First-order Taylor Approximation Research Articles

Prediction Intervals for ARIMA Models

The problem of constructing prediction intervals for linear time series (ARIMA) models is examined. The aim is to find prediction intervals that incorporate an allowance for sampling error associated with parameter estimates. The effect of constraints on parameters arising from stationarity and invertibility conditions is also incorporated. Two new methods, based on varying degrees of first-order Taylor approximations, are proposed. These are compared in a simulation study to two existing methods, a heuristic approach and the “plug-in” method whereby parameter values are set equal to their maximum likelihood estimates. A comparison of the four methods is also made for quarterly retail sales for 10 Organization for Economic Cooperation and Development countries. The new approaches provide a systematic improvement over existing methods.

Accuracy of error propagation exemplified with ratios of random variables

The method of error propagation provides a convenient tool for calculating mean and variance of a measurand from means and variances of primarily measured quantities. However, being based on a (usually first-order) Taylor approximation of the measurement function, it only yields approximate results with unknown accuracy. We develop a method for estimating the accuracy of (Nth-order) error propagation for an arbitrary number of correlated random quantities, and apply our findings to the ratio of two random variables (RVs). A comparison with some analytically solved expressions for certain probability density functions (PDFs) as well as with some computer simulations reveals the excellent quality of our estimates as long as the involved PDFs are not significantly skew. For the ratio of two RVs it turns out that conventional, first-order error propagation is safely applicable (with about 1% accuracy) as long as the denominator’s mean is larger than about 12 times its standard deviation. Using second-order error propagation, the approximation for the ratio’s mean can be refined, yielding 1% accuracy if the denominator’s mean is larger than about four times its standard deviation.

Geomagnetism, Earth rotation and the electrical conductivity of the lower mantle

A new approximation is derived for the electromagnetic torque acting on an electrically conducting mantle as a result of fluid flow in the core. The torque considered is that associated with the toroidal field induced in the mantle by advection of poloidal field (which is assumed to be frozen into an infinitely conducting liquid core) by the flow of liquid at the top of the core. The expression relies on the assumption that the mantle is an insulator apart from a ‘thin’ (with respect to the core radius) layer of finite conductance adjacent to the core-mantle boundary. This allows the toroidal field scalar in the mantle to be expressed as a first-order Taylor approximation. The time-dependent torque calculated at a sequence of epochs this century is compared with the torque which has previously been inferred from astronomical observations of the length of day. Although the initial results appear unpromising, a significant correlation exists when poorly determined components of the velocity field, which contribute substantially to variations in the calculated torque, are ignored. By regression analysis the conductance of the assumed thin layer is determined as 6.7 × 10 8 S and the lag of the electromagnetic torque behind the astronomical torque as 6 years, which is interpreted as the delay time for electromagnetic signals through the mantle. Finally, the implications of these results for the conductivity of the lower mantle are discussed. They imply that the bottom few hundred kilometres of the mantle probably have a conductivity of a few hundred to a few thousand siemens per metre; previously published lower-mantle conductivity models are examined in light of this conclusion. The close correlation of the variations in the calculated electromagnetic torque, which depend primarily on fluctuations in the core velocity field, with the length of day torque provides independent evidence that core flow is not steady on the decade time-scale.

Corrected sequential linear programming for sparse minimax optimization

We present a new algorithm for nonlinear minimax optimization which is well suited for large and sparse problems. The method is based on trust regions and sequential linear programming. On each iteration a linear minimax problem is solved for a basic step. If necessary, this is followed by the determination of a minimum norm corrective step based on a first-order Taylor approximation. No Hessian information needs to be stored. Global convergence is proved. This new method has been extensively tested and compared with other methods, including two well known codes for nonlinear programming. The numerical tests indicate that in many cases the new method can find the solution in just as few iterations as methods based on approximate second-order information. The tests also show that for some problems the corrective steps give much faster convergence than for similar methods which do not employ such steps.

Mathematical analysis of the operative temperature and energy budget

1. 1. An exact, explicit quartic solution to the energy budget can be used to calculate the operative temperature, the surface temperature of a wet organism, and the metabolic rate. 2. 2. All of the approximations and iteration methods arrive at solutions close to the proper root and do not converge around the other three physically unrealistic roots. 3. 3. A second-order Taylor approximation is satisfactory, but the first-order Taylor approximation is not. 4. 4. A new simple 4th order polynomial is introduced to estimate the saturated vapor pressure function. 5. 5. Partial differential analysis is used to show the errors associated with the use of mounts and mount analogs.

Computerized Method for Approximating the Variance of a Complicated Estimate

Abstract This article describes a generalized program for the computation of sampling errors. It employs computerized linearization of nonlinear estimates by the use of the first-order Taylor approximation. It can be used for any estimate derived from any “large” probability sample. In most instances the only inputs required are the weighted sample data and the form of the estimate whose precision is to be measured. In these cases, both the estimate and its sampling error can be produced with the same amount of data preparation and programming effort as is required to produce the estimate only.

On Multimode Oscillators with Constant Time Delay

Special emphasis is paid to narrow-band systems with a small number of modes, and to high switching speed. Sufficient stability conditions presented have been derived by use of a first-order Taylor approximation, which substantially reduces algebraic complexity. The system may deviate considerably from quasi-linearity. Graphical interpretation of stability and transients permits application to curve data and is believed to give an easily visualized explanation for experimental observations. Mode switching speeds of 7 mμsec could be achieved with an S-band bistable oscillator.