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The supplementary materials contain 7 figures and a discussion on the relationship
between residual cross-correlations and those of the innovations.
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preserved by the residuals.
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Conditions under which the cross-correlations between the innovations are asymptotically
Let 𝑌𝑖,𝑡 be the downstream measurement of the 𝑖th variable at time 𝑡, 𝑿𝑖,𝑡 the (𝑞 + 1)-
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dimensional vector covariates that include the corresponding upstream measurement 𝑋𝑖,𝑡 and its
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lags, i.e., 𝑿𝑖,𝑡 = (𝑋𝑖,𝑡 , 𝑋𝑖,𝑡−1 , … , 𝑋𝑖,𝑡−𝑞 ) , and that the following stochastic regression models
⊤
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hold (with the intercept and outliers omitted for simplicity):
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𝑌𝑖,𝑡 = 𝜷⊤
𝑖 𝑋𝑖,𝑡 + 𝜀𝑖,𝑡 ,
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where 𝑖 = 1, … , 𝑘 + 1, 𝜷𝑖 is a (𝑞 + 1)-dimensional coefficient vector and the regression errors
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𝜀𝑖,𝑡 follow an order 𝑝 autoregressive process, i.e.
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𝜀𝑖,𝑡 = 𝜙𝑖,1 𝜀𝑖,𝑡−1 + ⋯ + 𝜙𝑖,𝑝 𝜀𝑖,𝑡−𝑝 + 𝑎𝑖,𝑡 ,
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with the 𝑎’s being independent and identically distributed random variables of zero mean and
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positive variance 𝜎𝑖2 , and they are independent of past 𝑌’s and current and past 𝑋’s. The 𝑎’s are
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known as the innovations, and 𝑎𝑖,𝑡 may be used as the proxy for the measurement of the 𝑖th
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variable due to the processes occurring within the reservoir in the 𝑡th period. The main object of
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study consists of, say, regressing 𝑎𝑘+1,𝑡 on 𝑎𝑗,𝑡 , 𝑗 = 1, … , 𝑘. Since the 𝑎’s are unobservable, they
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will be replaced by the residuals, denoted by 𝑎̃𝑖,𝑡 , obtained from fitting the model defined by
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(A1) and (A2), separately for each the 𝑘 + 1 variables. We shall assume that the unknown
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parameters can be consistently estimated with the estimation error being of order 𝑂𝑝 (1/√𝑇) ,
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̃ 𝑖 , 𝑖 = 1, ⋯ , 𝑘 + 1 and
where 𝑇 is the sample size and the coefficient estimates are denoted by 𝜷
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𝜙̃𝑖,ℓ , ℓ = 1, ⋯ , 𝑝. For instance, this holds if the model defined by (A1) and (A2) is estimated by
(A1)
(A2)
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the method of conditional least squares, under some mild regularity conditions (Klimko and
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Nelson, 1978). We show below that the large-sample joint distribution of the regression
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coefficient estimates using the residuals is identical to that using the latent innovations under the
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following additional conditions:
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(C1) The vector process {(𝑌𝑖,𝑡 , 𝑿⊤
𝑖,𝑡 , 𝑖 = 1, … , 𝑘 + 1) , 𝑡 = ⋯ , −1, 0,1,2, ⋯ } is a stationary,
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ergodic process with finite second moments.
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We show the validity of the claim by proving below that
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∑𝑇𝑡=𝑝+1 𝑎̃𝑗,𝑡 𝑎̃𝑗,𝑡 = ∑𝑇𝑡=𝑝+1 𝑎𝑖,𝑡 𝑎𝑗,𝑡 + 𝑂𝑝 (1).
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The residuals 𝑎̃𝑖,𝑡 are computed by the following two equations where
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̃⊤
𝑌𝑖,𝑡 = 𝜷
𝑖 𝑿𝑖,𝑡 + 𝜀̃𝑖,𝑡 ,
(A4)
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𝜀̃𝑖,𝑡 = 𝜙̃𝑖,1 𝜀̃𝑖,𝑡−1 + ⋯ + 𝜙̃𝑖,𝑝 𝜀̃𝑖,𝑡−𝑝 + 𝑎̃𝑖,𝑡 ,
(A5)
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After some algebra, it can be shown that
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̃ 𝑖 )⊤ (𝑿𝑖,𝑡 − 𝜙̃𝑗,1 𝑿𝑖,𝑡−1 − ⋯ − 𝜙̃𝑗,𝑝 𝑿𝑖,𝑡−𝑝 ) + (𝜙𝑖,1 − 𝜙̃𝑖,1 )𝜀𝑖,𝑡−1 + ⋯
𝑎̃𝑖,𝑡 = 𝑎𝑖,𝑡 + (𝜷𝑖 − 𝜷
⊤
(A3)
+ (𝜙𝑖,𝑝 − 𝜙̃𝑖,𝑝 )𝜀𝑖,𝑡−𝑝 .
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Hence, ∑𝑇𝑡=𝑝+1 𝑎̃𝑗,𝑡 𝑎̃𝑗,𝑡 = ∑𝑇𝑡=𝑝+1 𝑎𝑖,𝑡 𝑎𝑗,𝑡 + 𝑂𝑝 (1), because, for instance, with 1 ≤ ℓ, 𝑞 ≤ 𝑝 ,
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̃ 𝑖 ) ∑𝑇𝑡=𝑝+1 𝑎𝑖,𝑡 𝑿𝑗,𝑡−ℓ = √𝑇(𝜷𝑖 − 𝜷
̃ 𝑖 ) ∑𝑇𝑡=𝑝+1 𝑎𝑗,𝑡 𝑿𝑗,𝑡−ℓ /√𝑇 = 𝑂𝑝 (1),
(𝜷𝑖 − 𝜷
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(𝜙𝑖,ℓ − 𝜙̃𝑖,ℓ )(𝜙𝑗,𝑞 − 𝜙̃𝑗,𝑞 ) ∑ 𝜀𝑖,𝑡−ℓ 𝜀𝑗,𝑡−𝑞
⊤
⊤
𝑇
𝑡=𝑝+1
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=
√𝑇(𝜙𝑖,ℓ − 𝜙̃𝑖,ℓ )√𝑇(𝜙𝑗,𝑞 − 𝜙̃𝑗,𝑞 ) ∑𝑇𝑡=𝑝+1 𝜀𝑖,𝑡−ℓ 𝜀𝑗,𝑡−𝑞
= 𝑂𝑝 (1),
𝑇
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as ∑𝑇𝑡=𝑝+1 𝑎𝑗,𝑡 𝑿𝑗,𝑡−ℓ /√𝑇 is asymptotically normally distributed by the martingale central limit
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theorem (Billingsley 2013, Theorem 18.1), and ∑𝑇𝑡=𝑝+1 𝜀𝑖,𝑡−ℓ 𝜀𝑗,𝑡−𝑞 /𝑇 approaches
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𝐸(𝜀𝑖,𝑡−ℓ 𝜀𝑗,𝑡−𝑞 ), with increasing sample size 𝑇, in view of (C1).
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Supplementary Figure captions:
Figure S1. Adjusted measurements as residuals of fitted transfer function models.
Figure S1. Model diagnostics of the transfer function model (6) fitted to the square root of N.
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The top sub-figure is the residual time plot, the middle sub-figure the residual autocorrelation
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function (ACF) and the bottom sub-figure shows the p-value of Ljung-Box test assessing the
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whiteness of the residuals based on the first 𝑘 lags of residual ACF, for 𝑘 ranging from 13 to 24.
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All p-values are greater than 5%, suggesting that the fitted transfer function model provides a
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good fit to the N data.
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Figure S3. Model diagnostics of the transfer function model (6) fitted to pH (P). Same
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convention as in Supplementary Figure 1. The p-values are slightly higher than 5%, indicating an
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adequate fit to the data.
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Figure S4. Model diagnostics of the transfer function model (6) fitted to total alkalinity (A).
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Same convention as in Supplementary Figure 1. The Ljung-Box tests suggest the model provides
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a good fit to the data.
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Figure S5. Model diagnostics of the transfer function model (6) fitted to total hardness (H).
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Same convention as in Supplementary Figure 1. The Ljung-Box tests suggest the model provides
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a good fit to the data.
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Figure S6. Model diagnostics of the transfer function model (6) fitted to the natural log of TSS.
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Same convention as in Supplementary Figure 1. The Ljung-Box tests suggest the model provides
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a good fit to the data.
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Figure S7. Model diagnostics for the final model (8) fitted with all data.
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Figure S1
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Figure 4
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Figure S2
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Figure S3
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Figure S4
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Figure S5
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Figure S6
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Figure S7
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