## 2 Scale-Space Methodology
Here, further, we assume that  
\[\bar{X}_{1}\sim N(\mu_{1},\sigma^{2}) \tag{1}\]  
for acquisition one and  
\[\bar{X}=\frac{1}{n-1}\sum_{k=2}^{n}\bar{X}_{k}\sim N\left(\mu,\frac{\sigma^{2 }}{n-1}\right), \tag{2}\]  
for the remaining \(n-1\) of the acquisitions. Here \(n\) is the total number of acquisitions. The normal assumption makes sense due to the central limit theorem since all \(\bar{X}_{k}\)'s, \(k=1,\ldots,n\) are averages over a large number of observations. Note that this means that we perform the training procedure by leavingone out cross validation. In the description above, the mean is chosen as parameter, but we have also implemented and performed our testing procedure for the median, the standard deviation and the range. We do this since these parameters can better describe certain aspects of a distribution and may therefore capture different types of changes. In practice, we will therefore typically test all these parameters for potential changes. For parameters other than the mean, Equation (1) will be an approximation that may be violated in practice. Equation (2) will, however, still be a reasonable approximation for all parameters due to the central limit theorem, but will sometimes only hold approximately. In our description below, we estimate \(\sigma^{2}\) by the standard estimator for variance using all acquisitions apart from the one left out. In the case acquisition 1 is left out, this means that \(\sigma^{2}\) is estimated by  
\[S^{2}=\frac{1}{n-2}\sum_{i=2}^{n}(\bar{X}_{i}-\bar{X})^{2}.\]  
The critical quantile at location \((d,s)\), when only using the Bonferroni correction, is then given by  
\[c(d,s)=\hat{t}_{\frac{s}{2p},n-2}, \tag{3}\]  
where \(\alpha=0.05\) is the significance-level and \(p\) the number of spectral channels of each spectral signature curve. In addition to the Bonferroni-corrected quantile in Equation (3), we tried here the so-called global quantile  
\[c(d,s)_{G}=\Phi^{-1}\left(\left(1-\frac{\alpha}{2}\right)^{\frac{1}{p\sum_{k =1}^{n}\theta_{k}}}\right), \tag{4}\]  
proposed in [12]. Here \(\Phi\) is the normal cumulative distribution function and \(n_{s}\) denotes the number of rows in the scale-space map. Moreover, \(\theta_{k}\) is given by  
\[\theta_{k}=2\Phi\left(\frac{\sqrt{3\log p}}{2s_{k}}\right)-1,\]  
where \(s_{k}\) is the scale in row \(k\). In the testing procedure, we test  
\[H_{0}:\mu_{1}=\mu\text{ against }H_{1}:\mu_{1}\
eq\mu\]  
using the test statistic  
\[T=\frac{\bar{X}_{\text{test}}-\bar{X}_{\text{train}}}{S\sqrt{1+\frac{1}{n-1}}},\]  
where \(H_{0}\) is rejected if  
\[|T|>c(d,s)\text{ or }|T|>c(d,s)_{G}.\]  
The algorithm is summarized in Algorithm 1 where \(Par\) is used to denote the parameter we are using in the tests. The outcomes of the tests are graphically summarized in a scale-space map, where the horizontal and vertical axes correspond to spectral frequency bands and scales, respectively. Thus, at each location \((d,s)\) we perform a test and the outcome is shown as a colored pixel, with red (blue) indicating a significant (not significant) difference at the position \(d\) for scale \(s\). To illustrate the method, consider the example introduced in Figure 1. Figure 2 shows the scale-space map produced by the procedure described above. The parameter used in this analysis was the range as it best detected the small change manually introduced to the data. Note how the map indicates a significant feature only for the smallest scales around the spectral channel given at point 80 on the horizontal axis. This is expected since the change is small and only present at one particular spectral channel for a single signature. ```
1:Initialization:Acquisitions that are correct under null hypothesis and the test-acquisitions are loaded. 2:procedureSS_CC_train()
3:Input: The loaded acquisitions that are correct under the null hypothesis. 4:Initialization: The significance level \(\alpha\) is chosen.