Parameter space

<h2 id="examples">Examples</h2>
<ul><li>A simple model of health deterioration after developing <a href="/facts/Lung_cancer/LHLdR454">lung cancer</a> could include the two parameters gender<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> and smoker/non-smoker, in which case the parameter space is the following set of four possibilities: {(Male, Smoker), (Male, Non-smoker), (Female, Smoker), (Female, Non-smoker)} .</li></ul>
<ul><li>The <a href="/facts/Logistic_map/wskv7mRQ">logistic map</a> 
 
 
 
 
 x
 
 n
 +
 1
 
 
 =
 r
 
 x
 
 n
 
 
 (
 1
 −
 
 x
 
 n
 
 
 )
 
 
 {\displaystyle x_{n+1}=rx_{n}(1-x_{n})}
 
 has one parameter, r, which can take any positive value. The parameter space is therefore <a href="/facts/Positive_real_numbers/VFXU56H9">positive real numbers</a>.</li></ul>
For some values of r, this function ends up cycling around a few values or becomes fixed on one value. These long-term values can be plotted against r in a <a href="/facts/Bifurcation_diagram/NIAKhdDr">bifurcation diagram</a> to show the different behaviours of the function for different values of r.
<ul><li>In a <a href="/facts/Sine_wave/mD1CB09M">sine wave</a> model 
 
 
 
 y
 (
 t
 )
 =
 A
 ⋅
 sin
 ⁡
 (
 ω
 t
 +
 ϕ
 )
 ,
 
 
 {\displaystyle y(t)=A\cdot \sin(\omega t+\phi ),}
 
 the parameters are <a href="/facts/Amplitude/aEckuXJM">amplitude</a> A > 0, <a href="/facts/Angular_frequency/VqLm3L3R">angular frequency</a> ω > 0, and <a href="/facts/Phase_(waves)/cpGtuVll">phase</a> φ ∈ S1. Thus the parameter space is 
 
 
 
 
 R
 
 +
 
 
 ×
 
 R
 
 +
 
 
 ×
 
 S
 
 1
 
 
 .
 
 
 {\displaystyle R^{+}\times R^{+}\times S^{1}.}
 
</li></ul>
<ul><li>In <a href="/facts/Complex_dynamics/NXHJfR3s">complex dynamics</a>, the parameter space is the <a href="/facts/Complex_plane/vE0QrxJp">complex plane</a> C = { z = x + y i : x, y ∈ R }, where i2 = −1.</li></ul>
The famous <a href="/facts/Mandelbrot_set/2NVUgrVG">Mandelbrot set</a> is a <a href="/facts/Subset/SJGERmbA">subset</a> of this parameter space, consisting of the points in the complex plane which give a <a href="/facts/Bounded_set/yCldjtoy">bounded set</a> of numbers when a particular <a href="/facts/Iterated_function/Vzsx64An">iterated function</a> is repeatedly applied from that starting point. The remaining points, which are not in the set, give an unbounded set of numbers (they tend to infinity) when this function is repeatedly applied from that starting point.
<ul><li>In <a href="/facts/Machine_learning/e0w0XJTu">machine learning</a>, <a href="/facts/Hyperparameter_(machine_learning)/jCxm3fTA">hyperparameters</a> are used to describe models. In <a href="/facts/Deep_learning/JLuwD3ea">deep learning</a>, the parameters of a deep network are called weights. Due to the layered structure of deep networks, their weight space has a complex structure and geometry.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a><a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> For example, in <a href="/facts/Multilayer_perceptrons/jkq4t7Gy">multilayer perceptrons</a>, the same function is preserved when <a href="/facts/Permutation/JSw9ukmv">permuting</a> the nodes of a hidden layer, amounting to permuting weight matrices of the network. This property is known as <a href="/facts/Equivariance/KkIkDwIH">equivariance</a> to permutation of deep weight spaces.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> The study seeks <a href="/facts/Hyperparameter_optimization/bzcJJrxs">hyperparameter optimization</a>.</li></ul>
<h2 id="history">History</h2>
Parameter space contributed to the liberation of <a href="/facts/Geometry/wTQf5ffQ">geometry</a> from the confines of <a href="/facts/Three-dimensional_space/KAqKWUbS">three-dimensional space</a>. For instance, the parameter space of <a href="/facts/Sphere_(geometry)/jywYLNM2">spheres</a> in three dimensions, has four dimensions—three for the sphere center and another for the radius. According to <a href="/facts/Dirk_Struik/D7cVVb9r">Dirk Struik</a>, it was the book Neue Geometrie des Raumes (1849) by <a href="/facts/Julius_Pl%25C3%25BCcker/QgoMT64G">Julius Plücker</a> that showed

...geometry need not solely be based on points as basic elements. Lines, planes, circles, spheres can all be used as the elements (Raumelemente) on which a geometry can be based. This fertile conception threw new light on both synthetic and algebraic geometry and created new forms of duality. The number of dimensions of a particular form of geometry could now be any positive number, depending on the number of parameters necessary to define the "element".<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>: 165 
The requirement for higher dimensions is illustrated by <a href="/facts/Pl%25C3%25BCcker_coordinates/SvB7rfeQ">Plücker's line geometry</a>. Struik writes

[Plücker's] geometry of lines in three-space could be considered as a four-dimensional geometry, or, as <a href="/facts/Felix_Klein/WmL9guqK">Klein</a> has stressed, as the geometry of a four-dimensional <a href="/facts/Quadric/L3cTApd4">quadric</a> in a five-dimensional space.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>: 168 
Thus the <a href="/facts/Klein_quadric/VexSnuRN">Klein quadric</a> describes the parameters of lines in space.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Sample_space/WVldkN4F">Sample space</a></li>
<li><a href="/facts/Configuration_space_(physics)/LglVrWpq">Configuration space</a></li>
<li><a href="/facts/Data_analysis/CNtARxLW">Data analysis</a></li>
<li><a href="/facts/Dimensionality_reduction/XoRpcrB0">Dimensionality reduction</a></li>
<li><a href="/facts/Model_selection/R11QK2GC">Model selection</a></li>
<li><a href="/facts/Parametric_equation/5Uvrlgqf">Parametric equation</a></li>
<li><a href="/facts/Parametric_surface/GjtIdq1q">Parametric surface</a></li>
<li><a href="/facts/Phase_space/qPe4Fq57">Phase space</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Hayashi, Fumio (2000). Econometrics. Princeton University Press. p. 446. ISBN 0-691-01018-8. <a href="0-691-01018-8" target="_blank">0-691-01018-8</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Hayashi, Fumio (2000). Econometrics. Princeton University Press. p. 446. ISBN 0-691-01018-8. <a href="0-691-01018-8" target="_blank">0-691-01018-8</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Gasperino, J.; Rom, W. N. (2004). "Gender and lung cancer". Clinical Lung Cancer. 5 (6): 353–359. doi:10.3816/CLC.2004.n.013. PMID 15217534. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Navon, Aviv; Shamsian, Aviv; Achituve, Idan; Fetaya, Ethan; Chechik, Gal; Maron, Haggai (2023-07-03). "Equivariant Architectures for Learning in Deep Weight Spaces". Proceedings of the 40th International Conference on Machine Learning. PMLR: 25790–25816. arXiv:2301.12780. <a href="https://proceedings.mlr.press/v202/navon23a.html" target="_blank">https://proceedings.mlr.press/v202/navon23a.html</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Hecht-Nielsen, Robert (1990-01-01), Eckmiller, Rolf (ed.), "ON THE ALGEBRAIC STRUCTURE OF FEEDFORWARD NETWORK WEIGHT SPACES", Advanced Neural Computers, Amsterdam: North-Holland, pp. 129–135, ISBN 978-0-444-88400-8, retrieved 2023-12-01 <a href="978-0-444-88400-8" target="_blank">978-0-444-88400-8</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Navon, Aviv; Shamsian, Aviv; Achituve, Idan; Fetaya, Ethan; Chechik, Gal; Maron, Haggai (2023-07-03). "Equivariant Architectures for Learning in Deep Weight Spaces". Proceedings of the 40th International Conference on Machine Learning. PMLR: 25790–25816. arXiv:2301.12780. <a href="https://proceedings.mlr.press/v202/navon23a.html" target="_blank">https://proceedings.mlr.press/v202/navon23a.html</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Dirk Struik (1967) A Concise History of Mathematics, 3rd edition, Dover Books <a href="/wiki/Dirk_Struik" target="_blank">/wiki/Dirk_Struik</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Dirk Struik (1967) A Concise History of Mathematics, 3rd edition, Dover Books <a href="/wiki/Dirk_Struik" target="_blank">/wiki/Dirk_Struik</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
</ol>

Parameter space open-in-new

Parameter space