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FSS_FractionScoreSkill

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思想

问题

http://shuwenlovestudy.top/FSS_FractionScoreSkill%20/%E5%BC%95%E5%85%A5%E9%97%AE%E9%A2%98.png

对于如上预测场和观测场，虽然降水都分布在7个降水格点上，但位置均未能一一对应，所以其TS评分等评分为0（没有降水预报技巧），但从概率角度看，观测与预报的降水发生概率是一样的，7/49，具有相同的降水预报面积。

FSS

作者：Robert、Lean（2008）

作用：考察不同尺度内降水发生概率

具体公式

按步骤

  1. Brier评分的变形：比较预报与观测的降水频率 FBS (Fraction Brier Score)

    公式：$FBS = \frac{1}{N} \displaystyle \sum_{N}(P_{fcst} - P_{obs})^2$

    $P_{fcst}$、$P_{obs}$：每个领域尺度内 预报与预测降水发生概率（0 - 1）

    N：网格数量

  2. 方差技巧评分：获取正定的（特征值均为正）评分

    公式：$FSS = 1 - \frac{FBS} {\frac{1}{N}(\displaystyle \sum_N P_{fcst}^2 + \displaystyle \sum_N P_{obs}^2)}$

    FSS

    1. 范围（0 - 1）

      0 ：完全不匹配

      1：完美匹配

    2. 特点：
      领域尺度增加 -> FSS评分增大

      当$P_{fcst} = bP_{obs}$，FSS评分向 $2b/(b^2+1)$ 渐近

简写

$FSS = 1 - \frac{\frac{1}{N} \displaystyle \sum_{N}(P_{fcst} - P_{obs})^2} {\frac{1}{N}(\displaystyle \sum_N P_{fcst}^2 + \displaystyle \sum_N P_{obs}^2)}$

代码

R语言

代码

链接：https://rdrr.io/cran/verification/src/R/fss.R#heading-0

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fss <- function(obs, pred, w = 0, FUN = mean, ...){
### compare matrixes of forecast of observed values and forecast.
### values can be calcuated using different windows.

### with a window size of 0, obs is returned.
obs.matrix <- matrix.func(DAT = obs, w = w, FUN = FUN) 
  
### with a window size of 0, obs is returned.
frcs.matrix <- matrix.func(DAT = pred, w = w, FUN = FUN)   

if(nrow(obs)!= nrow(pred) &  ncol(obs)!=  nrow(obs) ) stop("Observation matrix and forecast matrix different sizes")
  
n   <- prod(dim(obs.matrix))  ### number of gridpoints

N   <- sum((obs.matrix-frcs.matrix)^2, na.rm = TRUE)/n ### numerator
D   <- (sum(obs.matrix^2, na.rm = TRUE) +sum(frcs.matrix^2, na.rm = TRUE))/n ### denominator

FSS <- 1 - N/D  
return(FSS)
}

matrix.func <- function(DAT, w = 0, FUN = mean, ...){

### w is the '' radius'' of window.  eg. w = 2, defines a 5 by 5 square
  
### define function
FUN <- match.fun(FUN)

### define output dimension
II <- nrow(DAT) - 2*w   ## output row dimension
JJ <- ncol(DAT) - 2*w  

if(JJ<=0|II <= 0) {stop("The window exceeds the size of the observation" ) } 
OUT <- matrix(NA, nrow= II, ncol = JJ)

for(i in 1:II){
for(j in 1:JJ){
sub <- DAT[ i :(i + 2*w ),
             j :(j + 2*w ) ] # subset data

OUT[i,j] <- FUN(sub,...)
}  ## close J
}  ## close I

return(OUT)
}  ## close functionr

测试

链接：https://rdrr.io/cran/verification/man/fss.html

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library(verification)

grid<- list( x= seq( 0,5,,100), y= seq(0,5,,100)) 
obj<-Exp.image.cov( grid=grid, theta=.5, setup=TRUE)
look<- sim.rf( obj)
look[look < 0] <- 0

look2 <- sim.rf( obj)
look2[look2<0] <- 0

fss(look, look2, w=5)


## Not run: 
#  The following example replicates Figure 4 in Roberts and Lean (2008).
####      examples

LAG <- c(1,3,11,21)
box.radius <- seq(0,24,2)

OUT <- matrix(NA, nrow = length(box.radius), ncol = length(LAG) )

for(w in 1:4){

FRCS <- OBS <- matrix(0, nrow = 100, ncol = 100)

obs.id        <- 50
OBS[,obs.id]  <- 1

FRCS[, obs.id + LAG[w]] <- 1

for(i in 1:length(box.radius)){

OUT[i, w] <- fss(obs = OBS, pred = FRCS, w = box.radius[i] )

} ### close i
} ### close w

b <- mean(OBS) ## base rate

fss.uniform  <- 0.5 + b/2
fss.random   <- b

matplot(OUT, ylim = c(0,1), ylab = "FSS", xlab = "grid squares", 
type = "b", lty = 1, axes = FALSE , lwd = 2)

abline(h = c(fss.uniform, fss.random), lty = 2)  
axis(2)
box()
axis(1, at = 1:length(box.radius), lab = 2*box.radius + 1)
grid()

legend("bottomright", legend = LAG, col = 1:4, pch = as.character(1:4), 
 title = "Spatial Lag", inset = 0.02, lwd = 2 )

## End(Not run)

结果截图

http://shuwenlovestudy.top/FSS_FractionScoreSkill%20/r%E8%AF%AD%E8%A8%80%E7%BB%93%E6%9E%9C%E5%9B%BE.png

Python

代码

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import numpy as np 

def func(data, w = 0):
    row = data.shape[0]
    col = data.shape[1]
    
    ii = row - 2 * w
    jj = col - 2 * w
    
    if(ii <= 0 or jj <=0):
        print("error!")
        
    out = np.zeros((ii, jj))
    
    for i in range(1, ii):
        for j in range(1, jj):
            
            sub = data[i : (i + 2 * w + 1), j : (j + 2 * w + 1)]
            
            out[i, j] = np.mean(sub)
    return out

def fss(obs, fc, w = 0):
    obs_m = func(obs, w)
#     print(obs_m[0][16])
#     print(obs_m[0][17])
#     print(obs_m[0][18])
    fc_m = func(fc, w)
#     print(obs_m)
#     print(fc_m)
    n = obs_m.shape[0] * obs_m.shape[1]
    N = ((obs_m - fc_m) ** 2).sum() / n
#     print(N.shape)
    D = ((obs_m ** 2).sum() + (fc_m ** 2).sum()) / n
#     print(n)
#     print(N)
#     print(D)
    return  1 - N / D

测试

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import matplotlib.pyplot as plt

lag = [1, 3, 11, 21]
radius = np.arange(0, 24 + 2, 2)

out = np.zeros((len(radius), len(lag)))

frcs = np.zeros((100, 100))
obs = np.zeros((100, 100))

id = 50 - 1
obs[ : , id] = 1

for l in range(0, len(lag)):
    for r in range(0, len(radius)):   
        frcs[ : , id + lag[l - 1]] = 0
        frcs[ : , id + lag[l]] = 1
        out[r][l] = fss(obs, frcs, radius[r])
b = np.mean(obs)
uniform = 0.5 + b / 2
random = b

plt.plot(out)
plt.ylabel("FSS")
plt.xlabel("grid squares")

结果截图

http://shuwenlovestudy.top/FSS_FractionScoreSkill%20/python%E7%BB%93%E6%9E%9C%E5%9B%BE.png

坐标可以自定义修改，值是一样的

http://shuwenlovestudy.top/FSS_FractionScoreSkill%20/%E7%BB%93%E6%9E%9C%E7%9F%A9%E9%98%B5.png

下载

Roberts2008原文

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