In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a knee of a curve (or elbow of a curve) is a point where the curve visibly bends, specifically from high slope to low slope (flat or close to flat), or in the other direction. This is particularly used in
optimization
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criteria, from some set of available alternatives. It is generally divided into two subfiel ...
, where a knee point is the optimum point for some decision, for example when there is an
increasing function
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of ord ...
and a trade-off between the benefit (vertical
''y'' axis) and the cost (horizontal
''x'' axis): the knee is where the benefit is no longer increasing rapidly, and is no longer worth the cost of further increases – a cutoff point of
diminishing returns
In economics, diminishing returns means the decrease in marginal (incremental) output of a production process as the amount of a single factor of production is incrementally increased, holding all other factors of production equal ('' ceter ...
.
In
heuristic
A heuristic or heuristic technique (''problem solving'', '' mental shortcut'', ''rule of thumb'') is any approach to problem solving that employs a pragmatic method that is not fully optimized, perfected, or rationalized, but is nevertheless ...
use, the term may be used informally, and a knee point identified visually, but in more formal use an explicit
objective function
In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost ...
is used, and depends on the particular optimization problem. A knee may also be defined purely geometrically, in terms of the
curvature
In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ...
or the
second derivative
In calculus, the second derivative, or the second-order derivative, of a function is the derivative of the derivative of . Informally, the second derivative can be phrased as "the rate of change of the rate of change"; for example, the secon ...
.
Definitions
The knee of a curve can be defined as a
vertex of the graph. This corresponds with the graphical intuition (it is where the curvature has a maximum), but depends on the choice of scale.
The term "knee" as applied to curves dates at least to the 1910s,
and is found more commonly by the 1940s, being common enough to draw criticism.
The unabridged ''
Webster's Dictionary
''Webster's Dictionary'' is any of the US English language dictionaries edited in the early 19th century by Noah Webster (1758–1843), a US lexicographer, as well as numerous related or unrelated dictionaries that have adopted the Webster's n ...
'' (1971 edition) gives definition 3h of ''knee'' as:
Criticism
Graphical notions of a "knee" of a curve, based on curvature, are criticized due to their dependence on the coordinate scale: different choices of scale result in different points being the "knee". This criticism dates at least to the 1940s, being found in , who criticize:
Detection methods
The Kneedle algorithm The algorithm detects the best balanced tradeoff based on the mathematical
curvature
In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ...
concept, which is defined and well studied for continuous functions. Alternatively, the kneepointDetection() function from the SamSPECTRAL
R package can be used to find the knee point, where is a "phase change" in the data, by fitting two lines using linear regression.
Applications
*
Elbow method
*
Maximum power point tracking
Maximum power point tracking (MPPT), or sometimes just power point tracking (PPT), is a technique used with variable power sources to maximize energy extraction as conditions vary. The technique is most commonly used with photovoltaic (PV) solar s ...
References
*
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Curvature (mathematics)
Mathematical optimization
Operations research