The aggregation function of an mm() block

Selects the function f that reduces the weighted member records to a group-level feature — the f in the framework's \(\theta^{micro,f}(M_{it})\). fn("sum") (the default) is the additive weighted mean; the other named types are "emergent" features of the whole member set. fn() also accepts a one-sided formula written in a restricted expression DSL, so users can define their own reductions.

Named types:

fn("sum")                          # A_x  = sum_k w_k x_k          (default)
fn("var")                          # V_x  = sum_k w_k (x_k - A_x)^2
fn("var", moment = 3)              # higher central moments
fn("hhi")                          # C    = sum_k w_k^2  (weights only)
fn("effn")                         # 1 / C
fn("entropy")                      # -sum_k w_k log w_k
fn("threshold", c = 0.7, kappa = 10)  # T(c) = sum_k w_k ilogit(kappa (x_k - c))
fn("threshold", c = est())         # estimate the cutpoint
fn("smax", kappa = 5)              # (1/kappa) log sum_k w_k exp(kappa x_k)
fn("smax", kappa = est())          # estimate the aggregation function itself:
                                   #   kappa<0 -> min, kappa->0 -> mean, kappa>0 -> max
fn("gmean", p = est())             # power/CES mean (sum_k w_k x_k^p)^(1/p); x > 0
fn("cov")                          # C_xz = sum_k w_k (x_k - A_x)(z_k - A_z); two attributes

Expression DSL: w() normalizes weights to a probability measure over the group's members; E(e) is the expectation \(\sum_k w_k e_k\) under that measure. Member-level quantities are the vars() attributes and the reserved symbol w; anything wrapped in E() is a group scalar. Whitelisted operations: + - * / ^, exp, log, ilogit, pow. Any symbol that is not a data column or reserved word is a free parameter with a default prior (one rule, shared with w); the build messages the detected parameters.

fn(~ E((x - E(x))^2))                       # variance, written out
fn(~ E(ilogit(kappa * (x - c))))            # threshold with free c, kappa
fn(~ E((x - E(x)) * (z - E(z))))            # covariance

Identification: internal parameters reach the outcome only through b * feature; if the feature's coefficient is ~0 they are unidentified. Dispersion functions need real within-group spread; threshold/tail functions need mass of x in the region they read. A free parameter that multiplies an attribute inside a nonlinear E() is confounded with the feature's coefficient — treat inner parameters as shape parameters (cutpoints, sharpness), not slopes.

Usage

fn(type = "sum", ...)

Arguments

type

A type string (one of "sum", "var", "hhi", "effn", "entropy", "threshold", "smax", "gmean", "cov") or a one-sided formula (~ E(...)).

...

Shape parameters for the named types: moment (var), c and kappa (threshold), kappa (smax), p (gmean). Each is a number (fixed) or est() (estimated).

Value

A bml_fn object.

See also

mm, w, est, prior