Correlation serial

The starting point is the (pseudo-) randomization function supplied with most computers it generates a rectangular distribution of events, that is, if called many times, every value between 0 and 1 has an equal probability of being hit. For our purposes, many a mathematician s restraint regarding randomization algorithms (the sequence of numbers is not perfectly random because of serial correlation, and repeats itself after a very large number of... 

The key to calculating the Durbin-Watson statistic is that prior to performing the calculation, the data must be put into a suitable order. The Durbin-Watson statistic is then sensitive to serial correlations of the ordered data. While the serial correlation is often thought of in connection with time series, that is only one of its applications. 

Draper and Smith [1] discuss the application of DW to the analysis of residuals from a calibration their discussion is based on the fundamental work of Durbin, et al in the references listed at the beginning of this chapter. While we cannot reproduce their entire discussion here, at the heart of it is the fact that there are many kinds of serial correlation, including linear, quadratic and higher order. As Draper and Smith show (on p. 64), the linear correlation between the residuals from the calibration data and the predicted values from that calibration model is zero. Therefore if the sample data is ordered according to the analyte values predicted from the calibration model, a statistically significant value of the Durbin-Watson statistic for the residuals in indicative of high-order serial correlation, that is nonlinearity. 

Residuals should not be serially correlated, as identified by visual inspection or by an appropriate statistical test (for example a Run s test). 

Kent, D.R., Muller, R.P., Anderson, A.G., Goddard, W.A., Feldmann, M.T. Efficient algorithm for "on-the-fly" error analysis of local or distributed serially correlated data. J. Comput. Chem. 2007, 28, 2309-16. 

Other problems pointed out by Box et al. [20] are serially correlated errors, dynamic relations and feedback. All the above problems can be overcome by the use of properly designed statistical experiments that employ features such as randomisation, blocking and other suitable controls. 

Simple but useful diagnosis tools are the residual plots discussed by Wood (ref. 6). If the residuals are of highly nonrandom structure, at least one of the assumptions is questionable. This nonrandomness implies that the elements of the residual sequence r, r2,. rnm are correlated. A measure of this serial correlation is the D-statistics proposed by Durbin and Wattson (ref. 7), and computed according to... 

J.Durbin and G.S. Wattsan, Testing for serial correlations in least squares regression, I., Biometrika, 37 (1950) 409-428. 

Like most price series, power price data exhibit serial correlation. Hence, the error term et is characterized by a so-called 1(0) process (integrated of order zero).9... 

Figure 9.19 Autocorrelations and cross-compartment serial correlations with increased values of delay t° = 1,2,4 (solid, dashed, and dotted lines, respectively). ...

Stage 1. The MeOH/H20/NaCl data are subjected to the correlation procedure described previously which gives values of the Wilson energy constants (Zi and Z2) and a new set of data for temperature and vapor composition that are internally consistent (see Table I). The small values of the standard deviation and the bias indicate good quality data in the salt effect field. For the analysis of serial correlation among the residuals we use the Durbin-Watson test (9). A run of positive or negative signs in the series of residuals is some indication that the model... 

Ideally, experimental data (intensity) should be randomly distributed both above and below the calculated intensity profile. If there are multiple sequences with all observed points above or all below the calculated intensity values, it is said that serial correlation occurs. In other words, the 4-statistic reflects correlation between adjacent least-squares residuals and it can be used as an indicator that refined parameters are unbiased. 

If successive residuals are positively serially correlated, that is, positively correlated in their sequence, d will be near zero. 

Whenever there are inertial elements (capacity) in a process such as storage tanks, reactors or separation columns, the observations from such processes exhibit serial correlation over time. Successive observations are related to... 

M Bagshaw and RA Johnson. The effect of serial correlation on the performance of CUSUM tests. Technometrics, 17 73-80, 1975. 

E Yashchin. Performance of CUSUM control schemes for serially correlated observations. Technometrics, 35 37-52, 1993. 

Figure 6.2 Plot of exponential, Gaussian, and power serial correlation functions as a function of distance between measurements. The underlying correlation between measurements was 0.8.

At this stage, time-varying random effects should be examined by plotting the residuals obtained using a simple OLS estimate of (3, ignoring any serial correlation or random effects in the model, against time. If this plot shows no trends and is of constant variance across time, no other random effects need to be included in the model, save perhaps a random intercept term. If, however, a systematic trend still exists in the plot then further random effects need to be included in the model to account for the trend. 

If the first four experimental points were run before the lunch break where cooler temperatures prevailed and the second four experimental points were run in the afternoon where warmer temperatures prevailed, then Xj would be confounded with ambient temperature. If a were judged to be significant, it would be impossible to know whether the effect was due to variations in Xj or variations in the temperature. Even worse, since we are unaware that ambient temperature affects the results, we would ascribe the results to a even though there may be no such association Therefore, randomizing the run order is an important prophylactic against serial correlation. 

Cognizance of experimental error is critical for understanding the meaning of the coefficients and conducting the experiments themselves. As a first principle, factorial experimental designs should be run in fully randomized order. This is necessary in order to break any serial correlations that may exist. For example, suppose the design of Table 3.1 were run in the order presented. Further suppose that ambient temperature is an important but unrealized factor affecting the responses. 

Any decent model used to analyse a series of n-of-1 trials will allow for at least three sources of variation pure between-patient variability, within-patient variability and a random effect for patient-by-treatment interaction. Since a series of measurements are being obtained, however, it may be inappropriate to assume that within-patient errors are independent (or more formally that the correlations between measures are equal). If patients are subject to spells of illness for example, then two measurements taken during two administrations of the same drug are more likely to be similar if the administrations are close together rather than far apart. This phenomenon can be referred to as serial correlation and is potentially a problem for the analysis of n-of-1 trials. (It would also be a problem for multiperiod cross-overs, the standard analysis of which, however, ignores the even more serious problem of patient-by-treatment interaction.)... 

Special Problems in Simple Linear Regression Serial Correlation and Curve Fitting... 

Whenever there is a time element in the regression analysis, there is a real danger of the dependent variable correlating with itself. In the literature of statistics, this phenomenon is termed autocorrelation or serial correlation in this text, we use the latter as descriptive of a situation in which the value, y is dependent on y, i, which, in turn, is dependent on y, 2. From a statistical perspective, this is problematic because the error term, e,-, is not independent—a requirement of the linear regression model. This interferes with least-squares calculation. 

When positive serial correlation is present (r > 0), the e, value will be small in pairwise size and positive errors will tend to remain positive and negative errors will tend to be negative, slowly oscillating between positive and negative values (Figure 3.1). The regression parameters, bo and b, can be thrown off and the error term estimated incorrectly. 

Negative serial correlation (Figure 3.1b) tends to display abrupt changes between c, and c, i, generally bouncing from positive to negative values. 

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