Curve fitting on batches in the tidyverse: R, dplyr, and broom

September 9, 2018
By

(This article was first published on R on Programming notes, and kindly contributed to R-bloggers)

I recently needed to fit curves on several sets of similar data, measured from different sensors. I found how to achieve this with dplyr, without needing to define outside functions or use for-loops. This approach integrates perfectly with my usual dplyr and ggplot2 workflows, which means it adapts to new data or new experimental conditions with no changes. Here are the ready-made recipes for any one else who may run into a similar problem.

First we’ll generate some artificial data so that you can follow along at home. Here’s an exponentially decaying value measured by two sensors at different locations:

# if needed
install.packages(c("tidyverse", "broom"))

library(tidyverse)
library(broom)

t = 1:100
y1 = 22 + (53 - 22) * exp(-0.02 * t) %>% jitter(10)
y2 = 24 + (60 - 22) * exp(-0.01 * t) %>% jitter(10)


df <- data.frame(t = t, y = y1, sensor = 'sensor1') %>% 
  rbind(. , data.frame(t = t, y = y2, sensor = 'sensor2'))


Our data frame looks like this:

df
t y sensor
1 52.25302 sensor1
2 51.71440 sensor1
3 51.13971 sensor1
98 38.13328 sensor2
99 38.17017 sensor2
100 37.72184 sensor2

And our data looks like this:

qplot(t, y, data = df, colour = sensor)

raw_data

I created the data in long format, as it works best with dplyr pipelines. You can read more about long format vs wide format data here.

TL;DR

I want to plot the fitted curve over my data points

augmented <- df %>% 
  group_by(sensor) %>% 
  do(fit = nls(y ~ a * t + b, data = .)) %>% 
  augment(fit)

qplot(t, y, data = augmented, geom = 'point', colour = sensor) +
  geom_line(aes(y=.fitted))

raw_data

I want to plot the residuals

qplot(t, .resid, data = augmented, colour = sensor)
qplot(.fitted, .resid, data = augmented, colour = sensor)

raw_data

raw_data

I want a table of my fit parameters for each condition:

df %>% 
  group_by(sensor) %>% 
  do(fit = nls(y ~ a * t + b, data = .)) %>% 
  tidy(fit) %>% 
  select(sensor, term, estimate) %>% 
  spread(term, estimate)
sensor a b
1 sensor1 -0.2495347 47.87376
2 sensor2 -0.2350898 59.77793

I want a less horrendous fit. How can I do better?

Try with an actual exponential. You’ll probably want a self-starting function to avoid the “singular gradient” error — read more in my post on the subject.

Explanation: intro to curve fitting in R

The goal of our fitting example was to find an estimate $\hat{y}(t) = a t + b$ that approximates our measured data $y$. In R, we can directly write that we want to approximate $y$ as a function $a \cdot t + b$, using the very intuitive built-in formula syntax:

y ~ a * t + b

In R’s documentation, $a$ and $b$ are the coefficients, $t$ is the independent variable and $y$ is the dependent variable — $t$ can go about its day without every knowing $y$ exists, but $y$ can’t progress without knowing the latest trend in $t$.

We fit these models using various fitting functions: lm or glm for linear models or nls for non-linear least squares for example. I’m using nls here because I can specify the names of my coefficients, which I find clearer to explain:

# Select the data from our first sensor:
sensor1 <- df %>% filter(sensor == "sensor1")

# Fit our model:
fit <- nls(y ~ a * t + b, data = sensor1)

# nls works best if we specify an initial guess for the coefficients, 
# the 'start' point for the optimisation:
fit <- nls(y ~ a * t + b, data = sensor1, start = list(a = 1, b = 10))

The fitting functions all return a fit object:

> summary(fit)
Formula: y ~ a * t + b

Parameters:
   Estimate Std. Error t value Pr(>|t|)    
a -0.249535   0.006377  -39.13   <2e-16 ***
b 47.873759   0.370947  129.06   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.841 on 98 degrees of freedom

Number of iterations to convergence: 1 
Achieved convergence tolerance: 1.571e-09

You can use one of these base R functions to extract useful data from the fit object:

  • coef(fit) returns the values of the coefficients $a$ and $b$.
  • predict(fit) returns $\hat{y}(t_i)$, i.e. it applies the fitted model to each of the original data points $t_i$. It can also be applied on new values of $t$.
  • resid(fit) returns the residuals $y_i – \hat{y}(t_i)$ at each point of the original data.

Try them out in the console:

> coef(fit)
         a          b 
-0.2495347 47.8737585

> predict(fit)
[1] 47.62422 47.37469 47.12515 46.87562 46.62609 46.37655 ...

> predict(fit, newdata = data.frame(t = 101:200))
[1] 22.67076 22.42122 22.17169 21.92215 21.67262 21.42309 ...

> resid(fit)
[1] 4.628798 4.339712 4.014558 3.576258 3.528138 3.001447 ...

Now these functions all return vectors, which work best with R’s native plotting functions. To integrate with dplyr and ggplot, we’d rather have data frames. This is where the broom package comes in.

Introducing broom

Broom is a separate R package that feeds on fit results and produces useful data frames. Install it directly within the R console if you haven’t already:

install.packages('broom')
library(broom)

It provides three useful functions, which are easier to demonstrate than to explain. All of them return data frames:

  • tidy extracts the fit coefficients and related information. The term column contains the name of your coefficient, and estimate contains its fitted value:
tidy(fit)
term estimate std.error statistic p.value
a -0.2495347 0.006377179 -39.12932 1.267667e-61
b 47.8737585 0.370946838 129.05827 3.123848e-111
  • augment is equivalent to predict and resid combined. The returned data frame contains columns t and y with your original data, as well as .fitted, the fitted curve, and .resid, the residuals:
augment(fit)
t y .fitted .resid
1 52.25302 47.62422 4.62879752
2 51.71440 47.37469 4.33971163
3 51.13971 47.12515 4.01455804
4 50.45188 46.87562 3.57625801

You can plot the ‘augmented’ fit with qplot:

qplot(t, y, data = augment(fit)) + geom_line(aes(y = .fitted))
qplot(t, .resid, data = augment(fit))

raw_data

raw_data

  • glance shows fit statistics:
glance(fit)
sigma isConv finTol logLik AIC BIC deviance df.residual
1.840841 TRUE 1.571375e-09 -201.906 409.8119 417.6274 332.0921 98

Using broom with dplyr

Since broom’s functions return data frames, they integrate naturally with magrittr pipelines (%>%) and dplyr. Let’s illustrate by grouping the data by sensor id, then producing a fit object for each group. The do function calls nls once for each group, then stores the return value in a column we named fit:

fitted <- df %>% 
  group_by(sensor) %>% 
  do(fit = nls(y ~ a * t + b, data = ., start = list(a = 1, b = 1)))
fitted
sensor fit
1 sensor1
2 sensor2

We now have one nls object per group, which can be fed into either tidy or augment. These functions act on each group separately:

fitted %>% 
  tidy(fit)
sensor term estimate std.error statistic p.value
sensor1 a -0.2495347 0.006377179 -39.12932 1.267667e-61
sensor1 b 47.8737585 0.370946839 129.05827 3.123849e-111
sensor2 a -0.2350898 0.003088217 -76.12478 5.400611e-89
sensor2 b 59.7779291 0.179634964 332.77446 1.931798e-151

From here, we can use select and spread to generate a more succinct table, as in the TL;DR example, or keep the table as-is to inspect the quality of the fit.

The result of augment is a long data frame that can be used immediately with qplot, also shown in the TL;DR section above:

fitted %>% 
  augment(fit)
sensor t y .fitted .resid
sensor1 1 52.41684 47.58220 4.834645067
sensor1 2 51.64323 47.33347 4.309757950
sensor1 3 51.09595 47.08475 4.011196179
sensor2 98 38.06029 36.72428 1.336007642
sensor2 99 37.86891 36.48926 1.379642201
sensor2 100 37.89936 36.25425 1.645117019

That’s it for broom and dplyr. The magic of this approach is that you can add a sensor to your input data then re-run all your code; your new curves will just appear on the plots. Similarly, you can change the formula of your fit, and your new coefficients will appear as a new column in the coefficients table. No need to change your code.

References

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