paprox: Calculating P-approximation coefficients
In capn: Capital Asset Pricing for Nature

Description Usage Arguments Details Value References See Also Examples

The function provides the P-approximation coefficients of the defined Chebyshev polynomials in aproxdef. For now, only unidimensional case is developed.

1	paprox(aproxspace, stock, sdot, dsdotds, dwds)

`aproxspace`	An approximation space defined by `aproxdef` function
`stock`	An array of stock, s
`sdot`	An array of ds/dt, \dot{s}=\frac{ds}{dt}
`dsdotds`	An array of d(sdot)/ds, \frac{d \dot{s}}{d s}
`dwds`	An array of dw/ds, \frac{dW}{ds}

The P-approximation is finding the shadow price of a stock, p from the relation:

p(s) = \frac{W_{s}(s) + \dot{p}(s)}{δ - \dot{s}_{s}},

where W_{s} = \frac{dW}{ds}, \dot{p}(s) = \frac{dp}{ds}, \dot{s}_{s} = \frac{d\dot{s}}{ds} , and δ is the given discount rate.

Consider approximation p(s) = \mathbf{μ}(s)\mathbf{β}, \mathbf{μ}(s) is Chebyshev polynomials and \mathbf{β} is their coeffcients. Then, \dot{p} = diag (\dot{s}) \mathbf{μ}_{s}(s)\mathbf{β} by the orthogonality of Chebyshev basis. Adopting the properties above, we can get the unknown coefficient vector β from:

\mathbf{μ}\mathbf{β} = diag ≤ft( δ - \dot{s}_{s} \right)^{-1} ≤ft( W_{s} + diag (\dot{s}) \mathbf{μ}_{s} \mathbf{β} \right) , and thus,

\mathbf{β} = ≤ft( diag ≤ft( δ - \dot{s}_{s} \right) \mathbf{μ} - diag (\dot{s}) \mathbf{μ}_{s} \right)^{-1} W_{s} .

In a case of over-determined (more nodes than approaximation degrees),

≤ft( ≤ft( diag ≤ft( δ - \dot{s}_{s} \right) \mathbf{μ} - diag (\dot{s}) \mathbf{μ}_{s} \right)^{T} ≤ft( diag ≤ft( δ - \dot{s}_{s} \right) \mathbf{μ} - diag (\dot{s}) \mathbf{μ}_{s} \right) \right)^{-1} ≤ft( diag ≤ft( δ - \dot{s}_{s} \right) \mathbf{μ} - diag (\dot{s}) \mathbf{μ}_{s} \right)^{T} W_{s}

For more detils see Fenichel et al. (2016).

A list of approximation resuts: deg, lb, ub, delta, and coefficients. Use results$item
(or results[["item"]]) to import each result item.

`degree`	degree of Chebyshev polynomial
`lowerB`	lower bound of Chebyshev nodes
`upperB`	upper bound of Chebyshev nodes
`delta`	discount rate
`coefficient`	Chebyshev polynomial coefficients

Fenichel, Eli P. and Joshua K. Abbott. (2014) "Natural Capital: From Metaphor to Measurement." Journal of the Association of Environmental Economists. 1(1/2):1-27.
Fenichel, Eli P., Joshua K. Abbott, Jude Bayham, Whitney Boone, Erin M. K. Haacker, and Lisa Pfeiffer. (2016) "Measuring the Value of Groundwater and Other Forms of Natural Capital." Proceedings of the National Academy of Sciences .113:2382-2387.

aproxdef,psim

## 1-D Reef-fish example: see Fenichel and Abbott (2014)
data("GOM")
nodes <- chebnodegen(param$nodes,param$lowerK,param$upperK)
simuDataP <- cbind(nodes,sdot(nodes,param),
                   dsdotds(nodes,param),dwds(nodes,param))
Aspace <- aproxdef(param$order,param$lowerK,param$upperK,param$delta)
pC <- paprox(Aspace,simuDataP[,1],simuDataP[,2],
             simuDataP[,3],simuDataP[,4])