区间数据分析
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• [x1, x2], a closed interval, includes both x1 and x2; • [x1, x2), a left-closed, right-open interval, includes x1 and excludes x2; • (x1, x2], a left-open, right-closed interval, excludes x1 and includes x2; • (x1, x2), an open interval, excludes both x1 and x2.
Modeling with interval time series
• Ichino-Yaguchi distance Let w ([x]) = x2 - x1 denote the
length of an interval and r= [0, 0.5] be a control parameter. The IchinoYaguchi distance dIY ([x],[y]) is based on set operators, and is then dened as:
S&P500 Daily Range
Modeling with interval time series
• Dieference In classic time series, this is measured through squaring or
taking the absolute value of the diference between those two values. Subtraction can be defened using the principles of interval arithmetic so that, for any two intervals [x1, x2] and [y1, y2]: • [x1,x2]- [y1, y2] = [x1- y2, x2-y1] • In this case, [x1, x2] = [y1, y2]. • Therefore:[x1, x2] -[x1, x2] = [x1- x2, x2- x1] = [-2r, 2r] • This implies that the distance will only evaluate to [0; 0] in the case of degenerate intervals (r = 0). Therefore, it is an inaccurate measure of distance in the general case.
Байду номын сангаас
Properties of intervals
• Short notation: In order to shorten the notation of intervals,
an interval observation at time t, that is [y1,y2]t = <yc, yr>t, will sometimes be referred to as [y]t. The interval time series made of interval observations in periods 1 to T, that is {[y]t |1 ≤t ≤ T}, will be designated by {[y] T }.
• dH ([x],[y]) = max (丨x1- y1丨,丨x2 -y2丨)
•
= 丨xc -yc丨+丨xr -yr丨
• This defintion matches that of the distance for two real numbers if applied to degenerate intervals, but is also 0 whenever two intervals are identical.
• addition: [x1, x2] + [y1, y2] = [x1 + y1, x2 + y2]; • multiplication: [x1,x2] ·[y1, y2] = [x1·y1, x2 ·y2], if x1, y1 ≥0.
Properties of intervals
• Elementary functions Let f : R → R be a function in one variable,
• <xc, xr> + <0,K> = [xc-xr, xc + xr] + [K,K] = [xc- (xr + K) , xc + (xr + K)] = <xc, xr + K>
• K<xc, xr>- (K-1)xc = K [xc- xr, xc + xr]- (K- 1) [xc, xc] = [xc –Kxr, xc + Kxr] = <xc,Kxr>
Modeling with interval time series
• Hausdorff distance Let [x] = [x1, x2] = <xc, xr> and [y] = [y1, y2] =<yc, yr> be two intervals. Then the Hausdorff distance dH ([x], [y]) is defined in one of two equivalent ways:
Properties of intervals
• Set operators Given that intervals are sets, the set
operators' rules also apply to intervals, namely: • intersection: [x1, x2] ∩ [y1, y2] = [max(x1, y1), min(x2, y2)] • union: [x1, x2] ∪ [y1, y2] = [min(x1, y1), max(x2, y2)]
monotonic in the interval [x1, x2]. The image of the interval can be calculated by applying the function to the endpoints: • f ([x1, x2]) =[min{f(x1), f(x2)}, max{f(x1), f(x2)}] • opposite: -[x1,x2] = [-x2,-x1]; • inverse: 1/[x1,x2] =[1/x2,1/x1], if x2 < 0 or x1 > 0; • exponential: e[x1,x2] = [ex1 ,ex2 ]; • logarithm: log[x1,x2] = [log x1, log x2], if x1 > 0.
2015 SIDM
Interval Time Series
Repoter: yang ying
Properties of intervals
• Denition: An interval is a convex set of numbers, that is,
given two numbers in the set, any numbers lying between them also belong to the set. • 实数 R 上（或复数 C 上）的向量空间中，如果集合 S 中任 两点的连线上的点都在 S 内，则称集合 S 为凸集。 • 性质：一个集合是凸集，当且仅当集合中任意两点的连线 全部包含在该集合内。
Properties of intervals
• Radius transformations: Growing the radius of the
interval could be accomplished by adding an interval that is symmetric around the origin, given K > 0:
Properties of intervals
• Spherical notation: In a metric space, a ball is the space
inside a sphere. By considering R a 1-dimensional metric space, intervals can be viewed as balls centered at xc and with radius xr. • Let [x1, x2] be an interval, then an equivalent definition of it would be[xc-xr, xc + xr] so that: • xc =(x1 + x2)/2; xr =(x2-x1)/2 • The interval [x1, x2], also noted [x], described by its center xc and radius xr, can also be written in the form <xc , xr>
• dIY ([x],[y]) = w ([x] ∪[y]) -w ([x] ∩ [y])+ r(2 w ([x] ∩[y]) -w ([x])- w ([y]))
Properties of intervals
• Center transformations: Shifting the center of the
interval is accomplished through a translation, that is, by adding a constant to the interval, since a constant is a degenerate interval: • <xc, xr> + K = [xc-xr, xc + xr] + [K,K] • = [(xc + K) –xr, (xc + K) + xr] • = <xc + K, xr>
Properties of intervals
• Simple interval arithmetic Let [x1, x2] and [y1, y2] be
two intervals. Then, an arithmetic operation <op> can be defined as:
• [x1,x2] <op> [y1, y2] = {x <op> y | x∈ [x1, x2] and y ∈ [y1,y2]} = [min(x1<op>y1, x1<op>y2, x2<op>y1, x2<op>y2), max(x1<op>y1, x1<op>y2, x2<op>y1, x2<op>y2)]
Properties of intervals
• Standard notation: The written representation of an interval
varies to denote the inclusion of extreme values. Let x1 and x2 respectively be lower and upper boundaries to an interval, then:
• Connectedness: By definition, the union of two intervals is
an interval if and only if it is not the union of non-empty disjoint intervals, therefore intervals are connected subsets of R. This is implies that the image of an interval by a continuous function is also an interval.
Modeling with interval time series
• Ichino-Yaguchi distance Let w ([x]) = x2 - x1 denote the
length of an interval and r= [0, 0.5] be a control parameter. The IchinoYaguchi distance dIY ([x],[y]) is based on set operators, and is then dened as:
S&P500 Daily Range
Modeling with interval time series
• Dieference In classic time series, this is measured through squaring or
taking the absolute value of the diference between those two values. Subtraction can be defened using the principles of interval arithmetic so that, for any two intervals [x1, x2] and [y1, y2]: • [x1,x2]- [y1, y2] = [x1- y2, x2-y1] • In this case, [x1, x2] = [y1, y2]. • Therefore:[x1, x2] -[x1, x2] = [x1- x2, x2- x1] = [-2r, 2r] • This implies that the distance will only evaluate to [0; 0] in the case of degenerate intervals (r = 0). Therefore, it is an inaccurate measure of distance in the general case.
Байду номын сангаас
Properties of intervals
• Short notation: In order to shorten the notation of intervals,
an interval observation at time t, that is [y1,y2]t = <yc, yr>t, will sometimes be referred to as [y]t. The interval time series made of interval observations in periods 1 to T, that is {[y]t |1 ≤t ≤ T}, will be designated by {[y] T }.
• dH ([x],[y]) = max (丨x1- y1丨,丨x2 -y2丨)
•
= 丨xc -yc丨+丨xr -yr丨
• This defintion matches that of the distance for two real numbers if applied to degenerate intervals, but is also 0 whenever two intervals are identical.
• addition: [x1, x2] + [y1, y2] = [x1 + y1, x2 + y2]; • multiplication: [x1,x2] ·[y1, y2] = [x1·y1, x2 ·y2], if x1, y1 ≥0.
Properties of intervals
• Elementary functions Let f : R → R be a function in one variable,
• <xc, xr> + <0,K> = [xc-xr, xc + xr] + [K,K] = [xc- (xr + K) , xc + (xr + K)] = <xc, xr + K>
• K<xc, xr>- (K-1)xc = K [xc- xr, xc + xr]- (K- 1) [xc, xc] = [xc –Kxr, xc + Kxr] = <xc,Kxr>
Modeling with interval time series
• Hausdorff distance Let [x] = [x1, x2] = <xc, xr> and [y] = [y1, y2] =<yc, yr> be two intervals. Then the Hausdorff distance dH ([x], [y]) is defined in one of two equivalent ways:
Properties of intervals
• Set operators Given that intervals are sets, the set
operators' rules also apply to intervals, namely: • intersection: [x1, x2] ∩ [y1, y2] = [max(x1, y1), min(x2, y2)] • union: [x1, x2] ∪ [y1, y2] = [min(x1, y1), max(x2, y2)]
monotonic in the interval [x1, x2]. The image of the interval can be calculated by applying the function to the endpoints: • f ([x1, x2]) =[min{f(x1), f(x2)}, max{f(x1), f(x2)}] • opposite: -[x1,x2] = [-x2,-x1]; • inverse: 1/[x1,x2] =[1/x2,1/x1], if x2 < 0 or x1 > 0; • exponential: e[x1,x2] = [ex1 ,ex2 ]; • logarithm: log[x1,x2] = [log x1, log x2], if x1 > 0.
2015 SIDM
Interval Time Series
Repoter: yang ying
Properties of intervals
• Denition: An interval is a convex set of numbers, that is,
given two numbers in the set, any numbers lying between them also belong to the set. • 实数 R 上（或复数 C 上）的向量空间中，如果集合 S 中任 两点的连线上的点都在 S 内，则称集合 S 为凸集。 • 性质：一个集合是凸集，当且仅当集合中任意两点的连线 全部包含在该集合内。
Properties of intervals
• Radius transformations: Growing the radius of the
interval could be accomplished by adding an interval that is symmetric around the origin, given K > 0:
Properties of intervals
• Spherical notation: In a metric space, a ball is the space
inside a sphere. By considering R a 1-dimensional metric space, intervals can be viewed as balls centered at xc and with radius xr. • Let [x1, x2] be an interval, then an equivalent definition of it would be[xc-xr, xc + xr] so that: • xc =(x1 + x2)/2; xr =(x2-x1)/2 • The interval [x1, x2], also noted [x], described by its center xc and radius xr, can also be written in the form <xc , xr>
• dIY ([x],[y]) = w ([x] ∪[y]) -w ([x] ∩ [y])+ r(2 w ([x] ∩[y]) -w ([x])- w ([y]))
Properties of intervals
• Center transformations: Shifting the center of the
interval is accomplished through a translation, that is, by adding a constant to the interval, since a constant is a degenerate interval: • <xc, xr> + K = [xc-xr, xc + xr] + [K,K] • = [(xc + K) –xr, (xc + K) + xr] • = <xc + K, xr>
Properties of intervals
• Simple interval arithmetic Let [x1, x2] and [y1, y2] be
two intervals. Then, an arithmetic operation <op> can be defined as:
• [x1,x2] <op> [y1, y2] = {x <op> y | x∈ [x1, x2] and y ∈ [y1,y2]} = [min(x1<op>y1, x1<op>y2, x2<op>y1, x2<op>y2), max(x1<op>y1, x1<op>y2, x2<op>y1, x2<op>y2)]
Properties of intervals
• Standard notation: The written representation of an interval
varies to denote the inclusion of extreme values. Let x1 and x2 respectively be lower and upper boundaries to an interval, then:
• Connectedness: By definition, the union of two intervals is
an interval if and only if it is not the union of non-empty disjoint intervals, therefore intervals are connected subsets of R. This is implies that the image of an interval by a continuous function is also an interval.