Probability of precipitation: Difference between revisions

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In the [[mathematics|mathematical]] subfield of [[numerical analysis]], an '''I-spline'''<ref>{{cite journal | title = On Polya frequency functions. IV. The fundamental spline functions and their limits | last=Curry | first =H.B. | coauthors=Schoenberg, I.J. | year=1966 | volume=17 | pages=71&ndash;107 | journal=J. Analyse Math. | doi = 10.1007/BF02788653}}</ref><ref>{{cite journal | last=Ramsay | first=J.O. | journal=Statistical Science | year = 1988 | volume=3 | pages=425&ndash;441 | title = Monotone Regression Splines in Action | jstor=2245395 | doi=10.1214/ss/1177012761 | issue=4}}</ref> is a monotone [[Spline (mathematics)|spline]] function.


[[Image:Ispline order3.svg|thumb|325px|An ''I-spline'' family of order three with four interior knots.]]


==Definition==
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A family of ''I-spline'' functions of order ''k'' with ''n'' free parameters is defined in terms of the [[M-spline]]s ''M''<sub>''i''</sub>(''x''|''k'',&nbsp;''t'')
 
:<math>
I_i(x|k,t) = \int_L^x M_i(u|k,t)du,
</math>
 
where ''L'' is the lower limit of the domain of the splines.
 
Since M-splines are non-negative, ''I-splines'' are monotonically non-decreasing.
 
==Computation==
 
Let ''j'' be the index such that ''t''<sub>''j''</sub>&nbsp;≤&nbsp;''x''&nbsp;<&nbsp;''t''<sub>''j''+1</sub>. Then ''I''<sub>''i''</sub>(''x''|''k'',&nbsp;''t'') is zero if ''i''&nbsp;>&nbsp;''j'', and equals one if ''j''&nbsp;&minus;&nbsp;''k''&nbsp;+&nbsp;1&nbsp;>&nbsp;''i''.  Otherwise,
 
:<math>
I_i(x|k,t) = \sum_{m=i}^j (t_{m+k+1}-t_m)M_m(x|k+1,t)/(k+1).
</math>
 
==Applications==
 
''I-splines'' can be used as basis splines for regression analysis and [[data transformation (statistics)|data transformation]] when monotonicity is desired (constraining the regression coefficients to be non-negative for a non-decreasing fit, and non-positive for a non-increasing fit).
 
==References==
<references/>
 
[[Category:Splines]]
 
 
{{mathapplied-stub}}

Revision as of 22:03, 10 February 2014


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