Category Archives: Error Propagation

Calculating the Sensitivity of a Transfer Function to Independant Variables Using the Taylor Series

In this post we discussed the use of the Taylor Series to evaluate the propagated uncertainties in measurements to a calculated result. We can also use a similar approach to determine the sensitivity of a transfer function to know perturbations in the independent variables.

Frustum_750In this case we will look at the volume of a right conical frustum or a truncated cone. The volume can be calculated from the three variables: , and shown in the figure. The volume is given as

The first step is to calculate the partial derivatives of with respect to the independent variables.

The total uncertainty in the volume, , is given in terms of the uncertainties in the independent variables, , and .

Each term of the form is the contribution to the total perturbation of the function by the perturbation of the independent variable, . Therefore we can calculate the sensitivities, , of the volume to small changes in each of the three variables , and , by calculating the percentage contribution of each term to the total perturbation. Typically, we wish to also define the direction that the pertubations in the independent variables will affect the total, so we remove the absolute values and evaluate the partial derivatives while maintaining their signs.

It is also evident that .


Example:

Suppose we have a design for a conical frustum with nominal dimensions of 0.500 in, 0.375 in and 1.250 in. Our manufacturing process can hold to ±.002 in, to ± .007 in and to ± .010 in. We then have the following inputs to our formulae.

From this we can calculate the nominal volume of our manufactured part.

Next we can calculate the partial derivatives.

And having the partial derivatives, we can calculate the total propagated uncertainty in the nominal volume.

From this we can obtain the predicted volume of the manufactured part subject to the manufacturing tolerances: 0.757 ± .021 in3. We can then calculate the relative sensitivity of the total volume to the manufacturing tolerances specified.

It is important to note that these results are only valid for the initial conditions specified. If the nominal values for the dimensions or the tolerances change, both the uncertainties and sensitivities will change.


Caveat lector — All work and ideas presented here may not be accurate and should be verified before application.

Basic Error Propagation Through the Use of Taylor Series

In courses on experimentation, propagated errors are typically treated through the use of a Taylor series expansion to evaluate the total contribution of individual measurement uncertainties to a final calculated result. As an example, suppose we wish to experimentally determine the acceleration of a body due to gravity. We could take an object and drop it a measured distance while recording the elapsed time. From basic physics we know that the distance traveled is proportional to the time squared and that the proportionality constant is or,

Solving for the acceleration, we obtain,

which is a function of two measured variables, the distance travelled and the elapsed time. Both of these measurements, no matter how carefully obtained, will have some uncertainty. Suppose we measure the distance travelled with a ruler that has graduations every 1 inches and the time with a stopwatch with a resolution of 0.1 seconds. With both of these instruments it is evident that we cannot measure the quantity to a higher resolution than the instrument provides, therefore it is typical to take the total uncertainty in the measurement as the least significant digit in the scale, centered on the measurement value. This would equate to uncertainties in the measurements of ± .5 inches and ± .05 seconds.

Let’s say we dropped the object from a height of 36 feet and measured the elapsed time as 1.5 seconds. From the above equation we would find the acceleration to be 32 . But how carefully did we measure? Was the distance exactly 36 feet (432 inches), or was it 432.3 inches? Was the time 1.48 seconds? As long as these uncertainties are within the predefined ranges established above, we can calculate the total uncertainty in the measurement of the acceleration.

In this post we discussed the approximation of any function by a Taylor series expansion about a specific point. We can apply that technique to determine how the value of the acceleration may vary with perturbations in the input values of time and distance about the measured point. For the simplest implementation, we restrict ourselves to the first order terms of the expansion[1].

Recal that the Taylor series of a function about the point is given as

Expanding this to the first order terms yields

rewriting as

We can now see that the left side of the equation evaluates to the change in the function corresponding to a perturbation of and by a small amount and . Examining the partial derivative terms, we can see that we are multiplying the rate of change of the function in a single variable to a change in that variable from the interested point . Since we are interested in small perturbations of and about the point , we will denote these changes and . The change of the function under these perturbations we will denote .

Substituting we obtain

Lastly, since there should be no preference for the uncertainty to be in the positive or negative direction, we take the absolute value of the derivative terms and require that our perturbations be defined as positive,

This can be generalized to a function in any number of variables as

Returning to our example, to find the total uncertainty in the calculated acceleration, we simply need to determine the partial derivatives of the function in the independent variables,

and insert them into our formula

Evaluating with our collected data

or

Therefore our calculated acceleration should be given as .

It should be noted that the calculated uncertainty is the worst case situation that is possible under the individual assumptions in the treatment. Alternate treatments based on statistical treatments may be more realistic. Also, another useful application of these methods is to define the sensitivity of the dependant variable to inputs to the function. These will be address in subsequent posts.

Caveat lector — All work and ideas presented here may not be accurate and should be verified before application.


[1]
2nd order and higher terms include the square and higher powers of the perturbation amount. Assuming that the perturbation is small, the square of this small perturbation is much smaller still, and higher order terms become negligible. If the relative size of the perturbation to the curvature of the function is in doubt, the magnitude of the 2nd and higher order terms should be checked.

Taylor Series Approximation of a Function

Under certain conditions we may approximate an analytic function about a specified point on the function by an infinite series. The most useful series for our purposes is the Taylor series.

The Taylor series of a function about the point is given as

The equation above allows us to approximate the value of the function as an infinite series for any point sufficiently close to while only knowing the value of the function and its’ derivatives at . As an example, consider the equation. This is an inverted “W”-shaped function with roots at 0,. We are interested in a region centered about , so we may begin by evaluating the above equation with and for increasing values of .

For ,

For ,

For ,

etc.

For our 4th order function, any values of will result in the derivative being equal to . Therefore, we have a finite number of terms in the full Taylor series expansion up to a maximum of . The following graph shows the original function and the Taylor series approximations for . In this case, the series obtained when is algebraically equivalent to our original function.

graph

The Taylor series may also be defined for multivariate functions. This allows us to expand the usefulness of the series to functions of multiple variables, and as we shall see later, allow us to predict the function values for small deviations about a nominal point, as well as ascribe sensitivity of the function to the independent variables.

The Taylor series of a function about the point is given as

We’ll use this later as we begin to discuss error propagation.