Why do I get linear model when I tried to fit exponential model?

Question

I was wondering why do I get linear model when I'm using exponential model, y = a * exp(-b*-x), to fit my data.

Here is my code:

ff <- function(x,a,b){a * exp(-b*-x)}
fit2 <- nls(y ~ ff(x,a,b) , data = newdat, start =c(a=107.4623,b=-0.0037)

The graph below is mydata with the exponential fit (prediction of fit2) in purple curve. The green curve is what I though it would be, it is Smooth.splines fit.

Result from fit2:

Nonlinear regression model
  model: dif2 ~ ff(age, a, b)
   data: newdat
         a          b 
109.743680  -0.003793 
 residual sum-of-squares: 2585

Number of iterations to convergence: 2 
Achieved convergence tolerance: 1.446e-06

Here is my data:

   ID  x   y
    1 18 106.47
    1 19 100.35
    1 20 97.4
    1 21 101.03
    1 22 100.3
    1 23 99.06
    1 24 100.81
    2 18 101.95
    2 19 100.69
    2 20 100.89
    3 14 105.87
    3 15 107.44
    3 16 103.05
    3 17 104.86
    3 18 101.86
    3 19 101.48
    3 20 102.77
    3 21 99.63
    3 22 100.21
    3 23 101.28
    3 24 98.77
    3 25 99.91
    4 17 102.42
    4 18 101.85
    4 19 101.31
    5 18 101.24
    5 19 102.27
    5 20 100.03
    5 21 101.53
    6 20 98.08
    6 21 101.2
    6 22 103.16
    6 23 98.3
    6 24 102.21
    6 25 100.18
    6 27 95.28
    6 28 102.05
    6 29 100.72
    6 30 101.4
    7 13 111.3
    7 14 106.55
    7 15 103.23
    7 16 102.31
    7 17 101.11
    7 18 101.52
    7 19 100.14
    8 18 101.05
    8 19 98.15
    8 20 100.55
    8 21 101.62
    8 22 101.04
    8 23 98.22
    9 18 102.87
    9 19 101.46
    9 20 101.07
    9 21 101.32
    10 20 101.93
    10 21 101.73
    10 22 100.24
    11 19 99.75
    11 20 101.35
    11 21 99.34
    11 22 100.12
    12 18 103.34
    12 19 109.52
    12 20 106.98
    12 21 105.21
    12 22 98.87
    12 23 103.81
    12 24 100.38
    12 25 100.12
    12 26 99.7
    12 27 101.16
    12 28 99.02
    12 29 100.15
    12 30 97.32
    13 13 116.43
    13 14 111.75
    13 15 107.42
    13 16 103.5
    13 17 103.37
    13 18 100.66
    13 19 100.73
    13 20 100.84
    13 21 100.05
    14 18 101.66
    14 19 99.9
    14 20 101.4
    14 21 99.86
    14 22 100.82
    15 15 101.27
    15 16 100.01
    15 17 104.27
    16 19 100.26
    16 20 104.13
    17 18 106.12
    18 21 101.18
    18 22 99.51
    18 23 100.59
    19 18 100
    19 19 100.81
    19 20 99.37
    19 21 102.6
    20 22 102.18
    20 23 104.5
    20 24 100.74
    21 22 103.74
    21 23 98.66
    21 24 100.65
    21 25 99.63
    22 24 102.59
    22 25 94.62
    22 26 103.85
    23 20 100.7
    23 21 101.38
    23 22 102.36
    23 23 99.56
    23 24 100
    24 18 101.16
    24 19 99.64
    25 21 96.9
    25 22 109.3
    25 23 101.4
    25 24 98.04
    25 25 99.28
    25 26 99.63
    25 27 101.29
    25 28 100.08
    26 14 109
    26 15 112.37
    26 16 102.4
    26 17 102.15
    26 18 100.82
    27 18 101.14
    27 19 101.38
    28 17 105.09
    28 18 101.74
    28 19 100.2
    29 19 102.11
    29 20 100.57
    29 21 100.91
    29 22 99.61
    29 23 99.99
    30 18 99.81
    30 19 102.07
    31 19 100.75
    31 21 95.43
    32 23 99.73
    32 24 100.8
    32 25 100.1
    32 26 100.88
    32 27 97.73
    32 28 100.36
    33 22 99.4
    33 24 101.46
    33 18 97.65
    33 25 102.75
    33 26 97.7
    33 27 100.67
    34 21 98.27
    34 22 100.42
    34 23 101.16
    34 24 100.13
    34 25 98.55
    35 17 107.46
    35 18 100.22
    35 19 102.03
    35 20 101.52
    35 21 102.05
    35 22 102.46
    35 23 101.56
    35 24 96.88
    35 25 98.97
    35 26 101.68
    35 28 94.12
    36 20 98.63
    36 21 101.59
    36 22 98.76
    37 19 101.9
    37 20 98.66
    37 21 100.19
    37 22 100.03
    37 23 99.97
    38 15 104.32
    38 16 102.98
    38 17 103.4
    38 18 102.78
    38 19 101.73
    38 20 95.57
    39 22 101.5
    39 23 98.37
    39 24 100.4
    39 25 100.79
    40 19 102.93
    40 20 100.88
    40 21 99
    40 22 99.66
    41 21 107.08
    41 22 93.08
    41 24 100.91
    41 25 107.24
    41 26 99.8
    42 14 109.82
    42 15 106.09
    42 16 106.32
    42 17 102.8
    42 18 100.21
    42 19 102.08
    42 21 99.22
    42 22 100.13
    42 23 101.63
    43 16 100.95
    43 17 100.6
    43 18 101.81
    43 19 102.78
    43 20 98.43
    43 23 101.4
    43 24 103.12
    43 25 99.31
    43 26 100.47
    43 27 99.67
    43 28 98.75
    43 29 95.68
    44 23 103.78
    44 24 100.38
    44 25 99.39
    44 26 100.87
    44 27 99.64
    44 28 98.39
    44 29 97.62
    45 18 100.47
    45 19 101.41
    45 20 99.33
    45 21 101.08
    45 22 100.08
    45 23 100.22
    45 24 99.67
    45 25 100.45
    45 26 102.4
    45 27 95.7
    46 20 101.35
    46 21 98.73
    46 22 109.29
    46 23 100.04
    46 24 95.74
    46 25 100.44
    46 26 98.72
    47 19 100.51
    47 20 99.88
    47 21 101.7
    47 22 101.94
    47 23 100.72
    47 24 98.73
    47 25 102.16
    47 26 100.25
    47 27 95.1
    47 28 103.08
    48 25 105.21
    48 26 100.48
    48 27 98.07
    48 28 99.88
    48 29 95.61
    49 16 111.35
    49 17 92.43
    49 18 112.04
    49 19 100.8
    49 20 95.36
    49 21 103.13
    49 22 102.16
    49 23 98.81
    49 25 98.86
    49 26 99.93
    49 27 95.26
    50 23 98.15
    50 24 105.93
    50 25 99.01
    50 26 99.34
    50 27 93.68
    50 28 105.35
    51 24 100.96
    51 25 100.53
    51 26 99.2
    51 27 100.52
    51 28 100.86
    52 25 101.38
    52 26 98.45
    52 27 100.32
    52 28 99.24
    52 29 102.74
    53 24 101.37
    53 25 99.75
    53 27 96.31
    53 28 100.67
    54 22 98.09
    54 23 100.55
    54 24 100.25
    54 25 101.54
    54 26 98.48
    54 27 102.76
    54 28 98.5
    54 30 99.85
    55 22 103.87
    55 23 94.37
    55 24 105.12
    56 18 101.23
    56 19 99.26
    56 20 102.63
    56 21 100.75
    56 23 101.5
    56 24 99.14
    56 27 95.11
    57 16 107.57
    57 17 101.75
    57 18 107.18
    57 19 100.23
    57 20 105.48
    57 21 103.1
    57 22 100.45
    57 23 99.28
    57 24 100.52
    57 25 98.69
    58 27 103.13
    58 28 97.86
    58 29 101.33
    58 30 98.33
    58 32 102.14
    58 34 94.47
    58 35 98.29
    59 19 97.6
    59 20 98.93
    59 22 101.35
    59 23 93.88
    60 20 99.62
    60 22 97.36
    60 23 102.94
    60 24 98.98
    60 25 99.47
    61 18 100.15
    61 19 101.92
    61 20 101.34
    61 21 98.87
    61 22 97.68
    61 23 99.92
    61 24 100.78
    61 25 98.21
    62 20 102.7
    62 21 99.7
    62 22 100.17
    62 23 99.62
    62 24 100.59

Answer 1

I can't say precisely why your loess fit differs from the exponential fit -- that's more less "because it does, because they're different" -- but the reason that your exponential fit looks so linear, and why it looks so different from your plotted function, is that over the range of the data it is very close to linear. The parameter is -0.0037, the range of the data is about delta-x=20, so the curve only falls by about 7%. For this range, the expansion $\exp(bx) = 1+bx + O((bx)^2)$ works pretty well.

Update: your original (slightly mangled) nls fit was $y=a \exp(bx)+d$, i.e. there was an additive term. This makes all the difference:

dd <- read.table("expreg.dat",header=TRUE)
m0 <- nls(y~a*exp(b*x),dd,
          start=list(a=100,b=-0.1))
coef(m0)
##             a             b 
## 109.743701855  -0.003793346 

m1 <- nls(y~a*exp(b*x)+d,dd,
          start=list(a=100,b=-0.1,d=60))
coef(m1)
##            a            b            d 
## 1401.8573693   -0.3599526   99.7863827 

For what it's worth, you can fit the model without the additive term using glm(...,family=gaussian(link="log")), which is convenient for incorporating in ggplots.

dd$pred <- predict(m1)
library("ggplot2"); theme_set(theme_bw())
g0 <- ggplot(dd,aes(x,y))+geom_point(aes(colour=factor(ID)))+
    geom_smooth(method="glm",family=gaussian(link="log"))+
        scale_colour_discrete(guide="none")
g2 <- g0 + geom_line(aes(y=pred),colour="red")

Now plot over a wider range:

g1 <- g0 + expand_limits(x=c(0,240),y=c(0,120))+
    geom_smooth(method="glm",family=gaussian(link="log"),
                fullrange=TRUE)

Answer 2

ff <- function(x,a,b){a * exp(-b*-x)+d}
fit2 <- nls(y ~ ff(x,a,b) , data = newdat, start =c(a=107.4623,b=-0.0037)

In your function d is not defined! I had to remove d from the ff function, and I ended up estimating only a and b. In this way I obtained the following fit...

Formula: y ~ a * exp(-b * -x)

Parameters:
Estimate Std. Error t value Pr(>|t|)    
a 109.743724   0.924359  118.72   <2e-16 ***
b  -0.003793   0.000379  -10.01   <2e-16 ***
---
Signif. codes:  
0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.725 on 348 degrees of freedom

Number of iterations to convergence: 2 
Achieved convergence tolerance: 1.105e-06

How can your code work if d is not defined inside the function ff and you are not passing it as argument?