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samplealign

Align two data sets containing sequential observations by introducing gaps

Syntax

[I, J] = samplealign(X, Y)

[I, J] = samplealign(X, Y, ...'Band', BandValue, ...)
[I, J] = samplealign(X, Y, ...'Width', WidthValue, ...)
[I, J] = samplealign(X, Y, ...'Gap', GapValue, ...)
[I, J] = samplealign(X, Y, ...'Quantile', QuantileValue, ...)
[I, J] = samplealign(X, Y, ...'Distance', DistanceValue, ...)
[I, J] = samplealign(X, Y, ...'Weights', WeightsValue, ...)
[I, J] = samplealign(X, Y, ...'ShowConstraints', ShowConstraintsValue, ...)
[I, J] = samplealign(X, Y, ...'ShowNetwork', ShowNetworkValue, ...)
[I, J] = samplealign(X, Y, ...'ShowAlignment', ShowAlignmentValue, ...)

Input Arguments

X, YMatrices of data where rows correspond to observations or samples, and columns correspond to features or dimensions. X and Y can have a different number of rows, but they must have the same number of columns. The first column is the reference dimension and must contain unique values in ascending order. The reference dimension could contain sample indices of the observations or a measurable value, such as time.
BandValue

Either of the following:

  • Scalar.

  • Function specified using @(z), where z is the mid-point between a given observation in one data set and a given observation in the other data set.

BandValue specifies a maximum allowable distance between observations (along the reference dimension only) in the two data sets, thus limiting the number of potential matches between observations in two data sets. If S is the value in the reference dimension for a given observation (row) in one data set, then that observation is matched only with observations in the other data set whose values in the reference dimension fall within S ± BandValue. Then, only these potential matches are passed to the algorithm for further scoring. Default BandValue is Inf.

WidthValue

Either of the following:

  • Two-element vector, [U, V]

  • Scalar that is used for both U and V

WidthValue limits the number of potential matches between observations in two data sets; that is, each observation in X is scored to the closest U observations in Y, and each observation in Y is scored to the closest V observations in X. Then, only these potential matches are passed to the algorithm for further scoring. Closeness is measured using only the first column (reference dimension) in each data set. Default is Inf if 'Band' is specified; otherwise default is 10.

GapValue

Any of the following:

  • Cell array, {G, H}, where G is either a scalar or a function handle specified using @(X), and H is either a scalar or a function handle specified using @(Y). The functions @(X) and @(Y) must calculate the penalty for each observation (row) when it is matched to a gap in the other data set. The functions @(X) and @(Y) must return a column vector with the same number of rows as X or Y, containing the gap penalty for each observation (row).

  • Single function handle specified using @(Z), which is used for both G and H. The function @(Z) must calculate the penalty for each observation (row) in both X and Y when it is matched to a gap in the other data set. The function @(Z) must take as arguments X and Y. The function @(Z) must return a column vector with the same number of rows as X or Y, containing the gap penalty for each observation (row).

  • Scalar that is used for both G and H.

GapValue specifies the position-dependent terms for assigning gap penalties. The calculated value, GPX, is the gap penalty for matching observations from the first data set X to gaps inserted in the second data set Y, and is the product of two terms: GPX = G * QMS. The term G takes its value as a function of the observations in X. Similarly, GPY is the gap penalty for matching observations from Y to gaps inserted in X, and is the product of two terms: GPY = H * QMS. The term H takes its value as a function of the observations in Y. By default, the term QMS is the 0.75 quantile of the score for the pairs of observations that are potential matches (that is, pairs that comply with the 'Band' and 'Width' constraints). Default GapValue is 1.

QuantileValueScalar between 0 and 1 that specifies the quantile value used to calculate the term QMS, which is used by the 'Gap' property to calculate gap penalties. Default is 0.75.
DistanceValue

Function handle specified using @(R,S). The function @(R,S) must:

  • Calculate the distance between pairs of observations that are potential matches.

  • Take as arguments, R and S, matrices that have the same number of rows and columns, and whose paired rows represent all potential matches of observations in X and Y respectively.

  • Return a column vector of positive values with the same number of elements as rows in R and S.

Default is the Euclidean distance between the pairs.

    Caution   All columns in X and Y, including the reference dimension, are considered when calculating distances. If you do not want to include the reference dimension in the distance calculations, use the 'Weight' property to exclude it.

WeightsValue

Either of the following:

  • Logical row vector with the same number of elements as columns in X and Y, that specifies columns in X and Y.

  • Numeric row vector with the same number of elements as columns in X and Y, that specifies the relative weights of the columns (features).

This property controls the inclusion/exclusion of columns (features) or the emphasis of columns (features) when calculating the distance score between observations that are potential matches, that is, when using the 'Distance' property. Default is a logical row vector with all elements set to true.

    Tip   Using a numeric row vector for WeightsValue and setting some values to 0 can simplify the distance calculation when the data sets have many columns (features).

    Note:   The weight values are not considered when using the 'Band', 'Width', or 'Gap' property.

ShowConstraintsValueControls the display of the search space constrained by the specified 'Band' and 'Width' input parameters, thereby giving an indication of the memory required to run the algorithm with the specific 'Band' and 'Width' parameters on your data sets. Choices are true or false (default).
ShowNetworkValueControls the display of the dynamic programming network, the match scores, the gap penalties, and the winning path. Choices are true or false (default).
ShowAlignmentValueControls the display of the first and second columns of the X and Y data sets in the abscissa and the ordinate respectively, of a two-dimensional plot. Choices are true, false (default), or an integer specifying a column of the X and Y data sets to plot as the ordinate.

Output Arguments

IColumn vector containing indices of rows (observations) in X that match to a row (observation) in Y. Missing indices indicate that row (observation) is matched to a gap.
JColumn vector containing indices of rows (observations) in Y that match to a row (observation) in X. Missing indices indicate that row (observation) is matched to a gap.

Description

[I, J] = samplealign(X, Y) aligns the observations in two matrices of data, X and Y, by introducing gaps. X and Y are matrices of data where rows correspond to observations or samples, and columns correspond to features or dimensions. X and Y can have different number of rows, but must have the same number of columns. The first column is the reference dimension and must contain unique values in ascending order. The reference dimension could contain sample indices of the observations or a measurable value, such as time. The samplealign function uses a dynamic programming algorithm to minimize the sum of positive scores resulting from pairs of observations that are potential matches and the penalties resulting from the insertion of gaps. Return values I and J are column vectors containing indices that indicate the matches for each row (observation) in X and Y respectively.

    Tip   If you do not specify return values, samplealign does not run the dynamic programming algorithm. Running samplealign without return values, but setting the 'ShowConstraints', 'ShowNetwork', or 'ShowAlignment' property to true, lets you explore the constrained search space, the dynamic programming network, or the aligned observations, without running into potential memory problems.

[I, J] = samplealign(X, Y, ...'PropertyName', PropertyValue, ...) calls samplealign with optional properties that use property name/property value pairs. You can specify one or more properties in any order. Each PropertyName must be enclosed in single quotation marks and is case insensitive. These property name/property value pairs are as follows:


[I, J] = samplealign(X, Y, ...'Band', BandValue, ...)
specifies a maximum allowable distance between observations (along the reference dimension only) in the two data sets, thus limiting the number of potential matches between observations in the two data sets. If S is the value in the reference dimension for a given observation (row) in one data set, then that observation is matched only with observations in the other data set whose values in the reference dimension fall within S ± BandValue. Then, only these potential matches are passed to the algorithm for further scoring. BandValue can be a scalar or a function specified using @(z), where z is the mid-point between a given observation in one data set and a given observation in the other data set. Default BandValue is Inf.

This constraint reduces the time and memory complexity of the algorithm from O(MN) to O(sqrt(MN)*K), where M and N are the number of observations in X and Y respectively, and K is a small constant such that K<<M and K<<N. Adjust BandValue to the maximum expected shift between the reference dimensions in the two data sets, that is, between X(:,1) and Y(:,1).

[I, J] = samplealign(X, Y, ...'Width', WidthValue, ...) limits the number of potential matches between observations in two data sets; that is, each observation in X is scored to the closest U observations in Y, and each observation in Y is scored to the closest V observations in X. Then, only these potential matches are passed to the algorithm for further scoring. WidthValue is either a two-element vector, [U, V] or a scalar that is used for both U and V. Closeness is measured using only the first column (reference dimension) in each data set. Default is Inf if 'Band' is specified; otherwise default is 10.

This constraint reduces the time and memory complexity of the algorithm from O(MN) to O(sqrt(MN)*sqrt(UV)), where M and N are the number of observations in X and Y respectively, and U and V are small such that U<<M and V<<N.

    Note:   If you specify both 'Band' and 'Width', only pairs of observations that meet both constraints are considered potential matches and passed to the algorithm for scoring.

    Tip   Specify 'Width' when you do not have a good estimate for the 'Band' property. To get an indication of the memory required to run the algorithm with specific 'Band' and 'Width' parameters on your data sets, run samplealign, but do not specify return values and set 'ShowConstraints' to true.

[I, J] = samplealign(X, Y, ...'Gap', GapValue, ...) specifies the position-dependent terms for assigning gap penalties.

GapValue is any of the following:

  • Cell array, {G, H}, where G is either a scalar or a function handle specified using @(X), and H is either a scalar or a function handle specified using @(Y). The functions @(X) and @(Y) must calculate the penalty for each observation (row) when it is matched to a gap in the other data set. The functions @(X) and @(Y) must return a column vector with the same number of rows as X or Y, containing the gap penalty for each observation (row).

  • Single function handle specified using @(Z), that is used for both G and H. The function @(Z) must calculate the penalty for each observation (row) in both X and Y when it is matched to a gap in the other data set. The function @(Z) must take as arguments X and Y. The function @(Z) must return a column vector with the same number of rows as X or Y, containing the gap penalty for each observation (row).

  • Scalar that is used for both G and H.

The calculated value, GPX, is the gap penalty for matching observations from the first data set X to gaps inserted in the second data set Y, and is the product of two terms: GPX = G * QMS. The term G takes its value as a function of the observations in X. Similarly, GPY is the gap penalty for matching observations from Y to gaps inserted in X, and is the product of two terms: GPY = H * QMS. The term H takes its value as a function of the observations in Y. By default, the term QMS is the 0.75 quantile of the score for the pairs of observations that are potential matches (that is, pairs that comply with the 'Band' and 'Width' constraints).

If G and H are positive scalars, then GPX and GPY are independent of the observation where the gap is being inserted.

Default GapValue is 1, that is, both G and H are 1, which indicates that the default penalty for gap insertions in both sequences is equivalent to the quantile (set by the 'Quantile' property, default = 0.75) of the score for the pairs of observations that are potential matches.

    Note:   GapValue defaults to a relatively safe value. However, the success of the algorithm depends on the fine tuning of the gap penalties, which is application dependent. When the gap penalties are large relative to the score of the correct matches, samplealign returns alignments with fewer gaps, but with more incorrectly aligned regions. When the gap penalties are smaller, the output alignment contains longer regions with gaps and fewer matched observations. Set 'ShowNetwork' to true to compare the gap penalties to the score of matched observations in different regions of the alignment.

[I, J] = samplealign(X, Y, ...'Quantile', QuantileValue, ...) specifies the quantile value used to calculate the term QMS, which is used by the 'Gap' property to calculate gap penalties. QuantileValue is a scalar between 0 and 1. Default is 0.75.

    Tip   Set QuantileValue to an empty array ([]) to make the gap penalities independent of QMS, that is, GPX and GPY are functions of only the G and H input parameters respectively.

[I, J] = samplealign(X, Y, ...'Distance', DistanceValue, ...) specifies a function to calculate the distance between pairs of observations that are potential matches. DistanceValue is a function handle specified using @(R,S). The function @(R,S) must take as arguments, R and S, matrices that have the same number of rows and columns, and whose paired rows represent all potential matches of observations in X and Y respectively. The function @(R,S) must return a column vector of positive values with the same number of elements as rows in R and S. Default is the Euclidean distance between the pairs.

    Caution   All columns in X and Y, including the reference dimension, are considered when calculating distances. If you do not want to include the reference dimension in the distance calculations, use the 'Weight' property to exclude it.

[I, J] = samplealign(X, Y, ...'Weights', WeightsValue, ...) controls the inclusion/exclusion of columns (features) or the emphasis of columns (features) when calculating the distance score between observations that are potential matches, that is when using the 'Distance' property. WeightsValue can be a logical row vector that specifies columns in X and Y. WeightsValue can also be a numeric row vector with the same number of elements as columns in X and Y, that specifies the relative weights of the columns (features). Default is a logical row vector with all elements set to true.

    Tip   Using a numeric row vector for WeightsValue and setting some values to 0 can simplify the distance calculation when the data sets have many columns (features).

    Note:   The weight values are not considered when computing the constrained alignment space, that is when using the 'Band' or 'Width' properties, or when calculating the gap penalties, that is when using the 'Gap' property.

[I, J] = samplealign(X, Y, ...'ShowConstraints', ShowConstraintsValue, ...) controls the display of the search space constrained by the input parameters 'Band' and 'Width', giving an indication of the memory required to run the algorithm with specific 'Band' and 'Width' on your data sets. Choices are true or false (default).

[I, J] = samplealign(X, Y, ...'ShowNetwork', ShowNetworkValue, ...) controls the display of the dynamic programming network, the match scores, the gap penalties, and the winning path. Choices are true or false (default).

[I, J] = samplealign(X, Y, ...'ShowAlignment', ShowAlignmentValue, ...) controls the display of the first and second columns of the X and Y data sets in the abscissa and the ordinate respectively, of a two-dimensional plot. Links between all the potential matches that meet the constraints are displayed, and the matches belonging to the output alignment are highlighted. Choices are true, false (default), or an integer specifying a column of the X and Y data sets to plot as the ordinate.

Examples

Warping a sine wave with a smooth function to more closely follow cyclical sunspot activity

  1. Load sunspot.dat, a data file included with the MATLAB® software, that contains the variable sunspot, which is a two-column matrix containing variations in sunspot activity over the last 300 years. The first column is the reference dimension (years), and the second column contains sunspot activity values. Sunspot activity is cyclical, reaching a maximum about every 11 years.

    load sunspot.dat
  2. Create a sine wave with a known period of sunspot activity.

    years = (1700:1990)';
    T = 11.038;
    f = @(y) 60 + 60 * sin(y*(2*pi/T));
  3. Align the observations between the sine wave and the sunspot activity by introducing gaps.

    [i,j] = samplealign([years f(years)],sunspot,'weights',...
                        [0 1],'showalignment',true);

  4. Estimate a smooth function to warp the sine wave.

    [p,s,mu] = polyfit(years(i),years(j),15);
    wy = @(y) polyval(p,(y-mu(1))./mu(2));
  5. Plot the sunspot cycles, unwarped sine wave, and warped sine wave.

    years = (1700:1/12:1990)';
    figure
    plot(sunspot(:,1),sunspot(:,2),years,f(years),wy(years),...
         f(years))
    legend('Sunspots','Unwarped Sine Wave','Warped Sine Wave')
    title('Smooth Warping Example')

Recovering a nonlinear warping between two signals containing noisy Gaussian peaks

  1. Create two signals with noisy Gaussian peaks.

    rand('twister',5489)
    peakLoc = [30 60 90 130 150 200 230 300 380 430];
    peakInt = [7 1 3 10 3 6 1 8 3 10];
    time = 1:450;
    comp = exp(-(bsxfun(@minus,time,peakLoc')./5).^2);
    sig_1 = (peakInt + rand(1,10)) * comp + rand(1,450);
    sig_2 = (peakInt + rand(1,10)) * comp + rand(1,450);
  2. Define a nonlinear warping function.

    wf = @(t) 1 + (t<=100).*0.01.*(t.^2) + (t>100).*...
         (310+150*tanh(t./100-3));
  3. Warp the second signal to distort it.

    sig_2 = interp1(time,sig_2,wf(time),'pchip');
  4. Align the observations between the two signals by introducing gaps.

    [i,j] = samplealign([time;sig_1]',[time;sig_2]',...
                        'weights',[0,1],'band',35,'quantile',.5);
  5. Plot the reference signal, distorted signal, and warped (corrected) signal.

    figure
    sig_3 = interp1(time,sig_2,interp1(i,j,time,'pchip'),'pchip');
    plot(time,sig_1,time,sig_2,time,sig_3)
    legend('Reference','Distorted Signal','Corrected Signal')
    title('Non-linear Warping Example')

  6. Plot the real and the estimated warping functions.

    figure
    plot(time,wf(time),time,interp1(j,i,time,'pchip'))
    legend('Distorting Function','Estimated Warping')

References

[1] Myers, C.S. and Rabiner, L.R. (1981). A comparative study of several dynamic time-warping algorithms for connected word recognition. The Bell System Technical Journal 60:7, 1389–1409.

[2] Sakoe, H. and Chiba, S. (1978). Dynamic programming algorithm optimization for spoken word recognition. IEEE Trans. Acoustics, Speech and Signal Processing ASSP-26(1), 43–49.

See Also

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