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ALIGNCURVE procedure

Forms an optimal warping to align an observed series of observations with a standard series (D.B. Baird).

Options

PRINT = string tokens What to print (criterion, ss, warps); default * i.e. nothing
PLOT = string tokens What to plot (series, warping); default * i.e. no plots
WARPPENALTY = scalar The relative penalty to add to the criterion when jumping a unit in one series but not the other; default 1
MAXSTEP = scalar The largest jump that can be made between the two series at a single point; default 1
MAXDIFFERENCE = scalar Sets a limit on size of difference between the series to be squared and added to the criterion (differences greater than this are truncated to MAXDIFFERENCE, thus enabling the effects of outliers to be down-weighted); default * i.e. no limit
USEMEANS = string token Whether to use the means of points covered in one step, rather than the final value, when calculating the sums of squares between the two series (yes, no); default no
FORCEENDALIGNMENT = string token Whether to force the ends of the two series to align, so that warping happens only in the middle of the series (yes, no); default no
WINDOW = scalar Window number for the plots; default 1
KEYWINDOW = scalar Window for the key (zero for no key); default 2

Parameters

Y = variates Series to be aligned with the standard series
STANDARD = variates Standard series for each Y
WEIGHTS = variates Weights for the contribution of each point to the criterion; default * no weighting
UWARP = variates The warped positions of the unit numbers, required to align Y with STANDARD
YWARP = variates The warped series for Y, i.e. the optimally aligned y-values
CRITERIONVALUE = scalars The criterion value (as optimized during the alignment)
TITLE = text Title for the plots

Description

ALIGNCURVE is useful when you have a series of observations of a theoretical curve in which the x-axis may have been distorted by stretching or compression. The aim is to “warp” the units of the curve so that it optimally matches a standard series. (This can be used, for example, to align electrophoresis gels.) The observed and standard series are specified, in variates, by the Y and STANDARD parameters respectively.

The warping takes the form of one of the series jumping values, so it that makes several steps for a single step of the other series. There is no limitation on the number of times that this can happen along the series, but you can set option FORCEENDALIGNMENT=yes, to require the two series to match at their final values.

The dynamic warping algorithm that is used, seeks to optimize a criterion that is based on the sum of squares of residuals between the warped observed series and the standard series at each point, plus a penalty that depends on the amount of warping.

The WEIGHTS parameter can supply a variate of weights to use for the contribution of each point to the criterion. If the standard curve has non-constant variance around it, typically you would set the weights to be the reciprocal of the variance or equivalently, one over the square of the standard deviation. Suppose, for example, if the curve is the mean of a Poisson process. the mean is equal to the variance for a Poisson distribution, so the you could use weights of 1/STANDARD to allow for the non-constant variance. Alternatively, if the points on the curves were means of differing numbers of replicated observations, you could weight by the recriprocals of the replications. By default there is no weighting.

The MAXDIFFERENCE option sets a limit on contribution of each residual: differences greater than this are truncated (to MAXDIFFERENCE), thus enabling the effects of outliers to be down-weighted. By default there is no limit.

The WARPPENALTY option defines the relative penalty to be added to the criterion when the series are warped by one step in either direction; default 1. The penalty is given by WARPPENALTY multiplied by a scale factor equal to the average mean square of the initial (unwarped) differences between Y and STANDARD. (It is thus scale independent, so multiplying both Y and STANDARD by a constant will still give the same solution.) Setting WARPPENALTY to zero finds the optimally aligned series with no constraint on warping. Conversely, setting WARPPENALTY to a large value discourages warps, as a jump will be taken only if the decrease in the sum of squares is greater than the penalty.

The MAXSTEP option sets a limit on the number of units that can be jumped at any one point (default 1).

By default, the y-value after a jump is taken to be the value at the end of the jump, but you can set option USEMEANS=yes to use the mean of the values at the end and intervening (jumped) points.

The UWARP parameter can save a variate containing the warped units, i.e. the unit of Y that corresponds (after warping) to each unit of STANDARD. The YWARP parameter can save the y-values of the optimally aligned series, and the CRITERIONVALUE parameter can save the optimal criterion value.

The PRINT option controls printed output, with settings:

    criterion to print the smoothing weight, maximum step size and optimal criterion value,
    warps to print the location of the warps,
    ss to print sums of squares before and after alignment.

By default nothing is printed.

The PLOT option controls the plots that are displayed, with settings:

    series plots the aligned and standard series,
    warping plots the steps in the warping that have been used to align the two series.

By default nothing is plotted. The WINDOW option specifies the window to use for the plots (default 1), and the KEYWINDOW option specifies the window for their keys (default 2). You can supply a title for the plots using the TITLE parameter.

Options: PRINT, PLOT, WARPPENALTY, MAXSTEP, MAXDIFFERENCE, USEMEANS, FORCEENDALIGNMENT, WINDOW, KEYWINDOW.

Parameters: Y, STANDARD, WEIGHTS, UWARP, YWARP, CRITERIONVALUE, TITLE.

Action with RESTRICT

Any restrictions are ignored.

See also

Procedures: BASELINE, PEAKFINDER.

Commands for: Calculations and manipulation.

Example

CAPTION    'ALIGNCURVE examples'; STYLE=meta
CALCULATE  X = !(1...1000)/80
&          S = SIN(X)
&          Y = SIN(1.2*X)        " Y has an increased rate from S " 
ALIGNCURVE [PRINT=criterion,ss; PLOT=series,warping; WARPPENALTY=0;\
           FORCEENDALIGNMENT=yes] Y; STANDARD=S;\
           UWARP=WarpedunitsY; YWARP=WWarpedY; CRITERIONVALUE=CY
CALCULATE  Z = SIN(X + SIN(X/2)) " Z has an varying rate from S "
ALIGNCURVE [PRINT=criterion,ss; PLOT=series,warping; WARPPENALTY=0]\
           Z; STANDARD=S; UWARP=WarpedunitsZ; YWARP=WarpedZ; CRITERIONVALUE=CZ
" penalize warps "
ALIGNCURVE [PRINT=criterion,ss,warps; PLOT=series,warping; WARPPENALTY=1]\
           Z; STANDARD=S; UWARP=WarpedunitsZp; YWARP=WarpedZp;\
           CRITERIONVALUE=CZp; TITLE='Z aligned to S, penalty=1'
Updated on March 11, 2019

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