fit exponential or hockey stick decay

Question

I'm trying to find an equation to fit my data. I recognize the shape to be y=-exp(x) but nls(y~-a*exp(x*b)) with various parameter start values fail. y is negative so the "easy" fit of log(y)~log(a)+b x doesn't work well. I tried log(y+2)~a+b x to make everything positive but that didn't result in the right shape.

Can someone provide some help for fitting this data? Thanks!

y=swediff

x=avgdate

dat2=structure(list(swediff = c(0.0379635202678687, 0.0845477936160927, 
0.146010217481196, 0.0416237104326292, 0.0659140490644253, 0.134535534695029, 
0.0095147654468483, 0.238456044233877, 0.276025694437364, 0.29435448415394, 
0.00301157777812485, 0.19171002685605, 0.277759059448242, 0.00400780564144798, 
0.342605838471721, 0.236804884903432, 0.151048712082562, 0.188620966368049, 
-0.0615972418208484, -0.00184933102124457, -0.0163171325413688, 
0.00370250929658511, 0.30014673206306, 0.135354035472228, 0.00699671782210069, 
0.0174510674253347, -0.0145499677497698, 0.0113155610814752, 
-0.0683884523999768, 0.20157093417998, 0.186320361855075, -0.115609443650563, 
0.069177592825418, -0.0161221161393796, 0.150181081582068, 0.0632121126749741, 
0.0769960292118834, 0.061783685314432, 0.0442014176783082, -0.00990798027657931, 
-0.00186219548019918, 0.0274216740478325, 0.118878480695049, 
0.0592089915185285, -0.00823096478874009, 0.120750948230554, 
0.278594307094423, -0.0111994006625954, 0.0379360193757585, 6.29460162030332e-05, 
0.0602068958909111, 0.173755367986025, 0.135902420389977, 0.124863098282806, 
0.0706487190649132, 0.284186487140106, 0.395475978216811, 0.207437141623493, 
0.125297332063683, 0.173198938943319, 0.286364692763195, 0.187181622294097, 
0.0155249739392275, 0.0284811383447792, 0.3302152196822, 0.279809724823687, 
0.137578862533244, 0.116541887214791, 0.0092271882470954, -0.244303611220416, 
-0.0553353805122039, 0.000651263069993122, 0.095154425644194, 
0.208713883186232, 0.188079755592783, 0.205982363732731, 0.0411534316934769, 
0.0245409797776046, 0.309614117523271, 0.355492037841145, 0.288275778246439, 
0.321807979017761, 0.287947161484914, 0.156217353669998, 0.172890522219893, 
-0.0179090707980069, 0.0179643473340213, 0.25572307647916, 0.129243837667331, 
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0.167357636244551, 0.196797358553372, 0.0360344219433584, 0.104666349576576, 
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0.216084208573312, 0.0706260521616783, 0.0756847641524316, 0.0770749593558848, 
0.117128900658644, 0.176013651395269, -0.0908276818873188, -0.264146913474664, 
-0.397227204225366, -0.520243226629073, -0.315371107607145, -0.317452140016727, 
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-0.637940256805166, -0.27643564304224, -0.0101249183839163, 0.286080813914053, 
0.185985595857955, 0.0607903622684369, 0.0739324892640134, 0.0613910749095987, 
0.081924014991581, 0.191652099453867, 0.0671457700438239, -0.239590260546166, 
-0.153731321264683, -0.925001640233871, -0.768208799308804, -0.953874558691501, 
-0.879388655106864, -0.731569006141193, -0.80662535340525, -0.546541080560938, 
-0.0290726402048612, 0.0899201018225928, -0.427683149429864, 
-0.642914036196173, -0.00975702612958795, 0.108679758607406, 
0.115330348818079, 0.0636578050454333, 0.12389689549272, 0.0784572603799192, 
0.0984567282801977, -0.192624142725435, -0.512093030098511, -0.478389465413485, 
-0.621953473157772, -0.14258209361421, 0.215855068127397, 0.10933805943701, 
0.0371862858446161, 0.147743925023497, 0.110106354151748, 0.167891959372529, 
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0.118485470250309, 0.123284157640724, 0.00262964701743046, -0.324425563039869, 
0.0535619813150857, 0.172683623868712, 0.127449390525177, 0.268301759367799, 
-0.00433449818031201, -0.371211977194441, -0.457630978639089, 
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-0.310697158309397, -0.146761733349917, 0.0236095942741553, 0.00995413363968556, 
-0.220037507344027, -0.298201547051362, -0.228524953982665, 0.139997538150043, 
-0.0164089841500689, 0.0480316934068684, -0.0202200927067654, 
0.210784261132325, 0.235666847665425, 0.187862324743091, -0.0954145478677223, 
-0.198055375090692, -0.299881653134342, -0.693717473430792, -0.826874682094633, 
-0.537211297019843, 0.0727237791693081, 0.252505366529933, -0.317726197889352, 
-0.279117149104267, -0.0368974550397417, 0.292430819817608, 0.14796346005382, 
0.037310174880905, 0.111369531415814, 0.222644841684763, 0.232955939088023, 
0.0985704990997788, -0.045553014664072, -0.207731060229671, -0.229935960514404, 
-0.686597725938266, -0.247064364067584, -0.371521331373238, -0.430762168759812, 
-0.196945305918869, 0.215658033923907, 0.390228305203605, 0.300921043791892, 
0.043076468096275, -0.00079320334809857, 0.252832538725812, -0.0267246881907509, 
-0.0904732083287169, -0.254702182910177, -0.289199140111472, 
-0.903663808758846, -0.730480757062446, -0.399450175857138, -0.204946388704593, 
0.167241850751596, 0.108062173538109, 0.158279079258542, 0.110014597255262, 
-0.0307183404739954, -0.105426447364233, 0.133090339859816, -0.178600042668314, 
-0.271699090774754, -0.342264904547807, -1.09265736425287, -0.699172072891869, 
-0.0979254975665448, 0.268596423677819, 0.155665651298694, 0.0992990830653245, 
0.277629262139193, 0.176822125972084, -0.103178451026634, -0.461191269578747, 
-0.216966326644849, -0.1084886076781, -0.154140082268591, -0.652188752264078, 
-0.657623380576153, -0.0784736154290889, 0.0742726185810828, 
0.0709763425947463, 0.294940653726301, 0.293652811592831, 0.0640793559532362, 
-0.0367757730565124, -0.415018906491611, -0.538095246521145, 
-0.18261089894835, -0.566401860686253, -0.194554300183061, -0.0327973145856545, 
0.106726178908186, 0.0748878637317865, 0.0234611871797437, 0.218270627297708, 
-0.15143333931606, -0.028720599313675, -0.604914220546465, -0.484120047066584, 
-0.629905362306152, -0.363942052067613, -0.1558791539258, 0.0559939704995262, 
-0.267565269947471, -0.507879399175005, -0.163173363661096, -0.0301926306852119, 
0.0141783247953152, -0.078676708722438), avgdate = c(107.642857142857, 
108.142857142857, 111.928571428571, 101.857142857143, 80.1428571428571, 
78.1428571428571, 120.214285714286, 114.785714285714, 110.428571428571, 
112.5, 111.571428571429, 103.214285714286, 68.1428571428571, 
85, 79.6428571428571, 78.3571428571429, 100, 99.6428571428571, 
78.2142857142857, 116.285714285714, 82.5714285714286, 84.4285714285714, 
87, 65.7142857142857, 54.8571428571429, 77.2142857142857, 113.214285714286, 
99.7142857142857, 98.4285714285714, 116.428571428571, 128.928571428571, 
102.785714285714, 96.5, 121.071428571429, 115.214285714286, 116.5, 
114.785714285714, 53, 45.2142857142857, 39.5, 72.5714285714286, 
84.4285714285714, 112.571428571429, 74.3571428571429, 96.8571428571429, 
118.571428571429, 159.642857142857, 164.214285714286, 20.2857142857143, 
49.1428571428571, 59.7142857142857, 43.5, 60.8571428571429, 117, 
64.6428571428571, 31.2857142857143, 16.5714285714286, 16.5714285714286, 
32.3571428571429, 33.5, 42.0714285714286, 47.4285714285714, 68.7142857142857, 
113.071428571429, 135, 154.214285714286, 159.714285714286, 167.5, 
167.642857142857, 143.642857142857, 97.4285714285714, 40.0714285714286, 
22.2142857142857, 18.5714285714286, 39.6428571428571, 66.2857142857143, 
80.3571428571429, 56.6428571428571, 28.7142857142857, 30.8571428571429, 
19.8571428571429, 18.7857142857143, 85, 138.142857142857, 121.357142857143, 
104.785714285714, 64.4285714285714, 147.785714285714, 161.285714285714, 
171.214285714286, 173.357142857143, 168.214285714286, 170.5, 
172.5, 33.5, 47.2857142857143, 134.214285714286, 101.571428571429, 
66.7857142857143, 62.8571428571429, 75.8571428571429, 57.3571428571429, 
135.785714285714, 131.357142857143, 140.214285714286, 167.214285714286, 
171.857142857143, 157, 120.928571428571, 92.2857142857143, 90.2142857142857, 
127.5, 137.285714285714, 167.785714285714, 170.857142857143, 
169.714285714286, 180.714285714286, 174.857142857143, 191.071428571429, 
162.857142857143, 156.142857142857, 143.642857142857, 112.071428571429, 
155.571428571429, 155.928571428571, 156.714285714286, 168.142857142857, 
173.5, 180.428571428571, 175.642857142857, 152.642857142857, 
144.214285714286, 119, 141.428571428571, 164.714285714286, 175.142857142857, 
166.428571428571, 129.571428571429, 143.071428571429, 162.571428571429, 
NA, 187.357142857143, 167.285714285714, 159.642857142857, 152.785714285714, 
174.142857142857, 180.357142857143, 180.357142857143, 173.642857142857, 
177.642857142857, 170.357142857143, 158.928571428571, 141.357142857143, 
117.571428571429, 136.571428571429, 118.714285714286, 165.785714285714, 
169.5, 171.357142857143, 196.785714285714, 198.428571428571, 
195.428571428571, 193.642857142857, 185, 191.928571428571, NA, 
195.642857142857, 169, 168.642857142857, 162.928571428571, 147, 
136.928571428571, 93.6428571428571, 105.285714285714, 130.428571428571, 
139.714285714286, 145.214285714286, 160.928571428571, 164.714285714286, 
171.785714285714, 174.5, 185, 187.071428571429, 191.5, 190.214285714286, 
NA, 173.357142857143, 171.285714285714, 165.214285714286, 186.642857142857, 
178.714285714286, 169.928571428571, 132.428571428571, 133.357142857143, 
137.5, 141.357142857143, 152.642857142857, 151.428571428571, 
157.142857142857, 172.142857142857, 166.714285714286, 187.785714285714, 
179.857142857143, 164.428571428571, 141.357142857143, 140.357142857143, 
174.357142857143, 142.214285714286, 126.285714285714, 150.571428571429, 
154.571428571429, 128.857142857143, 141.785714285714, 163.5, 
169, 165.714285714286, 175.071428571429, 170.285714285714, 180.071428571429, 
180.5, 178.357142857143, 167.285714285714, 152.357142857143, 
136.714285714286, 154.5, 162.571428571429, 168.714285714286, 
164.428571428571, 164, 129.214285714286, 134.142857142857, 152.428571428571, 
169, 169.5, 174.857142857143, 174.857142857143, 177.285714285714, 
188.071428571429, 191.214285714286, 184.5, 176.214285714286, 
170.142857142857, 162.428571428571, 174, 173.428571428571, 167.785714285714, 
165.785714285714, 137.714285714286, 147, 145.642857142857, 141.571428571429, 
150.5, 158.428571428571, 172.428571428571, 174.285714285714, 
171.285714285714, 185.285714285714, 190.857142857143, 177.214285714286, 
166.571428571429, 167.571428571429, 169.428571428571, 159.357142857143, 
156.357142857143, 142.571428571429, 136.928571428571, 158.714285714286, 
159.071428571429, 146.928571428571, 149.214285714286, 152.285714285714, 
159.928571428571, 162.785714285714, 174, 179.285714285714, 181.571428571429, 
176.428571428571, 176.071428571429, 169.285714285714, 150.428571428571, 
134.785714285714, 144, 152.714285714286, 163.785714285714, 159.428571428571, 
172.714285714286, 164.642857142857, 167.857142857143, 165.642857142857, 
179.571428571429, 179.142857142857, 144.428571428571, 176.357142857143, 
152.285714285714, 132, 129.142857142857, 139.071428571429, 161.142857142857, 
171.785714285714, 161.642857142857, 171.357142857143, 172.571428571429, 
172.428571428571, 180.285714285714, 180.857142857143, 175.142857142857, 
164.5, 135.357142857143, 132, 144, 147.714285714286, 166.928571428571, 
178.571428571429, 176.071428571429, 173.785714285714, 169.857142857143, 
183.785714285714, 193.857142857143, 169.714285714286, 160.928571428571, 
144, 148.785714285714, 150, 160.428571428571, 157.785714285714, 
174.357142857143, 175.571428571429, 171.785714285714, 180.142857142857, 
180.857142857143, 172.928571428571, 168.214285714286, 158.5, 
162.214285714286, 166.928571428571, 169.071428571429, 168.5, 
178.428571428571, 188.285714285714, 178.142857142857, 175.785714285714, 
170.214285714286, 154.571428571429, 170.142857142857, 165.214285714286, 
168.642857142857, 167.214285714286, 154.071428571429, 168.357142857143
)), row.names = c(NA, -349L), class = c("tbl_df", "tbl", "data.frame"
), .Names = c("swediff", "avgdate"))

Answer 1

It's little wonder it doesn't converge. Your model doesn't allow the fit to go above 0.

You need an asymptote parameter in your nls model rather than forcing it to be 0.

If you try something like

nls(y~c-a*exp(x*b),start=list(a=.1,b=.1,c=0.2))

you get a reasonable fit:

A statistician will probably ask you why you need a functional form; typical nonparametric regression / smoothing approaches* should describe the relationship as well or slightly better; you could use cross-validation to choose the df.

* I'd expect that a natural cubic spline with say 3 df plus the constant term would do pretty well (it has an extra parameter so that would be little surprise if it does better than the nls exponential fit)

---

You can do a broken line fit easily with the package segmented :

 library(segmented)
 linmdl <- lm(y~x)
 segmentedmdl <- segmented(linmdl, seg.Z = ~x, psi=150)
 summary(segmentedmdl)

        ***Regression Model with Segmented Relationship(s)***

Call: 
segmented.lm(obj = linmdl, seg.Z = ~x, psi = 150)

Estimated Break-Point(s):
    Est.  St.Err 
154.173   1.962 

Meaningful coefficients of the linear terms:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)  0.1518234  0.0427016   3.555  0.00043 ***
x           -0.0003317  0.0003784  -0.877  0.38134    
U1.x        -0.0191520  0.0015109 -12.676       NA    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.1909 on 342 degrees of freedom
Multiple R-Squared: 0.5835,  Adjusted R-squared: 0.5799 

Convergence attained in 2 iterations with relative change 1.425652e-16 

As you see it placed the break at about 154, producing this fit: