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In the [[mathematics|mathematical]] subfield of [[numerical analysis]], an '''M-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 | issue=4 | pages=425&ndash;441 | title = Monotone Regression Splines in Action | jstor=2245395 | doi=10.1214/ss/1177012761}}</ref> is a non-negative [[Spline (mathematics)|spline]] function.


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


==Definition==
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A family of ''M-spline'' functions of order ''k'' with ''n'' free parameters is defined by a set of knots ''t''<sub>1</sub> &nbsp;≤&nbsp;''t''<sub>2</sub> &nbsp;≤&nbsp; ... &nbsp;≤&nbsp; ''t''<sub>''n''+''k''</sub> such that  
 
* ''t''<sub>1</sub>&nbsp;=&nbsp;...&nbsp;=&nbsp;''t''<sub>''k''</sub>
* ''t''<sub>''n''+1</sub>&nbsp;=&nbsp;...&nbsp;=&nbsp;''t''<sub>''n''+''k''</sub>
* ''t''<sub>''i''</sub>&nbsp;<&nbsp;''t''<sub>''i''+''k''</sub> for all ''i''
 
The family includes ''n'' members indexed by ''i''&nbsp;=&nbsp;1,...,''n''.
 
==Properties==
 
An ''M-spline'' ''M''<sub>''i''</sub>(''x''|''k'',&nbsp;''t'') has the following mathematical properties
 
* ''M''<sub>''i''</sub>(''x''|''k'',&nbsp;''t'') is non-negative
 
* ''M''<sub>''i''</sub>(''x''|''k'',&nbsp;''t'') is zero unless ''t''<sub>''i''</sub>&nbsp;≤&nbsp;''x''&nbsp;<&nbsp;''t''<sub>''i''+''k''</sub>
 
* ''M''<sub>''i''</sub>(''x''|''k'',&nbsp;''t'') has ''k''&nbsp;&minus;&nbsp;2 continuous derivatives at interior knots ''t''<sub>''k''+1</sub>, ..., ''t''<sub>''n''&minus;1</sub>
 
* ''M''<sub>''i''</sub>(''x''|''k'',&nbsp;''t'') integrates to 1
 
==Computation==
 
''M-splines'' can be efficiently and stably computed using the following recursions:
 
For ''k''&nbsp;=&nbsp;1,
 
:<math>
M_i(x|1,t) = \frac{1}{t_{i+1}-t_i}
</math>
 
if ''t''<sub>''i''</sub>&nbsp;≤&nbsp;''x''&nbsp;<&nbsp;''t''<sub>''i''+1</sub>, and ''M''<sub>''i''</sub>(''x''|1,''t'')&nbsp;=&nbsp;0 otherwise.
 
For ''k''&nbsp;>&nbsp;1,
 
:<math>
M_i(x|k,t) = \frac{k\left[(x-t_i)M_i(x|k-1,t) + (t_{i+k}-x)M_{i+1}(x|k-1,t)\right]}{(k-1)(t_{i+k}-t_i)}.
</math>
 
==Applications==
 
''M-splines'' can be integrated to produce a family of monotone splines called [[I-spline]]s.  ''M-splines'' can also be used directly as basis splines for regression analysis involving positive response data (constraining the regression coefficients to be non-negative).
 
==References==
<references/>
 
[[Category:Splines]]
 
 
{{mathapplied-stub}}

Latest revision as of 20:39, 5 January 2015


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