This bio-recipe explains what a linear classification (or a linear discrimination) is and shows how to compute them using Darwin. It also explains some of the algorithms used for the computation and how to gain some confidence on the results.

Classification in our definition aims to assign
observations into classes. This can be used to establish a rule
(a classifier) to assign a new observation into a class.
In other words,
previously know information is used to define the classifier and
classification deals with assigning a new point in a
vector space to a class separated by a boundary.
Linear classification provides a mathematical
formula to predict a binary result.
This result is typically a true/false (positive/negative) result
but it could be any other pair of characters.
Without loss of generality we will assume that our result is
a boolean variable, that is it takes the values true/false.
To do this prediction we use a linear formula over the given input data,
from here on referred to as *inputs*. Other names for inputs are
predictors, independent variables and source variables.
The linear form is computed over the inputs and its result
is compared against a basis constant.
Depending on the result of the comparison we predict true/false.
In terms of equations this can be stated as the discriminator:

where *a*_{1}, *a*_{2}, ... are
the variables corresponding to one observation and
*x*_{1}, *x*_{2}, ... together with
*x*_{0} are the solution vector plus basis constant.
Corresponding to each input vector *a* there is
a dependent boolean variable*b*. This is from now on called the *output*.
Other terms for the *outputs* is responses, dependent variable and target variables.
Normally we have many (*m*) data points (a row vector *a* of
inputs and a corresponding output *b*).
It is convenient to place all these data in matrices and vectors
as it is the practice in linear algebra.
*A* will denote the matrix of the inputs, *A*
is of dimension *m* by *n*.
The *m* boolean outputs will be stored in a vector *B*.
The problem of linear classification can be stated
now in terms of these matrices and vectors as: find *x*
and *x*_{0} such that

The equivalence is understood as follows,
every row of the left-hand-side is a relation, which for
the given data will be true/positive or false/negative.
This has to match the corresponding *B* value.
In Statistics, linear classification is sometimes also called
linear discrimination or supervised learning.

In general, depending on the data, or as soon as we
have more data points than columns (*m* > *n*) the
problem does not have a solution.
That is, we cannot find a vector *x* that satisfies all the
true values.
In this case we try to find a solution which matches the
vector *B* as "best" as possible.
Best matching means either a minimum number of errors,
or a weighted minimum of errors (usually a weight is
assigned to the `true` errors and another weight
is assigned to the `false` errors).

The main idea of obtaining a good classification
is to be able to use it to predict the output for
other (new) data points.
In this context, the discriminator is a prediction formula or
a model, *A* and *B* are the training data and *x*,
*x*_{0} are the parameters of the model fitted
to the training data.
The following picture shows an example of the main
components of linear classification:

where the true (positives) are marked with
a green "T" and the false (negatives) are marked with a red "F".
The inputs has two independent variables, *a*_{1}
and *a*_{2} called features, which we use to position
the points in the plane.
There are 11 positives and 12 negatives.
A linear classification in two dimensions will be a straight line.
In this case the straight line drawn is a very good classification,
it separates most of the points correctly.
In particular, all the positives are on one side and all the negatives
but one are on the other side.
So this classification has one "false positive" and no "false negatives".
(A false positive is a data point which is classified as positive and
it is negative and conversely for false negatives.)

Classical examples of linear classification problems
are medical diagnosis problems: we have individual data for a large
number (*m*) of patients;
some patients develop some particular disease, some don't.
The purpose of classification in this case is to
find a linear formula on the data of the patient that can
describe the condition (diseased/healthy).
Once that we have found a classification, we must validate it.
This means testing that it has predicting power, and how accurate
it appears to be.
Notice that finding a good classification does NOT guarantee that
it will be a good predictor.

Finally, if the outputs are numerical i.e. it is not binary, then classification is not appropriate. See function approximation (SVD, BestBasis, etc.) for that purpose.

Darwin has a main function to compute linear classifications called "LinearClassify". The information computed is stored in a class called "LinearClassification".

?LinearClassify

Function LinearClassify - Linear form which does pos/neg classification Calling Sequence: LinearClassify(A,accept,mode,WeightNeg) Parameters: Name Type Description ----------------------------------------------------------------------------- A matrix(numeric) an n x m matrix of independent variables accept {list,procedure} positive/negative determination mode anything optional, algorithm selection, defaults to Svd WeightNeg positive optional, weight of negatives Returns: LinearClassification Synopsis: Computes a vector X such that the values A[i]*X >= X0 classify the positive/negatives. Such a vector is called a linear discriminant in statistics. A special way of computing such a vector is called the Fisher linear discriminant. LinearClassify normally produces results which are much better than the Fisher linear discriminant. LinearClassify returns a data structure called LinearClassification which contains the vector X and the splitting value, called X0, the weights of positive and negatives, the score obtained and the worst mis-classifications. The third argument, if present, directs the function to use a particular method of computation. The methods are mode description ------------------------------------------------------------------------- BestBasis equivalent to BestBasis(10) BestBasis(k) Least Squares to 0-1 using SvdBestBasis of size k Svd (default mode) equivalent to Svd(1e-5) Svd(bound) Least Squares to 0-1 using Svd, with svmin=bound Svd(First(k)) LS to 0-1 using Svd so that k sing values are used CenterMass Direction between the pos/neg center of masses Variance Variance Discrimination of each variable Fisher Fisher linear discriminant Logistic Steepest descent optimization using the logistic function CrossEntropy Steepest descent optimization using cross entropy Best equivalent to Best(10) Best(n) A combination of methods found most effective by experimentation. n determines the amount of optimization. <default> Svd(1e-5) In practice the best results are obtained with Best(n). Svd(1e-12) and CrossEntropy are also very effective. Once a LinearClassification has been computed, it can be improved or refined with the functions: function description -------------------------------------------------------------------- LinearClassification_refine find the center of the min in each dim LinearClassification_refine2 Svd applied to a hyperswath LinearClassification_refine3 Minimize in a random direction LinearClassification_refine4 Svd on progressively smaller swaths Unless you use the Best option, no matter how good the initial results are, it always pays to do some refinement steps. In particular refine2 and refine4 give very good refinements. Examples: > A := [[0,3], [8,5], [10,7], [5,5], [7,4], [7,9]]: > lc := LinearClassify( A, [0,1,1,0,1,0] ): > print(lc); solution vector is X = [0.1945, -0.1364] discriminator is A[i] * X > 0.553293 6 data points, 3 positive, 3 negative positives weigh 1, negatives 1, overall misclassifications 0 Highest negative scores: [] Lowest positive scores: [] See Also: ?LinearClassification ?Stat ?SvdBestBasis ?LinearRegression ?SvdAnalysis ------------

?LinearClassification

Class LinearClassification - results of a linear classification Template: LinearClassification(X,X0,WeightPos,WeightNeg,NumberPos,NumberNeg, WeightedFalses,HighestNeg,LowestPos) Fields: Name Type Description ---------------------------------------------------------------------- X list(numeric) solution vector X0 numeric threshold value WeightPos numeric weight of the positives WeightNeg numeric weight of the negatives NumberPos posint number of positives NumberNeg posint number of negatives WeightedFalses numeric weighted misclassifications HighestNeg list([posint, numeric]) highest scoring negatives LowestPos list([posint, numeric]) lowest scoring positives Returns: LinearClassification Methods: DirOpt2 DirOpt3 DirOpt4 print refine type Synopsis: Data structure which holds the result of a linear classification. A linear classification is defined by a vector X, such that the internal product of every data point A[i] with X can be compared against a threshold and decide whether the data point is a positive or a negative. I.e. A[i] X < X0 implies a negative and A[i] X >= X0 a positive. See also: ?LinearClassify ------------

User who are only interested in using this function, should ignore all the choices for mode and use "Best". This uses a very effective combination of optimizations. The examples in the next two sections will also be useful for the practitioner.

Notice that finding a linear classification is a very difficult problem. It is NP-complete when the matrix A is composed of 0's and 1's. (The two-dimensional example is misleadingly simple, in higher dimensions it becomes basically intractable.) So our algorithms are going to be approximate by force.

The following example shows how to construct data that has a desired level of classification. We can submit this data to our function and see how well it does in discovering the relations. Synthetic examples are excellent to test the quality of the algorithms for linear classification. Real data is interesting for other obvious reasons, but when we want to evaluate how the algorithms to find classifications perform, it should be with synthetic data of this nature.

We will use a matrix *A* of random
normally distributed numbers.

m := 100; n := 10; A := CreateArray( 1..m, 1..n ): for i to m do for j to n do A[i,j] := Rand(Normal) od od: # print 3 rows, to see if they are ok print( A[1..3] );

m := 100 n := 10 -1.3202 -0.6293 0.5911 0.0651 -0.0350 0.0246 -1.9268 0.1629 -0.2779 -0.3022 -0.1996 0.1359 1.3616 2.4518 0.0184 -1.4088 -0.8077 -0.0589 -2.4263 -0.8450 -0.0705 -0.6531 -1.1668 0.6330 -0.5435 1.6289 -0.1584 -0.6618 0.1331 -0.6210

Now we have to decide which is going to be
our linear relation.
Once that we have it, we use it to compute the vector *B*.
We can do this deterministically or adding some noise.
For this example we will add random noise.
We will also compute how many entries are correct under
our (known) linear formula.

B := CreateArray( 1..m ): for i to m do B[i] := 3*A[i,3] - 4*A[i,5] + Rand() > 0.1 od: # print the first 3 to check print( B[1..3] ); correct := 0: for i to m do if ( 3*A[i,3] - 4*A[i,5] > 0.1 ) = B[i] then correct := correct+1 fi od; lprint( correct );

[true, true, false] 99

Now we are ready to compute the linear classification:

lc := LinearClassify( A, B, Best ): print(lc);

solution vector is X = [-0.7431, -0.6004, 7.2707, 0.5679, -10, -0.6192, -0.3624, -1.6222, -0.6198, -0.1855] discriminator is A[i] * X > -0.731373 100 data points, 39 positive, 61 negative positives weigh 1, negatives 0.639344, overall misclassifications 0 Highest negative scores: [] Lowest positive scores: []

At first glance the results appear very different from the expected, but this is not so. The solution vector accepts an arbitrary scaling, and it is scaled so that its largest component be 10 in absolute value. If we scale it back, we find:

lc[X] * 4/10;

[-0.2972, -0.2402, 2.9083, 0.2272, -4, -0.2477, -0.1450, -0.6489, -0.2479, -0.07419078]

and we can see that the main components are the third and the fifth and they have the right values. The rest of the differences can be explained by the fact that the noise that we added was probably too large, and our formula produced 1 errors. The classification found by the program has 0 misclassifications, so it is clearly better. The rest of the columns were used to compensate for the extra noise.

We can now use the linear classification to predict
the output *b* for a new randomly generated point

new:=[]: to n do new:=append(new, Rand()) od: new*lc[X]>=lc[X0];

false

We will now run a more realistic example based on
properties of proteins.
Our goal is to derive a classification scheme that can determine
if a protein is an outer membrane protein or not.
To do this we will use the data in SwissProt, which we have to load
first.
We will search for the string "outer membrane" in the entire
database with the command `SearchDb.`

SwissProt := '~cbrg/DB/SwissProt.Z': ReadDb(SwissProt): memb := [SearchDb('outer membrane')]: lm := length(memb);

Reading 169638448 characters from file /home/cbrg/DB/SwissProt.Z Pre-processing input (peptides) 163235 sequences within 163235 entries considered lm := 942

There are 942 matching sequences. This is a good size for our sample. Next we should select a similar size sample with entries which are not membrane proteins. We will do this by selecting random entries from the database and requiring that the word "membrane" does not appear in their definition (notice that SearchString returns -1 when it does not match).

non_memb := []: while length(non_memb) < lm do e := Rand(Entry); if SearchString( membrane, e ) < 0 then non_memb := append(non_memb,e) fi od:

As features we will use the amino acid frequencies of these sequences. This will give us 20 values per protein. Technically, all the frequencies add to 1, so we need only 19, but it does little harm to linear classification to have extra variables. We concatenate the membrane entries followed by the non-membrane entries and then compute the amino acid frequencies for all of them. We will place this information in the matrix A. We write a function to compute the amino acid frequencies of a given entry.

AAFreq := proc( e ) F := CreateArray(1..20); s := Sequence(e); for i to length(s) do j := AToInt(s[i]); if j > 0 and j < 21 then F[j] := F[j]+1 fi od; F / sum(F) end: A := [ seq( AAFreq(i), i = [ op(memb), op(non_memb) ] )]:

We are almost ready to compute the classification, we just need to construct the vector B, which in this case will have its first half filled with "true" and its second half with "false".

B := [seq(true,i=1..lm), seq(false,i=1..lm)]: lc2 := LinearClassify( A, B, Best ): print( lc2 );

solution vector is X = [2.3323, -1.5549, 2.7910, -0.3047, -9.8599, 3.3371, -4.3826, 0.3360, -8.9198, -2.7926, -0.2509, 0.1157, -2.8817, -1.6977, -4.9915, 3.2286, 2.3616, 10, 3.5917, 0.05464701] discriminator is A[i] * X > -0.000236669 1884 data points, 942 positive, 942 negative positives weigh 1, negatives 1, overall misclassifications 302 Highest negative scores: [[1.5261, 1464], [1.0716, 1718], [1.0574, 1726], [1.0468, 1509], [1.0177, 1374], [0.9426, 1023], [0.8751, 1355], [0.8003, 1112], [0.7304, 1815], [0.7025, 1883], [0.6449, 1669], [0.6324, 1053], [0.6042, 1423], [0.5885, 1339], [0.5798, 1201], [0.5740, 1719], [0.5281, 1124], [0.5250, 1501], [0.5205, 1829], [0.5188, 1824]] Lowest positive scores: [[-1.7366, 462], [-1.6998, 459], [-1.5951, 463], [-1.5743, 460], [-1.5443, 461], [-0.9745, 803], [-0.9461, 804], [-0.9185, 381], [-0.9105, 805], [-0.6499, 806], [-0.6129, 252], [-0.5489, 485], [-0.5421, 881], [-0.4941, 486], [-0.4894, 91], [-0.4893, 56], [-0.4880, 235], [-0.4811, 930], [-0.4659, 271], [-0.4552, 321]]

This appears to be a mediocre classification with 302 misclassifications. We print a table with the coefficients of the classification for each amino acid. Remember that the coefficients are multiplied by the frequency of the amino acid to evaluate the discriminator. So a positive coefficient means that abundance of that amino acid suggest being and "outer membrane" protein. A negative coefficient, the converse. The magnitude of the coefficients indicate their relative importance.

for i to 20 do printf( '%s %6.2f\n', IntToAAA(i), lc2[X,i] ) od;

Ala 2.33 Arg -1.55 Asn 2.79 Asp -0.30 Cys -9.86 Gln 3.34 Glu -4.38 Gly 0.34 His -8.92 Ile -2.79 Leu -0.25 Lys 0.12 Met -2.88 Phe -1.70 Pro -4.99 Ser 3.23 Thr 2.36 Trp 10.00 Tyr 3.59 Val 0.05

Naturally, this is just a first approximation, if we were to pursue membrane classification we should add more features which may be related with the property in the hope of improving the results.

We will now use the linear classification for membrane proteins to see if it will correctly assign another membrane protein.

new:=AAFreq(Entry(4799)): new*lc2[X]>=lc2[X0];

false

Lets try the same for a non-membrane protein

new:=AAFreq(Entry(1801)): new*lc2[X]>=lc2[X0];

false

In this section we give a very brief overview of the main problems/ideas behind the algorithms for computing linear classifications. There are many books and papers written on the subject, far too many to cover here. The algorithms for finding good linear classifications typically have 3 components:

A | An initial solution |

B | A refinement procedure to improve solutions |

C | A method to introduce some randomness. |

The construction of the initial solution consists of
the vector *x* and *x*_{0}.
As it turns out to be, computing *x*_{0} is not
very difficult, it just requires the sorting of the vector
*Ax* plus an additional linear pass.
So the difficulty of the problem lies in the computation of
the direction *x*.
The refinement algorithms, if they have some randomness,
can be applied repeatedly and hence produce better solutions.
As we have various refinements algorithms, we can also
alternate with the different refinements.

The different methods are described in detail and are complemented with extensive simulation results. One clear conclusion is that none of the methods to build an initial classification is uniformly better than the others. As a matter of fact, they all pale against an arbitrary method and a few steps of refinement.

It should be noticed that the linear discriminator de facto includes a constant term. So adding a constant term in the input variables is of no use.

A linear classification remains unchanged if it is
multiplied by an arbitrary positive constant.
(all the coefficients in *x* and *x*_{0} are
multiplied by the same constant.)
To make results more readable and comparable, it is customary to
normalize the result, so that the norm has some property.
Darwin chooses to make the largest component of *x* equals
to 10 in absolute value.

Similarly, multiplying a particular input
variable by an arbitrary constant has no effect (the corresponding
value of *x* is divided by this constant).
Adding an arbitrary constant to an input variable has no
effect either, its value is absorbed by the constant *x*_{0}.
So we can apply a linear transformation of the input variables
without any harm on the linear classification itself.
In this case, it is sometimes reasonable to transform all input
variables so that they have average 0 and variance 1.
A secondary effect of this transformation is that the magnitudes
of the *x* values are now comparable.

A primitive form of non-linear classification can be obtained by including the input variables and some of their powers, for example their squares. The benefits of this are not always positive. There is one case where this helps, which is when a classification depends on a particular input variable being inside a range. Purely linear classification will never be able to cope with a range, but including the square of the variable will produce a linear classification that resolves this problem.

© 2005 by Gaston Gonnet, Informatik, ETH Zurich

Last updated on Fri Sep 2 16:55:12 2005 by GhG

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