Easy error propagation in R
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In a previous post I demonstrated how to use R’s simple built-in symbolic engine to generate Jacobian and (pseudo)-Hessian matrices that make non-linear optimization perform much more efficiently. Another related application is Gaussian error propagation.
Say you have data from a set of measurements in variables x
and y
where you know the corresponding measurement errors (dx
and dy
, typically the standard deviation or error from a set of replicates or a calibration curve). Next you want to create a derived value defined by an arbitrary function z = f(x,y)
. What would the corresponding error in value of z
, i.e. dz = df
, be?
If the function f(x,y)
is a simple sum or product, their are simple equations for determining df
. However, if f(x,y)
is something more complex, like:
you’ll need to use a bit of calculus, specifically the chain rule:
Applying the above equation allows for the derivation of Gaussian error propagation for any arbitrary function. So how does one do this in R? Again, the D()
function and R expression()
objects come to our rescue.
Say the definition of z (ergo f(x,y)
) is defined in an R formula
:
<span class="pun">></span><span class="pln"> f </span><span class="pun">=</span><span class="pln"> z </span><span class="pun">~</span><span class="pln"> </span><span class="pun">(</span><span class="pln">x</span><span class="pun">-</span><span class="pln">y</span><span class="pun">)/(</span><span class="pln">x</span><span class="pun">+</span><span class="pln">y</span><span class="pun">)^</span><span class="lit">2</span>
If you probe the structure of a formula object you get:
<span class="pun">></span><span class="pln"> str</span><span class="pun">(</span><span class="pln">f</span><span class="pun">)</span><span class="pln"><br /></span><span class="typ">Class</span><span class="pln"> </span><span class="str">'formula'</span><span class="pln"> length </span><span class="lit">3</span><span class="pln"> z </span><span class="pun">~</span><span class="pln"> </span><span class="pun">(</span><span class="pln">x </span><span class="pun">-</span><span class="pln"> y</span><span class="pun">)/(</span><span class="pln">x </span><span class="pun">+</span><span class="pln"> y</span><span class="pun">)^</span><span class="lit">2</span><span class="pln"><br /> </span><span class="pun">..-</span><span class="pln"> attr</span><span class="pun">(*,</span><span class="pln"> </span><span class="str">".Environment"</span><span class="pun">)=<</span><span class="pln">environment</span><span class="pun">:</span><span class="pln"> R_GlobalEnv</span><span class="pun">></span>
What’s key is the “length 3
” bit:
<span class="pun">></span><span class="pln"> f</span><span class="pun">[[</span><span class="lit">1</span><span class="pun">]];</span><span class="pln"> f</span><span class="pun">[[</span><span class="lit">2</span><span class="pun">]];</span><span class="pln"> f</span><span class="pun">[[</span><span class="lit">3</span><span class="pun">]]</span><span class="pln"><br /></span><span class="str">`~`</span><span class="pln"><br />z<br /></span><span class="pun">(</span><span class="pln">x </span><span class="pun">-</span><span class="pln"> y</span><span class="pun">)/(</span><span class="pln">x </span><span class="pun">+</span><span class="pln"> y</span><span class="pun">)^</span><span class="lit">2</span>
The code above shows us that a formula
object can be subsetted into its constituent parts:
- the formula operator:
~
- the left-hand side (LHS) of the formula:
z
- the right-hand side (RHS) of the formula:
(x - y)/(x + y)^2
The class()
of the RHS is a call
, which is close enough to an R expression
that both all.vars()
and D()
work as expected to generate the mathematical expressions for the partial derivatives with respect to each variable:
<span class="pun">></span><span class="pln"> all</span><span class="pun">.</span><span class="pln">vars</span><span class="pun">(</span><span class="pln">f</span><span class="pun">[[</span><span class="lit">3</span><span class="pun">]])</span><span class="pln"><br /></span><span class="pun">[</span><span class="lit">1</span><span class="pun">]</span><span class="pln"> </span><span class="str">"x"</span><span class="pln"> </span><span class="str">"y"</span><span class="pln"><br /><br /></span><span class="pun">></span><span class="pln"> lapply</span><span class="pun">(</span><span class="pln">all</span><span class="pun">.</span><span class="pln">vars</span><span class="pun">(</span><span class="pln">f</span><span class="pun">[[</span><span class="lit">3</span><span class="pun">]]),</span><span class="pln"> </span><span class="kwd">function</span><span class="pun">(</span><span class="pln">v</span><span class="pun">)</span><span class="pln"> D</span><span class="pun">(</span><span class="pln">f</span><span class="pun">[[</span><span class="lit">3</span><span class="pun">]],</span><span class="pln"> v</span><span class="pun">))</span><span class="pln"><br /></span><span class="pun">[[</span><span class="lit">1</span><span class="pun">]]</span><span class="pln"><br /></span><span class="lit">1</span><span class="pun">/(</span><span class="pln">x </span><span class="pun">+</span><span class="pln"> y</span><span class="pun">)^</span><span class="lit">2</span><span class="pln"> </span><span class="pun">-</span><span class="pln"> </span><span class="pun">(</span><span class="pln">x </span><span class="pun">-</span><span class="pln"> y</span><span class="pun">)</span><span class="pln"> </span><span class="pun">*</span><span class="pln"> </span><span class="pun">(</span><span class="lit">2</span><span class="pln"> </span><span class="pun">*</span><span class="pln"> </span><span class="pun">(</span><span class="pln">x </span><span class="pun">+</span><span class="pln"> y</span><span class="pun">))/((</span><span class="pln">x </span><span class="pun">+</span><span class="pln"> y</span><span class="pun">)^</span><span class="lit">2</span><span class="pun">)^</span><span class="lit">2</span><span class="pln"><br /><br /></span><span class="pun">[[</span><span class="lit">2</span><span class="pun">]]</span><span class="pln"><br /></span><span class="pun">-(</span><span class="lit">1</span><span class="pun">/(</span><span class="pln">x </span><span class="pun">+</span><span class="pln"> y</span><span class="pun">)^</span><span class="lit">2</span><span class="pln"> </span><span class="pun">+</span><span class="pln"> </span><span class="pun">(</span><span class="pln">x </span><span class="pun">-</span><span class="pln"> y</span><span class="pun">)</span><span class="pln"> </span><span class="pun">*</span><span class="pln"> </span><span class="pun">(</span><span class="lit">2</span><span class="pln"> </span><span class="pun">*</span><span class="pln"> </span><span class="pun">(</span><span class="pln">x </span><span class="pun">+</span><span class="pln"> y</span><span class="pun">))/((</span><span class="pln">x </span><span class="pun">+</span><span class="pln"> y</span><span class="pun">)^</span><span class="lit">2</span><span class="pun">)^</span><span class="lit">2</span><span class="pun">)</span>
These expressions need to be modified a bit – i.e. in this case they need to be multiplied by dx
and dy
, respectively and then squared. What’s returned from D()
is a call
object, so the elements above need to be converted to character
to manipulate them accordingly. This is done with deparse()
.
<span class="pun">></span><span class="pln"> lapply</span><span class="pun">(</span><span class="pln">all</span><span class="pun">.</span><span class="pln">vars</span><span class="pun">(</span><span class="pln">f</span><span class="pun">[[</span><span class="lit">3</span><span class="pun">]]),</span><span class="pln"> </span><span class="kwd">function</span><span class="pun">(</span><span class="pln">v</span><span class="pun">)</span><span class="pln"> deparse</span><span class="pun">(</span><span class="pln">D</span><span class="pun">(</span><span class="pln">f</span><span class="pun">[[</span><span class="lit">3</span><span class="pun">]],</span><span class="pln"> v</span><span class="pun">)))</span><span class="pln"><br /></span><span class="pun">[[</span><span class="lit">1</span><span class="pun">]]</span><span class="pln"><br /></span><span class="pun">[</span><span class="lit">1</span><span class="pun">]</span><span class="pln"> </span><span class="str">"1/(x + y)^2 - (x - y) * (2 * (x + y))/((x + y)^2)^2"</span><span class="pln"><br /><br /></span><span class="pun">[[</span><span class="lit">2</span><span class="pun">]]</span><span class="pln"><br /></span><span class="pun">[</span><span class="lit">1</span><span class="pun">]</span><span class="pln"> </span><span class="str">"-(1/(x + y)^2 + (x - y) * (2 * (x + y))/((x + y)^2)^2)"</span>
The final error propagation expression is created with a bit of string manipulation:
<span class="pun">></span><span class="pln"> sprintf</span><span class="pun">(</span><span class="str">'sqrt(%s)'</span><span class="pun">,</span><span class="pln"> <br /> paste</span><span class="pun">(</span><span class="pln"><br /> sapply</span><span class="pun">(</span><span class="pln">all</span><span class="pun">.</span><span class="pln">vars</span><span class="pun">(</span><span class="pln">f</span><span class="pun">[[</span><span class="lit">3</span><span class="pun">]]),</span><span class="pln"> </span><span class="kwd">function</span><span class="pun">(</span><span class="pln">v</span><span class="pun">)</span><span class="pln"> </span><span class="pun">{</span><span class="pln"><br /> sprintf</span><span class="pun">(</span><span class="str">'(d%s*(%s))^2'</span><span class="pun">,</span><span class="pln"> v</span><span class="pun">,</span><span class="pln"> deparse</span><span class="pun">(</span><span class="pln">D</span><span class="pun">(</span><span class="pln">f</span><span class="pun">[[</span><span class="lit">3</span><span class="pun">]],</span><span class="pln"> v</span><span class="pun">)))</span><span class="pln"><br /> </span><span class="pun">}),</span><span class="pln"> <br /> collapse</span><span class="pun">=</span><span class="str">'+'</span><span class="pln"><br /> </span><span class="pun">)</span><span class="pln"><br /> </span><span class="pun">)</span><span class="pln"><br /></span><span class="pun">[</span><span class="lit">1</span><span class="pun">]</span><span class="pln"> </span><span class="str">"sqrt((dx*(1/(x + y)^2 - (x - y) * (2 * (x + y))/((x + y)^2)^2))^2+(dy*(-(1/(x + y)^2 + (x - y) * (2 * (x + y))/((x + y)^2)^2)))^2)"</span>
Now that we’ve got the basics down, let’s test this out with some data …
<span class="pun">></span><span class="pln"> </span><span class="kwd">set</span><span class="pun">.</span><span class="pln">seed</span><span class="pun">(</span><span class="lit">0</span><span class="pun">)</span><span class="pln"><br /></span><span class="pun">></span><span class="pln"> data </span><span class="pun">=</span><span class="pln"> data</span><span class="pun">.</span><span class="pln">frame</span><span class="pun">(</span><span class="pln"><br /> x </span><span class="pun">=</span><span class="pln"> runif</span><span class="pun">(</span><span class="lit">5</span><span class="pun">),</span><span class="pln"> <br /> y </span><span class="pun">=</span><span class="pln"> runif</span><span class="pun">(</span><span class="lit">5</span><span class="pun">),</span><span class="pln"> <br /> dx </span><span class="pun">=</span><span class="pln"> runif</span><span class="pun">(</span><span class="lit">5</span><span class="pun">)/</span><span class="lit">10</span><span class="pun">,</span><span class="pln"> <br /> dy </span><span class="pun">=</span><span class="pln"> runif</span><span class="pun">(</span><span class="lit">5</span><span class="pun">)/</span><span class="lit">10</span><span class="pln"><br /> </span><span class="pun">)</span><span class="pln"><br /></span><span class="pun">></span><span class="pln"> data<br /> x y dx dy<br /></span><span class="lit">1</span><span class="pln"> </span><span class="lit">0.8966972</span><span class="pln"> </span><span class="lit">0.2016819</span><span class="pln"> </span><span class="lit">0.006178627</span><span class="pln"> </span><span class="lit">0.07698414</span><span class="pln"><br /></span><span class="lit">2</span><span class="pln"> </span><span class="lit">0.2655087</span><span class="pln"> </span><span class="lit">0.8983897</span><span class="pln"> </span><span class="lit">0.020597457</span><span class="pln"> </span><span class="lit">0.04976992</span><span class="pln"><br /></span><span class="lit">3</span><span class="pln"> </span><span class="lit">0.3721239</span><span class="pln"> </span><span class="lit">0.9446753</span><span class="pln"> </span><span class="lit">0.017655675</span><span class="pln"> </span><span class="lit">0.07176185</span><span class="pln"><br /></span><span class="lit">4</span><span class="pln"> </span><span class="lit">0.5728534</span><span class="pln"> </span><span class="lit">0.6607978</span><span class="pln"> </span><span class="lit">0.068702285</span><span class="pln"> </span><span class="lit">0.09919061</span><span class="pln"><br /></span><span class="lit">5</span><span class="pln"> </span><span class="lit">0.9082078</span><span class="pln"> </span><span class="lit">0.6291140</span><span class="pln"> </span><span class="lit">0.038410372</span><span class="pln"> </span><span class="lit">0.03800352</span>
and with a little help from dplyr
:
<span class="pun">></span><span class="pln"> library</span><span class="pun">(</span><span class="pln">dplyr</span><span class="pun">)</span><span class="pln"><br /></span><span class="pun">></span><span class="pln"> data </span><span class="pun">%>%</span><span class="pln"><br /></span><span class="pun">+</span><span class="pln"> mutate_</span><span class="pun">(.</span><span class="pln">dots</span><span class="pun">=</span><span class="pln">list</span><span class="pun">(</span><span class="pln"><br /></span><span class="pun">+</span><span class="pln"> </span><span class="com"># compute derived value</span><span class="pln"><br /></span><span class="pun">+</span><span class="pln"> z </span><span class="pun">=</span><span class="pln"> deparse</span><span class="pun">(</span><span class="pln">f</span><span class="pun">[[</span><span class="lit">3</span><span class="pun">]]),</span><span class="pln"><br /></span><span class="pun">+</span><span class="pln"> <br /></span><span class="pun">+</span><span class="pln"> </span><span class="com"># generates a mathematical expression to compute dz</span><span class="pln"><br /></span><span class="pun">+</span><span class="pln"> </span><span class="com"># as a character string</span><span class="pln"><br /></span><span class="pun">+</span><span class="pln"> dz </span><span class="pun">=</span><span class="pln"> sapply</span><span class="pun">(</span><span class="pln">all</span><span class="pun">.</span><span class="pln">vars</span><span class="pun">(</span><span class="pln">f</span><span class="pun">[[</span><span class="lit">3</span><span class="pun">]]),</span><span class="pln"> </span><span class="kwd">function</span><span class="pun">(</span><span class="pln">v</span><span class="pun">)</span><span class="pln"> </span><span class="pun">{</span><span class="pln"><br /></span><span class="pun">+</span><span class="pln"> dfdp </span><span class="pun">=</span><span class="pln"> deparse</span><span class="pun">(</span><span class="pln">D</span><span class="pun">(</span><span class="pln">f</span><span class="pun">[[</span><span class="lit">3</span><span class="pun">]],</span><span class="pln"> v</span><span class="pun">))</span><span class="pln"><br /></span><span class="pun">+</span><span class="pln"> sprintf</span><span class="pun">(</span><span class="str">'(d%s*(%s))^2'</span><span class="pun">,</span><span class="pln"> v</span><span class="pun">,</span><span class="pln"> dfdp</span><span class="pun">)</span><span class="pln"><br /></span><span class="pun">+</span><span class="pln"> </span><span class="pun">})</span><span class="pln"> </span><span class="pun">%>%</span><span class="pln"><br /></span><span class="pun">+</span><span class="pln"> paste</span><span class="pun">(</span><span class="pln">collapse</span><span class="pun">=</span><span class="str">'+'</span><span class="pun">)</span><span class="pln"> </span><span class="pun">%>%</span><span class="pln"><br /></span><span class="pun">+</span><span class="pln"> sprintf</span><span class="pun">(</span><span class="str">'sqrt(%s)'</span><span class="pun">,</span><span class="pln"> </span><span class="pun">.)</span><span class="pln"><br /></span><span class="pun">+</span><span class="pln"> </span><span class="pun">))</span><span class="pln"><br /> x y dx dy z dz<br /></span><span class="lit">1</span><span class="pln"> </span><span class="lit">0.8966972</span><span class="pln"> </span><span class="lit">0.2016819</span><span class="pln"> </span><span class="lit">0.006178627</span><span class="pln"> </span><span class="lit">0.07698414</span><span class="pln"> </span><span class="lit">0.57608929</span><span class="pln"> </span><span class="lit">0.14457245</span><span class="pln"><br /></span><span class="lit">2</span><span class="pln"> </span><span class="lit">0.2655087</span><span class="pln"> </span><span class="lit">0.8983897</span><span class="pln"> </span><span class="lit">0.020597457</span><span class="pln"> </span><span class="lit">0.04976992</span><span class="pln"> </span><span class="pun">-</span><span class="lit">0.46718831</span><span class="pln"> </span><span class="lit">0.03190297</span><span class="pln"><br /></span><span class="lit">3</span><span class="pln"> </span><span class="lit">0.3721239</span><span class="pln"> </span><span class="lit">0.9446753</span><span class="pln"> </span><span class="lit">0.017655675</span><span class="pln"> </span><span class="lit">0.07176185</span><span class="pln"> </span><span class="pun">-</span><span class="lit">0.33019871</span><span class="pln"> </span><span class="lit">0.01978697</span><span class="pln"><br /></span><span class="lit">4</span><span class="pln"> </span><span class="lit">0.5728534</span><span class="pln"> </span><span class="lit">0.6607978</span><span class="pln"> </span><span class="lit">0.068702285</span><span class="pln"> </span><span class="lit">0.09919061</span><span class="pln"> </span><span class="pun">-</span><span class="lit">0.05778613</span><span class="pln"> </span><span class="lit">0.07604809</span><span class="pln"><br /></span><span class="lit">5</span><span class="pln"> </span><span class="lit">0.9082078</span><span class="pln"> </span><span class="lit">0.6291140</span><span class="pln"> </span><span class="lit">0.038410372</span><span class="pln"> </span><span class="lit">0.03800352</span><span class="pln"> </span><span class="lit">0.11809201</span><span class="pln"> </span><span class="lit">0.02424023</span>
Taking this a step further, this method can be wrapped in a chainable function that determines the name of new variables from the LHS of a formula argument:
<span class="pln">mutate_with_error </span><span class="pun">=</span><span class="pln"> </span><span class="kwd">function</span><span class="pun">(.</span><span class="pln">data</span><span class="pun">,</span><span class="pln"> f</span><span class="pun">)</span><span class="pln"> </span><span class="pun">{</span><span class="pln"><br /> exprs </span><span class="pun">=</span><span class="pln"> list</span><span class="pun">(</span><span class="pln"><br /> </span><span class="com"># expression to compute new variable values</span><span class="pln"><br /> deparse</span><span class="pun">(</span><span class="pln">f</span><span class="pun">[[</span><span class="lit">3</span><span class="pun">]]),</span><span class="pln"><br /><br /> </span><span class="com"># expression to compute new variable errors</span><span class="pln"><br /> sapply</span><span class="pun">(</span><span class="pln">all</span><span class="pun">.</span><span class="pln">vars</span><span class="pun">(</span><span class="pln">f</span><span class="pun">[[</span><span class="lit">3</span><span class="pun">]]),</span><span class="pln"> </span><span class="kwd">function</span><span class="pun">(</span><span class="pln">v</span><span class="pun">)</span><span class="pln"> </span><span class="pun">{</span><span class="pln"><br /> dfdp </span><span class="pun">=</span><span class="pln"> deparse</span><span class="pun">(</span><span class="pln">D</span><span class="pun">(</span><span class="pln">f</span><span class="pun">[[</span><span class="lit">3</span><span class="pun">]],</span><span class="pln"> v</span><span class="pun">))</span><span class="pln"><br /> sprintf</span><span class="pun">(</span><span class="str">'(d%s*(%s))^2'</span><span class="pun">,</span><span class="pln"> v</span><span class="pun">,</span><span class="pln"> dfdp</span><span class="pun">)</span><span class="pln"><br /> </span><span class="pun">})</span><span class="pln"> </span><span class="pun">%>%</span><span class="pln"><br /> paste</span><span class="pun">(</span><span class="pln">collapse</span><span class="pun">=</span><span class="str">'+'</span><span class="pun">)</span><span class="pln"> </span><span class="pun">%>%</span><span class="pln"><br /> sprintf</span><span class="pun">(</span><span class="str">'sqrt(%s)'</span><span class="pun">,</span><span class="pln"> </span><span class="pun">.)</span><span class="pln"><br /> </span><span class="pun">)</span><span class="pln"><br /> names</span><span class="pun">(</span><span class="pln">exprs</span><span class="pun">)</span><span class="pln"> </span><span class="pun">=</span><span class="pln"> c</span><span class="pun">(</span><span class="pln"><br /> deparse</span><span class="pun">(</span><span class="pln">f</span><span class="pun">[[</span><span class="lit">2</span><span class="pun">]]),</span><span class="pln"><br /> sprintf</span><span class="pun">(</span><span class="str">'d%s'</span><span class="pun">,</span><span class="pln"> deparse</span><span class="pun">(</span><span class="pln">f</span><span class="pun">[[</span><span class="lit">2</span><span class="pun">]]))</span><span class="pln"><br /> </span><span class="pun">)</span><span class="pln"><br /><br /> </span><span class="pun">.</span><span class="pln">data </span><span class="pun">%>%</span><span class="pln"><br /> </span><span class="com"># the standard evaluation alternative of mutate()</span><span class="pln"><br /> mutate_</span><span class="pun">(.</span><span class="pln">dots</span><span class="pun">=</span><span class="pln">exprs</span><span class="pun">)</span><span class="pln"><br /></span><span class="pun">}</span>
Thus, adding new derived variables and propagating errors accordingly becomes relatively easy:
<span class="pun">></span><span class="pln"> </span><span class="kwd">set</span><span class="pun">.</span><span class="pln">seed</span><span class="pun">(</span><span class="lit">0</span><span class="pun">)</span><span class="pln"><br /></span><span class="pun">></span><span class="pln"> data </span><span class="pun">=</span><span class="pln"> data</span><span class="pun">.</span><span class="pln">frame</span><span class="pun">(</span><span class="pln">x</span><span class="pun">=</span><span class="pln">runif</span><span class="pun">(</span><span class="lit">5</span><span class="pun">),</span><span class="pln"> y</span><span class="pun">=</span><span class="pln">runif</span><span class="pun">(</span><span class="lit">5</span><span class="pun">),</span><span class="pln"> dx</span><span class="pun">=</span><span class="pln">runif</span><span class="pun">(</span><span class="lit">5</span><span class="pun">)/</span><span class="lit">10</span><span class="pun">,</span><span class="pln"> dy</span><span class="pun">=</span><span class="pln">runif</span><span class="pun">(</span><span class="lit">5</span><span class="pun">)/</span><span class="lit">10</span><span class="pun">)</span><span class="pln"><br /></span><span class="pun">></span><span class="pln"> data<br /> x y dx dy<br /></span><span class="lit">1</span><span class="pln"> </span><span class="lit">0.8966972</span><span class="pln"> </span><span class="lit">0.2016819</span><span class="pln"> </span><span class="lit">0.006178627</span><span class="pln"> </span><span class="lit">0.07698414</span><span class="pln"><br /></span><span class="lit">2</span><span class="pln"> </span><span class="lit">0.2655087</span><span class="pln"> </span><span class="lit">0.8983897</span><span class="pln"> </span><span class="lit">0.020597457</span><span class="pln"> </span><span class="lit">0.04976992</span><span class="pln"><br /></span><span class="lit">3</span><span class="pln"> </span><span class="lit">0.3721239</span><span class="pln"> </span><span class="lit">0.9446753</span><span class="pln"> </span><span class="lit">0.017655675</span><span class="pln"> </span><span class="lit">0.07176185</span><span class="pln"><br /></span><span class="lit">4</span><span class="pln"> </span><span class="lit">0.5728534</span><span class="pln"> </span><span class="lit">0.6607978</span><span class="pln"> </span><span class="lit">0.068702285</span><span class="pln"> </span><span class="lit">0.09919061</span><span class="pln"><br /></span><span class="lit">5</span><span class="pln"> </span><span class="lit">0.9082078</span><span class="pln"> </span><span class="lit">0.6291140</span><span class="pln"> </span><span class="lit">0.038410372</span><span class="pln"> </span><span class="lit">0.03800352</span><span class="pln"><br /><br /></span><span class="pun">></span><span class="pln"> data </span><span class="pun">%>%</span><span class="pln"> mutate_with_error</span><span class="pun">(</span><span class="pln">z </span><span class="pun">~</span><span class="pln"> </span><span class="pun">(</span><span class="pln">x</span><span class="pun">-</span><span class="pln">y</span><span class="pun">)/(</span><span class="pln">x</span><span class="pun">+</span><span class="pln">y</span><span class="pun">)^</span><span class="lit">2</span><span class="pun">)</span><span class="pln"><br /> x y dx dy z dz<br /></span><span class="lit">1</span><span class="pln"> </span><span class="lit">0.8966972</span><span class="pln"> </span><span class="lit">0.2016819</span><span class="pln"> </span><span class="lit">0.006178627</span><span class="pln"> </span><span class="lit">0.07698414</span><span class="pln"> </span><span class="lit">0.57608929</span><span class="pln"> </span><span class="lit">0.14457245</span><span class="pln"><br /></span><span class="lit">2</span><span class="pln"> </span><span class="lit">0.2655087</span><span class="pln"> </span><span class="lit">0.8983897</span><span class="pln"> </span><span class="lit">0.020597457</span><span class="pln"> </span><span class="lit">0.04976992</span><span class="pln"> </span><span class="pun">-</span><span class="lit">0.46718831</span><span class="pln"> </span><span class="lit">0.03190297</span><span class="pln"><br /></span><span class="lit">3</span><span class="pln"> </span><span class="lit">0.3721239</span><span class="pln"> </span><span class="lit">0.9446753</span><span class="pln"> </span><span class="lit">0.017655675</span><span class="pln"> </span><span class="lit">0.07176185</span><span class="pln"> </span><span class="pun">-</span><span class="lit">0.33019871</span><span class="pln"> </span><span class="lit">0.01978697</span><span class="pln"><br /></span><span class="lit">4</span><span class="pln"> </span><span class="lit">0.5728534</span><span class="pln"> </span><span class="lit">0.6607978</span><span class="pln"> </span><span class="lit">0.068702285</span><span class="pln"> </span><span class="lit">0.09919061</span><span class="pln"> </span><span class="pun">-</span><span class="lit">0.05778613</span><span class="pln"> </span><span class="lit">0.07604809</span><span class="pln"><br /></span><span class="lit">5</span><span class="pln"> </span><span class="lit">0.9082078</span><span class="pln"> </span><span class="lit">0.6291140</span><span class="pln"> </span><span class="lit">0.038410372</span><span class="pln"> </span><span class="lit">0.03800352</span><span class="pln"> </span><span class="lit">0.11809201</span><span class="pln"> </span><span class="lit">0.02424023</span>
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