6. SPATIAL DECOMPOSITION
6.2 Examples of decomposition versus time
Figure 6.1: 2-dimensional display of the centers of mass. The colour of the points changes with time: purple represents the beginning of the run, then it follows the colours of the rainbow. Red is the end of the run. Most of this run was dominated by losses of type B2H. Points closer to zero for B1/B2 separation or closer to−0.5 for H/V separation show a less clear separation.
6.2 Examples of decomposition versus time
In this section, several examples of the results of the decompositions of complete LHC fills will be presented(cf. fig.6.2, 6.3 and 6.5). The factors associated to the six vectors are plotted versus time, as well as the error on the decomposition. All vectors are normalised before decomposition, so the recomposition should be also normalised. This means that when one scenario is dominating the loss and the associated factor increases, the other factors will decrease, even if the corresponding loss hasn’t: only the proportion has changed. Every time, the two decomposition algorithms are presented, SVD and G-S. The factors associated to TCTs (cf. § 4.4.5) are represented on a separated plot
6. SPATIAL DECOMPOSITION
(cf. fig. 6.4).
6.2.1 Error propagation for the factors
It was suggested that an error on each decomposition factor could be calculated, in addition to the error on the decomposition(cf. § 3.5). This is achieved by calculating the error propagation in the factors for Gram-Schmidt.
Each of the Gram-Schmidt factors fi is defined as the result of the scalar product between the corresponding reference vector v~i and the vector of the current loss X.~ This can be expressed analytically:
fi=v~i·X~ = Xm j=1
vij ·xj (6.1)
where (xj) are the coordinates of X, and (v~ ij) the coordinates of the vector v~i.
The error onxj, written asσxj, is the error on the measure of the BLM. It is known to be proportional to the value of the loss:
σxj = 0.01xj (6.2)
The error on the vector v~i is written as:
σvij =q
sys2+ (0.01vij)2
where sys represents the systematic error on the reference vector. This is known: it is the standard deviation of each coordinate of the reference vector over all loss maps composing this vector. It was calculated when the reference vector was created. It corresponds to the length of the error bar in fig. 4.3, and is the RSD of fig. 4.11, fig. 4.12 and fig. 4.13 before being divided by the monitor signal. The second term is the error on the measure of the BLM. Both are independent: one is the error on the measure from the physical monitor, the other is the error over the different loss maps.
They are consequently summed in square.
Then, the squared error on the factor is calculated as the squared partial derivative with respect to each independent variableT:
σf2 =X
T
∂f
∂T 2
·σ2T
6.2 Examples of decomposition versus time
Figure 6.2: Top: SVD versus time, for transversal and longitudinal loss scenarios and the error on decomposition, for the stable beam of the 15thof October 2011. Bottom: Gram-Schmidt versus time. The factors are similar in both cases, and the times at which noticeable changes appear are exactly the same time. The factors for B1V and B2V in G-S are close to zero, because there is nothing left of the vector after the projections. As it is often the case, the factor for the L scenarios are not dominating.
6. SPATIAL DECOMPOSITION
Figure 6.3: Decomposition for transversal and longitudinal loss scenarios and the error on decomposition, for the stable beam of the 22nd of September 2011. Top: SVD versus time, bottom: Gram-Schmidt versus time. The factors are similar in both cases, and the times at which noticeable changes appear are exactly the same time. This is an example when the longitudinal factors actually dominate the decomposition: B1L, dark blue, higher factor at the end of the fill.
B2L is the lower green.
6.2 Examples of decomposition versus time
Figure 6.4: Factors calculated for each of the canonical vectors associated to a tertiary collimator (TCT), for the two decompositions of the stable beam of the 22ndof September 2011. The factor correspond to the part of the signal at the TCT that wasn’t reconstructed by the reference vectors.
Left: for SVD. Right: for Gram-Schmidt. Bottom: decomposition for Gram-Schmidt represented in log scale. The factors in both decompositions are equal to a precision of 10−4, and the times at which noticeable changes appear are exactly the same time. All factors seem to behave similarly (cf. bottom); the different offsets could be due to the different levels of debris coming from the experiments measured at each TCT.
6. SPATIAL DECOMPOSITION
Figure 6.5: Example of almost constant decomposition for transversal and longitudinal loss scenarios and the error on decomposition, for the stable beam of the 22ndof October 2011. Top:
SVD versus time, bottom: Gram-Schmidt versus time. This fill was dominated by horizontal losses for both beams. The factors are similar in both cases, and the times at which noticeable changes appear are exactly the same time.
6.2 Examples of decomposition versus time
which is calculated as:
σ2f = ∂f
∂vij
2
σv2ij+ ∂f
∂xj
2
σx2j =x2j ·σv2ij +vij2 ·σx2j
The error on each vector coordinate corresponds to an error on the measurement, so is also calculated fromeq. 6.2.
σf2 = Xm j=1
x2j· sys2+ (0.01vij)2
+vij2 ·(0.01xj)2 (6.3)
The error on each factor depends on the current loss vector X~ and and has to be calculated every second. The results are presented infig. 6.6.
The errors, calculated this way, are too small to affect the final decomposition. They are dominated by the error on the loss maps, and are at least 3 orders of magnitude lower than the values of the factors, so they will not be displayed on the results.
Figure 6.6: Error on the decomposition factors calculated by propagation, for the stable beam of the 22nd of October 2011(cf. fig. 6.5). The highest error corresponds to the reference vector which had the highest standard deviation: B2L. All the errors are lower than 3 orders of magnitude or more of the values of the factors.
6. SPATIAL DECOMPOSITION