[spa-temp.inf] Spatial-temporal information

Transcription

1 [spa-temp.inf] Spatial-temporal information VI

2 Table of Contents for Spatial-temporal information I. Spatial-temporal information VI - 1 A. Cohort-survival method VI - 1 B. Spatial vs. univariate forecasts VI - 2 C. Univariate forecast with intervention VI - 2 D. Spatial forecast with intervention VI - 4 E. Calibration of a spatial model VI - 4

3 I. Spatial-temporal information A. Cohort-survival method (Jha 1972) (a) Crude birth-rate of any region is defined as birth-per-person (or per 100 persons) in that period for that region. For example, if the number of births for York County is 2000 for five years ( ) and its average population over the period is , crude birth-rate for York in is 2000/210000=1/105, or we can say that crude birth-rate is one in 105 people. Crude death-rate is similar to crude birth-rate. If in the above example for the same period, the number of deaths for different reasons is 500, then crude death-rate for that period will be 500/210000= If the total number of people coming into York County in the five-year period ( ) is 1400 and the outgoing number from this region is 1295, then net migration will be =105. This will, of course, be net in-migration. (b) From the given Table, the number of children born in is 7. Probability of having a child for age-group is [7/( )][14/( )]=(0.137)(0.275)= Here ( )=51 is the total number of women of child-bearing age in 1945, ranging from 15 to 29 year old. Probability of having a child for the woman age-group is (0.137)(16/51)=0.043; and probability of having a child for the woman age-group is (0.137)(21/51)= For the desired growth-matrix, the top row of the matrix--the birth rates =( b ij )--may be calculated as follows and so on. Here b(10-14) is the fertility-rate of age-group (which is zero in our case), s ij =(s s 9 )/(s s 4 ) is the surviving ratio of cohort-group i (0-4 year) in group j (5-9 year), N f is the number of female children, while N c is the total number of children. As a degenerate case for the postnatal population, the probability of living from birth to the end of five-years is (s s 4 )/s(0), where s(0) is the number of births to begin with. We can now write the equation G N(t)=N(t+ t) as discussed in the "Interregional growth and distribution" section of the "Economics" chapter, where G is the 6 6 growth-matrix, N(t) is the femalepopulation in 1940, and N(t+ t) is the female-population in Here the surviving-ratio (s s 4 )/s(0) is given as 0.98 and the percentage of female-children is N f /N c =0.49. Over five-years, we have (5)(0.98)(0.49)=2.43 times as many female children, following this part of Equation (1) [(s s 4 )/s(0)](n f /N c ). In the absence of migration, the sub-diagonal elements of the growth-matrix G=[G ij ] can be calculated as follows. Starting with the surviving-ratio (s s 9 )/(s s 4 ) for the 5-9 year group, where the numerator is 10 and the denominator is also 10, the ratio is unity. Here are the detailed calculations for the remaining age-groups Age Group (i) calculation 1 12/14 14/15 16/18 21/22 - s i+1,i =G i+1,i VI - 1

4 Elements of the top row of the growth matrix, or the birth-rate for female-children b ij, are computed from Equation (1) as follows b 13 =[(0.86)(35/2)(2.43)]/1000=0.036 b 14 =[(32/2)+(0.93)(43/2)]2.43/1000=0.090 b 15 =[(43/2)+(0.89)(56/2)]2.43/1000=0.109 b 16 =(56/2)2.43/1000= The equation G N(t)=N(t+ t) now reads Thus the difference between the 1945 women-population totals--as given and as computed--is only (80)-( )=80-76=4, or 400 women. The difference is attributable to the 'truncated' first entry (or the 0-4 year group). B. Spatial vs. univariate forecasts. C. Univariate forecast with intervention (Wright 1995) In interpreting the pumping intervention, 0.0 is entered into the matrix if the intervention effect was less than that of the nominal rate (300 gallons [1.20 m 3 ] per minute). Our first effort is on well 1. The difference between two time-periods is a week in our study. The International Mathematical & Statistical Library (IMSL) is used for our time-series analysis. In this case, the sample autocorrelations and sample partial-autocorrelations are shown below in Figure 1 1. Inspection of the time series given in the problem statement suggests that the series is approximately stationary in the mean before any intervention. No differencing is warranted. Since both the autocorrelation and partial-autocorrelation function have one significant spike, an AR(1) or MA(1) model could be selected. Since neither the autocorrelation nor the partial-autocorrelation fluctuation show clear exponential-decay toward zero in a more precise plot, the choice between AR(1) and MA(1) is subjective. The AR(1) model is selected at this instance. Figure 1 - Autocorrelations and partial autocorrelations for the period before intervention (Wright 1995) Parameter coefficients for the selected model are then calculated. These estimates are used as pre-whitening parameters in the transfer-function computation, which is accomplished on the entire time- 1 Notice these digital graphic-plots have significant round-off-errors in the display. VI - 2

5 series. The intervention time-series is used as the input time-series and the entire (41 observations) timeseries as the output time-series. The transfer-function weights are calculated and displayed, along with the autoregressive and moving-average parameter-coefficients. Only four transfer-function weights are calculated for this work since it is not expected that the effect of an increased level of pumping would be significant after three weeks. The four weights are being applied to the level-of-intervention at zero time (immediately) and one to three time-periods later. Preliminary results show an autoregressive-coefficient φ 1 = and transfer-function weights d 0 = , d 1 = , d 2 = , and d 3 = The transfer-function weights can be interpreted as the expected increase in the level of arsenic-concentration measured in µg/l due to one-foot (0.3 m) increase in the draw-down, at a site downstream from the drawdown. As mentioned, the intervention is at the nominal pumping-rate of 300 gallons (1.20 m 3 ) per minute. Since the pre-whitening parameter-coefficients were computed using only the time-series prior to intervention, it is not expected that these first-obtained transfer-function weights will accurately reflect the final model. Additional iterations were performed until pre-whitening parameter-coefficients converged to one within ten-thousandths on successive iterations. If the user specifies another iteration should occur, the following procedure is carried out. The predicted noise-series that consist of the original time-series minus the expected impact of the intervention is used as the time series for which the pre-whitening estimates are based. The transfer function then uses the original time-series with the new-estimate pre-whitening parameter-coefficient to determine the new transfer-function weights. The precess is then repeated until the user is satisfied with convergence of the parameter-coefficient-estimates. The number of iterations varies, but is around 12 for this work. In order to assist the convergence of the algorithm, it is useful to select constant model-forms from one iteration to the next. For instance, the second iteration for the modelling of well 1 shows exponential decay in the autocorrelations, but two spikes in the partial autocorrelations, suggesting an AR(2) modelform. This determination requires judgment, since actual autocorrelation and autocorrelation functions rarely exhibit the precise shapes of their associated theoretical-postulates. The AR(2) form selected here is consistent with the first AR(1) model in that it is an extension of the first. The concern is that switching between AR and MA models on successive iterations may hamper solution-convergence. Figure 2 - Autocorrelations and partial autocorrelations for well 2128 (Wright 1995) The final iteration of the temporal modelling of well 1 displays autocorrelations and partial autocorrelations shown in Figure 2. The autocorrelation plot is interpreted as a sinusoidal-pattern dampening, though enough data are not available to make that judgment. The partial-autocorrelation plot is judged to have two spikes at lags one and two. This combination of autocorrelation and partialautocorrelation plots suggests an AR(2) process and the final autoregressive-coefficients are φ 1 = and φ 2 = However, different interpretations of the autocorrelation and partial-autocorrelation plots are possible. The first two autocorrelations could be considered spikes, thereby suggesting a moving-average process, MA(2). The algorithm was run fitting this tentative model. The MA(2)-model parametercoefficients are approximately equal to the AR(2) coefficients, but opposite in sign. The problem with the fit is in the estimation of the transfer-function weights. The AR(2) weights behave as expected, dying completely out by the fourth term d 0 = , d 1 = , d 2 = , and d 3 = The MA(2) weights, however, suggest a strong influence even at the third week after the intervention occurs d 3 =98. Since the MA(2) model does not provide any variance-reduction over the AR(2) model, MA(2) is no longer a candidate model for consideration. VI - 3

6 Another possible interpretation of the autocorrelation plots is that of a mixed process. Several mixed-process models were fit, but the ARMA(2,2) provided the most insight. In only two iterations of fitting the ARMA(2,2) model, it became apparent that only two of the four estimated-coefficients would be at all significant the φ 2 term representing the two-lag autoregressive parameter and the θ 1 term representing the one-lag moving-average term. The estimated coefficients are φ 2 =0.263 and θ 1 = with transfer-function weights of d 0 = , d 1 = , d 2 = , d 3 = and a constant-term of The model shows very similar response-weight as the AR(2)-model, and the random-shock variance of the estimated models is nearly equal (120.6). Given the equal model-fit, the AR(2) model is selected as the most parsimonious since only two coefficients are estimated, as opposed to the ARMA(2,1),φ 1 =0 model, in which the φ 1 term still forces the loss of a degree-of-freedom. After convergence, the final residuals are plotted along with their autocorrelations and partial autocorrelations. Figure 3(a) suggests the variance is greater during periods of intervention (t 26) than periods prior to the intervention. This is likely due to two reasons. First, the arsenic concentration threshold of 10.0 µg/l produced many concentration values of 10.0 µg/l, thereby reducing the variance for the pre-intervention series. Second, the data series reflecting a nominal-level of arsenic-concentration is likely to have a small variance compared to the same data with additional arsenic imposed upon it from some outside source. The reason is that the amount of additional arsenic imposed on the system contain its own variability. The autocorrelations (Figure 3(b)) and partial correlations (Figure 3(c)) similarly suggest that additional parameters could be warranted. However, it has been shown that additional parameters do not aid this model. The other candidate model, ARMA(2,1) with φ 1 =0, exhibits similar residual autocorrelation and partial-autocorrelation patterns. The most parsimonious model for well 1, therefore, is the AR(2) model Z t = Z t Z t X t X t X t-2 +A t where Z t is the original time-series, and X t is the intervention time-series. The complete model has an R 2 of 0.745, accounting for 74.5 percent of the total variance of the original time-series. The model is relatively simple to use since past values of A t are not required to be determined, as in a moving-average model. Similar arguments can be followed in justifying the other two time-series for the remaining two wells, wells 2 and 3 Z t = Z t Z t X t X t X t-2 +A t Z t = Z t Z t X t X t x t-2 +A t. Figure 3 - Residuals, autocorrelations and partial correlations (Wright 1995) D. Spatial forecast with intervention. E. Calibration of a spatial model (Cliff & Ord 1981) Table I gives some more details on the three sets of estimates for a (linear) model linking insurgent (Huk) control to the cultural, demographic, economic, and physical exogenous-variables for each municipality. The non-spatial model is the standard linear-model, fitted by ordinary least-squares without any spatialautoregressive component. This includes the case where there are autocorrelated residuals. In the second VI - 4

7 part of the Table, two spatial models are estimated. In the first of these, the simultaneous scheme has been fitted by maximum likelihood as follows. Table I - Alternative estimations for the model of Huk insurgent (Doreian 1981) 1. Non-spatial model 2. Spatial model OLS Simultaneous - MLE Conditional - OLS The asymptotic formulas were used for the simultaneous scheme in regression with autocorrelated residuals (w (l) ) T (W (l) y) ~Y denotes the regressor for y i ; all developments in the spatial model degenerate into the non-spatial model when one set W=I and w to the corresponding vector with a single unitary entry. We use the model Y=FWY+Zb+e, where Y is endogenous variables and Z the exogenous variables. The error-term e is normally distributed, or N(0,σ 2 I). It can be shown that the log-likelihood is If we let Y'=Y-FWY, the maximum-likelihood estimators are given by is that value of which maximizes (4) The value must then be substituted back into Equations (3). Notice the objective-function (4) is of the same form as that for the purely autoregressive-scheme. To simplify the numerical details, we may write n's 2 =n' as (5) where Y*=(W (l) y) ~Y, and and so on. The best method to estimate the coefficients is to evaluate by a search on function (4), and then to solve Equations (2) and (3) directly. VI - 5

8 In the second spatial-model, ordinary least-squares has been used to estimate the parameters of the following conditional model. where (w (i) ) T and Z i T are the ith rows of W and Z respectively, and y i * denotes y after deletion of y i. When the Y are taken to be normal, this specification yields the joint-distribution Y~N( Zb,σ 2 ) in which =I- FW. Maximum-likelihood estimation for this model is awkward, because the iterative procedure requires the inversion of at each stage. However the form of Equation (6) is such that ordinary least-squares (OLS) provides consistent estimators for and. The OLS results for the spatial model have not been identified previously in terms of a conditional scheme. The most noteworthy features of the results are the inflated parameter-estimates and the greatly increased standard-errors of these estimates. They are induced by their variance-explained, which increases from 73% to 80%. Both spatial schemes suggest a strong element of geographical interaction. Despite their differences in formulation, both arrive at similar estimates for. Thus the presence of spatial interaction is well established. The estimated standard-errors are somewhat higher for the conditional scheme than for the simultaneous version. This could be due to the inefficient procedure used for the conditional model, or it could reflect a difference between the two formulations. The only striking difference between the standard errors is for the spatial-interaction coefficient itself. Here the use of asymptotic expression may account for part of the difference. Further work is required to establish the adequacy of the asymptotic results for these models. Now turn to the regression element of the spatial model, it can be seen that only the variables PF, PO, and PM are significant at the 10% level, whereas all variables appear significant in the non-spatial version. However, the sugar-cane variable PS alone has a coefficient less than its standard error, and it would appear that this term accounts spatial interaction in the non-spatial model. (6) VI - 6

9 Listing of Figures for Spatial-temporal information Figure 1 - Autocorrelations and partial autocorrelations for the period before intervention (Wright 1995) VI - 2 Figure 2 - Autocorrelations and partial autocorrelations for well 2128 (Wright 1995) VI - 3 Figure 3 - Residuals, autocorrelations and partial correlations (Wright 1995) VI - 4 VI - 7

10 AUTOCORRELATION PLOT 1. - a / * * * * * c lag PARTIAL A/C PLOT p 1. - a r t i a * * * * l a / c lag Figure 6-1. Autocorrelations and partial autocorrelations for the period before intervention (Wright 1995)

11 AUTOCORRELATION PLOT 1. - a / * * * * * c lag PARTIAL A/C PLOT p 1. - a r t i a * * * * * l a / c lag Figure 6-2. Autocorrelations and partial autocorrelations for well 2128

12 (a) TIME SERIES PLOT a r s e n i c u g / * * * *** * * l 0. - *** **** **** **** **** * ** * **** ** * * time period (b) AUTOCORRELATION PLOT 1. - * a / * * * * c (c) PARTIAL A/C PLOT p 1. - a r t * i a * * * * l a / c lag lag Figure 6-3. Residuals, autocorrelations and partial correlations (Wright 1995)