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#### Table of Contents

# Spectrum Analysis

## Theory

### Spectrum

The differential flux \(\phi(E)\) per area, time and energy interval is defined as \[\phi(E) = \frac{dN}{dA\cdot dt\cdot dE}\]

Often \(\phi(E)\) is also referred to as \(\frac{dN}{dE}\) as observation time and effective collection area is a constant.

For an observation with an effective observation time \(\Delta T=\sum\delta t_i\), this yields in a given Energy interval \(\Delta E\): \[\phi(\Delta E) = \frac{1}{A_0\cdot \Delta T}\frac{N(\Delta E)}{\epsilon(\Delta E)\cdot \Delta E}\]

For simplicity, in the following, \(\Delta E\) will be replaced by just \(E\) but always refers to to a given energy interval. If data is binned in a histogram, the relation between the x-value for the bin \(E_\textrm{x}\) and the corresponding interval \(\Delta E_\textrm{x}\) is not well defined. Resonable definitions are the bin center (usually in logarithmic bins) or the average energy.

### Efficiency

The total area \(A_0\) and the corresponding efficiency \(\epsilon(E)\) are of course only available for simulated data. For simulated data, \(A_0\) is the production area and \(\epsilon(E)\) the corresponding energy dependent efficiency of the analysis chain. For a given energy bin, the efficiency is then defined as

\[\epsilon(E) = \frac{N_\textrm{exc}(E)}{N_0(E)} \]

where \(N_0\) is the number of simulated events in this energy bin and \(N=N_{exc}\) the number of *excess* events that are produced by the analysis chain.

Note that the exact calculation of the efficiency \(\epsilon(\Delta E)\) depends on prior knowledge of the correct source spectrum \(N_0(E)\). Therefore, it is strictly speaking only correct if the simulated spectrum and the real spectrum are identical. As the real spectrum is unknown, special care has to be taken of the systematic introduced by the assumption of \(N_0(E)\).

### Effective Collection Area

The effective area is then defined as \(A_\textrm{eff}(E)=\epsilon(E)\cdot A_0\). Note that at large distances \(R_0\) the efficiency \(\epsilon(R_0)\) vanishes, so that the effective area is an (energy dependent) constant while \(A_0=\pi R_0^2\) and the efficiency \(\epsilon(E)\) are mutually dependent.

### Excess and Error

The number of excess events, for data and simulations, is defined as

\[N_\textrm{exc} = N_\textrm{sig} - \hat N_\textrm{bg}\]

where \(N_\textrm{sig}\) is the number of events identified as potential gammas from the source direction ('on-source') and \(N_\textrm{bg}\) the number of gamma-like events measured 'off-source'. Note that for Simulations, \(\hat N_\textrm{bg}\) is not necessarily zero for wobble-mode observations as an event can survive the analysis for on- and off-events, if this is not prevented by the analysis (cuts).

The average number of background events \(\hat N_\textrm{bg}\) is the total number of background events \(N_\textrm{bg}\) from all off-regions times the corresponding weight \(\frac{1}{5}\) (often referred to as \(\alpha\)). For five off-regions, this yields

\[\hat N_\textrm{bg} = \frac{N_\textrm{bg}}{5}\]

Assuming Gaussian errors, the statistical error is thus

\[\sigma^2(N_\textrm{exc}) = \left(\frac{dN_\textrm{exc}}{dN_\textrm{sig}}\right)^2\sigma^2(N_\textrm{sig}) + \left(\frac{dN_\textrm{exc}}{d\hat N_\textrm{bg}}\right)^2\sigma^2(\hat N_\textrm{bg})= \sigma^2(N_\textrm{sig}) + \frac{1}{5^2}\sigma^2(N_\textrm{bg})\]

For data this immediately resolves to

\[\sigma^2(N_\textrm{exc}) = N_\textrm{sig} + \frac{1}{5^2}N_\textrm{bg} \]

with the Poisson (counting) error \(\sigma^2(N_\textrm{sig,bg}) = N_\textrm{sig,bg}\).

### Weights and Error

In the following \(N\) refers to a number of simulated events and \(M\) to a number of measured (excess) events.

\(n(\delta E, \delta\theta)\) is the number of produced events in the energy interval \(E\in\delta E=[e_\textrm{min};e_\textrm{max}]\) and the zenith angle interval \(\theta\in\delta\theta=[\theta_\textrm{min};\theta_\textrm{max}]\). The weighted number of events \(n'(\delta E, \delta\theta)\) in that interval is then

\[n'(\delta E, \delta\theta) = \sum_{i=0...n}^{E\in\delta E}\sum_{j=0...n}^{\theta\in\delta\theta} \omega(E_i, \theta_j)\]

Since \(\omega(E, \theta) = \rho(E)\cdot \tau(\theta) \) with \(\rho(E)\) the spectral weight to adapt the spectral shape of the simulated spectrum to the real (measured) spectrum of the source and \(\tau(\theta)\) the weight to adapt to oberservation time versus zenith angle.

\[n'(\delta E, \delta\theta) = \sum_{i=0...n}^{E\in\delta E}\sum_{j=0...n}^{\theta\in\delta\theta} \rho(E_i)\cdot\tau(\theta_j) = \sum_{i=0...n}^{E\in\delta E}\rho(E_i)\cdot\sum_{j=0...n}^{\theta\in\delta\theta} \tau(\theta_j)\]

The weighted number of produced events \(N'(\Delta E, \Delta\Theta)\) in the total energy interval \(E\in\Delta E=[E_\textrm{min};E_\textrm{max}]\) and the total zenith angle interval \(\theta\in\Delta\Theta=[\Theta_\textrm{min};\Theta_\textrm{max}]\) is then

\[N'(\Delta E, \Delta\Theta) = \sum_{i=0...N}^{E\in\Delta E}\sum_{j=0...N}^{\theta\in\Delta\Theta} \rho(E_i)\cdot\tau(\theta_j) = \sum_{i=0...N}^{E\in\Delta E}\rho(E_i)\cdot\sum_{j=0...N}^{\theta\in\Delta\Theta} \tau(\theta_j)\]

with \(N(\Delta E, \Delta\Theta)\) being the total number of produced events.

For a sum of weights, e.g. \(N' = \sum_N \rho_i\tau_i\) the corresponding error is \[\sigma^2(N')^2 = \sum_N\left[\rho_i\cdot\sigma(\tau_i)+\tau_i\cdot\sigma(\rho_i)\right]^2\]

As the energy is well defined, \(\sigma(\rho_i)=0\) and thus

\[\sigma^2(N')^2 = \sum_N\rho_i^2\cdot\sigma^2(\tau_i)\]

The weights are defined as follow:

\[\rho(E) = \rho_0\frac{\phi_\textrm{src}(E)}{\phi_0(E)}\]

where \(\phi_0(E)\) is the simulated spectrum and \(\phi_\textrm{src}\) the (unknown) real source spectrum. \(\rho_0\) is a normalization constant. The zenith angle weights \(\tau(\delta\theta)\) in the interval \(\delta\theta\) are defined as

\[\tau(\delta\theta) = \tau_0\frac{\Delta T(\delta\theta)}{N(\delta\theta)}\]

where \(N(\delta\theta)\) is the number of produced events in the interval \(\delta\theta\) and \(\Delta T(\delta\theta)\) is the total observation time in the same zenith angle interval. \(\tau_0\) is the normalization constant. The error on the weight \(\tau=\tau(\delta\theta)\) in each individual \(\theta\)-bin with \(\Delta T=\Delta T(\delta\theta)\) and \(N=N(\delta\theta)\)is then

\[\sigma^2(\tau) = \left[\frac{d\tau}{d\Delta T}\sigma(\Delta T)\right]^2+\left[\frac{d\tau}{dN}\sigma(N)\right]^2= \tau^2\cdot\left[\left(\frac{\sigma(\Delta T)}{\Delta T}\right)^2+\left(\frac{\sigma(N)}{N}\right)^2\right]\]

While \(\sigma(\Delta T)/\Delta T \approx 1\textrm{s}/5\textrm{min}\) is given by the data acquisition and 1s per 5min run, \(\sigma(N)/N=1/\sqrt{N}\) is just the statistical error of the number of events.

As the efficiency \(\epsilon(\delta E)\) for an energy interval \(\delta E\) is calculated as

\[\epsilon(\delta E) = \epsilon(\delta E, \Delta\Theta) = \frac{N'_\textrm{exc}(\delta E, \Delta\Theta)}{N'_\textrm{src}(\delta E,\Delta\Theta)}\]

and \(N'_\textrm{exc}(\delta E,\Delta\Theta)\) and \(N'_\textrm{src}(\delta E,\Delta\Theta)\) are both expressed as the sum given above, the constants \(\rho_0\) and \(\tau_0\) cancel.

The differential flux in an energy interval \(\delta E\) is then given as

\[\phi(\delta E) = \phi(\delta E,\Delta\Theta) = \frac{1}{A_0\cdot \Delta T}\frac{M_\textrm{exc}(\delta E)}{\epsilon(\delta E)\cdot \delta E} = \frac{M_\textrm{exc}(\delta E)}{N'_\textrm{exc}(\delta E)}\cdot \frac{N'_\textrm{src}(\delta E)}{A_0\cdot\Delta T\cdot \delta E}\]

Where \(A_0\) is total area of production and \(\Delta T\) the total observation time. The number of measured excess events is in that energy interval is \(M_\textrm{exc}(\delta E)=M_\textrm{exc}(\delta E, \Delta\Theta)\).

Using Gaussian error propagation, the error in a given energy interval \(\delta\) is then given by

\[\sigma^2(\phi) = \left(\frac{1}{A_0\cdot \Delta T\cdot\delta E}\right)^2\cdot\left[\left(\frac{d\phi'}{dM_\textrm{exc}}\right)^2\sigma^2(M_\textrm{exc}) + \left(\frac{d\phi'}{dN'_\textrm{exc}}\right)^2\sigma^2(N'_\textrm{exc}) + \left(\frac{d\phi'}{dN'_\textrm{src}}\right)^2\sigma^2(N'_\textrm{src})\right]\]

with \(\phi':=A_0\cdot\Delta T\cdot\delta E\cdot\phi\).

\[\rightarrow\quad\sigma^2(\phi) = \phi^2 \cdot\left[\left(\frac{\sigma(M_\textrm{exc})}{M_\textrm{exc} }\right)^2 + \left(\frac{\sigma(N'_\textrm{exc})}{N'_\textrm{exc}}\right)^2 + \left(\frac{\sigma(N'_\textrm{src})}{N'_\textrm{src}}\right)^2\right]\]

## Conceptual Example

### Define Binnings

### Get Data File List

A list with file IDs containing the events to be analyed is required a.t.m. The following query retrieves such a list and fills a temporary table (`DataFiles`) with the IDs.

CREATE TEMPORARY TABLE DataFiles ( FileId INT UNSIGNED NOT NULL, PRIMARY KEY (FileId) USING HASH ) ENGINE=Memory AS ( SELECT FileId FROM factdata.RunInfo WHERE %0:where ORDER BY FileId )

`%0:where` is a placeholder, for example for

fZenithDistanceMean<30 AND fThresholdMinSet<350 AND fSourceKEY=5 AND fRunTypeKEY=1 AND fNight>20161201 AND fNight<20170201 AND fR750Cor>0.9e0*fR750Ref

### Get Observation Time

The following query bins the effective observation time of the runs listed above in zenith angle bins and stores the result in a temporary table (`ObservationTime`). Note that the result contains only those bins which have entries.

CREATE TEMPORARY TABLE ObservationTime ( `.theta` SMALLINT UNSIGNED NOT NULL, OnTime FLOAT NOT NULL, PRIMARY KEY (`.theta`) USING HASH ) ENGINE=Memory AS ( SELECT INTERVAL(fZenithDistanceMean, %0:bins) AS `.theta`, SUM(TIME_TO_SEC(TIMEDIFF(fRunStop,fRunStart))*fEffectiveOn) AS OnTime FROM DataFiles LEFT JOIN factdata.RunInfo USING (FileId) GROUP BY `.theta` ORDER BY `.theta` )

`%0:bins` is a placeholder for the bin boundaries, e.g. `5, 10, 15, 20, 25, 30` (five bins between 5° and 30° plus underflow and overflow).

### Get Monte Carlo File List

The next query obtains all Monte Carlo runs which have their ThetaMin or ThetaMax within one of the bins obtained in the previous query (so all MC runs that correspond to bins in which data is available). Strictly speaking, this step is not necessary, but it accelerats further processing. In addition (here as an example) only runs with even FileIDs are obtained as test-runs (assuming that odd runs were used for training). The resulting FileIDs are stored in a temporary table (`MonteCarloFiles`).

CREATE TEMPORARY TABLE MonteCarloFiles ( FileId INT UNSIGNED NOT NULL, PRIMARY KEY (FileId) USING HASH ) ENGINE=Memory AS ( SELECT FileId FROM ObservationTime LEFT JOIN BinningTheta ON `.theta`=bin LEFT JOIN factmc.RunInfoMC ON (ThetaMin>=lo AND ThetaMin<hi) OR (ThetaMax>lo AND ThetaMax<=hi) WHERE PartId=1 AND FileId%%2=0 ORDER BY FileId )

### Get Zenith Angle Histogram

The following table creates a temporaray table (`EventCount`) internally which bins the MonetCarlo files from the file list in MonteCarloFiles in zenith angle bins. This temporary table ois then joined with table containing the binning for the data files (EventCount) and for each bin, the ratio (ZdWeight) and the corresponding error (ErrZdWeight) is calculated (assuming an error on the on-time of 1s per 5min). To have also the in edges in the same table, the binning is joined as well.

CREATE TEMPORARY TABLE ThetaHist ( `.theta` SMALLINT UNSIGNED NOT NULL, lo DOUBLE NOT NULL COMMENT 'Lower edge of zenith distance bin in degree', hi DOUBLE NOT NULL COMMENT 'Upper edge of zenith distance bin in degree', CountN INT UNSIGNED NOT NULL, OnTime FLOAT NOT NULL, ZdWeight DOUBLE NOT NULL COMMENT 'tau(delta theta)', ErrZdWeight DOUBLE NOT NULL COMMENT 'sigma(tau)', PRIMARY KEY (`.theta`) USING HASH ) ENGINE=Memory AS ( WITH EventCount AS ( SELECT INTERVAL(DEGREES(Theta), %0:bins) AS `.theta`, COUNT(*) AS CountN FROM MonteCarloFiles LEFT JOIN factmc.OriginalMC USING(FileId) GROUP BY `.theta` ) SELECT `.theta`, lo, hi, CountN, OnTime, OnTime/CountN AS ZdWeight, (OnTime/CountN)*SQRT(POW(1/300, 2) + 1/CountN) AS ErrZdWeight FROM ObservationTime LEFT JOIN EventCount USING(`.theta`) LEFT JOIN BinningTheta ON `.theta`=bin ORDER BY `.theta` )

`%0:bins` is a placeholder for the bin boundaries, e.g. `5, 10, 15, 20, 25, 30` (five bins between 5° and 30° plus underflow and overflow). It should be identical to the binning used for data files in DataFiles.

### Analysis Query

The analysis of Data and MontaCarlo files must be done totally identical to produce reasonable results. Therefore, the exact same query (or code) should be used for the analysis of both. In this example, the query has to provide two columns, `Weight` and `LogEnergyEst` for all gamma-line events. The weight must be +1 for events in the on-region and -0.2 for an event in the off-region (corresponding to the number of five wobble positions). LogEnergyEst must contain \(\log_{10}\) of the estimated energy for each event. Only events surviving background suppression and spatial (theta) cuts should be considered.

WITH Table0 AS ( SELECT %0:columns -- this could be removed if we can join events via the same columns (without CorsikaNumResuse) Weight, Size, NumUsedPixels, NumIslands, Leakage1, MeanX, MeanY, CosDelta, SinDelta, M3Long, SlopeLong, Width/Length AS WdivL, PI()*Width*Length AS Area, cosa*X - sina*Y AS PX, cosa*Y + sina*X AS PY FROM %1:files LEFT JOIN %2:runinfo USING (FileId) LEFT JOIN %3:events USING (FileId) -- This could be replaced by a user uploaded temporary table LEFT JOIN %4:positions USING (FileId, EvtNumber) CROSS JOIN Wobble WHERE NumUsedPixels>5.5 AND NumIslands<3.5 AND Leakage1<0.1 ), Table1 AS ( SELECT %0:columns Weight, Size, CosDelta, SinDelta, M3Long, SlopeLong, Leakage1, WdivL, MeanX - PX/1.02e0 AS DX, MeanY - PY/1.02e0 AS DY FROM Table0 WHERE Area < LOG10(Size)*898e0 - 1535e0 ), Table2 AS ( SELECT %0:columns Weight, Size, CosDelta, SinDelta, DX, DY, M3Long, SlopeLong, Leakage1, WdivL, SQRT(DX*DX + DY*DY) AS Norm FROM Table1 ), Table3 AS ( SELECT %0:columns Weight, Size, M3Long, SlopeLong, Leakage1, WdivL, Norm, LEAST(GREATEST((CosDelta*DY - SinDelta*DX)/Norm, -1), 1) AS LX, SIGN(CosDelta*DX + SinDelta*DY) AS Sign FROM Table2 ), Table5 AS ( SELECT %0:columns Weight, Size, Leakage1, WdivL, LX, Norm *0.0117193246260285378e0 AS Dist, M3Long *Sign*0.0117193246260285378e0 AS M3L, SlopeLong*Sign/0.0117193246260285378e0 AS Slope FROM Table3 ), Table6 AS ( SELECT %0:columns Weight, Size, WdivL, Dist, LX, M3L, Slope, 1.39252e0 + 0.154247e0*Slope + 1.67972e0*(1-1/(1+4.86232e0*Leakage1)) AS Xi FROM Table5 ), Table7 AS ( SELECT %0:columns Weight, Size, Dist, LX, IF (M3L<-0.07 OR (Dist-0.5e0)*7.2e0-Slope<0, -Xi, Xi) * (1-WdivL) AS Disp FROM Table6 ) SELECT %0:columns Weight, (Disp*Disp + Dist*Dist - 2*Disp*Dist*SQRT(1-LX*LX)) AS ThetaSq, %5:estimator AS LogEnergyEst FROM Table7 HAVING ThetaSq<0.024

The placeholder `%0:column` is currently used to request MonteCarlo true values (such as Energy) in addition to the anaylsis values only for the analysis of simulated data. The must be no comma behind! In additon, `%1:files` is a placeholder for the table containing the `FileId`s to analyze, `%2:runinfo` for the table with the run info data, `%3:events` for the table with the image parameters and `%4:positions` for the table with the source positions in the camera.

### Analyze Data

### Analyze Monte Carlo Data

### Summarize Corsika Production

CREATE TEMPORARY TABLE SimulatedSpectrum ( `.energy` SMALLINT UNSIGNED NOT NULL COMMENT 'Bin Index [MC Energy]', CountN DOUBLE NOT NULL, CountW DOUBLE NOT NULL, CountW2 DOUBLE NOT NULL, PRIMARY KEY (`.energy`) USING HASH ) ENGINE=Memory AS ( SELECT INTERVAL(LOG10(Energy), %0:energyest) AS `.energy`, COUNT(*) AS CountN, SUM( (%2:spectrum)/pow(Energy, SpectralIndex) ) AS CountW, SUM(POW((%2:spectrum)/pow(Energy, SpectralIndex),2)) AS CountW2 FROM MonteCarloFiles LEFT JOIN factmc.RunInfoMC USING (FileId) LEFT JOIN factmc.OriginalMC USING (FileId) GROUP BY `.energy` ORDER BY `.energy` )