NONLINEAR TIME SERIES/SYSTEM ANALYSIS (STATE-SPACE, DYNAMICS, INTERVENTION, RESILIENCE, KALMAN)

A discussion of nonlinear dynamics, demonstrated by the familiar automobile, is followed by the development of a systematic method of analysis of a possibly nonlinear time series using difference equations in the general state-space format. This format allows recursive state-dependent parameter estimation after each observation thereby revealing the dynamics inherent in the system in combination with random external perturbations.^ The one-step ahead prediction errors at each time period, transformed to have constant variance, and the estimated parametric sequences provide the information to (1) formally test whether time series observations y(,t) are some linear function of random errors (ELEM)(,s), for some t and s, or whether the series would more appropriately be described by a nonlinear model such as bilinear, exponential, threshold, etc., (2) formally test whether a statistically significant change has occurred in structure/level either historically or as it occurs, (3) forecast nonlinear system with a new and innovative (but very old numerical) technique utilizing rational functions to extrapolate individual parameters as smooth functions of time which are then combined to obtain the forecast of y and (4) suggest a measure of resilience, i.e. how much perturbation a structure/level can tolerate, whether internal or external to the system, and remain statistically unchanged. Although similar to one-step control, this provides a less rigid way to think about changes affecting social systems.^ Applications consisting of the analysis of some familiar and some simulated series demonstrate the procedure. Empirical results suggest that this state-space or modified augmented Kalman filter may provide interesting ways to identify particular kinds of nonlinearities as they occur in structural change via the state trajectory.^ A computational flow-chart detailing computations and software input and output is provided in the body of the text. IBM Advanced BASIC program listings to accomplish most of the analysis are provided in the appendix. ^

Inherited and noninherited sources of variation in metastasis and the growth rate of tumors: Implications for tumor evolution and therapy

I have undertaken measurements of the genetic (or inherited) and nongenetic (or noninherited) components of the variability of metastasis formation and tumor diameter doubling time in more than 100 metastatic lines from each of three murine tumors (sarcoma SANH, sarcoma SA4020, and hepatocarcinoma HCA-I) syngeneic to C3Hf/Kam mice. These lines were isolated twice from lung metastases and analysed immediately thereafter to obtain the variance to spontaneous lung metastasis and tumor diameter doubling time. Additional studies utilized cells obtained from within 4 passages of isolation. Under the assumption that no genetic differences in metastasis formation or diameter doubling time existed among the cells of a given line, the variance within a line would estimate nongenetic variation. The variability derived from differences between lines would represent genetic origin. The estimates of the genetic contribution to the variation of metastasis and tumor diameter doubling time were significantly greater than zero, but only in the metastatic lines of tumor SANH was genetic variation the major source of metastatic variability (contributing 53% of the variability). In the tumor cell lines of SA4020 and HCA-I, however, the contribution of nongenetic factors predominated over genetic factors in the variability of the number of metastasis and tumor diameter doubling time. A number of other parameters examined, such as DNA content, karyotype, and selection and variance analysis with passage in vivo, indicated that genetic differences existed within the cell lines and that these differences were probably created by genetic instability. The mean metastatic propensity of the lines may have increased somewhat during their isolation and isotransplantation, but the variance was only slightly affected, if at all. Analysis of the DNA profiles of the metastatic lines of SA4020 and HCA-I revealed differences between these lines and their primary parent tumors, but not among the SANH lines and their parent tumor. Furthermore, there was a direct correlation between the extent of genetic influence on metastasis formation and the ability of the tumor cells to develop resistance to cisplatinum. Thus although nongenetic factors might predominate in contributing to metastasis formation, it is probably genetic variation and genetic instability that cause the progression of tumor cells to a more metastatic phenotype and leads to the emergence of drug resistance. ^