*4.2. Multivariate Real Quantity*

To illustrate the ability of DCCs to present information for a multivariate real quantity, consider the example of simultaneous measurement of resistance and reactance from clause 9.4 of GUMS2 [11]. In this example, the resistance and reactance of a circuit element are determined by measuring the amplitude of a sinusoidally-alternating potential difference across its terminals, the amplitude of the alternating current passing through it, and the phase angle of the alternating potential difference relative to the alternating current. There are three output quantities: the resistance, the reactance and the impedance.

For this paper, the calculation stage has been implemented using a fixed number *M* = 10<sup>6</sup> of trials. Using the component structure of Table 8, summary information from the calculation stage can be encapsulated as follows:

```
<!-- MCM, 1e6 samples - Summary information -->
<si:list>
  <si:listUnit>\ohm</si:listUnit>
  <si:real>
    <si:value>127.732</si:value>
  </si:real>
  <si:real>
    <si:value>219.847</si:value>
  </si:real>
```

```
<si:real>
    <si:value>254.260</si:value>
  </si:real>
  <si:ellipsoidalRegion>
    <si:covarianceMatrix>
      <si:column>
        <si:value>0.003364</si:value>
        <si:unit>\ohm\ohm</si:unit>
        <si:value>-0.04090216</si:value>
        <si:unit>\ohm\ohm</si:unit>
        <si:value>-0.02688114</si:value>
        <si:unit>\ohm\ohm</si:unit>
      </si:column>
      <si:column>
        <si:value>-0.04090216</si:value>
        <si:unit>\ohm\ohm</si:unit>
        <si:value>0.058081</si:value>
        <si:unit>\ohm\ohm</si:unit>
        <si:value>0.2281734312</si:value>
        <si:unit>\ohm\ohm</si:unit>
      </si:column>
      <si:column>
        <si:value>-0.02688114</si:value>
        <si:unit>\ohm\ohm</si:unit>
        <si:value>0.2281734312</si:value>
        <si:unit>\ohm\ohm</si:unit>
        <si:value>0.037249</si:value>
        <si:unit>\ohm\ohm</si:unit>
      </si:column>
    </si:covarianceMatrix>
    <si:coverageFactor>2.80</si:coverageFactor>
    <si:coverageProbability>0.95</si:coverageProbability>
  </si:ellipsoidalRegion>
</si:list>
```
Using the component structure of Table 9, the full set of values **y***k*, *k* = 1, ... , *M*, of the output quantity returned by the Monte Carlo calculation can be encapsulated as follows, showing only the first (*k* = 1) and final (*k* = *M*) values:

```
<!-- MCM, 1e6 samples - Output quantity values -->
<si:list>
  <si:list>
    <si:listUnit>\ohm</si:listUnit>
    <si:real>
      <si:value>127.763</si:value>
    </si:real>
    <si:real>
      <si:value>219.303</si:value>
    </si:real>
    <si:real>
      <si:value>254.397</si:value>
    </si:real>
  </si:list>
    ...
  <si:list>
```

```
<si:listUnit>\ohm</si:listUnit>
    <si:real>
      <si:value>127.838</si:value>
    </si:real>
    <si:real>
      <si:value>220.055</si:value>
    </si:real>
    <si:real>
      <si:value>253.699</si:value>
    </si:real>
  </si:list>
</si:list>
```
When written to file (without any spaces or indentation), the full set of values in the format above takes up approximately 194 MB (and requires 12*M* + 2 lines).
