A computer program is a list of instructions that can be executed by a central processing unit. A program's execution is done in order for the CPU that is executing it to solve a specific problem and thus accomplish a specific result. While simple processors are able to execute instructions one after another,
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processors are capable of executing a variety of different instructions at once.
Program flow may be influenced by special 'jump' instructions that transfer execution to an instruction other than the numerically following one. Conditional jumps are taken (execution continues at another address) or not (execution continues at the next instruction) depending on some condition.
A much more readable rendition of machine language, called
assembly language
, uses
mnemonic codes
to refer to machine code instructions, rather than using the instructions' numeric values directly. For example, on the
Zilog Z80
processor, the machine code
00000101
, which causes the CPU to decrement the
B
processor register
, would be represented in assembly language as
DEC B
.
The MIPS architecture provides a specific example for a machine code whose instructions are always 32 bits long. The general type of instruction is given by the op (operation) field, the highest 6 bits. J-type (jump) and I-type (immediate) instructions are fully specified by op . R-type (register) instructions include an additional field funct to determine the exact operation. The fields used in these types are:
rs , rt , and rd indicate register operands; shamt gives a shift amount; and the address or immediate fields contain an operand directly.
For example, adding the registers 1 and 2 and placing the result in register 6 is encoded:
Load a value into register 8, taken from the memory cell 68 cells after the location listed in register 3:
Jumping to the address 1024:
In some
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, the machine code is implemented by an even more fundamental underlying layer called
microcode
, providing a common machine language interface across a line or family of different models of computer with widely different underlying
dataflows
. This is done to facilitate
porting
of machine language programs between different models. An example of this use is the IBM
System/360
family of computers and their successors. With dataflow path widths of 8 bits to 64 bits and beyond, they nevertheless present a common architecture at the machine language level across the entire line.
The estimation, from the data, of the terms in this equation is explained in the Additional file
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. Below we will interpret the direction of the
p
-vector,
b
, as the direction in expression space that best characterizes the differential expression, and show how the components of this vector can be used to identify differentially expressed genes. However, first we note a few potential issues: the calculation involves the inverse of a very large
p
×
p
matrix which is not only expensive to compute but also the elements must be estimated from a relatively small sample-size (
p
>>
N
), which means that the matrix is singular and this leads to large variance in the results even when using the generalized inverse.
where
MathML
is the estimated covariance matrix, and
σ
^{
2
}
is the scalar covariance (see Additional file
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for elaboration). The inclusion of a constant on the diagonal resolves the singularity problem, and the modulation of the off-diagonal terms helps to reduce noise arising from the estimation of covariance from few samples.
Then the contribution of each MathML to this sum can be interpreted as quantifying the relative contribution of each component to the total differential expression giving the significance of the corresponding gene. The above interpretation provides a quantitative measure of the relative, but not absolute, significance of each gene to the differential expression, and as such can be used to rank the genes in order of significance. However, we also want to identify a shortlist of significant DEGs. This could be done completely within the framework we have outlined by using a L _{ 1 } regularization scheme in place of that used in the shrinkage equation above; such a penalty results in automatic feature selection because many components fall to zero; the genes corresponding to the features retained would then comprise the DEGs. An alternative method to deriving a significance threshold is described below.
We generate synthetic normalized expression data which incorporates multivariate structure. The multivariate structure of real biological expression data is not fully known, we therefore use a simple approach which incorporates some of the best established properties of such data: 1) large number of features (genes) with a relatively small number of samples; 2) significant dependencies between the expression levels of the genes, leading to dimensionality which is much smaller than the number of features. In addition to these properties we require control over the number and identity of genes which are differentially expressed between two datasets. There are a number of ways that datasets with these properties may be generated, but we chose the simplest, with the fewest free parameters. In a nutshell, we use a multi-variate normal distribution distributed throughout a random subspace of the full expression space, the dimension of which reflects the dimension of the dataset. By ensuring that this subspace spans a predefined vector of differentially expressed genes we can perturb the mean of the normal distribution, preserving the covariance matrix, to generate data with pre-defined differentially expressed genes. An explicit description of the algorithm follows:
A polyline can be constructed given one of the following inputs:
Note that degenerate polylines are not supported. That is, it is impossible to construct a polyline that contains a segment of length zero, or an isolated point. Finally, a polyline is continuous and well-oriented ; that is, the target of the i th segment is the source of the i+1 st segment. For example, the general polyline
can be represented by one of the following two
Also, note, that a single polyline can be split into several x -monotone polylines, and that the number of intersection points (or overlapping sections) between two polylines can also be large.
Technically speaking, it is possible to construct a general polyline that is neither well-oriented nor continuous. However, it is impossible to use such polylines for the purpose of computing an arrangement.
You can traverse over the range of defining segments of a given polyline. The
first
and
past-the-end
iterators can be obtained through the access functions of the polyline
begin_segments()
and
end_segments()
, respectively. The vertices of an
x
-monotone curve are always stored in a strongly monotonic lexicographical order. In other words,
x
-monotone polylines can be directed
either
left-to-right
or
right-to-left. If the macro
CGAL_ALWAYS_LEFT_TO_RIGHT
is set to 1, then the
x
-monotone polylines are always directed from left-to-right (only proposed for backward compatibility).
The polyline-traits class does not perform any geometric operations directly. Instead, it solely relies on the functionality of the segment traits. For example, when we need to determine the position of a point with respect to an x -monotone polyline, we use binary search to locate the relevant segment that contains the point in its x -range. Then, we compute the position of the point with respect to this segment. Thus, operations on x -monotone polylines of size m typically take O(\log m) time.
You are free to choose the underlying segment traits class. Your decision could be based, for example, on the number of expected intersection points; see Section
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. Moreover, it is possible to substitute the
SubcurveTraits_2
template parameter with a traits class that handles segments with some additional data attached to each individual segment; see Section
Traits-Class Decorators
. This makes it possible to associate different data objects with the different segments that compose a polyline.
The following example program constructs an arrangement of three polylines, as depicted in Figure 34.13 . Note that most points defining the polylines are not associated with arrangement vertices. The arrangement vertices are either the endpoints of each x -monotone polyline or the intersection points between two polylines: