Helper functions for polynomials created by poly().
Install the latest release of polypoly from CRAN:
install.packages("polypoly")Or install the developmental version from Github:
# install.packages("devtools")
devtools::install_github("tjmahr/polypoly")The poly() function in the stats package creates a matrix of
(orthogonal) polynomials over a set of values. The code below shows some
examples of these matrices.
# orthogonal polynomials
m <- poly(1:6, degree = 3, simple = TRUE)
m
#> 1 2 3
#> [1,] -0.5976143 0.5455447 -0.3726780
#> [2,] -0.3585686 -0.1091089 0.5217492
#> [3,] -0.1195229 -0.4364358 0.2981424
#> [4,] 0.1195229 -0.4364358 -0.2981424
#> [5,] 0.3585686 -0.1091089 -0.5217492
#> [6,] 0.5976143 0.5455447 0.3726780
# the terms are uncorrelated. that's why they are "orthogonal".
zapsmall(cor(m))
#> 1 2 3
#> 1 1 0 0
#> 2 0 1 0
#> 3 0 0 1
# raw polynomials
m <- poly(1:6, degree = 3, simple = TRUE, raw = TRUE)
m
#> 1 2 3
#> [1,] 1 1 1
#> [2,] 2 4 8
#> [3,] 3 9 27
#> [4,] 4 16 64
#> [5,] 5 25 125
#> [6,] 6 36 216
# raw polynomials are highly correlated.
round(cor(m), 2)
#> 1 2 3
#> 1 1.00 0.98 0.94
#> 2 0.98 1.00 0.99
#> 3 0.94 0.99 1.00This package provides some helpful functions for working with these matrices.
This package provides a tidying function poly_melt().
library(polypoly)
xs <- 1:40
poly_mat <- poly(xs, degree = 5)
poly_melt(poly_mat)
#> # A tibble: 200 × 3
#> observation degree value
#> <int> <chr> <dbl>
#> 1 1 1 -0.267
#> 2 2 1 -0.253
#> 3 3 1 -0.240
#> 4 4 1 -0.226
#> 5 5 1 -0.212
#> 6 6 1 -0.199
#> 7 7 1 -0.185
#> 8 8 1 -0.171
#> 9 9 1 -0.158
#> 10 10 1 -0.144
#> # … with 190 more rowsThe returned dataframe has one row per cell of the original matrix.
Essentialy, the columns of the matrix are stacked on top of each other
to create a long dataframe. The observation and degree columns
record each values’ original row number and column name, respectively.
Plot a matrix with poly_plot().
poly_plot(poly_mat)
We can also plot raw polynomials, but that display is less useful because the x-axis corresponds to the row number of polynomial matrix.
poly_raw_mat <- poly(-10:10, degree = 3, raw = TRUE)
poly_plot(poly_raw_mat)
We can make the units clearer by using by_observation = FALSE so that
the x-axis corresponds to the first column of the polynomial matrix.
poly_plot(poly_raw_mat, by_observation = FALSE)
poly_plot() returns a plain ggplot2 plot, so we can further customize
the output. For example, we can use ggplot2 to compute the sum of the
individual polynomials and re-theme the plot.
library(ggplot2)
poly_plot(poly_mat) +
stat_summary(
aes(color = "sum"),
fun = "sum",
geom = "line",
size = 1
) +
theme_minimal()
For total customization, poly_plot_data() will return the dataframe
that would have been plotted by poly_plot().
poly_plot_data(poly_mat, by_observation = FALSE)
#> # A tibble: 200 × 4
#> observation degree value `degree 1`
#> <int> <fct> <dbl> <dbl>
#> 1 1 1 -0.267 -0.267
#> 2 1 2 0.328 -0.267
#> 3 1 3 -0.360 -0.267
#> 4 1 4 0.369 -0.267
#> 5 1 5 -0.360 -0.267
#> 6 2 5 -0.0831 -0.253
#> 7 2 1 -0.253 -0.253
#> 8 2 2 0.278 -0.253
#> 9 2 3 -0.249 -0.253
#> 10 2 4 0.180 -0.253
#> # … with 190 more rowsThe ranges of the terms created by poly() are sensitive to repeated
values.
# For each column in a matrix, return difference between max and min values
col_range <- function(matrix) {
apply(matrix, 2, function(xs) max(xs) - min(xs))
}
p1 <- poly(0:9, degree = 2)
p2 <- poly(rep(0:9, 18), degree = 2)
col_range(p1)
#> 1 2
#> 0.9908674 0.8703883
col_range(p2)
#> 1 2
#> 0.2335497 0.2051525Thus, two models fit with y ~ poly(x, 3) will not have comparable
coefficients when the number of rows changes, even if the unique values
of x did not change!
poly_rescale() adjusts the values in the polynomial matrix so that the
linear component has a specified range. The other terms are scaled by
the same factor.
col_range(poly_rescale(p1, scale_width = 1))
#> 1 2
#> 1.0000000 0.8784105
col_range(poly_rescale(p2, scale_width = 1))
#> 1 2
#> 1.0000000 0.8784105
poly_plot(poly_rescale(p2, scale_width = 1), by_observation = FALSE)
poly_add_columns() adds orthogonal polynomial transformations of a
predictor variable to a dataframe.
Here’s how we could add polynomials to the sleepstudy dataset.
df <- tibble::as_tibble(lme4::sleepstudy)
print(df)
#> # A tibble: 180 × 3
#> Reaction Days Subject
#> <dbl> <dbl> <fct>
#> 1 250. 0 308
#> 2 259. 1 308
#> 3 251. 2 308
#> 4 321. 3 308
#> 5 357. 4 308
#> 6 415. 5 308
#> 7 382. 6 308
#> 8 290. 7 308
#> 9 431. 8 308
#> 10 466. 9 308
#> # … with 170 more rows
poly_add_columns(df, Days, degree = 3)
#> # A tibble: 180 × 6
#> Reaction Days Subject Days1 Days2 Days3
#> <dbl> <dbl> <fct> <dbl> <dbl> <dbl>
#> 1 250. 0 308 -0.495 0.522 -0.453
#> 2 259. 1 308 -0.385 0.174 0.151
#> 3 251. 2 308 -0.275 -0.0870 0.378
#> 4 321. 3 308 -0.165 -0.261 0.335
#> 5 357. 4 308 -0.0550 -0.348 0.130
#> 6 415. 5 308 0.0550 -0.348 -0.130
#> 7 382. 6 308 0.165 -0.261 -0.335
#> 8 290. 7 308 0.275 -0.0870 -0.378
#> 9 431. 8 308 0.385 0.174 -0.151
#> 10 466. 9 308 0.495 0.522 0.453
#> # … with 170 more rowsWe can optionally customize the column names and rescale the polynomial terms.
poly_add_columns(df, Days, degree = 3, prefix = "poly_", scale_width = 1)
#> # A tibble: 180 × 6
#> Reaction Days Subject poly_1 poly_2 poly_3
#> <dbl> <dbl> <fct> <dbl> <dbl> <dbl>
#> 1 250. 0 308 -0.5 0.527 -0.458
#> 2 259. 1 308 -0.389 0.176 0.153
#> 3 251. 2 308 -0.278 -0.0878 0.381
#> 4 321. 3 308 -0.167 -0.264 0.338
#> 5 357. 4 308 -0.0556 -0.351 0.131
#> 6 415. 5 308 0.0556 -0.351 -0.131
#> 7 382. 6 308 0.167 -0.264 -0.338
#> 8 290. 7 308 0.278 -0.0878 -0.381
#> 9 431. 8 308 0.389 0.176 -0.153
#> 10 466. 9 308 0.5 0.527 0.458
#> # … with 170 more rowsWe can confirm that the added columns are orthogonal.
df <- poly_add_columns(df, Days, degree = 3, scale_width = 1)
zapsmall(cor(df[c("Days1", "Days2", "Days3")]))
#> Days1 Days2 Days3
#> Days1 1 0 0
#> Days2 0 1 0
#> Days3 0 0 1This package also (accidentally) works on splines. Splines are not officially supported, but they could be an avenue for future development.
poly_plot(splines::bs(1:100, 10, intercept = TRUE))
poly_plot(splines::ns(1:100, 10, intercept = FALSE))
This section illustrates a use case that may or may not be included in the package someday: Visualizing the weighting of polynomial terms from a linear model. For now, here’s how to do that task with this package.
Suppose we want to model some change over time using a cubic polynomial. For example, the growth of trees.
library(lme4)
#> Loading required package: Matrix
df <- tibble::as_tibble(Orange)
df$Tree <- as.character(df$Tree)
df
#> # A tibble: 35 × 3
#> Tree age circumference
#> <chr> <dbl> <dbl>
#> 1 1 118 30
#> 2 1 484 58
#> 3 1 664 87
#> 4 1 1004 115
#> 5 1 1231 120
#> 6 1 1372 142
#> 7 1 1582 145
#> 8 2 118 33
#> 9 2 484 69
#> 10 2 664 111
#> # … with 25 more rows
ggplot(df) +
aes(x = age, y = circumference, color = Tree) +
geom_line()
We can bind the polynomial terms onto the data and fit a model.
df <- poly_add_columns(Orange, age, 3, scale_width = 1)
model <- lmer(
scale(circumference) ~ age1 + age2 + age3 + (age1 + age2 + age3 | Tree),
data = df
)
#> boundary (singular) fit: see help('isSingular')
summary(model)
#> Linear mixed model fit by REML ['lmerMod']
#> Formula: scale(circumference) ~ age1 + age2 + age3 + (age1 + age2 + age3 |
#> Tree)
#> Data: df
#>
#> REML criterion at convergence: -11.7
#>
#> Scaled residuals:
#> Min 1Q Median 3Q Max
#> -1.54075 -0.42547 0.08553 0.49188 1.37391
#>
#> Random effects:
#> Groups Name Variance Std.Dev. Corr
#> Tree (Intercept) 0.12584 0.35474
#> age1 0.38473 0.62027 0.99
#> age2 0.00997 0.09985 -0.63 -0.49
#> age3 0.01070 0.10343 -0.89 -0.95 0.19
#> Residual 0.01688 0.12991
#> Number of obs: 35, groups: Tree, 5
#>
#> Fixed effects:
#> Estimate Std. Error t value
#> (Intercept) -1.867e-14 1.602e-01 0.000
#> age1 2.719e+00 2.852e-01 9.533
#> age2 -1.520e-01 7.995e-02 -1.901
#> age3 -2.529e-01 8.085e-02 -3.128
#>
#> Correlation of Fixed Effects:
#> (Intr) age1 age2
#> age1 0.950
#> age2 -0.348 -0.266
#> age3 -0.502 -0.529 0.062
#> optimizer (nloptwrap) convergence code: 0 (OK)
#> boundary (singular) fit: see help('isSingular')How do we understand the contribution of each of these terms? We can recreate the model matrix by attaching the intercept term to a polynomial matrix.
poly_mat <- poly_rescale(poly(df$age, degree = 3), 1)
# Keep only seven rows because there are 7 observations per tree
poly_mat <- poly_mat[1:7, ]
pred_mat <- cbind(constant = 1, poly_mat)
pred_mat
#> constant 1 2 3
#> [1,] 1 -0.54927791 0.5047675 -0.2964647
#> [2,] 1 -0.29927791 -0.1637435 0.4264971
#> [3,] 1 -0.17632709 -0.3312105 0.2957730
#> [4,] 1 0.05591335 -0.3573516 -0.2139411
#> [5,] 1 0.21096799 -0.1635520 -0.3699278
#> [6,] 1 0.30727947 0.0419906 -0.2489830
#> [7,] 1 0.45072209 0.4690995 0.4070466Weight the predictors using the model fixed effects.
weighted <- pred_mat %*% diag(fixef(model))
colnames(weighted) <- colnames(pred_mat)And do some tidying to plot the two sets of predictors.
df_raw <- poly_melt(pred_mat)
df_raw$predictors <- "raw"
df_weighted <- poly_melt(weighted)
df_weighted$predictors <- "weighted"
df_both <- rbind(df_raw, df_weighted)
# Only need the first 7 observations because that is one tree
ggplot(df_both[df_both$observation <= 7, ]) +
aes(x = observation, y = value, color = degree) +
geom_line() +
facet_grid(. ~ predictors) +
labs(color = "term")
The linear trend drives the growth curve. The quadratic and cubic terms
make tiny contributions. We can see that the intercept term does nothing
(because we used scale() in the model).
Hmmm… perhaps we need to find a better example dataset for this example.
If you searched for help on poly(), see also: