The Swiss National Forest Inventory Data Analysis System

Abstract

The NAtional Forest Inventory Data Analysis System (NAFIDAS) is a multitier application to store and analyse Swiss National Forest Inventory (NFI) data. It provides a basis to publish findings about the current state and development of the Swiss forest on a regional and national level. The system supports the complete chain of data processing of a sample-based forest inventory, starting with the upload of field data and ending with the publication of tables and maps on the Internet. It also provides the necessary tools for administering and maintaining the system and offers access to lists of variables, derivations and dependencies, as well as to documentation of code and production processes. NAFIDAS creates concise result tables and maps with reproducible statistics on NFI data including standard errors and allows the publication of multilingual results based on the most recent NFI data. The efficient long-term storage facilities of raw and derived inventory data form the backbone of the system.

Appendix

1.1 Parameterisation of Data Analysis

On the first parameterisation page, the parameters type of analysis, inventory period(s), target variable , sampling grid(s) and reference domain are determined by the user (Fig. 20.3). Valid combinations are queried from an administrative database table, which has to be maintained manually.

On the following page, one to three classification units can be selected. Which of these is valid depends not only on the parameters already selected but also, in particular, on the hierarchy of the database tables the target variable is stored in. In some cases, a specific classification unit is also mandatory. Furthermore, if the difference between two inventories is to be analysed, only classification units are considered where the lookup values are the same.

On the third page, the method of double sampling stratification can be selected, as well as the reference unit , which depends on the type of analysis and the inventory period. The user can select a second target variable and other optional parameters, such as the number of decimal places. The choice of the second target variable has to follow certain rules because not every kind of ratio makes sense or is even valid (Table 20.7).

Table 20.7 Enhanced information on the DAA parameter (minimum set required). IDs refer to a derivation variable (column in an analysis table). Further information about the parameters based on derivation is given in Table 20.3

1.2 Equations for Point Estimators and Standard Errors

All DAA equations and explanations are taken from Lanz (2017, Chap. 7), with double sampling for stratification under simple random sampling, with stratified sampling of first-phase plots.

General Equations

• (96) $\hat{Y}={\lambda}_F^{-1}{\sum}_{k=1}^L{\lambda}_k{\hat{Y}}_k$

• (97) $\hat{\mathit{\operatorname{var}}}\left\langle \hat{Y}\right\rangle ={\lambda}_F^{-2}{\sum}_{k=1}^L{\lambda}_k^2\hat{\mathit{\operatorname{var}}}\left\langle {\hat{Y}}_k\right\rangle$

• (98) ${\hat{Y}}_k={\dot{n}}_k^{-1}{\sum}_{k.e=1}^H{\dot{n}}_{k.e}{\hat{Y}}_{k.e}$

• (99) ${\hat{Y}}_{k.e}={\ddot{n}}_{k.e}^{-1}\sum \limits_{j\in {\ddot{S}}_{k.e}}{y}_j$

• (100) $\hat{\mathit{\operatorname{var}}}\left\langle {\hat{Y}}_k\right\rangle ={\dot{n}}_k^{-1}{\left({\dot{n}}_k-1\right)}^{-1}\left[{\sum}_{k.e=1}^{k.H}{\dot{n}}_{k.e}\left({\dot{n}}_{k.e}-1\right){\ddot{n}}_{k.e}^{-1}{\hat{\sigma}}_{k.e}^2\left\langle y\right\rangle +{\sum}_{k.e=1}^{k.H}{\dot{n}}_{k.e}{\left({\hat{Y}}_{k.e}-{\hat{Y}}_k\right)}^2\right]$

• (101) ${\hat{\sigma}}_{k.e}^2\left\langle y\right\rangle ={\left({\ddot{n}}_{k.e}-1\right)}^{-1}\sum \limits_{j\epsilon {\ddot{S}}_{k.e}}{\left({y}_j-{\overline{y}}_{k.e}\right)}^2$ with ${\overline{y}}_{k.e}={\hat{Y}}_{k.e}$

• (103) $\hat{R}={\hat{Y}}_X{\hat{Y}}_Z^{-1}$

• (104a) $=\left({\lambda}_F^{-1}{\sum}_{k=1}^L{\lambda}_k{\hat{Y}}_{X.k}\right){\left({\lambda}_F^{-1}{\sum}_{k=1}^L{\lambda}_k{\hat{Y}}_{Z.k}\right)}^{-1}$

• (104b) $=\left({\sum}_{k=1}^L{\lambda}_k{\hat{Y}}_{X.k}\right){\left({\sum}_{k=1}^L{\lambda}_k{\hat{Y}}_{Z.k}\right)}^{-1}$ (which corresponds to ${\hat{R}}_k$if L = 1)

• (105) $\hat{\mathit{\operatorname{var}}}\left\langle \hat{R}\right\rangle ={\hat{Y}}_Z^{-2}{\lambda}_F^{-2}{\sum}_{k=1}^L{\lambda}_k^2\hat{\mathit{\operatorname{var}}}\left\langle {\hat{U}}_k\right\rangle$

• (106) ${\hat{Y}}_{X.k}={\dot{n}}_k^{-1}{\sum}_{k.e=1}^{k.H}{\dot{n}}_{k.e}{\hat{Y}}_{X.k.e}$

• (107) ${\hat{Y}}_{Z.k}={\dot{n}}_k^{-1}{\sum}_{k.e=1}^{k.H}{\dot{n}}_{k.e}{\hat{Y}}_{Z.k.e}$

• (108) ${\hat{Y}}_{X.k.e}={\ddot{n}}_{k.e}^{-1}\sum \limits_{j\in {\ddot{S}}_{k.e}}{x}_j$

• (109) ${\hat{Y}}_{Z.k.e}={\ddot{n}}_{k.e}^{-1}\sum \limits_{j\in {\ddot{S}}_{k.e}}{z}_j$

• (110) $\hat{\mathit{\operatorname{var}}}\left\langle {\hat{U}}_k\right\rangle ={\dot{n}}_k^{-1}{\left({\dot{n}}_k-1\right)}^{-1}\left[{\sum}_{k.e=1}^{k.H}{\dot{n}}_{k.e}\left({\dot{n}}_{k.e}-1\right){\ddot{n}}_{k.e}^{-1}{\hat{\sigma}}_{k.e}^2\left\langle u\right\rangle +{\sum}_{k.e=1}^{k.H}{\dot{n}}_{k.e}{\left({\hat{U}}_{k.e}-{\hat{U}}_k\right)}^2\right]$

• (111) ${\hat{U}}_{k.e}={\ddot{n}}_{k.e}^{-1}\sum \limits_{j\in {\ddot{S}}_{k.e}}{u}_j$

• (112) ${\hat{U}}_k={\dot{n}}_k^{-1}{\sum}_{k.e=1}^{k.H}{\dot{n}}_{k.e}{\hat{U}}_{k.e}$

• (113) ${\hat{\sigma}}_{k.e}^2\left\langle u\right\rangle ={\left({\ddot{n}}_{k.e}-1\right)}^{-1}\sum \limits_{j\epsilon {\ddot{S}}_{k.e}}{\left({u}_j-{\overline{u}}_{k.e}\right)}^2$

• (114) ${u}_j={x}_j-{z}_j\ {\hat{R}}_k$

• (115) ${\overline{u}}_{k.e}={\ddot{n}}_{k.e}^{-1}\sum \limits_{j\in {\ddot{S}}_{k.e}}{u}_j$ with ${\overline{u}}_{k.e}={\hat{U}}_{k.e}$ and, in general, ${\overline{u}}_{k.e}\ne 0$

Variation Between Sampling Units

All variance estimators contain a term ${\left(n-1\right)}^{-1}{\sum}_{j=1}^n{\left({u}_j-\overline{u}\right)}^2$. For the variance of a mean, one finds u_j = y_j and $\overline{u}=\hat{Y}$ (with $\hat{Y}\ne 0$), the sum term of the variance can be written as:

• (146) ${\sum}_{j=1}^n{\left({u}_j-\overline{u}\right)}^2\dot{=}{\sum}_{j=1}^n{\left({y}_j-\hat{Y}\right)}^2={\sum}_{j=1}^n{y}_j^2-{n}^{-1}{\left({\sum}_{j=1}^n{y}_j\right)}^2$

For the variance of a ratio, u_j needs to be pre-calculated at the plot level. Alternatively, the variance terms can be expressed as a function of the original local densities Yx and Yz. Then ${u}_j={x}_j-{z}_j\hat{R}$, $\overline{u}\ne 0$ because $\hat{R}$ is the global ratio. The sum term of the variance of a ratio of means estimator can then be written as:

• (148) ${\sum}_{j=1}^n{\left({u}_j-\overline{u}\right)}^2\dot{=}{\sum}_{j=1}^n{\left(\left({x}_j-{z}_j\hat{R}\right)-\left({n}^{-1}{\sum}_{j=1}^n\left({x}_j-{z}_j\hat{R}\right)\right)\right)}^2=$

• $\left({\sum}_{j=1}^n{x}_j^2-{n}^{-1}{\left({\sum}_{j=1}^n{x}_j\right)}^2\right)-2\hat{R}\left({\sum}_{j=1}^n{x}_j{z}_j-{n}^{-1}\left({\sum}_{j=1}^n{x}_j\right)\left({\sum}_{j=1}^n{z}_j\right)\right)$

• $+{\hat{R}}^2\left({\sum}_{j=1}^n{z}_j^2-{n}^{-1}{\left({\sum}_{j=1}^n{z}_j\right)}^2\right)$

Variation Between Double Sampling Strata

Under two-phase sampling schemes, the variance estimators contain a term in the form ${\sum}_{e=1}^H{\alpha}_e{\left({\hat{U}}_e-\hat{U}\right)}^2$. A shorthand computation of the term is:

• (152) ${\sum}_{e=1}^H{\alpha}_e\ {\left({\hat{U}}_e-\hat{U}\right)}^2=\left({\sum}_{e=1}^H{\alpha}_e{\hat{U}}_e^2\right)-{\hat{U}}^2{\sum}_{e=1}^H{\alpha}_e$

because ${\sum}_{e=1}^H{\alpha}_e{\hat{U}}_e=\hat{U}{\sum}_{e=1}^H{\alpha}_e$.

Under single-plot double sampling with stratified sampling of first-phase sample plots, ${\alpha}_e={\dot{n}}_{k.e}$ and ${\sum}_{k.e=1}^{k.H}{\dot{n}}_{k.e}{\hat{Y}}_{k.e}={\hat{Y}}_k{\sum}_{k.e=1}^{k.H}{\dot{n}}_{k.e}$

The sum term can then be written as:

• (155) ${\sum}_{e=1}^H{\alpha}_e{\left({\hat{U}}_e-\hat{U}\right)}^2\dot{=}{\sum}_{e=1}^H{\dot{n}}_e{\left({\ddot{n}}_e^{-1}\sum \limits_{j\in {\ddot{S}}_{e.j}}{x}_j\right)}^2-{\dot{n}}^{-1}{\left(\sum \limits_{e=1}^H{\dot{n}}_e\left({\ddot{n}}_e^{-1}\sum \limits_{j\in {\ddot{S}}_{e.j}}{x}_j\right)\right)}^2$

• $+{\hat{R}}^2\left\{\sum \limits_{e=1}^H{\dot{n}}_e{\left({\ddot{n}}_e^{-1}\sum \limits_{j\in {\ddot{S}}_{e.j}}{z}_j\right)}^2-{\dot{n}}^{-1}{\left({\sum}_{e=1}^H{\dot{n}}_e\left({\ddot{n}}_e^{-1}{\sum}_{j\in {\ddot{S}}_{e.j}}{z}_j\right)\right)}^2\right\}$

• $-2\hat{R}\left\{{\sum}_{e=1}^H{\dot{n}}_e\left({\ddot{n}}_e^{-1}{\sum}_{j\in {\ddot{S}}_{e.j}}{x}_j\right)\left({\ddot{n}}_e^{-1}{\sum}_{j\in {\ddot{S}}_{e.j}}{z}_j\right)-{\dot{n}}^{-1}\left[\left({\sum}_{e=1}^H{\dot{n}}_e\left({\ddot{n}}_e^{-1}{\sum}_{j\in {\ddot{S}}_{e.j}}{x}_j\right)\right)\left({\sum}_{e=1}^H{\dot{n}}_e\left({\ddot{n}}_e^{-1}{\sum}_{j\in {\ddot{S}}_{e.j}}{z}_j\right)\right)\right]\right\}$