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WarpingCorrespondence

WarpingCorrespondence[s₁,s₂]

给出序列 s₁ 与 s₂ 间的时间翘曲（DTW）相似路径.

WarpingCorrespondence[s₁,s₂,win]

使用由 win 为局部搜索指定的窗口.

范例

打开所有单元关闭所有单元

基本范例 (1)

查找两个序列间的时间翘曲距离：

Wolfram Language code:

s1 = {0, 0, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6};
s2 = {1, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 7};

Wolfram Language code: {n, m} = WarpingCorrespondence[s1, s2]

绘制索引间的响应：

Wolfram Language code: ListLinePlot[{n, m}]

翘曲版本近似相等：

Wolfram Language code: ListLinePlot[{s1[[n]], s2[[m]]}]

范围 (9)

数据 (6)

查找两个实值向量间的时间翘曲距离：

Wolfram Language code: WarpingCorrespondence[{Pi, E, E + Pi, -Pi / E}, {Pi ^ E, E - Pi, -E * Pi, E ^ Pi}]//TextGrid

求两个向量序列间的时间翘曲响应：

Wolfram Language code:

s1 = {{11, 12}, {13, 14}, {15, 16}};
s2 = {{12, 13}, {14, 15}};

Wolfram Language code: WarpingCorrespondence[s1, s2]//TextGrid

计算两个布尔向量间的时间翘曲响应：

Wolfram Language code: s1 = PrimeQ /@ Range[1, 20]

Wolfram Language code: s2 = CoprimeQ[#, 6]& /@ Range[1, 20]

Wolfram Language code: WarpingCorrespondence[s1, s2]//ListLinePlot

布尔值向量的序列：

Wolfram Language code:

s1 = {{True, True, True}, {False, True, True}, {False, False, True}, {True, True, False}, {False, True, True}};
s2 = {{True, True, True}, {True, False, True}, {False, True, True}, {False, True, True}};

Wolfram Language code: WarpingCorrespondence[s1, s2]//TextGrid

带有兼容单位的量的序列：

Wolfram Language code:

s1 = QuantityArray[{1, 2.5, 4}, "Decimeters"];
s2 = {Quantity[0.1, "Kilometers"], Quantity[200, "Inches"]};

Wolfram Language code: WarpingCorrespondence[s1, s2]//TextGrid

二维量数组

Wolfram Language code: s1 = QuantityArray[{{1, 2.5}, {4, 5}, {5, 4}}, {"Seconds", "Minutes"}]

Wolfram Language code: s2 = Table[Quantity[(i + j) / 60, "Hours"], {i, 5}, {j, 2}]

Wolfram Language code: WarpingCorrespondence[s1, s2]//TextGrid

搜索窗 (3)

默认情况下，搜索不是局部受限的：

Wolfram Language code:

s1 = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13};
s2 = {1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 5, 5, 6, 7};

Wolfram Language code: WarpingCorrespondence[s1, s2]//TextGrid

Wolfram Language code: WarpingCorrespondence[s1, s2] == WarpingCorrespondence[s1, s2, Automatic]

指定用于搜索窗的半径：

Wolfram Language code: WarpingCorrespondence[s1, s2, 1]//TextGrid

搜索半径为 2 允许你找到最优对齐：

Wolfram Language code: WarpingCorrespondence[s1, s2, 2]//TextGrid

使用半径为 1 的带：

Wolfram Language code:

s1 = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13};
s2 = {1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10};

Wolfram Language code: WarpingCorrespondence[s1, s2, {"Band", 1}]//TextGrid

使用斜率为 3 的平行四边形：

Wolfram Language code: WarpingCorrespondence[s1, s2, {"Parallelogram", 3}]//TextGrid

对于等长的信号，"Band" 窗等同于 "SlantedBand" 窗：

Wolfram Language code:

s1 = RealDigits[Pi, 10, 25]//First;
s2 = RealDigits[E, 10, 25]//First;

Wolfram Language code: WarpingCorrespondence[s1, s2, {"Band", 3}] == WarpingCorrespondence[s1, s2, {"SlantedBand", 3}]

选项 (4)

DistanceFunction (3)

对于布尔序列，WarpingCorrespondence 使用 MatchingDissimilarity：

Wolfram Language code:

s1 = {{True, False, False}, {True, True, False}, {True, False, False}, {True, False, False}, {False, False, False}};
s2 = {{False, False, False}, {True, True, False}, {False, True, False}, {False, True, False}, {True, False, True}, {False, False, False}, {True, False, False}};

Wolfram Language code: WarpingCorrespondence[s1, s2]

Wolfram Language code: WarpingCorrespondence[s1, s2, DistanceFunction -> MatchingDissimilarity] == %

使用不同的距离函数求两序列间的响应：

Wolfram Language code:

v1 = {50, 60, 70, 71, 71, 73, 75, 80, 80, 80, 78, 76, 75, 73, 71, 71, 71, 73, 75, 76, 76, 68, 76, 76, 75, 73};
v2 = {61, 62, 62, 64, 69, 69, 73, 75, 79, 80, 79, 78, 76, 73, 72, 71, 70, 70, 69, 69, 69, 71, 73, 75, 76};
distances = {EuclideanDistance, BinaryDistance, CorrelationDistance};

Wolfram Language code: ListLinePlot[Transpose[WarpingCorrespondence[v1, v2, DistanceFunction -> #]]& /@ distances, PlotLegends -> distances]

对于一维信号，某些距离函数是等价的：

Wolfram Language code:

s1 = Range[10];
s2 = Range[15];

欧几里德、曼哈顿和棋盘距离总是相同的：

Wolfram Language code:

WarpingCorrespondence[s1, s2, DistanceFunction -> EuclideanDistance] == WarpingCorrespondence[s1, s2, DistanceFunction -> ManhattanDistance] == WarpingCorrespondence[s1, s2, DistanceFunction -> ChessboardDistance]

归一化的欧几里得距离和相关距离一样：

Wolfram Language code:

WarpingCorrespondence[s1, s2, DistanceFunction -> NormalizedSquaredEuclideanDistance] == WarpingCorrespondence[s1, s2, DistanceFunction -> CorrelationDistance]

对于布尔数据，匹配相异和二进制距离是一样的：

Wolfram Language code:

s1  = {True, False, True};
s2 = {False, False, True};

Wolfram Language code:

WarpingCorrespondence[s1, s2, DistanceFunction -> MatchingDissimilarity] == WarpingCorrespondence[s1, s2, DistanceFunction -> BinaryDistance]

Method (1)

比较灵活搜索的完整搜索结果：

Wolfram Language code:

query = Table[{Cos[a], Sin[a]}, {a, Range[10] * Pi / 5.}];
ref = PadLeft[PadRight[query, {15, 2}], {20, 2}];

Wolfram Language code: (openEnd = WarpingCorrespondence[ref, query, Method -> {"MatchingInterval" -> "FlexibleEnd"}])//TextGrid

Wolfram Language code: (full = WarpingCorrespondence[ref, query])//TextGrid

可视化比较结果：

Wolfram Language code:

ListLinePlot[{openEnd, full}, PlotLegends -> {"Relaxed DTW", "Strict DTW"}, 
	GridLines -> {Range[20], Range[10]}, PlotStyle -> {Thickness[.03], Automatic}, PlotRange -> {{0, 21}, {0, 11}}]

应用 (2)

通过绘制响应点间的线可视化时间翘曲响应：

Wolfram Language code:

timeWarpingPlot[s1_, s2_] := 
	Module[{n, m}, 
	{n, m} = WarpingCorrespondence[s1, s2];
	ListLinePlot[{s1, s2}, PlotMarkers -> Automatic, Epilog -> {RGBColor[0.560181, 0.691569, 0.194885], Line@Transpose[{{n, s1[[n]]}, 
{m, s2[[m]]}}]}]
	];

应用于某些数据：

Wolfram Language code: {s1, s2} = {{1, 2, 3, 4, 5, 9}, {11, 9, 8, 9, 12, 14}};

Wolfram Language code: WarpingCorrespondence[s1, s2]

Wolfram Language code: timeWarpingPlot[s1, s2]

比较 2017 年第一季度的 HPQ 股票价格与 2010 年至 2016 年的历史数据：

Wolfram Language code: recent = QuantityMagnitude[FinancialData["HPQ", {{2017, 1, 1}, {2017, 3, 31}}]["Values"]];

获得历史数据：

Wolfram Language code:

history = FinancialData["HPQ", {{2010, 1, 1}, {2016, 12, 31}}, "DateValue"];
{histDates, hist} = {history["Dates"], QuantityMagnitude[history["Values"]]};

求最佳匹配历史数据的子序列：

Wolfram Language code: {corrHist, corrRecent} = WarpingCorrespondence[hist, recent, Method -> {"MatchingInterval" -> "Flexible"}];

检测最类似于 2017 第一季度的历史间隔：

Wolfram Language code:

{m, n} = corrHist[[{1, -1}]];
histDates[[{m, n}]]

可视化比较最新数据和最佳历史匹配：

Wolfram Language code:

DateListPlot[{AssociationThread[Take[histDates[[m ;; n]], UpTo@Length[recent]], Take[recent, UpTo[n - m + 1]]], AssociationThread[histDates[[m ;; n]], hist[[m ;; n]]]}, ...]

根据历史数据预测未来30天的股价：

Wolfram Language code:

l = Length[recent];
colDat = ColorData[97];
offset = Last[recent] - hist[[n]];
hist30d = hist[[n ;; n + 30]];
ListLinePlot[{recent, {l + Range[31], hist30d + offset}, hist[[m ;; n]], {n - m + Range[31], hist30d}}, ...]

属性和关系 (7)

两序列不需要同等长度：

Wolfram Language code: WarpingCorrespondence[{1, 2, 3, 4, 5}, {1, 2, 3}]

两相等序列的翘曲路径是一条对角线：

Wolfram Language code:

c = WarpingCorrespondence[Range[10], Range[10]];
ListPlot[c]

搜索窗半径越小，计算速度越快：

Wolfram Language code:

s1 = Sin[Range[1000] / 50 * 2 * Pi];
s2 = Sin[Range[5, 1004] / 100 * 2 * Pi] + RandomReal[{-.1, .1}, 1000];

Wolfram Language code: w1 = WarpingCorrespondence[s1, s2, 10];//AbsoluteTiming//First

Wolfram Language code: w2 = WarpingCorrespondence[s1, s2];//AbsoluteTiming//First

可能存在计算质量的折衷：

Wolfram Language code: ListLinePlot[{Legended[Transpose[w1], "r = 10"], Legended[Transpose[w2], "optimal warping"]}]

当计算完整响应，WarpingCorrespondence 是对称的：

Wolfram Language code:

s1 = {1, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 7};
s2 = {0, 0, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6};
WarpingCorrespondence[s1, s2] == Reverse@WarpingCorrespondence[s2, s1]

当匹配子序列时，函数不是对称的：

Wolfram Language code:

WarpingCorrespondence[s1, s2, Method -> {"MatchingInterval" -> "Flexible"}] == Reverse@WarpingCorrespondence[s2, s1, Method -> {"MatchingInterval" -> "Flexible"}]

从一个序列增加或减去可能导致不同的响应：

Wolfram Language code:

s1 = {0, 0, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6};
s2 = {1, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 7};
res1 = WarpingCorrespondence[s1, s2];res2 = WarpingCorrespondence[s1, s2 - 2];
ListLinePlot[{res1, res2}, PlotLegends -> {"original signal", "translated signal"}]

同时从两个序列增加或减去不会影响结果，当距离函数是由范数（平移不变）产生：

Wolfram Language code:

s1 = {0, 0, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6};
s2 = {1, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 7};

这对于欧几里得、曼哈顿等是真的：

Wolfram Language code: WarpingCorrespondence[s1, s2] == WarpingCorrespondence[s1 - 3, s2 - 3]

使用不是平移不变的距离函数可能产生不同的响应：

Wolfram Language code:

res1 = WarpingCorrespondence[s1, s2, DistanceFunction -> BrayCurtisDistance];res2 = WarpingCorrespondence[s1 - 3, s2 - 3, DistanceFunction -> BrayCurtisDistance];
ListLinePlot[{res1, res2}, PlotLegends -> {"original signals", "translated signals"}]

与 WarpingDistance 的关系：

Wolfram Language code:

s1 = Sin[Range[100] / 50 * 2 * Pi];
s2 = Sin[Range[5, 104] / 100 * 2 * Pi] + RandomReal[{-.1, .1}, 100];
distance = EuclideanDistance;

Wolfram Language code: {n, m} = WarpingCorrespondence[s1, s2, DistanceFunction -> distance];

使用响应求时间翘曲序列：

Wolfram Language code:

l1 = s1[[n]];
l2 = s2[[m]];

WarpingDistance 给出响应元素间所有距离的总和：

Wolfram Language code: Total[MapThread[distance, {l1, l2}]] == WarpingDistance[s1, s2, DistanceFunction -> distance]

可能存在的问题 (3)

两个序列的所有元素必须相称：

Wolfram Language code: WarpingCorrespondence[{1, 2, 3}, {{1, 2}, {3, 4}, {5, 6}}]

两个序列的元素必须是同样类型：

Wolfram Language code: WarpingCorrespondence[{1, 2, 3}, {True, False, True}]

如果指定的窗太窄，会自动使用更大的窗：

Wolfram Language code:

s1 = {0, 0, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6};
s2 = {1, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 7};
WarpingCorrespondence[s1, s2, {"Band", 1}]

Top

		Automatic	完整搜索
		r	半径为的倾斜带窗
		{"SlantedBand",r}	半径为的倾斜带窗
		{"Band",r}	半径为 (Sakoe-Chiba) 的倾斜带窗
		{"Parallelogram",a}	放在原点的平行四边形窗，斜率为和 (Itakura)

	Automatic	自动确定距离函数
	EuclideanDistance	欧几里得距离
	ManhattanDistance	曼哈顿或“城区”距离
	BinaryDistance	如果元素相等则为 0；否则为 1
	ChessboardDistance	切比雪夫或超 (sup) 标准距离
	SquaredEuclideanDistance	平方欧几里得距离
	NormalizedSquaredEuclideanDistance	正则平方欧几里得距离
	CosineDistance	角余弦距离
	CorrelationDistance	相关系数距离
	BrayCurtisDistance	Total[Abs[u-v]]/Total[Abs[u+v]]
	CanberraDistance	Total[Abs[u-v]/(Abs[u]+Abs[v])]
	MatchingDissimilarity	匹配布尔向量间的相异

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WarpingCorrespondence

更多信息和选项

范例

基本范例 (1)

范围 (9)

数据 (6)

搜索窗 (3)

选项 (4)

DistanceFunction (3)

Method (1)

应用 (2)

属性和关系 (7)

可能存在的问题 (3)

文本

CMS

APA

BibTeX

BibLaTeX

	EuclideanDistance	数值数据
	MatchingDissimilarity	布尔数据

	Automatic	完全匹配
	"Flexible"	两端均有弹性
	"FlexibleEnd"	只在间隔的一端有弹性

WarpingCorrespondence

更多信息和选项

范例

基本范例 (1)

范围 (9)

数据 (6)

搜索窗 (3)

选项 (4)

DistanceFunction (3)

Method (1)

应用 (2)

属性和关系 (7)

可能存在的问题 (3)

参见

相关指南

历史

文本

CMS

APA

BibTeX

BibLaTeX