### 统计代写|化学计量学作业代写chemometrics代考|Parametric Time Warping

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|化学计量学作业代写chemometrics代考|Parametric Time Warping

In PTW, one approximates the time axis of the reference signal by applying a polynomial transformation of the time axis of the sample (Eilers 2004):
$$\hat{S}\left(t_{k}\right)=S\left(w\left(t_{k}\right)\right)$$
where $\hat{S}\left(t_{k}\right)$ is the value of the warped signal at time point $t_{k}$, where $k$ is an index. The warping function, $w$, is given by:
$$w(t)=\sum_{j=0}^{J} a_{j} t^{j}$$
with $J$ the maximal order of the polynomial. In general, only low-order polynomials are used. Since neighboring points on the time axis will be warped with almost the same amount, peak shape distortions are limited. Thus, the method finds the set of coefficients $a_{0}, \ldots, a_{J}$ that minimizes the difference between the sample $S$ and reference $R$, using whatever difference measure is desired. Especially for higher-degree warpings there is a real possibility that the optimization ends in a local optimum, and it is usually a good idea to use several different starting values.

This procedure is very suitable for modelling gradual changes, such as the slow deterioration of chromatographic columns, so that measurements taken days or weeks apart can still be made comparable. For situations where a few individual peak shifts have to be corrected (e.g., pH-dependent shifting of patterns in NMR spectra), the technique is less ideal (Giskeødegård et al. 2010).

The original implementation of ptw (corresponding to function argument mode = “backward”) predicts, for position $i$, which point $j$ in the signal will end up at position $i$. This is somewhat counterintuitive, and in later versions (from version $1.9 .1$ onwards) the default mode is “forward”, basically predicting the position of point $i$ after warping. The interpretation of the coefficients in the two modes is the same, just with reversed signs.

## 统计代写|化学计量学作业代写chemometrics代考|Dynamic Time Warping

Dynamic Time Warping (DTW), implemented in package dtw (Giorgino 2009), provides a similar approach, constructing a warping function that provides a mapping from the indices in the query signal to the points in the reference signal ${ }^{1}$ :
$>$ warpfun. dtw <- dtw (ssamp, sref)

plot (warpfun. dtw)
$>$ abline ${0,1$, col $=$ “gray”, lty $=2}$
The result is shown in the left plot of Fig.3.11. Here, the warping function is not restricted to be a polynomial, as in PTW.

A horizontal segment indicates that several points in the query signal are mapped to the same point in the reference signal; the axis of the query signal is compressed by elimination (or rather, averaging) of points. Similarly, vertical segments indicate a stretching of the query signal axis by the duplication of points. Note that these horizontal and vertical regions in the warping function of Fig.3.11 may also lead to peak shape distortions.

DTW chooses the warping function that minimizes the (weighted) distance between the warped signals.

## 统计代写|化学计量学作业代写chemometrics代考|Practicalities

In almost all cases, a set of signals should be aligned in such a way that all features of interest are at the same positions in every trace. One strategy is to use the column means of the data matrix as a reference. This is only possible with very small shifts and will lead to peak broadening. Simply taking a random record from the set as a reference is better but still may be improved upon-it usually pays to perform some experiments to see which reference would lead to the smallest distortion of the other signals, while still leading to good alignment. If the number of samples is not too large, one can perform all possible combinations and see which one comes out best. Careful data pretreatment is essential-baselines may severely influence the results and should be removed before alignment. In fact, one of the motivations of the CODA algorithm is to select traces that do not contain a baseline (Windig et al. 1996). Another point of attention is the fact that features can have intensities differing several orders in magnitude. Often, the biggest gain in the alignment optimization is achieved by getting the prominent features in the right location. Sometimes, this dominance leads to suboptimal alignments. Also differences in intensity between sample and reference signals can distort the results. Methods to cope with these phenomena will be treated in Sect. 3.5. Finally, it has been shown that in some cases results can be improved when the signals are divided into segments which are aligned individually (Wang and Isenhour 1987). Especially with more constrained warping methods like PTW this adds flexibility, but again, there is a danger of warping too much and mapping features onto the wrong locations. Especially in cases where there may be differences between the samples (control versus diseased, for instance) there is a risk that a biomarker peak, present only in one of the two classes, is incorrectly aligned. This, again, is all the more probable when that particular peak has a high intensity.

Packages dtw and ptw are by no means alone in tackling alignment. We already mentioned the VPdtw package: in addition, several Bioconductor packages, such as PROcess and xcms, implement both general and more specific alignment procedures, in most cases for mass-spectrometry data or hyphenated techniques like LC-MS.

## 统计代写|化学计量学作业代写chemometrics代考|Parametric Time Warping

ptw 的原始实现（对应于函数参数 mode = “backward”）预测，对于位置一世, 哪一点j在信号将结束在位置一世. 这有点违反直觉，在以后的版本中（来自版本1.9.1onwards) 默认模式是“forward”，基本上是预测点的位置一世翘曲后。两种模式中系数的解释是相同的，只是符号相反。

## 统计代写|化学计量学作业代写chemometrics代考|Dynamic Time Warping

>经趣。dtw <- dtw(ssamp, sref)

>下线0,1$,C这l$=$“Gr一种是”,l吨是$=2

DTW 选择最小化扭曲信号之间（加权）距离的扭曲函数。

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## MATLAB代写

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