3
1
1
2
1
za
2
1
1
1
1
1
4
1
1
2
zb
1
2
1
1
3
2
1
1
1
2
zc
1
1
1
2
2
1
1
1
1
1
zd
3
3
3
1
1
1
1
2
ze
1
1
2
1
3
2
1
1
2
1
zf
1
2
1
1
2
1
2
2
2
1
2
1
pTree masks of the 3 z1_clusters (obtained by ORing)
9 11 12 13
zd
3
4
7 10 12 13
1
1
2
9 11 12
0
0
2
9 11 12
za
zf
z1
1 2 3 4 5 6 7 8 9 a b
1 1=q
2
3
3 2
4
4
5
5
6
7
f
8
9
6
p
d
a
b
b
F=2 c e
c
F=1 Fp=MN,q=z1=0
d
a
e
8
f 7
9
8
9 10 11
1
2
z11
0
0
0
0
0
0
1
1
1
1
1
1
1
1
0
z12
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
z13
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
What have we learned?
What is the DPPd FAUST CLUSTER algorithm?
X2=SubCluster2
D=MedianMean, d1≡D/|D| is a good start.
But first, Variance-Gradient hill-climb it. (Median means Vector of Medians).
For X2=SubCluster2 use a d2 which is perpendicular to d1?
In high dimensions, there are many perpendicular directions.
GV hill-climb d2=D2/|D2| (D2=MedianX2-MeanX2) constrained to
be  to d1, i.e., constrained to d2od1=0 (in addition to d2od2=1.
We may not want to constrain this second hill-climb to unit
vectors perpendicular to d1. It might be the case that the gap gets
wider using a d2 which is not perpendicular to d1?
SubCluster1
GMP: Gradient hill-climb (wrt d) VarianceDPPd
starting at d2=D2/|D2| where
d2≡Unitized( Vom{x-xod1|xX2} - Mean{x-xod1|xX2} ) Variance-Gradient hillclimbed subject only to dod=1
(We shouldn't constrain the 2nd hill-climb to d1od2=0 and subsequent hill-climbs to dkodh=0, h=2...k-1. (gap could be larger).
So the 2nd round starts at d2≡Unitized( Vom{x-xod1|xX2} - Mean{x-xod1|xX2} ) and hill-climbs subject only to dod=1)
GCCP: Gradient hill-climb (wrt d) VarianceDPPd
the
CCs
starting at d2=D2/|D2| where D2=CCi(X2)-CCj(X2), and hill-climbs subject to dod=1, where
are two of the Circumscribing rectangle's Corners (the CCs may be a faster calculations than Mean and Vom).
Taking all edges and diagonals of CCR(X) (the Coordinate-wise Circumscribing Rectangle of X) provides a grid of unit vectors. It is an equispaced grid iff we use a CCC(X) (Coordinate-wise Circumscribing Cube of X). Note that there may be many CCC(X)s. A canonical one is the
one that is furthest from the origin (take the longest side first. Extend each other side the same distance from the origin side of that edge.
A good choice may be to always take the longest side of CR(X) as D, D≡LSCR(X).
Should outliers on the (n-1)-dim-faces at the ends of LSCR(X) be removed first?
So remove all LSCR(X)-endface outliers until after removal the same side is still the LSCR(X). Then use that LSCR(X) as D.
C11 F-MN gp2
MVM
0
1
1
.11 .19 .96 .19 209
[0.12)
F-MN Ct gp8
1
1
1
.41 .91 0 232
.07 .15 .98 .12 588 -.02
1L 0H
0
1 12
2
3
1
gp3
-.01 .26 .97 .00 608 C1(F-MN)
_4L
2H
12
1
3
3
3
2
0
1
1
(F-MN) gp8
___
_
[12,28)
1L
2H
15
2
13
5
2
1
1
6
1
0
1
1
_2L
1H
28
1
2
6
1
2
2
5
1
1
4
1
_0L
2H
30
1
2
8
2
2
3
2
1
2
4
1
32
2
2
10
2
1
4
4
1
3
5
1
34
1
1
11
1
1
5
8
1
4
4
1
35
2
3
12 3L 45H 1
6
8
1
5
6
1
_ [28,46) 2L 6H
38 ___ 1
8
13 1L 11H 2
7
4
1
6
8
1
46
1
1
15
2
8
3
1
7
6
1
___
_
[46,57)
2L
2H
47
3 10
9
7
1
C121
max thin
8
4
1
57
1
1
10
1
1
0
1
1
9
5
1
58
1
1
11
4
1
C1 F-M Ct g3
1
6
1
10
2
1
59
1
1
12
6
1
0
1
1
2
5
1
11
3
1
60
1
2
13
4
1
1
2
1
3
3
1
12
7
1
62
1
2
14
2
1
3L
2H
2
2
3
4
3
1
13
4
1
64
1
1
15
3
1
5
1
1
5
8
1
14
3
1
65
1
1
16
3
1
6
1
1
6
8
1
15
2
1
66
1
1
17
2
1
7
4
1
7
4
1
16
2
1
67
4
1
18
2
1
8
2
2
8
7
1
17
3
1
68
2
1
19
3
1
10
2
1
9 23L3 25H1
18
4
1
69
1
1
20
4
1
11
1
2 10 6L1 21H1
19
3
1
70
1
2
21
6
1
13
2
1 11
5
1
20
4
1
72
3
1
22
4
1
14
1
1 12
6
1
21
1
1
73
1
1
23
1
1
15
1
1
13
3
1
22
7
1
74
3
1
24
2
1
16
5
2
14
2
1
23
2
1
75
2
1
25
4
1
C11
18
1
2
15
3
1
24
4
1
76
1
1
26
1
1
2010L 213H3 16
3
1
25
1
1
77
1
2
27
1
2
23
1C12 1 17
4
26
1
1
79
1
3
29
2
1
24
1
1
27
1
1
1
1
30
2
2 [0.35) C11 82
25
1
1 C12 F-M gp2
28
1
1
83
1
1
32
1
3
26
2
2
0
1
1
29
1
1
38L 68H 84
2
1
35
1
1
28
1
1
1
8
1
30
1
1
85
1
1
36
1
1
29
1
1
2
3
1
31
1
1
86
1
2
37
1
1
30
5
1
3
2
1
32
1
3
88
2
1
38
1
1
31
2
1
4
4
1
35
1
2
89
4
1
39
4
1
32
1
1
5 11
1
37
3
1
90
2
1
40
2
2
33
4
1
6
8
1
38
1
1
91
1
1
42
2
2
34
5
1
7
2
1
39
1
1
92
6
1
44
1
1
35
4
1
8
6
1
40
3
1
93
3
1
45
2
2
36
4
1
9
4
1
41
3
3
94
5
1
47
4
1
37
2
1
10
3
1
44
2
1
95
4
2
48
2
1
38
3
1
11
4
1
45
2
1
5
1
49
1
1 [35,53) C12 97
39
3
1 12
4
1
46
4
1
98
2
1
50
1
310L 13H
40
2
1 13
5
1
47
2
2
99
1
1
53
1
1
41
4
1 14
3
1
49
1
2
100
4
1
54
2
1
42
3
1 15
3
1
51
1
1
___ 2 [53,56) 3L 2H 101
7
1
55
43
5
1 16 29L 4 46H2
52
1
3
102
4
1
44
3
1 18
2
1
55
1
1 51L 83H
103
2
1
45
4
1 19
5
1
56
1
1 [0.66) C1
104
3
1
46
5
1
20
6
1
57
1
9
105
6
1
47
4
1
21
4
1
66 ___ 2
106
3
1
_1 [66,75) 2L 2H
48
3
1
22
1
1
67
2
8
107
8
1
49 11
1 23
2
1
75
1
4
108 10
1
50
5
1 24
3
1
79
2
1
109
2
1
7L
19H
51
3
1 25
3
3
80
1
2
110
4
1
52
5
1 28
2
1
82
2
1
111
5
1
2L
2H
53
4
1 29
2
2
83 ___ 1
2
112
2
1
_
[75,98)
2L
6H
54
4
1 31 0L1 1H
85
1 13
113
4
1
55
1
98
1
2
114 [57,115)
1
___
51L 83H C1
100
1
3
103 ___ 1 11
_
[98,115)
2L
2H
114
1
WINE GV
ACCURACY
GV
MVM
GM
GM
-.05 -.31 -.95 -.01 605
.01 -.27 -.96 -.0 608
XF-M gp3
-.11 -.02 -.86 .5 43
0
1 11
11 2L 10H 4C1 -.05 -.4 -.92 .01 68
C7F-M*3 g3
15
1
1
1
3
16 _0L12H13 C2 0
3
1
2
29
1
1
5
1
4
30
1
2
9
2
6
32
2
2
15 _2L 14H 3C71
34
1
1
_2L
5H
C3
18
1
2
35
2
4
1
1
39 _0L11H 8C4 20
21
1
1
47
2
1
48 _0L 2 1H 9C4 22 ___ 3 4L32H
25 ___ 2 0L32H
57
1
1
28
1
2
58
1
1
30
3
1
59
1
2
31
2
1
61
1
2
32
1
1
63
1
2
33
1
1
65
1
1
34
2
1
66
1
1
35
1
1
67
1
1
36 _2L3 12H3
68
5
1
39
2
1
69
2
1
1
1
70 _9L 1 7H 3C5 40
41 _1L2 4H 3
73
3
1
44
1
2
74
3
1
46
3
2
75
1
1
48
1
2
76
2
1
50
2
2
77
2
2
1
1
79 4L 18H 3C6 52
53
1
1
82
1
1
54
1
1
83
1
1
55
1
1
84
1
1
56
2
1
85
1
1
57
1
1
86
1
1
58
2
1
87
1
1
59
2
1
88
1
1
60
2
1
89
1
1
61
1
1
90
4
1
62
1
1
91
2
1
63
2
2
92
7
1
65
1
1
93
1
1
66
1
1
94
5
1
67
1
1
95
4
1
68
1
1
96
2
1
69
3
1
97
3
1
70
1
1
98
2
1
71
1
1
99
2
1
72
1
1
100 3
1
73
1
1
101 4
1
74
2
1
102 7
1
75
2
1
103 3
1
76
1
1
104 2
1
77
3
1
105 6
1
78
4
1
106 3
1
79
3
1
107 5
1
80
4
1
108 9
1
81
1
1
109 6
1
82
1
1
110 4
1
83
1
2
111 5
1
85
2
1
112 4
1
2
2
11338L4 68H 1C7 86
88
3
1
114 1
89
2
2
91
2
2C76
___
28L
44H
93
4
3
1L
WINE
62.7
66.7
81.3
-.08 .59 -.8 -.07 80
.08 .83 -.56 -.01 95
C5 g3
0
1
4
1
12 _3L1 0H
15
1
17
1
19 _1L1 2H
23
1
24
1
26
1
27
1
29
3
31
1
32
1
33 5L1 5H
4
8
3
2
2
4
1
2
1
2
2
1
1
.01 -.27 -.96 -.01 23
-.04 -.43 -.9 .03 24
C76*4 g3
___
0
1 _1L
31 0H
___ 1 _0L
31
3 1H
34
1
1
35
2
2
37
1
2
39
2
2
41
1
2
43
1
1 4L8H
44
1
2
46
3
3 C763
49
1
1
50
1
1
51
2
1
52
1
1
53
2
2
55
2
2
___
_
57
2
3 2L 9H
60
1
2
62
2
1
63
1
2
65
3
1
_3 1L 8H
66 ___ 2
69
1
1
70
1
1
71
2
2
73
2
1
74
1
1
75
2
1
76
3
1
77
2
1
78
2
1
79
3
1
80
3
2
82
1
2
84
1
2
86
2
1
87
1
1
88
1 17L2 15H
90
2
1
91
2
3 C766
94
2
2
96
2
1
_ 3L 3H
97 ___ 2
.05 .59 -.293 .75 18
-.1 .9 -.3 .1 34
C6*8 16
0
1
4
4_1L 22H 16
20
1 11
31_2L 1 37
68
1 15
83
1 15
98
1
8
106
1 11
117
1
1
118_1L 26H
.19 .8 -.54 .18 7
-.21 .7 -.7 -.09 9
C763F-M*8 g8
0 0L 2 2H16
16
1 13
29 _2L1 0H12
41
2
4
45
1
7
52
1
4
56
1
7
_2L
4H
63
1
8
_0L
2H
71
2
-.21 .34 -.91 .9 8
C766 *16 g4
0 _0L 1 1H30
30
1
2
32 _2L 1 0H 7
39
1
1
40
1
1
41
1
1
42 _3L 1 1H 4
46
1
2
48
1
2
_1L
3H
50
2
5
55
1
3
_2L
0H
58
1
7
65 _2L 1 0H 2
67
1
5
72
1
3
75
2
2
77
1
1
78
4
2
80
1
3
83
1
1
_4L
8H
84
2
4
88
1
1
89 _2L 1 0H11
100 1
4
_0L 1 2H11
104
115
_1L 1 0H
SEEDS
GV
219 31 14 29 akk
d1 d2 d3 d4 V(d
.98 .14 .06 .13 9
.98 .14 .06 .13 9
10(F-MN) gp6
0
2 1
1 10 1
2
5 1
1___6 [0,9) 0k 0r 18c C1
___3
9
3 1
10 10 1
11 10 1
___
12
2___6 [9,18) 1k 0r 24c C2
18
2 1
19
3 1
20
7 1
21
2 1
22
1 1
___
23
3___6 [18,29) 10k 0r 8c C3
29
6 1
30
4 1
31
7 1
___
32
1 ___6 [29,38) 18k 0r 0c C4
38
1 1
39
2 1
40
6 1
41
5 1
___
42
1 ___7 [38,49) 13k 2r 0c C5
49
3 1
50
1 2
52
7___1 [49,60) 7k 6r 0c C6
___
53
2 7
60
1 2
62
4 1
___
63
3___8 [60,71) 1k 7r 0c C7
71
5 1
72
2 2
___
74
1___6 [71,80) 0k 8r 0c C8
80
5 1
81
8 1
82
5___1 [80,92) 0k 21r 0c C9
___
83
3 9
___
92
2___
10 [92,102) 0k 2r 0c Ca
102
1 1
103
2 1
___
104
1___ [102,105) 0k 4r 0c Cb
C3 .97 .15 .09 .14 0
___0
___
10
___
20
30
___
31
___
40
___
50
___
61
___
70
0 .07 1 0 4 10F-M g9
2k 0r 0c
2___
10 [0,10)
3___
10 [10,20) 2k 0r 1c
3___
10 [20,30) 2k 0r 1c
4 1
1___9 [30,40) 4k 0r 1c
1___
10 [40,50) 0k 0r 1c
1___
11 [50,61) 0k 0r 1c
___
1 9 [61,70) 0k 0r 1c
2___ [70,71) 0k 0r 2c
256 36
10 32 akk
.98 .14 .04 .12 0
.00 -.00 .96 .29 3
C6 10(F-M) g12
0
3 10
___
10
1___
12 [0,22) 4k 0r 0c
22
3 10
32
3 9
41
2 7
___
48
1___ [22,49) 3k 6r 0c
ACCURACY
GV
MVM
GM
MVM
10(F-MN)gp6
0
2 1
1 10 1
2
5 1
___3
1___6 [0,9) 0k 0r 18c C1
9
3 1
10 10 1
11 10 1
___
12
2 ___6 [9,18) 1k 0r 24c C2
18
2 1
19
3 1
20
7 1
21
2 1
22
1 1
___
23
3___6 [18,29) 10k 0r 8c C3
29
6 1
30
4 1
31
7 1
___
32
1___6 [29,38) 18k 0r 0c C4
38
1 1
39
2 1
40
6 1
41
5 1
___
42
1___7 [38,49) 13k 2r 0c C5
49
3 1
50
1 2
52
7 1
___
53
2 ___7 [49,60) 7k 6r 0c C6
60
1 2
62
4 1
___
63
3 ___8 [60,71) 1k 7r 0c C7
71
5 1
72
2 2
___
74
1 ___6 [71,80) 0k 8r 0c C8
80
5 1
81
8 1
82
5 1
___
83
3 ___9 [80,92) 0k 21r 0c C9
___
92
2 ___
10 [92,102) 0k 2r 0c Ca
102
1 1
103
2 1
___
104
1 ___ [102,105) 0k 4r 0c Cb
C3
200(F-MN)gp12
0
2 12
12
3 12
___
24
3___
12 [0,35) 8k 0r 0c
___
36
5___
12 [35,48) 2k 0r 3c
48
1 12
___
60
1 ___
12 [48,72) 0k 0r 2c
72
1 40
___
112
2 ___ [72,113) 0k 0r 3c
C6
200(F-MN)gp12
0
3 12
___ 1 ___
12
38 [0,50)
___
50
3 ___
10 [50,60)
60
1 2
___
___
62
3 12 [60,74)
74
2 ___ [74,75)
___
4k 0r 0c
1k 0r 2c
1k 0r 3c
1k 0r 1c
SEEDS WINE
94
62.7
93.3 66.7
96
81.3
GM
.794 -.403 -.304 .337 6
0.957 .156 -.205 .132 9
10(F-MN) gp3
0
1
2
2
1
2
4
4
2
6
3
2
8
7
2
10
2
2
12
1
2
14
1
2
16 10
2
18 10
1
___
19
2 ___3 [0,22) 0k 0r 42c C1
22
2
1
23
2
1
24
1
1
25
1
2
27
4
2
29
4
2
31
4
2
___
33
2 ___5 [22,33) 10k 0r 8c C2
38
3
1
39
3
2
41
7
2
43
2
2
45
2
1
46
1
2
48
1
1
49
1
1
50
4
2
52
5
1
53
1 ___1 [33,57) 33k 2r 0c C3
___
54
3
3
57
2
2
59
3
2
61
3
1
62
1
2
64
3 ___2 [57,69) 6k 9r 0c C4
___
66
3
3
___
69
5 ___7 [69,76) 1k 4r 0c C6
76
1
2
78
2
2
80
2
2
82
4
2
84
1
2
86
1
2
88
4
1
89
1
1
90
8
2
___
___
92
5 11 [76,103) 0k 26r 0c C7
103
2
1
104
1
1
105
1
1
106
1
2
___
___ [103,109) 0k 6r 0c C8
108
1
-.577 .577 .577 .000 1
.119 .112 .986 .000 3
C2: 10(F-MN) gp10
0
1 10
10
2
1
11
3 10
___
21
3 ___
10 [0,31) 9k 0r 0c C21
___
31
5 ___
10 [31,41) 1k 0r 4c C22
41
1 10
51
1 11
62
1
1
___
63
1 ___ [41,64) 0k 0r 4c C23
-.832 -.282 .134
-.44
.00 -.87
C4: 10(F-MN) gp21
0
11
___
31
___
52
79
___
99
-.458
-.22
3 11
2 20
3 ___
21 [0,52) 1k 7r C41
3 ___
27 [52,79) 1k 2r C42
1 20
3 ___ [79100) 4k 0r C43
0
2
IRIS GM
.81 .28 -.28 .42 13...
.53 .23 .73 .37 39
MVM
C12 4*F-M g3 F-MN gp8
0
2
4
0
2
3
.88 .09 -.98 -.18 168
4
1
4
3
5
1
-.29 .13 -.88 -.36 417
8
2
2
4
5
1
-.36 .09 -.86 -.36 420
10
1
2
5 14
1
F-MN Ct gp5
12
1
2
6 11
1
14
1
3
0
1
3
7
6
1
-.36
.09
-.86
-.36
105
17
1
1
8
1
1
3
2
1
-.54 -0.17 -.76 -.33 118 18
1
2
9
5
1
4
1
2
C1 2*(F-M g3
20
1
1
10
1
5
0
2
4
6
1
1
50s
1i
C1
21
1
1
15
1
8
4
1
1
7
1
2
22
1
2
23
1 C2 2
5
1
1
9
2
1
24
3
1
25
2
2
6
1
5
25
1
2
10
1
2
11
1
2
27
1
2
13
1
3
27
1
1
29
1
12
3
1
16
1
2
28
1
2
1..
13
1
1
18
1
3
30 19e 1 1i 4
___
68
1
14
3
1
21
1
1
34
1
2 C22 4(F-) g4
22
1
1
15
4
1
0
1
6
36
2
2
23
1
2
16
2
1
6
1
4
38
1
3
25
2
1
___
6e
0i
10
1
2
41
2
3
17
3
1
26
2
2
12
1
4
44
1
2
18
1
1
28
2
1
...
46
2
2
29
1
2
19
6
1
33
2
1
48
1
2
31
3
1
20
3
1
___
34
1 18e 4
50
1
2
32
1
2
21
1
1
38
1
1
34
1
1
52
2
2
22
2
1
35
2
1
39
1C2213
54
1
1
36
2
1
...
55
1
1
23
2
1
___28i
37
4 C113
79 29e 1 14i 2
56
1
1
24
6
1
40
2
1
81
1 ___5
57
1
1
___
25
7
1
41
1
2
86
1
2
58
3
1
43
3
2
26
2
1
88
2
2
59
1
1
45
1
2
27
3
1
90
1
1
60
2
2
47
4
1
28
2
1
91
1
1
62
1
1
48
1
1
92
2
2
63
4
2
29
6
1
49
2
1
94
1
1
65
1
1
50
4
1
30
3
1
95
1
2
51
3
2
66
1
1
31
2
1
53
5
1
97
1
1
67
1
1
32
3
1
54
2
1
98
1
3
68
1
1
33
3
1
55
2
1
18e 69 11i 3C1233 101
2
1
56
1
1
34
3
1
2
4
72
1
2 102
57
3
2
106
1
1
74
1
2
35
3
1
59
3
2
107
1
2
76
1
2
36
5
1
61
2
2
109
1
1
78
1
1
63
1
1
37
1
1
___
3e
2i
110
2
1
79
1
3
64
1
1
38
2
1
2
6
82
1
2 111
65
2
2
39
1
1
1
1
84
1
1 117
67
1
1
___
40
2
1
1
1
68
1
1
85 0e 1 3i 4 118
69
2
1
1
1
89
1
1 119
41
1
2
70
2
1
___
26i
120
1
90
1
2
43
1
2
71
3
1
92
1
1
___
0e
4i
45
1
1
72
1
1
93
1
C221 8F-)g5
46
1
1
73
2
2
0
1
7
75
1
1
47
1
5
7
1
4.
76
1
1
___50e
49i
C1
-.034 .37 -.31 .87 4
___
52
1
8
11 3e 1
5
77
1
1
C123
12*F-M
g4
16
1
1
60
2
1
__
1i
.
78
1
1
0
1
6
17
1
3
79
1
1
___
61
3
1
6 1e 1 10 .
20
1
1
80
1
2
21
1
2
62
4
1
_46e
21i
C12
16
1
2
82
2
10
23
1
18
1
3
63
3
1
___ 5e 1 1i 1
92
1
2
24
5
21
1
1
64 13
1
29
1
3
94 4e 2C13 2
___
22
1
1
32
2
2
96
1
65 12
1
23
1
6
34
1
1
66
4
1
35
1
4.
29
1
3
___9e
39
3
5
67
5
1
32
1
3
44
1
3
68
2
2
35
1
5
47
2
3
___
50s
1i
C2
___
9e
1i
.
40
2
5
70
2
50
1
3
45
1
4
53
1
4
MVM
57
1
3
49
1
1
C2 2(F-)g4
60
1
3
_50 4e 2
4.
0
1
4
63
1 2i 1
54
1
2
4
1
1
__9e
64
2
5
___
3e
__
56 0e 1 2i 5.
5
1
4
69
2
1
61
2
1
9
1
3
70
1
3
73
1
1
62
1
2
...
74
1
1
47e69 40i 1C22 4
64
1
1
75
1
4
65
1
2
73
1
1
79
1
1
67
1
3
74
1
2
80
2
2
70
1
1
76
2
4
82
2
1
83
1
1
___ 2e 1 6i12.
71
80
1
4
84
1
2
83
1
1
84
1
2
86
1
4
84
1
1
86
2
5
90
1
5
___
___ 2e 1 1i
85
___
91 0e 1 11i
95 5e 1 11i
GV
.90 .24 .37 .04 180
.41 -.04 .84 .35 418
.36 -.08 .86 .36 420
F-MN Ct gp3
0
2
2
2
2
1
3
2
1
4
5
1
5
7
1
6 16
1
7
6
1
8
4
1
9
4
1
10
2
8
___50s
1i
18
1
5 C1
23
1
2
25
2
2
27
1
2
29
1
1
30
1
1
31
1
1
32
2
1
33
1
1
34
3
1
35
5
1
36
4
1
37
3
1
38
1
1
39
4
1
40
3
1
41
3
1
42
4
1
43
4
1
44
2
1
45
5
1
46
7
1
47
3
1
48
2
1
49
1
1
50
3
1
51
4
1
52
3
1
53
2
1
54
3
1
55
3
1
56
3
1
57 50e 1 40i 1 C2
___
58
4 9i 3 C3
61
2
1
62
1
1
63
1
2
65
1
1
66
1
1
67
2
3
70
1
C2
3762 808 2260 266 C23 F-M*3 g3
3847 818 2284 257
d1 d2 d3 d4
.96 .22 .06 -.14 15
.84 .18 .51 .06 64
___0 1e1 0i 6
.57 .22 .71 .34 82
6 1
2
.51 .22 .74 .38 83
8 1
4
(F-MN)*3
12 1
3
Ct gp3
15 1
1
0 1
2
16 1
2
2 1
1
18
__2e 2 5i 8
3 1
2
2
___ 5 4e 1 1i 15 C21 26 1
28 1
1
20 2
3
29
1
1
23 1
3
30 1
2
26 2
2
32 1
1
28 1
1
33 1
3
29 1
2
36 1
3
31 1
2
39 2
1
33 2
2
40 1
1
35 2
2
41 2
1
37 1
1
42
2
2
38 3
1
44 2
2
39 1
1
46 1
1
40 1
1
47 2
5
41 1
1
___4219e1 1i 4 C22 52 1
1
53 1
3
46 1
1
56 1
1
47 2
2
57 1
3
49 1
1
60 1
1
50 1
1
61 1
1
51 1
2
62
1
2
53 1
1
64
1
___ 16e 11i 6
54 2
2
70 2
5
56 1
2
75 1
2
58 1
1
77 2
3
59 2
2
___
80 1 6e 9
61 2
1
89 1
8
62 2
1
___97 12e
63 3
1
64 1
1
C221 8F- g5
65 2
2
0
1
7
___1e
67 3
1
7
1
4
68 2
1
___
11 2e 1
5
69 1
1
16
1
1
70 2
1
17
1
3
20
1
1
71 2
1
21
1
2
72 2
2
23
1
1
74 1
1
__
4e
1i
24
1
5
75 1
2
29
1
3
77 1
1
32
2
2
78 1
1
34
1
1
35
1
4
79 1
2
___9e 3
39
5
81 1
2
44
1
3
83 1
1
2
3
___8427e1 16i 3C23 47
50
1
3
87 2
1
53
1
4
88 1
1
57
1
3
60
1
3
89 1
1
63
1
1
90 2
1
__9e 2 2i 5
91 9i 1
1 C24 64
___
69
2
1
92 2
3
70
1
3
95 1
1
73
1
1
96 2
1
74
1
1
75
1
4
97 1
2
79
1
1
99 1
2
80
2
2
101 2 8i 2
___
82
2
1
103
1
3
83
1
1
___
106 3 3i 3
84
1
2
109 1
1
86
1
4
___
5e
10i
90
1
5
110 1_3i 1
___
___ 1 1i
95
111 1
ACCURACY IRIS SEEDS
GV
82.7 94
MVM
94
93.3
GM
94.7 96
WINE
62.7
66.7
81.3
MVM C11 F-/4 g4
GM
0
4
2
MVM
2
1
2
F-m/8 g4 C2
4
4
2
(F-)/4 gp4
0
1
2
6 25
2
2
1
1
0
1
1
8
2
1
3
1
2
___
1
1 2M 4
5
2
3
9
7
1
5
1
1
8
1
2
10
4
1
...
1s
10
1
1
46
4
3
11
9
2
C2111 0L 1 8M5 0H
C14943L 133M755H
13
3
1
16
1
2
56
1
2
6
1
C2218
1 2M 5 0H 14
58
1
3
15
4
1
1
1
C2323
61
1
4
65
1
1
24
1
1
16
1
3
66
1
3
25
2
1
19
5
4
69 7M 1 C2 2
26
2
1 C111 23 3L 2 23M3 49H
GV
___
71
1
6
27
1
2
26
5
1
77
1
3
29
4
1
80
1
3
27
4
1
2
1
g4 F-MN/8 30
83
1
3
28
9
1
2
1
___
4M 1 C314
0
1
2 31
86
1
1
29
5
2
100
1
3
2
1
2 32
3
2
103
1
2
31
6
1
4
1
2 33
...
1s
105
1
3
32
5
3
6
1
1 65
1
108
2
35
6
5
___6M 1C4 4
112
7
1
1
___
40 30L2 1M 4H
8
1
2
C231 g4 F-M/8
10
1
1
0
1
7
11
1
1
...
1s
12
1
1
12
1
2
13
1
3
14
6
1
16
2
3
15
7
4
19
1
2
C1F-/4 g4
19
1
1
___14M
0H
C1
21
1
5
0
1
1
20
3
1
C2
1
1
7
26
1
1
8
1
4
21
3
1
27
1
1
12
1
4.
___
4M
22
2
1
16
1
2
28
2
1 C23 g3 F-M/8
18
1
2
23
1
2
29
2
1 0
2
2
20
2
1
25
1
2
30
1
2 2
21
2
2
1
1
27
1
2
...
1s+2s
32
5
1 3
3
1
71
2
2
29
1
1
33
2
1 4
3
1
73
1
1
30
1
1
74
1
2
34
2
1 5
1
1
76
2
2
31
1
2
35
1
1 6
1
1
C1178 43L 223M4 53H
_30L
8H_
.
33
1
6
36
3
1 7
82
2
2
6
1
39 3L 1 2M 3
37
3
1 8
84
1
6
1
1
42
1
4
38
3
1 9
8
1
90
2
8
46
1
10
39
5
1 10
2
1
98
1
9
56
2
40
3
1 11
6
1
107 1 16
C232
g2
F-M/8
41
7
1 12
2
1
___6M1 2H
123
0
1
1
42
6
1 13
5
1
1
1
1
43
3
1 14
2
1
2
2
1
44
5
1 15 0L 2 32M3 13H 3 2L1 2M 2 1H
45
1
1 18 11L1 13M1 54H 5
2
1
46
3
1 19
6
1
1
7
1
7
2
1
47
3
1 20
1
1
8
2
1
48
4
1 21
3
1
C111
___1L
9
1 4M 7 3H
49
7
1 22
1
1
F-/4 g4
16
1
1
50
4
1 23
2
1
17
3
1
0 1 16
___
__1L
51
6
1 24
___1L
1M
4H
18
2
2
4
1
16 3 1
52 10
1 25
20
7
1
1
2
17 2 1
21
8
1
53
3
1 27
8
1
18 9 1
22
7
1
54
4
1 28
9
1
19 3 2
___ 3L 1 2M 2 18H
23
55
8
1 29
4
2
21 1L 5 6 21M
25
2
1
56
5
1 31
2
1
26
3
1
27 3 1
57
3
1 32
27
2
1
1
1
28 5 1
58
7
1 33
28
3
1
3
1
29 14 1
29
1
1
59
2
1 34
__
3
2
30 1L 12M8 20H
30
1
1
60
2
1 36
7
1
38 2 2
___4L 22M 2 8H
31
61
1
1 37 12
2
40 15 1
33
1
1
62
2
2 39
1
1
34
4
1
41 3 4
64
1
1 40
35
3
3 8H
___
1
1
45 3 2
65
2
1 41
38
3
1
1
1
47 2 19
39
8 2M11 9H
66
1
1 ___
42
6___ 6 1H. ___
66 3 21
50
2
1
67
2
48
1
2 2M 51
___
87 1 __ 31H
___1L 1
2H
CONCRETE
C2-.6 .2 -.07 .771 6882..
-.72 .19 -.40 .54 9251
.38 .14 -.79 .46 11781
X g4 (F-MN)/8
0
2
2
2
1
2
4
2
1
5
1
3
8
2
3
11
1
1
12
3
2
14
4
1
15
3
1
16
3
1
17
2
1
18
3
1
19
6
1
20
3
1
21
3
1
22
2
1
23
5
1
24
4
1
25
3
1
26
6
1
27
3
1
28
1
1
29
6
1
30
3
1
31
2
1
32
3
1
33
3
1
34
1
2
36
3
1
37
1
1
38
2
1
39
3
1
40
5
1
41
1
1
42
6
1
43
1
1
44
3
2
46
5
1
47
1
1
48
3
1
49
1
1
50
2
1
51
1
1
52
1
1
53
1
1
54
1
1
55
1
1
56
3
1
57
3
2
59
1
2
61
1
1
62
3
3
255H9 C2
43L65
38M
74
1
4
0L7814M1 0H 3 C1
81
1
2
83
1
3
86
1
2
88
1
2
90
1
1
91
1
4
95
1
2
97
1
1
98
1
2
100
1
4
104
1
3
ACCURACY
GV
MVM
GM
C2 gp8 (F-MN)/5
0
2
2
2
1
2
4
2
1
5
1
3
8
2
3
11
1
1
12
2
2
14
4
1
15
3
1
16
3
1
17
2
1
18
3
1
19
6
1
20
3
1
21
3
1
22
1
1
23
5
1
24
3
1
25
3
1
26
6
1
27
3
1
28
1
1
29
6
1
30
3
1
31
2
1
32
1
1
33
3
1
34
1
2
36
3
2
38
2
1
39
2
1
40
5
1
41
1
1
42
6
1
43
1
1
44
3
2
46
5
1
47
1
1
48
1
1
49
1
1
50
2
1
51
1
1
52
1
1
53
1
1
54
1
1
55
1
1
56
3
1
57
2
2
59
1
2
61
1
1
355H 3 C21
43L 62
28M
65
2
9
74
1
4
78
1
8
86
1
2
88
1
2
90
1
5
95
1
2
97
1
1
98
1
2
1 0H 4 C22
0L100
10M
104
1
CONCRETE
76
78.8
83
IRIS SEEDS
82.7 94
94
93.3
94.7 96
C21 g4 F-M/4
0
1
1
1
1
3
4
1
3
7
2
1
8
2
1
9
1
2
11
1
2
13
4
1
14
2
1
15
4
1
16
1
2
18
2
1
19
3
1
20
1
1
21
2
1
22
6
2
24
2
1
25
3
1
26
1
2
28
2
2
30
1
1
31
1
2
C21133
1
4
32L 3713M 10H 1
38
2
1
39
2
1
40
1
1
41
1
1
42
1
1
43
2
1
44
1
1
45
2
1
46
1
1
47
1
1
1
1
C21248
210H 2
7L 49
3M
51
2
4
55
1
1
56
8
1
57
4
1
58
4
1
59
2
1
60
1
1
61
1
2
63
5
2
65
1
2
67
2
1
68
1
3
71
1
1
72
4
1
8
1
C21373
4L 74 7M 538H1
75
1
8
83
3
1
84
3
1
2
1
C214 85
1
0L 86
99 5M 37H
WINE
62.7
66.7
81.3
C211 g5 F-M)/4
0
1
6
6
2
1
___5L
7
2
5.
12
1
1
13
4
1
14
1
1
15
4
2
17
1
1
18
2
1
19
2
2
21
2
1
22
3
1
23
1
1
24
3
4
__20L 15M14.
28
42
1
2
44
1
1
45
1
3
48
2
2
___
5L
1M
50
1
5.
55
1
2
57
1
1
___
58 2L 11M 5 .
63
1
1
64
1
7
71
1 11
82
1 16
___5L 2 1M
98
C212 g5 F-M/3
0
1 20
20
1
8
28
1
1
29
2
9
38
1 11
49
1
5
54
1 11
__6L 1 3M
65
10 .
75
2
3
__1L 12H11
78
89
1
7
96
1
2
98
1
2
100
1 11
111
2
1
___ 1 8H
112
ABALONE
GV
0.11
0.09
0.03
0.14
2
0.27
0.86
0.33
0.27
73
1.00
0.00
0.00
0.00
5
0.29
0.84
0.36
0.29
72
0.26
0.87
0.32
0.26
73
0.00
1.00
0.00
0.00
56
0.25
0.88
0.31
0.25
73
0.00
0.00
1.00
0.00
8
0.29
0.84
0.36
0.29
72
0.26
0.87
0.32
0.26
73
0.00
0.00
0.00
1.00
5
0.29
0.84
0.36
0.29
72
0.26
0.87
0.32
0.26
73
1.00
1.00
0.00
0.00
93
0.26
0.87
0.32
0.26
73
1.00
0.00
1.00
0.00
27
0.29
0.84
0.36
0.29
72
0.26
0.87
0.32
0.26
73
1.00
0.00
0.00
1.00
22
0.29
0.84
0.36
0.29
72
0.26
0.87
0.32
0.26
73
1.00
1.00
1.00
0.00
154
0.27
0.87
0.33
0.27
73
1.00
1.00
0.00
1.00
141
0.26
0.87
0.33
0.26
73
1.00
0.00
1.00
1.00
57
0.29
0.84
0.36
0.29
72
0.26
0.87
0.32
0.26
73
0.00
1.00
1.00
1.00
154
0.27
0.87
0.33
0.27
73
1.00
1.00
1.00
1.00
216
0.27
0.86
0.33
0.27
73
1.00
0.00
0.00
0.00 23
0.71
0.23
0.66
0.01 47
C1 g3 400*F-M
0
1
1
...
2L 1 0M 1 0H 6_
97 1L 1 19M 1H _
7
1
3
10
2
2
1.0 .00 .00 .00 10
12
3
2
.62 .41 .13 .65 46
14
3
1
.33 .29 .13 .89 56
7L15 3M 1 0H 3_
C2 g3 300*F-M
18
1
2
0
1
8
20
1
2
8
1
1
22
3
4
9
1
2
26
1
3
11
1
1
29
1
3
12
1
1
32
1
1
13
3
1
33
1
2
14
1
2
35
1
2
16
2
1
37
2
2
17
1
1
39
1
1
18
3
2
6L40 8M 1 0H 5_
20
2
1
45
2
2
2113M 1 5H 3_
47
1
1
24
1
1
48
2
1
25
1
2
49
1
2
27
2
1
51
1
1
28
1
1
52
2
1
29
2
1
53
2
1
30
1
1
54
2
2
31
1
2
56
1
2
33
2
1
58
3
1
34
1
1
59
1
1
35
1
2
60
1
2
37
1
1
62
2
1
38
3
1
63
1
1
39
1
1
2L 6421M2 1H3_
40
1
12M 7H 5_
67
4
1
45
1
1
68
1
1
46
1
2
69
2
1
48
1
6
1L70 7M 1
_3
54
1
4
73
1
2
58
1
1
75
2
1
59
1
3
76
2
2
62
1
1
78
1
1
63
1
1
79
2
2
64
1
4
81
1
1
68
1
1
82
1
1
6M 15H14
_
69
83
1
1
83
1
3
_
861M 11H23
109 1 1H
MVM
g3 200*F-M
0
1 1H11
11 1M 1 14
_
25
1 1H17
42
1
1
43
1
5
2M
1H
48
1
3_
51
1
2
5M 12H _
...
67
2
1
68
2
1
69
3
2
...
1s
30L 92 85M1 12H C1
C1 g3 100*F-M
0
1 1H 6
6
1
1
...
1s
20L 54 84M1 11H 2 C11
10L56 1M 2 0H 3...
71
2
C11 g3 400*F-M
0
1
1
1
1
4
5 2M 1 1H 3_
8
4
1
9 4M 1 1H 3_
12
2
2
..
17L 81 78M 2 9H 3 C111
3L 84
2
1
85
1
C111 g3 1500*F-M
0
1 15
15
1
5
3L20
_4
1
24
1
1
25
1
1
26 3M 1
3_
29
1
1
30
1
1
31
2
1
32
1
1
4L 33 3M 2 _3
36
1
2
38
3
1
39
2
2
41
2
1
42
1
1
43
2
2
45
1
2
47
3
1
48
1
2
50
1
1
3L5113M1 2H4
55
2
1
56
3
2
58
1
2
60
3
1
61
2
1
62
2
2
64
1
4L65 8M 2 4H 1
3
68
2
1
...
112
1
4
3L
51M
3H
GM
0.39
0.57
0.77
0.58
0.55
0.57
0.44
0.61
0.48
0.46
0.10
0.09
0.17
0.17
0.16
-0.72
-0.69
0.01
0.64
0.68
0.21
0.24
2.19
3.8
3.81
ACR
GV
MVM
GM
CONC
76
79
83
IRIS
83
94
95
SEEDS
94
93
96
WINE
63
67
81
ABAL
73
79
81
X g2 100(F-M)
0.25 0.30 -0.20 -0.90 0.18
3
2
3
-0.44 -0.37 -0.19 -0.79 0.81
6
1
2
-0.52 -0.42 -0.19 -0.72 0.83
8
1
1
C1 g3 300(F-M)
6L 9
2
3.
0
1
1
12 1M 1
3_
1
1
2
15
2
1
3
2
1
3L16
1
2.
4
1
1
18
2
1
5
1
1
19
1
1
6
2
1
20
2
1
7
1
3
21
3
1
10
1
1
22
2
1
11 7M 1 4H 3.
23
1
1
14
3
2
24
6
1
16
2
1
25
1
1
17
1
1
12L 26 7M 1
_2
18
2
2
28
3
1
20
1
2
29
2
1
22
1
1
3L 30 4M 2
_2
23
2
1
32
3
1
24
1
1
33
2
1
25
2
1
34
3
1
26
3
1
35
5
1
27
1
1
36
4
1
28
2
1
37
4
1
29
1
2
38
3
1
31
1
1
16M
8H
39
5
1
32
1
3C11
40
3
1
35
1
1
41
2
1
36
1
2
42
1
1
38
1
3
43
2
1
41
1
3
44
3
1
44
3
1
45
4
1
45
1
1
46
2
1
46
2
2
.55 .43 .14 .27 .38
47
3
1
48
1
1
48
3
1
49
1
1
C11 g3 1000(F-M)
49
1
1
50
2
2
50
3
1
52
2
1
0
1 10
51
1
1
53
1
1
10
1
7
52
1
1
54
1
1
17
1
2
53
7
1
55 17M1 2H 4.
19
1
8
54
4
1
59
2
1
27
1
9
1M
2H
_
55
3
1
60
1
4
36
1 0M11 6H _
56
3
1
64
1
1
57
4
1
65
1
1
47
2
2
58
2
1
66
1
1
49
1 1M 3 2H _
59
1
1
67
2
2
52
2
4
60
3
1
69
2
1
56
1
4
61
4
1
70
2
1
60
1
2
4L 62 72M2 15H 2 C1 71
2
2
62
1
2
10L64 1M 2 0H 1
73
1
1
64
1
7
65
1
1
74
1
1
71
3
1
66 3M 1 1H 2_
75
2
1
72
1
5
68
3
1
76
2
1
77
2
4
69
2
1
77
1
1
81
1
3
70
1
4
78
3
2
84
1
6
74
1
2
80
1
1
90
1 15H _
76
1
3
81
3
2
79
2
1
83
2
1
80
2
3
84
1
1
83
2
2
85
1
1
855M 110H4_
86
1
2
1M
_
89
1 13
88
1
1
1H
102 1
89
1
1
90
1
3L92 30M1 1H 2
KOSblogs
3364
3365
3366
3367
d=UnitSTDVec g>6*avg
3368
3369
3370
0.1=AvgGp
3371
3372
Row# Doc#
F 28.2=MxGp
3373
1 1791
5.67
Gap
0
3374
...
...
...
...
...
3375
3376
___8 3389
___
7.00 0.19
0 gap=.65 Ct=9 C1
3377
9 2397
7.65 0.65
1
3378
3379
10 2841
7.82 0.17
0
3380
...
...
...
...
...
3381
2621
2334 89.40 0.06
0 gap=.6 Ct=2613 C2 3382
___
___
3383
2622 1122 90.00 0.60
1
3384
2623
245 90.06 0.06
0
3385
3386
...
...
...
...
...
3123
___
3169 132.06 0.00
___
0 gap=.75 Ct= 502 C3 3387
3388
3389
3124
321 132.81 0.75
1
3390
3125 2047 133.05 0.24
0
3391
3392
...
...
...
...
...
___
___
3210
343 145.29 0.37
0 gap=.6 Ct= 87 C4 3393
3394
3395
3211 2475 145.89 0.60
1
3396
3212
458 146.10 0.21
0
3397
...
...
...
...
...
3398
___
___
3240
542 151.15 0.09
0 gap=.61 Ct=30 C5 3399
3400
3241 2569 151.76 0.61
1
3401
3402
3242 1143 151.92 0.15
0
3403
...
...
...
...
...
3404
___
___
3285
1803 157.97 0.00
0 gap=.73 Ct=45 C6 3405
3406
3286 2257 158.70 0.73
1
3407
3287 2723 158.77 0.07
0
3408
3409
...
...
...
...
...
___
___
3293
129 159.56 0.32
0 gap=.89 Ct=8 C7 3410
3411
3412
3294 2541 160.45 0.89
1
3413
3295 2870 160.48 0.03
0
3414
3415
...
...
...
...
...
___
___
3301
401 161.38 0.04
0 gap=.65 Ct=8 C8 3416
3417
3302 2918 162.03 0.65
1
3418
3419
3303
100 162.07 0.04
0
3420
...
...
...
...
...
3421
___
___
gp=.72
Ct=
11
C9
3422
3312 1157 164.54 0.08
0
3423
___
___
gp=.65
Ct=1
outlr
3313
185 165.26 0.72
1
3424
3425
3314
685 165.91 0.65
1
3426
3315 2948 166.25 0.34
0
3427
...
...
...
...
...
3428
___
___
3325
190 168.59 0.37
0 gp=.61 Ct=12 C11 3429
3430
3326
3327
3328
3329
3330
___
3331
3332
3333
...
___
3342
___
3343
3344
3345
___
3346
3347
3348
3349
___
3350
3351
3352
3353
3354
___
3355
3356
2498
264
1611
3052
1002
1628
1241
3155
...
861
2509
2293
1257
2776
1422
12
183
620
679
462
3404
1850
3342
1396
169.20
169.31
169.64
169.96
170.43
170.64
171.80
172.00
...
173.84
174.98
175.65
175.67
176.04
177.15
177.24
177.26
177.29
179.08
179.15
180.02
180.79
181.21
183.04
gp=1 Ct=8 C16
outliers. Some
of them are
substantial
64=#gaps
.6=GapThreshold
0.61
0.11
0.33
0.32
0.47
___
0.20
1.16
0.20
...
___
0.15
___
1.13
0.67
0.02
___
0.37
1.11
0.09
0.02
___
0.03
1.79
0.07
0.88
0.76
___
0.43
1.82
1
0
0
0
0
0
1
0
...
0
1
1
0
0
1
0
0
0
1
0
1
1
0
1
gp=1.2 Ct=6 C12
gp=1.1 Ct=11 C13
gap=.67 Ct=1 utlr
gp=1.1 Ct=3 C15
gp=1.8 Ct=4 C16
gp=1.8 Ct=5 otl;r
1804
3399
980
1518
2090
890
24
2435
804
930
1096
1441
2885
2315
699
2108
1316
991
1564
2800
880
2038
481
480
295
1234
2140
3353
3402
45
3017
3365
2436
553
2545
54
1933
3201
2895
446
2302
2873
3388
1509
32
3189
3228
2107
1150
2279
2289
2385
1037
201
1252
1739
2446
1637
3220
1304
2355
232
3411
1955
1832
1197
2852
185.38
186.38
186.68
187.84
188.45
189.10
189.74
189.77
190.14
190.24
191.30
191.39
191.86
191.91
192.04
194.34
195.58
195.85
196.05
196.37
196.62
196.75
197.09
197.85
198.38
200.42
201.46
202.36
202.64
202.86
204.63
207.54
207.77
209.73
210.52
213.63
214.58
216.16
217.18
217.83
218.43
219.47
223.00
225.98
229.46
231.30
231.43
232.39
232.79
236.69
237.43
238.03
245.93
246.72
249.23
250.34
257.59
258.64
260.55
262.67
271.20
293.86
299.23
303.42
328.03
335.83
364.01
. 0.56
1.00
0.30
1.15
0.61
0.65
0.65
0.03
0.36
0.11
1.06
0.09
0.47
0.05
0.13
2.30
1.24
0.27
0.20
0.32
0.25
0.13
0.34
0.76
0.53
2.04
1.04
0.90
0.28
0.21
1.77
2.91
0.24
1.96
0.79
3.11
0.95
1.57
1.02
0.65
0.61
1.04
3.52
2.99
3.48
1.84
0.13
0.96
0.40
3.90
0.74
0.60
7.90
0.79
2.51
1.11
7.26
1.05
1.91
2.12
8.53
22.66
5.37
4.19
24.61
7.81
28.18
0
MVM gaps>6*avg
1
0
AvgGp.0085
gp>6*avg
1
ROW
KOS
F
GAP
CT
1
1 1 1791 0.2270
---1 2 1317 0.2920 0.065
1
0
2668
1602
6.6576
0.007
2667
0
3090
1390 9.8504 0.004 422
0
1
3132
1546 10.278 0.012
42
0
3148
2662 10.507 0.021
16
0
3216
505 11.289 0.019
68
0
0
3264
2219 11.994 0.027
48
1
3291
231
12.445
0.039
27
1
3302
710 12.631 0.038
11
0
3317
220 12.934 0.023
15
0
0
3338
405 13.315 0.028
21
0
3355
194
13.693
0.009
17
0
3368
12 14.151 0.078
8
0
3378
2731 14.590 0.011
10
1
0
3392
1096 15.459 0.022
5
1
1
1
0
0
1 on 22 highest STD KOS wds
GV
1
d=(.46
.16 .03 .32 .71 .07 .06 .03 .09 .03 .10
0
1 .19 .04 .16 .14 .01 .02 .04 .02 .00 .02)
.10
1
1
1
Doc
F=DPPd Gap
24=MxGp
1
1
2682
0
1
2749
7.574 0.038
0 3029
1
1
2983
8.436 0.079
0
42
1
3402
8.629 0.052
0
2
1
1
864
9.184 0.053
0
10
1
2293
9.462 0.106
1
4
0
1
2994
13.45 0.055
0 316
0
1445
13.66 0.029
0
4
1
1
3399
14.05 0.099
0
6
0
185
14.21 0.156
1
1
1
1
2731
14.35 0.143
1
1
1
2948
14.65 0.066
0
5
1
1
1495
14.99 0.014
0
2
1
804
15.20 0.205
1
1
1
1
3177
15.42 0.034
0
6
1
1
1316
15.61 0.024
0
2
1
1335
16.01 0.028
0
3
1
1
1637
16.35 0.330
1
1
1
880
16.86 0.039
0
3
1
1509
2885
446
1197
3189
1252
17.03
17.21
18.07
18.65
19.30
20.65
0.176
0.177
0.863
0.005
0.644
1.352
1
1
1
0
1
1
Cluster size:
d=USTD
MVM
GV
10
7
3
11
8
3
15
8
4
16
9
4
17
11
4
21
11
5
27
12
6
42
30
6
48
45
10
68
87
42
422
502
316
2667
2613
3029
1
1
1
4
1
1
d=e841 (highest STD).
DOC W=841
1716
0
...
... 1379 C0
2427
0
2750
54
2293
183
2870
1222
3217
1519
13
13
13
13
13
13
13
13 8 C13
2164
1656
3244
1709
14 otlrs
14
14
14
185
401
414
893
15 otlrs
15
15
15
1027
...
3427
1
...
1 743
1
...
2519
2
...
2 470
C2
868
...
3224
3
...
3 274
C3
1882
...
3257
4
...
4 175 C4
1434
...
910
5
...
5 127 C5
2731
1396
3220
3190
16 otlrs
16
16
16
2753
...
549
6
...
6 75 C6
1832
17 otlr
1186
...
1015
7
...
7 79 C7
2852
3201
1234
18 otlrs
18
18
503
...
3156
8
...
8 43 C8
3189
19 otlr
1524
22 otlr
1529
24 otlr
2971
...
2182
9
...
9 39 C9
1197
25 otlr
2868
...
1316
10
...
10 32 C10
201
27 otlr
1150
29 otlr
2648
...
336
11
...
11 18 C11
1335
34 otlr
2983
...
3177
12
...
12 14 C12
C1
UCUC(1101) 0.58
0.69
0.65
0.60
0.55
0.51
0.49
0.46
0.45
0.43
0.42
0.42
0.41
UCUC(1011) 0.58
0.82
0.93
0.97
0.99
1.00
0.99
0.98
0.94
0.89
0.82
0.75
0.68
0.62
0.57
0.53
0.50
0.47
0.45
0.44
0.43
0.42
0.41
0.41
UCUC(0111) 0.00
-0.15
-0.34
-0.46
-0.47
-0.45
-0.44
-0.43
-0.42
-0.41
-0.41
UCUC(1111)
0.50
0.83
0.95
0.97
0.95
0.90
0.83
0.76
0.69
0.63
0.58
0.54
0.50
0.48
0.46
0.44
0.43
0.42
0.41
0.41
akk
0.17
0.06
-0.04
-0.12
-0.19
-0.24
-0.27
-0.30
-0.32
-0.34
-0.35
-0.36
-0.37
-0.38
MVM
0.00
0.28
0.18
0.06
-0.04
-0.12
-0.19
-0.24
-0.27
-0.30
-0.33
-0.34
-0.35
-0.36
-0.37
GV using a grid (Unitized Corners of Unit Cube + Diagonal of the Variance Matrix + Mean-to-Vector_of_Medians)
CONC
UCUC(1000)
d1
d2
d3
d4
1.00
0.99
0.97
0.93
0.87
0.80
0.73
0.66
0.61
0.56
0.52
0.49
0.47
0.45
0.44
0.43
0.42
0.41
0.00
-0.23
-0.12
-0.01
0.09
0.16
0.22
0.26
0.29
0.32
0.33
0.35
0.36
0.37
0.37
0.00
-0.10
-0.17
-0.23
-0.27
-0.30
-0.32
-0.34
-0.35
-0.36
-0.37
-0.37
0.00
0.08
0.16
0.22
0.26
0.29
0.32
0.34
0.35
0.36
0.37
0.37
0.71
0.94
0.88
0.82
0.74
0.68
0.62
0.57
0.53
0.50
0.47
0.45
0.44
0.43
0.42
0.41
0.41
0.40
0.00
-0.05
-0.03
-0.00
0.03
0.05
0.08
0.09
0.11
0.12
0.13
0.13
0.14
0.14
0.14
0.14
0.14
0.14
1.00
0.33
0.21
0.19
0.19
0.18
0.17
0.17
0.16
0.16
0.16
0.15
0.15
0.15
0.15
0.00
-0.18
-0.18
-0.17
-0.17
-0.16
-0.16
-0.16
-0.15
-0.15
-0.15
-0.15
0.00
0.16
0.17
0.17
0.17
0.16
0.16
0.16
0.15
0.15
0.15
0.15
0.71
0.02
0.02
0.05
0.07
0.09
0.10
0.12
0.12
0.13
0.13
0.14
0.14
0.14
0.14
0.14
0.15
0.15
0.00
-0.11
-0.21
-0.32
-0.42
-0.51
-0.58
-0.63
-0.67
-0.70
-0.72
-0.73
-0.74
-0.75
-0.75
-0.76
-0.76
-0.76
0.00
-0.78
-0.82
-0.83
-0.83
-0.82
-0.81
-0.81
-0.80
-0.79
-0.79
-0.78
-0.78
-0.78
-0.78
1.00
0.93
0.86
0.83
0.81
0.80
0.79
0.79
0.78
0.78
0.78
0.78
0.00
-0.44
-0.69
-0.77
-0.79
-0.79
-0.79
-0.79
-0.78
-0.78
-0.78
-0.78
0.00
-0.29
-0.40
-0.49
-0.57
-0.62
-0.66
-0.69
-0.71
-0.73
-0.74
-0.75
-0.75
-0.76
-0.76
-0.76
-0.77
-0.77
0.00
0.06
0.13
0.19
0.26
0.31
0.36
0.39
0.42
0.43
0.45
0.46
0.46
0.47
0.47
0.47
0.47
0.48
0.00
0.49
0.52
0.52
0.52
0.52
0.51
0.50
0.50
0.50
0.49
0.49
0.49
0.49
0.48
0.00
-0.31
-0.44
-0.48
-0.49
-0.50
-0.49
-0.49
-0.49
-0.49
-0.49
-0.48
1.00
0.88
0.68
0.57
0.53
0.51
0.50
0.49
0.49
0.49
0.49
0.48
0.00
0.18
0.24
0.30
0.35
0.38
0.41
0.43
0.44
0.45
0.46
0.47
0.47
0.47
0.47
0.48
0.48
0.48
UCUC(1010) 0.71
0.69
0.62
0.52
0.40
0.27
0.15
0.03
0.00
-0.18
-0.20
-0.21
-0.21
-0.21
-0.20
-0.20
0.71
0.67
0.68
0.72
0.76
0.80
0.82
0.83
0.00
-0.21
-0.33
-0.41
-0.46
-0.50
-0.51
-0.52
On these pages we
display the variance
hill-climb for each
of the four datasets
(Concrete, IRIS,
Seeds, Wine) for a
grid of starting unit
vectors, d.
UCUC(0100)
I took the
circumscribing unit
non-negative cube
and used all the
Unitized diagonals.
UCUC(0010)
In low dimension
(all dimension=4
here) this grid is
very nearly a
uniform grid.
UCUC(0001)
Note that this will
work less and less
well as the
dimension grows.
UCUC(1100)
In all cases, the
same local max
and nearly the
same unit vector
are reached.
VAR
-0.06
-0.14
10249
-0.20
10585
-0.25
10947
-0.28
11370
-0.31
11796
-0.33
12168
-0.35
12453
-0.36
12649
-0.37
12776
-0.37
12855
-0.38
12902
-0.38
12929
12945
UCUC(1001) 0.71
12954
0.78
12960
0.74
12963
0.68
12965
0.62
12966
0.57
0.53
795
0.50
11645
0.47
12191
0.45
12469
0.44
12660
0.43
12783
0.42
12859
0.41
12904
0.41
12931
0.40
12946
12955
UCUC(0110) 0.00
12960
-0.19
12963
-0.25
12965
-0.28
12966
-0.31
-0.33
9950
-0.35
12279
-0.36
12749
-0.37
12865
-0.37
12911
12935
UCUC(0101) 0.00
12949
0.01
12956
0.09
12961
0.16
12964
0.22
12965
0.26
12966
0.29
0.32
6686
0.34
10572
0.35
12435
0.36
12816
0.37
12901
0.37
12932
12947
UCUC(0011) 0.00
12955
-0.06
12960
-0.16
12963
-0.23
12965
-0.28
12966
-0.31
-0.33
4968
-0.34
11266
-0.35
11709
-0.36
12096
-0.37
12400
-0.38
12614
12754
UCUC(1110) 0.58
12841
0.76
12894
0.72
12924
0.65
12942
0.56
12953
0.44
12959
0.32
12962
0.19
12964
0.07
12965
-0.04
12966
-0.12
12967
-0.18
-0.23
9007
-0.27
10074
-0.30
10486
-0.32
10867
-0.34
11289
-0.35
11721
-0.36
12106
-0.37
12408
-0.38
-0.19
-0.18
-0.17
-0.17
-0.16
-0.16
-0.16
-0.16
-0.15
-0.15
-0.15
-0.15
-0.15
0.00
0.05
0.07
0.09
0.10
0.12
0.12
0.13
0.13
0.14
0.14
0.14
0.14
0.14
0.15
0.15
0.71
-0.13
-0.17
-0.16
-0.16
-0.16
-0.16
-0.15
-0.15
-0.15
0.71
0.20
0.18
0.18
0.17
0.17
0.16
0.16
0.16
0.15
0.15
0.15
0.15
0.00
-0.09
-0.15
-0.17
-0.16
-0.16
-0.16
-0.16
-0.15
-0.15
-0.15
-0.15
0.58
-0.15
-0.19
-0.20
-0.20
-0.21
-0.21
-0.21
-0.20
-0.19
-0.18
-0.18
-0.17
-0.17
-0.16
-0.16
-0.16
-0.15
-0.15
-0.15
-0.15
0.83
0.82
0.82
0.81
0.80
0.79
0.79
0.79
0.78
0.78
0.78
0.78
0.77
0.00
-0.32
-0.50
-0.60
-0.65
-0.69
-0.71
-0.73
-0.74
-0.75
-0.75
-0.76
-0.76
-0.76
-0.77
-0.77
0.71
0.94
0.86
0.82
0.80
0.79
0.79
0.78
0.78
0.78
0.00
-0.54
-0.73
-0.79
-0.80
-0.80
-0.80
-0.79
-0.79
-0.78
-0.78
-0.78
-0.78
0.71
0.89
0.97
0.90
0.84
0.81
0.80
0.79
0.78
0.78
0.78
0.78
0.58
0.62
0.61
0.64
0.69
0.74
0.78
0.81
0.83
0.83
0.83
0.82
0.81
0.80
0.80
0.79
0.79
0.78
0.78
0.78
0.78
-0.52
-0.52
-0.51
-0.51
-0.50
-0.50
-0.49
-0.49
-0.49
-0.49
-0.48
-0.48
-0.48
0.71
0.53
0.44
0.42
0.42
0.43
0.45
0.45
0.46
0.47
0.47
0.47
0.47
0.48
0.48
0.48
0.00
-0.25
-0.41
-0.47
-0.49
-0.49
-0.49
-0.49
-0.49
-0.48
0.71
0.81
0.65
0.56
0.53
0.51
0.50
0.50
0.49
0.49
0.49
0.49
0.48
0.71
0.45
-0.02
-0.33
-0.44
-0.48
-0.49
-0.49
-0.49
-0.49
-0.49
-0.48
0.00
-0.14
-0.27
-0.36
-0.41
-0.46
-0.49
-0.51
-0.52
-0.52
-0.52
-0.51
-0.51
-0.50
-0.50
-0.49
-0.49
-0.49
-0.49
-0.49
-0.48
12619
12758
12843
12895
12925
12943
12953
12959
12962
12964
12965
12966
12967
9105
11499
12306
12601
12753
12841
12894
12924
12942
12953
12959
12962
12964
12965
12966
12967
3491
12162
12806
12915
12942
12953
12959
12963
12964
12966
4926
11209
12473
12765
12861
12907
12932
12947
12955
12960
12963
12965
12966
4951
6835
10755
12547
12876
12934
12951
12958
12962
12964
12965
12966
4647
9784
10149
10422
10750
11149
11582
11988
12319
12559
12719
12820
12881
12917
12938
12950
12957
12961
12964
12965
12966
0.58
0.10
0.10
0.11
0.12
0.13
0.13
0.14
0.14
0.14
0.14
0.14
0.15
0.00
-0.10
-0.12
-0.11
-0.10
-0.08
-0.06
-0.04
-0.01
0.02
0.05
0.07
0.09
0.10
0.12
0.12
0.13
0.13
0.14
0.14
0.14
0.14
0.14
0.15
0.58
0.02
-0.08
-0.12
-0.13
-0.14
-0.14
-0.14
-0.14
-0.14
-0.15
0.50
-0.04
-0.06
-0.04
-0.01
0.01
0.04
0.07
0.09
0.10
0.11
0.12
0.13
0.13
0.14
0.14
0.14
0.14
0.14
0.15
0.05
-0.19
-0.19
-0.18
-0.18
-0.17
-0.17
-0.16
-0.16
-0.16
-0.15
-0.15
-0.15
-0.15
-0.00
-0.19
-0.20
-0.20
-0.19
-0.18
-0.18
-0.17
-0.17
-0.16
-0.16
-0.16
-0.15
-0.15
-0.15
0.00
-0.43
-0.58
-0.66
-0.70
-0.72
-0.73
-0.74
-0.75
-0.76
-0.76
-0.76
-0.76
0.58
0.46
0.32
0.20
0.11
0.02
-0.08
-0.18
-0.29
-0.40
-0.49
-0.56
-0.62
-0.66
-0.69
-0.71
-0.73
-0.74
-0.75
-0.75
-0.76
-0.76
-0.76
-0.76
0.58
0.71
0.86
0.88
0.81
0.77
0.76
0.76
0.76
0.76
0.76
0.50
0.32
0.09
-0.09
-0.24
-0.36
-0.47
-0.55
-0.61
-0.65
-0.69
-0.71
-0.73
-0.74
-0.75
-0.75
-0.76
-0.76
-0.76
-0.76
0.98
0.93
0.88
0.84
0.83
0.81
0.80
0.80
0.79
0.79
0.78
0.78
0.78
0.78
0.00
0.49
0.71
0.79
0.82
0.82
0.82
0.81
0.80
0.80
0.79
0.79
0.78
0.78
0.78
0.58
0.57
0.48
0.45
0.45
0.45
0.46
0.46
0.47
0.47
0.47
0.48
0.48
0.58
0.33
0.12
0.02
-0.00
0.01
0.05
0.11
0.18
0.24
0.30
0.35
0.38
0.41
0.43
0.44
0.45
0.46
0.47
0.47
0.47
0.47
0.48
0.48
0.58
0.68
0.37
-0.09
-0.33
-0.42
-0.45
-0.47
-0.47
-0.48
-0.48
0.50
0.46
0.28
0.20
0.21
0.25
0.30
0.34
0.38
0.41
0.43
0.44
0.45
0.46
0.46
0.47
0.47
0.47
0.48
0.48
0.01
-0.30
-0.44
-0.49
-0.50
-0.50
-0.50
-0.50
-0.49
-0.49
-0.49
-0.49
-0.49
-0.48
-0.01
-0.80
-0.65
-0.58
-0.54
-0.53
-0.52
-0.51
-0.50
-0.50
-0.49
-0.49
-0.49
-0.49
-0.49
6756
11945
12599
12784
12864
12908
12933
12947
12956
12960
12963
12965
12966
6414
8390
9506
9889
10069
10254
10508
10851
11263
11695
12084
12391
12609
12751
12839
12892
12924
12942
12953
12959
12962
12964
12965
12966
3102
5237
7997
11648
12756
12928
12955
12962
12964
12965
12966
4385
8393
9943
10663
11151
11601
12007
12334
12569
12726
12824
12883
12918
12939
12951
12958
12962
12964
12965
12966
9327
11888
12502
12715
12822
12882
12918
12939
12951
12958
12962
12964
12965
12966
1
10378
11773
12296
12563
12724
12823
12883
12918
12939
12951
12958
12962
12964
12965
GV using a grid (Unitized Corners of Unit Cube + Diagonal of the Variance Matrix + Mean-to-Vector_of_Medians)
IRIS
d1
d2
d3
d4
VAR
SEEDS
d1
d2
d3
d4
VAR
UCUC(1000) 1.00
0.45
0.36
UCUC(0100) 0.00
-0.10
-0.34
UCUC(0010) 0.00
0.35
0.00
-0.03
-0.08
0.00
0.83
0.86
0.00
0.34
0.36
68
415
420
UCUC(1000) 1.00
0.97
0.00
0.16
0.00
-0.11
0.00
0.14
8
9
1.00
0.48
0.10
0.00
-0.82
-0.86
0.00
-0.30
-0.36
19
334
420
UCUC(0100) 0.00
0.96
1.00
0.23
0.00
-0.14
0.00
0.13
0
9
0.00
-0.09
1.00
0.86
0.00
0.35
311
420
UCUC(0001) 0.00
0.34
0.00
-0.08
0.00
0.85
1.00
0.39
58
420
0.00
-0.07
-0.15
-0.16
1.00
0.93
0.55
0.27
0.00
-0.00
-0.09
-0.12
2
4
8
9
0.00
0.15
0.00
-0.00
1.00
0.19
0
9
UCUC(1100) 0.71
0.53
0.37
UCUC(1010) 0.71
0.38
0.71
0.12
-0.07
0.00
0.78
0.86
0.00
0.33
0.36
39
390
420
0.71
0.17
0.00
-0.12
0.00
0.13
6
9
0.00
-0.07
0.71
0.85
0.00
0.35
316
420
0.00
0.16
0.16
0.71
0.20
-0.05
0.00
0.15
0.14
4
8
9
UCUC(1001) 0.71
0.40
0.36
UCUC(0110) 0.00
0.37
0.36
UCUC(0101) 0.00
0.41
0.37
UCUC(0011) 0.00
0.35
0.00
-0.05
-0.08
0.00
0.84
0.86
0.71
0.36
0.36
114
419
420
0.00
0.16
0.00
-0.10
0.71
0.14
5
9
0.71
-0.04
-0.08
0.71
0.86
0.86
0.00
0.36
0.36
133
419
420
0.71
0.06
-0.08
0.00
0.82
0.86
0.71
0.40
0.36
27
410
420
0.71
0.06
0.04
0.11
0.16
0.16
0.71
0.98
0.94
0.69
0.18
-0.06
0.00
0.08
0.10
0.14
0.15
0.14
1
2
3
5
8
9
0.71
0.20
0.00
-0.08
0.71
0.15
0
9
0.00
-0.09
0.71
0.86
0.71
0.36
312
420
UCUC(1110) 0.58
0.40
0.36
UCUC(1101) 0.58
0.43
0.37
UCUC(1011) 0.58
0.37
0.58
-0.04
-0.08
0.58
0.85
0.86
0.00
0.35
0.36
193
419
420
0.00
-0.01
-0.03
-0.10
-0.15
-0.16
0.71
0.99
1.00
0.86
0.44
0.23
0.71
0.09
0.05
-0.03
-0.10
-0.13
1
2
3
5
8
9
0.58
0.01
-0.08
0.00
0.83
0.86
0.58
0.36
0.36
72
414
420
0.58
0.17
0.16
0.58
0.15
-0.07
0.00
0.15
0.14
4
8
9
0.00
-0.07
0.58
0.85
0.58
0.36
349
420
0.58
0.17
0.00
-0.10
0.58
0.14
5
9
UCUC(0111) 0.00
0.36
0.58
-0.05
0.58
0.85
0.58
0.37
185
420
0.00
0.16
0.16
0.58
0.17
-0.06
0.58
0.16
0.14
4
8
9
UCUC(1111) 0.50
0.90
0.41
0.36
akk
0.90
0.41
0.36
MVM -0.00
0.35
0.50
0.24
-0.04
-0.08
0.50
0.37
0.84
0.86
0.50
0.04
0.35
0.36
243
180
418
420
0.58
0.11
0.15
0.16
0.58
0.80
0.31
-0.02
0.58
0.14
0.15
0.14
1
4
8
9
0.24
-0.04
-0.08
0.37
0.84
0.86
0.04
0.35
0.36
180
418
420
UCUC(0010) 0.00
-0.36
-0.82
-0.94
UCUC(0001) 0.00
0.97
UCUC(1100) 0.71
0.97
UCUC(1010) 0.71
0.96
0.97
UCUC(1001) 0.71
0.97
UCUC(0110) 0.00
0.19
0.33
0.70
0.96
0.97
UCUC(0101) 0.00
0.97
UCUC(0011) 0.00
0.08
-0.07
-0.51
-0.88
-0.95
UCUC(1110) 0.58
0.96
0.97
UCUC(1101) 0.58
0.97
UCUC(1011) 0.58
0.96
0.97
UCUC(0111) 0.00
0.56
0.92
0.98
UCUC(1111) 0.50
0.97
0.97
akk
0.98
0.50
0.17
0.16
0.50
0.13
-0.07
0.50
0.15
0.14
4
8
9
0.14
0.06
0.13
9
-0.04
-0.09
0.05
0.86
0.01
0.36
1
420
0.36
-0.15
0.27
0.22
-0.30
-0.13
4
9
MVM
-0.62
-0.95
WINE
d1
UCUC(1000) 1.00
0.40
0.02
UCUC(0100) 0.00
-0.00
-0.01
UCUC(0010) 0.00
-0.01
UCUC(0001) 0.00
-0.20
-0.02
UCUC(1100) 0.71
0.02
-0.01
UCUC(1010) 0.71
-0.01
UCUC(1001) 0.71
0.46
0.02
UCUC(0110) 0.00
-0.01
UCUC(0101) 0.00
-0.01
-0.01
UCUC(0011) 0.00
-0.02
UCUC(1110) 0.58
-0.01
-0.01
UCUC(1101) 0.58
0.02
-0.01
UCUC(1011) 0.58
-0.01
UCUC(0111) 0.00
-0.01
UCUC(1111) 0.50
-0.01
-0.01
akk
0.07
-0.01
MVM -0.13
0.01
2
d2
d3
d4
VAR
0.00
-0.06
-0.25
0.00
-0.91
-0.97
0.00
-0.07
-0.01
4
497
608
1.00
0.49
0.28
0.00
0.87
0.96
0.00
0.00
0.00
82
577
608
0.00
0.25
1.00
0.97
0.00
0.00
567
608
0.00
0.17
0.26
0.00
0.84
0.96
1.00
0.47
0.01
1
455
608
0.71
0.51
0.29
0.00
0.86
0.96
0.00
-0.00
0.00
42
570
608
0.00
0.25
0.71
0.97
0.00
0.00
277
608
0.00
0.00
-0.25
0.00
-0.88
-0.97
0.71
0.12
-0.00
2
447
608
0.71
0.31
0.71
0.95
0.00
0.00
472
608
0.71
0.48
0.28
0.00
0.88
0.96
0.71
0.01
0.00
42
578
608
0.00
0.25
0.71
0.97
0.71
0.01
287
608
0.58
0.31
0.27
0.58
0.95
0.96
0.00
0.00
0.00
310
607
608
0.58
0.50
0.29
0.00
0.86
0.96
0.58
0.01
0.00
29
572
608
0.00
0.25
0.58
0.97
0.58
0.01
186
608
0.58
0.30
0.58
0.95
0.58
0.01
317
608
0.50
0.31
0.27
0.50
0.95
0.96
0.50
0.01
0.00
234
607
608
0.15
0.26
0.98
0.97
0.12
0.00
588
608
-1.00
-0.27
-3.07
-0.96
-0.03
-0.00
6314
608
As we all know, Dr. Ubhaya is the best Mathematician on campus and he is attempting to prove three things: 1. That a GV-hill-climb that does
not reach the global max Variance is rare indeed. 2. That one is guaranteed to reach the global maximum with at least one of the coordinate unit
vectors (so a 90 degree grid will always suffice). 3. That akk will always reach the global max.
Finding round clusters that aren't DPPd separable? (no linear gap)
Find the golf ball? Suppose we have a white mask pTree.
No linear gaps exits to reveal it.
Search a grid of d-tubes until a DPPd gap is found in the interior of the tube
(Form mask pTree for interior of the d-tube.
Apply DPPd that mask to reveal interior gaps.)
Look for conical gaps (fix the the cone point at the middle of tube) over all cone angles
(look for an interval of angles with no points).
Notice that this method includes DPPd since a gap for a cone angle of 90 degrees is linear.
d
Width  24 so compute all pTree combinations
down to p4 and p'4
d=M-p
FAUST Gap Revealer
Z
p= z1
z2
z3
z4
z5
z6
z7
z8
z9
za
zb
zc
zd
ze
zf
1 1
3 1
2 2
3 3
6 2
9 3
15 1
14 2
15 3
13 4
10 9
11 10
9 11
11 11
7 8
F=zod
11
27
23
34
53
80
118
114
125
114
110
121
109
125
83
p6 p5 p4 p3 p2 p1 p0
0 0 0 1 0 1 1
0 0 1 1 0 1 1
0 0 1 0 1 1 1
0 1 0 0 0 1 0
0 1 1 0 1 0 1
1 0 1 0 0 0 0
1 1 1 0 1 1 0
1 1 1 0 0 1 0
1 1 1 1 1 0 1
1 1 1 0 0 1 0
1 1 0 1 1 1 0
1 1 1 1 0 0 1
1 1 0 1 1 0 1
1 1 1 1 1 0 1
1 0 1 0 0 1 1
p6' p5' p4' p3' p2' p1' p0'
1
1
1
0
1
0
0
1
1
0
0
1
0
0
1
1
0
1
0
0
0
1
0
1
1
1
0
1
1
0
0
1
0
1
0
0
1
0
1
1
1
1
0
0
0
1
0
0
1
0
0
0
1
1
0
1
0
0
0
0
0
1
0
0
0
0
1
1
0
1
0
0
1
0
0
0
1
0
0
0
0
1
1
0
0
0
1
0
0
1
0
0
0
0
0
0
1
0
0
1
0
1
1
0
0
1 z1 z2
z7
2
z3
z5
z8
3
z4
z6
z9
4
za
5
M
6
7
8
zf
9
zb
a
zc
b
zd ze
c
0 1 2 3 4 5 6 7 8 9 a b c d e f
p6'p5'p4'
0 0 0
1
p6' &p5' &p4'
0
1
1
0 [0,15]=[0,16)
1
1
0 [000 0000, 000 1111]=
1
0 is a 24 thinning.
0 has 1 point, z1. This1
0
1
1
0
z1
o
d=11
is
only
5
units
0
1
0
1 from the right
0
0
1 edge, so z1 is not declared
an outlier)
0
0
0
0
0 Next, we check the 0
min dis from the
1
0
0
right
edge
of
the
next
0
0 interval to see
1
1is actually  24
0 if z1's right-side gap0
0
0
1
(the
calculation
of
the
0
0
1 min is a pTree
0
1 process - no x looping required!)
C=5
C=3
C=1
p6'p5'p4
p6'p5'p4
p6'p5 p4'
p6'p5 p4'
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
1 [001 0000, 001 1111]1 = [16,32). The
1
1
[010 0000 , 010 1111]
= [32,48).
0 minimum, z3od=23 is1
0 7 units from the
1
1
1
0
02 of 32, so z4
1
1
1
0
0
z4
od=34 is within1
1
1
0
0
0 left edge, 16, so z1 has
0 only a 5+7=12
is not declared an0
1
1
0
0 unit gap on its right (not
0 a 24 gap). So
1
1anomaly.
1
1
0
0
0
04
1
1
1 z1 is not declared a 20
1 (and is declared a 0
0
0
1
1
1 24 inlier).
1
0
0
0
0
1
1
0
0
0
0
1
1
1
1
0
0
0
0
1
1
0
0
0
0
1
1
1
1
0
0
0
0
1
1
0
0
0
0
1
1
C=5
C=5
C=3
C=2
C=2
C=1
p6 p5'p4'
p6 &p5'&p4'
0
0
1
1
0
0
1 [100 0000 , 100 1111]=
1
[64, 80).
0
0
1
1
4
1 This is clearly a 2 0
1gap.
0
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
0
0
1
1
1
1
0
0
0
0
1
1
0
0
1
1
p6
0
1
0
1
0
1
0
1
0
1
1
0
1
0
1
0
1
0
1
0
1
0
0
1
1
0
1
C10
C=2
C=0
p6'p5 p4
p6'p5 p4
0
0
1
1
1
1
0 [011 0000, 011 1111]
0 = [ 48, 64).
1
0 z5od=53 is 19 from1
0z4od=34 (>24)
0
0
1
1
1 but 11 from 64. But1 the next int
1
0 [64,80) is empty z51
0is 27 from its
1
1
0
0
right
nbr.
z5
is
declared
an outlier
1
1
0
0
0
0
1 and we put a subcluster
1
cut thru z5
0
0
1
1
0
0
1
1
0
0
1 [111 0000 , 111 1111]=
1
[112,128)
0
0
1 z7od=118
1 od=114
z8
0
0
1
1
1 z9od=125
1 od=114
0
0
za
C=5 zcod=121
ze
od=125
C=2
C=1 No 24 gaps. But we can consult
SpS(d2(x,y) for actual distances:
p5'p4
p6 p5'p4
0
1
0
1
[101 0000 , 101 1111]=
[80, 96).
0
1
z6od=80, zfod=83 0
1
X1 X2 dX1X2 0
X1 X2 dX1X2
z7 z8
1.4 1
z9 z10 2.2
0
z7 z9
2.0 1
z9 z11 7.8
0
z7 z10 3.6 1
z9 z12 8.1
1
0
z7 z11 9.4 z9 z13 10.0
1
0
z7 z12 9.8 1
z9 z14 8.9
z7 z13 11.7 0
1
z10 z11 5.8
z7 z14 10.8 0
0
1
z10 z12 6.3
1
z8 z9
1.4 0
1
z8 z10 2.2 z10 z13 8.1
C10 z8 z11 8.1
z10 z14 7.3
C=2 z8 z12 8.5
C=2 z8 z13 10.3
z8 z14 9.5
p6 p5 p4'
p6 p5 p4'
0
0
1
1
0
[110 0000 , 11001111]= [96,112).
0
0
zbod=110, zdod=109.
So both
0
0
1
1
0
1
1 outliers (gap16
{z6,zf} declared0
1
1
0
0
both sides.
1
1
0
0
X1 X2 dX1X2
1
1
0
0
1
1
0
0
z11 z12 1.4
1
1
0
0
z11
z13
2.2
1
1
z11 z14 2.2
1
1
0
0
1
1
z12 z13 2.2
1
1
0
0
1
1
0
0
z12 z14 1.0
C10
C=8
C=2
z13 z14
2.0
p6 p5 p4
p6 p5 p4
0
0
0
0
1
1
0
0
1
1
that there are
0
1 Which reveals 0
1
0
0
1 no 24 gaps in this
1
1
1
0 subcluster.
0
1
1
1
1
1 And, incidentally,
1 it reveals
1
1
1
1
0 a 5.8 gap between
0 {7,8,9,a}
1 and {b,c,d,e} but
1 that
1
1
0
0
1 analysis is messy
1 and the
1
1
0 gap would be revealed
0
by
C10
the
next
x
o
fM
round
on
this
C=8
C=6 sub-cluster anyway.
FAUST Tube Clustering: (This method attempts to build tubular-shaped gaps around clusters)
Allows for a better fit around convex clusters that are elongated in one direction (not round).
Exhaustive Search for all tubular gaps:
It takes two parameters for a pseudo- exhaustive search (exhaustive modulo a grid width).
1. A StartPoint, p (an n-vector, so n dimensional)
2. A UnitVector, d (a n-direction, so n-1 dimensional - grid on the surface of sphere in Rn).
q
Gaps in dot product
lengths [projections]
on the line.
Then for every choice of (p,d) (e.g., in a grid of points in R2n-1) two functionals are used to
enclose subclusters in tubular gaps.
a. SquareTubeRadius functional, STR(y) = (y-p)o(y-p) - ((y-p)od)2
b. TubeLength functional, TL(y) = (y-p)od
y
tube
cap gap
width
Given a p, do we need a full grid of ds (directions)? No! d and -d give the same TL-gaps.
Given d, do we need a full grid of p starting pts? No! All p' s.t. p'=p+cd give same gaps.
Hill climb gap width from a good starting point and direction.
MATH: Need dot product projection length and dot product projection distance (in red).
y
yo f
|f|
f
|f|
p
tube radius
gap width
(yof) f o y - (yof) f
=
fo f
fo f
(yof)2 + (yof)2 fof
dot product projection distance
(yof)2 squared = yoy - 2 fof
(fof)2
Squared y fon f Proj Dis = yoy (yof)2 + (yof)2
fof
squared = yoy - 2
fof
fo f
y - yo f f
|f| |f|
y - (yof) f
fo f
dot prod proj len
Squared y-p on q-p Projection Distance = (y-p)o(y-p) 1st
= yoy -2yop + pop - ( yo(q-p) - p o(q-p
|q-p|
|q-p|
2
squared is y -
( (y-p)o(q-p) )2
(q-p)o(q-p)
For the dot product length projections (caps) we already needed:
(y-p)o M-p
|M-p|
= ( yo(M-p) - po M-p )
|M-p|
|M-p|
That is, we needed to compute the green constants and the blue and red dot product functionals in an optimal way (and then do the PTreeSet
additions/subtractions/multiplications). What is optimal? (minimizing PTreeSet functional creations and PTreeSet operations.)
Cone Clustering:
x=s2
cone=.1
w maxs-to-mins
cone=.939
w naaa-xaaa
cone=.95
(finding cone-shaped clusters)
39
40
41
44
45
46
47
52
59
60
61
62
63
64
65
66
67
69
70
14
16
18
19
20
22
23
24
25
26
27
28
29
30
31
32
34
35
36
37
38
39
40
41
46
47
48
49
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
12
1
13
2
14
1
15
2
16
1
17
1
18
4
19
3
20
2
21
3
22
5
23
6 i21
24
5
25
1
27
1
28
1
29
2
30
2 i7
41/43 e
so picks e
x=s1
cone=1/√2
60
61
62
63
64
65
66
67
69
3
4
3
10
15
9
3
1
2
50
x=s2
cone=1/√2
x=s2
cone=.9
47
59
60
61
62
63
64
65
66
67
69
70
59
60
61
62
63
64
65
66
67
69
70
1
2
4
3
6
10
10
5
4
4
1
1
51
2
3
3
5
9
10
5
4
4
1
1
47
w maxs
cone=.707
0
2
F=(y-M)o(x-M)/|x-M|-mn 8 1
10
3
restricted to a cosine cone 12 2
13
1
on IRIS
14
3
15
1
16
3
17
5
18
3
19
5
x=i1
20
6
cone=.707 21
2
22
4
x=e1
34
1
23
3
cone=.707 35
1
24
3
36
2
25
9
33
1
37
2
26
3
36
2
38
3
27
3
37
2
39
5
28
3
38
3
40
4
29
5
39
1
42
6
30
3
40
5
43
2
31
4
41
4
44
7
32
3
42
2
45
5
33
2
43
1
47
2
34
2
44
1
48
3
35
2
45
6
49
3
36
4
46
4
50
3
37
1
47
5
51
4
38
1
48
1
52
3
40
1
49
2
53
2
41
4
50
5
54
2
42
5
51
1
55
4
43
5
52
2
56
2
44
7
54
2
57
1
45
3
55
1
58
1
46
1
57
2
59
1
47
6
58
1
60
1
48
6
60
1
61
1
49
2
62
1
62
1
51
1
63
1
63
1
52
2
64
1
64
1
53
1
65
2
66
1
55
1
60
75
137
2
1
1
1
1
1
1
1 i39
2
4
3
6
10
10
5
4
4
1
1
59
w maxs
cone=.93
8
1
13
1
14
3
16
2
17
2
18
1
19
3
20
4
21
1
24
1
25
4
26
1
27
2
29
2
37
1
27/29
i10
e21 e34
i7
are i's
w maxs
cone=.925
8
1 i10
13
1
14
3
16
3
17
2
18
2
19
3
20
4
21
1
24
1
25
5
26
1 e21 e34
27
2
28
1
29
2
31
1 e35
37
1 i7
31/34 are i's
1 i25
1 i40
2 i16 i42
2 i17 i38
2 i11 i48
2
1
4 i34 i50
3 i24 i28
3 i27
5
3
2
2
3
4
3
4
2
2
2
3
1
2
1
2
1
1 i39
1
2
1
1
8
5
4
7
4
5
5
1
3
1
1
1
114
14 i and 100 s/e.
So picks i as 0
w xnnn-nxxx
cone=.95
8
2
10
2
11
2
12
4
13
2
14
4
15
3
16
8
17
4
18
7
19
3
20
5
21
1
22
1
23
1
34
1
43/50
i22 i50
Gap in dot product
projections onto the
cornerpoints line.
Cosine cone gap
(over some  angle)
Corner points
w aaan-aaax
cone=.54
7
3 i27 i28
8
1
9
3
10 12 i20 i34
11
7
12 13
13
5
14
3
15
7
19
1
20
1
21
7
22
7
23 28
24
6
100/104 s or e
so 0 picks i
i28
i24 i27 i34
i39
e so picks out e
Cosine conical gapping seems
quick and easy (cosine = dot
product divided by both
lengths.
Length of the fixed vector, x-M,
is a one-time calculation.
Length y-M changes with y so
build the PTreeSet.
rotation d toward a higher F-STD or grow 1 gap using support pairs:
F-slices are hyperplanes (assuming F=dotd) so it would makes sense to try to "re-orient" d so that the gap grows.
Instead of taking the "improved" p and q to be the means of the entire n-dimensional half-spaces which is cut by the
gap (or thinning), take as p and q to be the means of the F-slice (n-1)-dimensional hyperplanes defining the gap or
thinning. This is easy since our method produces the pTree mask the sequence of F-values and the sequence of counts
of points that give us those value that we use to find large gaps in the first place.
d2-gap >> than d1=gap (still not optimal.) Weight mean by the dist from gap? (d-barrel radius)
In this example it seems to make for a larger gap, but what weightings should be used? (e.g., 1/radius2) (zero
weighting after the first gap is identical to the previous). Also we really want to identify the Support vector pair of the
gap (the pair, one from one side and the other from the other side which are closest together) as p and q (in this case, 9
and a but we were just lucky to draw our vector through them.) We could check the d-barrel radius of just these gap
slice pairs and select the closest pair as p and q???
"Gap Hill Climbing": mathematical analysis
Dot F p=aaan q=aaax
0
6
1 28
2
7 C1<7 (50 Set)
3
7
4
1
5
1
9
7
10
3 7<C2<16 (4i, 48e)
11
5
12 13
13
8
14 12
15
4
16
2
17 12
18
5 C3>16 (46i, 2e)
19
6
20
6
21
3
22
8 hill-climb gap at 16
23
3
24
3 w half-space avgs.
C2uC3 p=avg<16 q=avg>16
0
1
1
1
No conclusive gaps Sparse Lo end: Check [0,9]
2
2
0
1
2
2
3
7
7
9
9
3
1
i39 e49 e8 e44 e11 e32 e30 e15 e31
7
2
i39
0 17 21 21 24 22 19 19 23
9
2
e49 17
0
4
4
7
8
8
9
9
10
2
e8
21
4
0
1
5
7
8 10
8
11
3
e44 21
4
1
0
4
6
8
9
7
12
3
e11 24
7
5
4
0
7
9 11
7
13
2
e32 22
8
7
6
7
0
3
6
1
14
5
e30 19
8
8
8
9
3
0
4
4
15
1
e15 19
9 10
9 11
6
4
0
6
16
3
e31 23
9
8
7
7
1
4
6
0
17
3
i39,e49,e11 singleton outliers. {e8,i44} doubleton outlier set
18
2
19
2
20
4
21
5
There is a thinning at 22 and it is the same one but it is
22
2
not more prominent. Next we attempt to hill-climb the
23
5
24
9
gap at 16 using the mean of the half-space boundary.
25
1
(i.e., p is avg=14; q is avg=17.
26
1
27
3
28
2
Sparse Hi end: Check [38,47] distances
29
1
38 39 42 42 44 45 45 47 47
30
3
i31 i8 i36 i10 i6 i23 i32 i18 i19
31
5
i31
0
3
5 10
6
7 12 12 10
32
2
i8
3
0
7 10
5
6 11 11
9
33
3
i36
5
7
0
8
5
7
9 10
9
34
3
i10 10 10
8
0 10 12
9
9 14
35
1
i6
6
5
5 10
0
3
9
8
5
36
2
i23
7
6
7 12
3
0 11 10
4
37
4
i32 12 11
9
9
9 11
0
4 13
38
1
i18 12 11 10
9
8 10
4
0 12
39
1
i19 10
9
9 14
5
4 13 12
0
42
2
i10,i18,i19,i32,i36 singleton outliers {i6,i23} doubleton outlier
44
1
45
2
47
2
f
e
d
c
b
a
9
8
7
6
5
4
3
2
1
0
0 1 2 3 4 5 6 7 8 9 a b c d e f
1 0
2 3
4 5 6
7 8 =p
9
a
j
q=b
d
k
c
e
l
q
r
f
g
o
p
h
i
C123 p avg=14 q avg=17
0
1
32
2
2
3
33
3
3
2
34
4
4
4
35
1
5
7
36
3
6
4
37
4
7
8
38
2
8
2
39
2
9 11
40
5
10
4
41
3
12
3
42
3
13
1
43
6
20
1
44
8
21
1
45
1
22
2
46
2
23
1
47
1
27
2
48
3
28
1
49
3
29
1
51
7
30
2
52
2
31
4
53
54
55
56
57
58
61
63
64
66
67
m n
s
2
3
1
3
3
1
2
2
1
1
1
f
e
d
c
b
a
9
8
7
6
5
4
3
2
1
0
0 1 2 3 4 5 6 7 8 9 a b c
1
2 3
4 5 6
p
7 8
9
a
b
d
j
k
qc
e
q
f
Here, gap between C1,C2 is more pronounced Why?
Thinning C2,C3 more obscure?
It did not grow gap wanted to grow (tween C2 ,C3.
CAINE 2013 Call for Papers 26th International Conference on Computer Applications in Industry and Engineering September 25{27, 2013, Omni Hotel, Los Angles,
Califorria, USA Sponsored by the International Society for Computers and Their Applications (ISCA) CAINE{2013 will feature contributed papers as well as workshops
and special sessions. Papers will be accepted into oral presentation sessions. The topics will include, but are not limited to, the following areas:
Agent-Based Systems Image/Signal Processing Autonomous Systems Information Assurance Big Data Analytics Information Systems/Databases
Bioinformatics, Biomedical Systems/Engineering Internet and Web-Based Systems
Computer-Aided Design/Manufacturing Knowledge-based Systems
Computer Architecture/VLSI Mobile Computing Computer Graphics and Animation Multimedia Applications Computer Modeling/Simulation Neural Networks
Computer Security Pattern Recognition/Computer Vision Computers in Education Rough Set and Fuzzy Logic Computers in Healthcare Robotics
Computer Networks Fuzzy Logic
Control Systems Sensor Networks Data Communication Scientic Computing
Data Mining Software Engineering/CASE
Distributed Systems Visualization Embedded Systems Wireless Networks and Communication
Important Dates: Workshop/special session proposal . . May 2.5,.2.013
Full Paper Submis . .June 5,.2013.
Notice Accept ..July.5 , 2013.
Pre-registration & Camera-Ready Paper Due . . . ..August 5, 2013.
Event Dates . . .Sept 25-27, 2013
SEDE Conf is interested in gathering researchers and professionals in the domains of SE and DE to present and discuss high-quality research results and outcomes in their
fields. SEDE 2013 aims at facilitating cross-fertilization of ideas in Software and Data Engineering, The conference topics include, but not limited to:
. Requirements Engineering for Data Intensive Software Systems. Software Verification and Model of Checking. Model-Based Methodologies. Software Quality and
Software Metrics. Architecture and Design of Data Intensive Software Systems. Software Testing. Service- and Aspect-Oriented Techniques. Adaptive Software Systems
. Information System Development. Software and Data Visualization. Development Tools for Data Intensive. Software Systems. Software Processes. Software Project Mgnt
. Applications and Case Studies. Engineering Distributed, Parallel, and Peer-to-Peer Databases. Cloud infrastructure, Mobile, Distributed, and Peer-to-Peer Data Management
. Semi-Structured Data and XML Databases. Data Integration, Interoperability, and Metadata. Data Mining: Traditional, Large-Scale, and Parallel. Ubiquitous Data
Management and Mobile Databases. Data Privacy and Security. Scientific and Biological Databases and Bioinformatics. Social networks, web, and personal information
management. Data Grids, Data Warehousing, OLAP. Temporal, Spatial, Sensor, and Multimedia Databases. Taxonomy and Categorization. Pattern Recognition, Clustering,
and Classification. Knowledge Management and Ontologies. Query Processing and Optimization. Database Applications and Experiences. Web Data Mgnt and Deep Web
May 23, 2013 Paper Submission Deadline
June 30, 2013 Notification of Acceptance
July 20, 2013 Registration and Camera-Ready Manuscript
Conference Website: http://theory.utdallas.edu/SEDE2013/
ACC-2013 provides an international forum for presentation and discussion of research on a variety of aspects of advanced computing and its applications, and
communication and networking systems. Important Dates
May 5, 2013 - Special Sessions Proposal
June 5, 2013 - Full Paper Submission
July 5, 2013 - Author Notification
Aug. 5, 2013 - Advance Registration & Camera Ready Paper Due
CBR International Workshop Case-Based Reasoning CBR-MD 2013 July 19, 2013, New York/USA Topics of interest include (but are not limited to):
CBR for signals, images, video, audio and text Similarity assessment Case representation and case mining Retrieval and indexing Conversational CBR Metalearning for model improvement and parameter setting for processing with CBR Incremental model improvement by CBR Case base maintenance for systems Case
authoring Life-time of a CBR system Measuring coverage of case bases Ontology learning with CBR
Submission Deadline: March 20th, 2013
Notification Date: April 30th, 2013
Camera-Ready Deadline: May 12th, 2013
Workshop on Data Mining in Life Sciences DMLS Discovery of high-level structures, incl e.g. association networks Text mining from biomedical literatur Medical
images mining Biomedical signals mining Temporal and sequential data mining Mining heterogeneous data Mining data from molecular biology, genomics,
proteomics, pylogenetic classification With regard to different methodologies and case studies: Data mining project development methodology for biomedicine
Integration of data mining in the clinic Ontology-driver data mining in life sciences Methodology for mining complex data, e.g. a combination of laboratory test results,
images, signals, genomic and proteomic samples Data mining for personal disease management Utility considerations in DMLS, including e.g. cost-sensitive learning
Submission Deadline: March 20th, 2013 Notification Date: April 30th, 2013 Camera-Ready Deadline: May 12th, 2013 Workshop date: July 19th, 2013
Workshop on Data Mining in Marketing DMM'2013 In business environment data warehousing - the practice of creating huge, central stores of customer data that can
be used throughout the enterprise - is becoming more and more common practice and, as a consequence, the importance of data mining is growing stronger. Applications in
Marketing Methods for User Profiling Mining Insurance Data
E-Markteing with Data Mining
Logfile Analysis
Churn Management Association Rules for
Marketing Applications
Online Targeting and Controlling
Behavioral Targeting
Juridical Conditions of E-Marketing, Online Targeting and so one
Controll of
Online-Marketing Activities
New Trends in Online Marketing Aspects of E-Mailing Activities and Newsletter Mailing
Submission Deadline: March 20th, 2013 Notification Date: April 30th, 2013 Camera-Ready Deadline: May 12th, 2013 Workshop date: July 19th, 2013
Workshop Data Mining in Ag DMA 2013 Data Mining on Sensor and Spatial Data from Agricultural Applications Analysis of Remote Sensor Data
Feature
Selection on Agricultural Data Evaluation of Data Mining Experiments
Spatial Autocorrelation in Agricultural Data
Submission Deadline: March 20th, 2013 Notification Date: April 30th, 2013 Camera-Ready Deadline: May 12th, 2013 Workshop date: July 19th, 2013
Hierarchical
Clustering
Any maximal anti-chain
(maximal set of nodes s.t
no 2 directly connected)
is a clustering.
(dendogram offers many
DE

A

FG

BC
F

B

But horizontal antichains are clusterngs
from top down
(or bottom up)
method(s).
DEFG
ABC

D
E


C

G

GV
F=(DPP-MN)/4 Concrete(C, W, FA, A)
0
1
1
1
5
1
6
1
7
1
8
4
med=14
9
1
10
1
11
2
12
1
13
5
14
1
15
3
med=18
16
3
17
4
18
1
19
3
20
9
21
4
22
3
23
7
24
2 med=40
25
4
26
8
27
7
28
7
med=56
29 10
30
3
31
1
32
3
33
6
med=61
34
4
35
5
37
2
38
2
40
1
42
3
43
1
44
1
45
1
46
4
______
CLUS 4 gap=7
49
1
56
1 [52,74) 0L 7M 0H CLUS_3
58
1
61
1
65
1
66
1
69
1
______
gap=6
71
1
77
1 [74,90) 0L 4M 0H CLUS_2
80
1
83
1[0.90) 43L 46 M 55H
________
86
1 [90,113) 0L 6M 0H CLUS_1gap=14
100
1
103
1
105
1
108
2
112
1
_____________At this level,
FinalClus1={17M}
0 errors
C1 C2 C3 C4
med=10
med=9
med=17
med=21
med=23
med=34
med=33
med=57
med=62
med=71
med=71
med=86
CLUS 4 (F=(DPP-MN)/2, Fgap2
_______
0L 0M 3H CLUS 4.4.1 gap=7
0
3 =0
0L 0M 4H CLUS 4.4.2 gap=2
7
4 =7
9
1 [8,14] 1L 5M 22H CLUS 4.4.3 1L+5M err H
10 12
11
8
gap=3
12
7
______
0L 0M 4H
CLUS 4.3.1 gap=3
15
4 =15
0L 0M 10H CLUS 4.3.2 gap=3
18 10 =18
21
3
22
7
______
23
2 [20,24) 0L 10M 2H CLUS 4.7.2 gap=2
25
2 [24,30) 10L 0M 0H CLUS_4.7.1
26
3
27
1
28
2
gap=2
29
1
31
3
CLUS 4.2.1 gap=2
32
1 [30,33] 0L 4M 0H
0L 2M 0H
CLUS 4.2.2 gap=6
34
2 =34
______
0L 4M 0H CLUS_4.2.3 gap=7
40
4 =40
0L 3M 0H CLUS_4.2.4 gap=5
47
3 =47
52
1
53
3
54
3
55
4
56
2
57
3
______
gap=2
58
1 [50,59) 12L 1M 4H CLUS 4.8.1
8L 0M 0H CLUS_4.8.2
60
2 [59,63)
61
2
gap=2
62
4
______
2L 0M 2H CLUS 4.6.1
gap=3
64
4 =64
[66,70)
10L
0M
0H
CLUS
4.6.2
67
2
gap=3
68
1
71
7 [70,79) 10L 0M 0H CLUS_4.5
______
gap=7
72
3
5L 0M 0H CLUS_4.1.1
gap=6
79
5 =79
85
1 [74,90) 2L 0M 1H CLUS_4.1 1 Merr in L
87
2
Median=0 Avg=0
Median=7 Avg=7
Median=11 Avg=10.7
Median=15 Avg=15
Median=18 Avg=18
Median=22 Avg=22 2H errs in L
Median=26 Avg=26
Median=31
Median=34
Median=40
Median=47
Avg=32.3
Avg=34
Avg=40
Avt=47
Accuracy=90%
Median=55 Avg=55 1M+4H errs in L
Median=61.5 Avg=61.3
Median=64 Avg=64 2 H errs in L
Median=67 Avg=67.3
Median=71 Avg=71.7
Median=79 Avg=79
Median=87 Avg=86.3
Suppose we know (or want) 3 clusters, Low, Medium and High Strength. Then we find
Suppose we know that we want 3 strength clusters, Low, Medium and High. We can use an antichain that gives us exactly 3 subclusters two ways, one show in brown and the other in purple
Which would we choose? The brown seems to give slightly more uniform subcluster sizes.
Brown error count: Low (bottom) 11, Medium (middle) 0, High (top) 26, so 96/133=72% accurate.
The Purple error count: Low 2, Medium 22, High 35, so 74/133=56% accurate.
What about agglomerating using single link agglomeration (minimum pairwise distance?
Agglomerate (build dendogram) by iteratively gluing together clusters with min Median separation.
Should I have normalize the rounds?
Should I have used the same Fdivisor and made sure the range of values was the same in 2nd round
as it was in the 1st round (on CLUS 4)?
Can I normalize after the fact, I by multiplying 1st round values by 100/88=1.76?
Agglomerate the 1st round clusters and then independently agglomerate 2nd round clusters?
CONCRETE
GV
Agglomerating using single link (min pairwise distance = min gap size! (glue min-gap adjacent clusters 1st)
CLUS 4 (F=(DPP-MN)/2, Fgap2
_______
0L 0M 3H CLUS 4.4.1 gap=7
0
3 =0
0L 0M 4H CLUS 4.4.2 gap=2
7
4 =7
9
1 [8,14] 1L 5M 22H CLUS 4.4.3 1L+5M err H
10 12
11
8
gap=3
12
7
______
0L 0M 4H
CLUS 4.3.1 gap=3
15
4 =15
0L 0M 10H CLUS 4.3.2 gap=3
18 10 =18
21
3
22
7
______
23
2 [20,24) 0L 10M 2H CLUS 4.7.2 gap=2
25
2 [24,30) 10L 0M 0H CLUS_4.7.1
26
3
27
1
28
2
gap=2
29
1
31
3
CLUS 4.2.1 gap=2
32
1 [30,33] 0L 4M 0H
0L 2M 0H
CLUS 4.2.2 gap=6
34
2 =34
______
0L 4M 0H CLUS_4.2.3 gap=7
40
4 =40
0L 3M 0H CLUS_4.2.4 gap=5
47
3 =47
52
1
53
3
54
3
55
4
56
2
57
3
______
gap=2
58
1 [50,59) 12L 1M 4H CLUS 4.8.1
8L 0M 0H CLUS_4.8.2
60
2 [59,63)
61
2
gap=2
62
4
______
2L 0M 2H CLUS 4.6.1
gap=3
64
4 =64
[66,70)
10L
0M
0H
CLUS
4.6.2
67
2
gap=3
68
1
71
7 [70,79) 10L 0M 0H CLUS_4.5
______
gap=7
72
3
5L 0M 0H CLUS_4.1.1
gap=6
79
5 =79
85
1 [74,90) 2L 0M 1H CLUS_4.1 1 Merr in L
87
2
Median=0 Avg=0
Median=7 Avg=7
Median=11 Avg=10.7
Median=15 Avg=15
Median=18 Avg=18
Median=22 Avg=22 2H errs in L
Median=26 Avg=26
Median=31
Median=34
Median=40
Median=47
Avg=32.3
Avg=34
Avg=40
Avt=47
Accuracy=90%
Median=55 Avg=55 1M+4H errs in L
Median=61.5 Avg=61.3
Median=64 Avg=64 2 H errs in L
Median=67 Avg=67.3
Median=71 Avg=71.7
Median=79 Avg=79
Median=87 Avg=86.3
The first thing we can notice is that outliers mess up agglomerations which are supervised by knowledge of the number of subclusters expected.
Therefore we might remove outliers by backing away from all gap5 agglomerations, then looking for a 3 subcluster max anti-chains.
What we have done is to declare F<7 and F>84 as extreme tripleton outliers sets; and F=79. F=40 and F=47 as singleton outlier sets because they
are F-gapped by at least 5 (which is actually 10) on either side.
The brown gives more uniform sizes. Brown errors: Low (bottom) 8, Medium (middle) 12 and High (top) 6, so 107/133=80% accurate.
The one decision to agglomerate C4.7.1 to C4.7.2 (gap=3) instead of C4.3.2 to C4.7.2 (gap=3) lots of error. C4.7.1 and C4.7.2 are problematic
since they are separate out, but in increasing F order, it's H M L M L, so if we suspected this pattern we would look for 5 subclusters.
The 5 orange errors in increasing F-order are: 6, 2, 0, 0, 8 so 127/133=95% accurate.
If you have ever studied concrete, you know it is a very complex material. The fact that it clusters out with a F-order pattern of HMLML is just
bizarre! So we should expect errors.
CONCRETE