Interval Spanned by $(n+1)$ points in $mathbb{R}$
Suppose that we have a real valued function $f(x)$ that has local support, i.e. it's non-zero just for some values of $x$.
If you are familiar with B-splines, this function $f$ can be interpreted as a B-spline of order $n$ "that is non-zero in the interval spanned by $(n+1)$ knots".The Elements of Statistical Learning (Hastie, et.at.,p.188)
My question is, isn't all the real line spanned by two points? I've always seen "span" as something related to vectors, not to merely scalars.
So what's wrong with the "local support" condition of the B-splines? Is this a improper use of the term "span"?
real-analysis linear-algebra spline
edited Nov 27 '18 at 17:51
daw
24.1k1544
asked Nov 27 '18 at 16:02
Ramiro ScorolliRamiro Scorolli
655113
add a comment |
Suppose that we have a real valued function $f(x)$ that has local support, i.e. it's non-zero just for some values of $x$.
If you are familiar with B-splines, this function $f$ can be interpreted as a B-spline of order $n$ "that is non-zero in the interval spanned by $(n+1)$ knots".The Elements of Statistical Learning (Hastie, et.at.,p.188)
My question is, isn't all the real line spanned by two points? I've always seen "span" as something related to vectors, not to merely scalars.
So what's wrong with the "local support" condition of the B-splines? Is this a improper use of the term "span"?
real-analysis linear-algebra spline
edited Nov 27 '18 at 17:51
daw
24.1k1544
asked Nov 27 '18 at 16:02
Ramiro ScorolliRamiro Scorolli
655113
1
All the real line is spanned by a single nonzero value. The support of each basis function is the convex hull of $d+2$ consecutive knots where $d$ is the spline degree. Your $n$ must be the spline order if each basis is defined by $n+1$ knots.
– Oppenede
Nov 27 '18 at 16:14
you are right, it's not degree, but order. edited
– Ramiro Scorolli
Nov 27 '18 at 16:16
When saying "the convex hull spanned by points" or "the interval spanned by numbers" the meaning of "span" is different from the linear algebra meaning. It is widely used though and I wouldn't call it improper.
– Oppenede
Nov 28 '18 at 9:30
add a comment |
Suppose that we have a real valued function $f(x)$ that has local support, i.e. it's non-zero just for some values of $x$.
If you are familiar with B-splines, this function $f$ can be interpreted as a B-spline of order $n$ "that is non-zero in the interval spanned by $(n+1)$ knots".The Elements of Statistical Learning (Hastie, et.at.,p.188)
My question is, isn't all the real line spanned by two points? I've always seen "span" as something related to vectors, not to merely scalars.
So what's wrong with the "local support" condition of the B-splines? Is this a improper use of the term "span"?
real-analysis linear-algebra spline
edited Nov 27 '18 at 17:51
daw
24.1k1544
asked Nov 27 '18 at 16:02
Ramiro ScorolliRamiro Scorolli
655113
Suppose that we have a real valued function $f(x)$ that has local support, i.e. it's non-zero just for some values of $x$.
If you are familiar with B-splines, this function $f$ can be interpreted as a B-spline of order $n$ "that is non-zero in the interval spanned by $(n+1)$ knots".The Elements of Statistical Learning (Hastie, et.at.,p.188)
My question is, isn't all the real line spanned by two points? I've always seen "span" as something related to vectors, not to merely scalars.
So what's wrong with the "local support" condition of the B-splines? Is this a improper use of the term "span"?
real-analysis linear-algebra spline
real-analysis linear-algebra spline
edited Nov 27 '18 at 17:51
daw
24.1k1544
asked Nov 27 '18 at 16:02
Ramiro ScorolliRamiro Scorolli
655113
edited Nov 27 '18 at 17:51
daw
24.1k1544
asked Nov 27 '18 at 16:02
Ramiro ScorolliRamiro Scorolli
655113
edited Nov 27 '18 at 17:51
daw
24.1k1544
edited Nov 27 '18 at 17:51
daw
24.1k1544
edited Nov 27 '18 at 17:51
daw
24.1k1544
24.1k1544
asked Nov 27 '18 at 16:02
Ramiro ScorolliRamiro Scorolli
655113
asked Nov 27 '18 at 16:02
Ramiro ScorolliRamiro Scorolli
655113
asked Nov 27 '18 at 16:02
Ramiro ScorolliRamiro Scorolli
655113
655113
1
All the real line is spanned by a single nonzero value. The support of each basis function is the convex hull of $d+2$ consecutive knots where $d$ is the spline degree. Your $n$ must be the spline order if each basis is defined by $n+1$ knots.
– Oppenede
Nov 27 '18 at 16:14
you are right, it's not degree, but order. edited
– Ramiro Scorolli
Nov 27 '18 at 16:16
When saying "the convex hull spanned by points" or "the interval spanned by numbers" the meaning of "span" is different from the linear algebra meaning. It is widely used though and I wouldn't call it improper.
– Oppenede
Nov 28 '18 at 9:30
add a comment |
1
All the real line is spanned by a single nonzero value. The support of each basis function is the convex hull of $d+2$ consecutive knots where $d$ is the spline degree. Your $n$ must be the spline order if each basis is defined by $n+1$ knots.
– Oppenede
Nov 27 '18 at 16:14
you are right, it's not degree, but order. edited
– Ramiro Scorolli
Nov 27 '18 at 16:16
When saying "the convex hull spanned by points" or "the interval spanned by numbers" the meaning of "span" is different from the linear algebra meaning. It is widely used though and I wouldn't call it improper.
– Oppenede
Nov 28 '18 at 9:30
1
1
All the real line is spanned by a single nonzero value. The support of each basis function is the convex hull of $d+2$ consecutive knots where $d$ is the spline degree. Your $n$ must be the spline order if each basis is defined by $n+1$ knots.
– Oppenede
Nov 27 '18 at 16:14
All the real line is spanned by a single nonzero value. The support of each basis function is the convex hull of $d+2$ consecutive knots where $d$ is the spline degree. Your $n$ must be the spline order if each basis is defined by $n+1$ knots.
– Oppenede
Nov 27 '18 at 16:14
you are right, it's not degree, but order. edited
– Ramiro Scorolli
Nov 27 '18 at 16:16
you are right, it's not degree, but order. edited
– Ramiro Scorolli
Nov 27 '18 at 16:16
When saying "the convex hull spanned by points" or "the interval spanned by numbers" the meaning of "span" is different from the linear algebra meaning. It is widely used though and I wouldn't call it improper.
– Oppenede
Nov 28 '18 at 9:30
When saying "the convex hull spanned by points" or "the interval spanned by numbers" the meaning of "span" is different from the linear algebra meaning. It is widely used though and I wouldn't call it improper.
– Oppenede
Nov 28 '18 at 9:30
add a comment |
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All the real line is spanned by a single nonzero value. The support of each basis function is the convex hull of $d+2$ consecutive knots where $d$ is the spline degree. Your $n$ must be the spline order if each basis is defined by $n+1$ knots.
– Oppenede
Nov 27 '18 at 16:14
you are right, it's not degree, but order. edited
– Ramiro Scorolli
Nov 27 '18 at 16:16
When saying "the convex hull spanned by points" or "the interval spanned by numbers" the meaning of "span" is different from the linear algebra meaning. It is widely used though and I wouldn't call it improper.
– Oppenede
Nov 28 '18 at 9:30