Piecewise Linear Segmentation by Dynamic Programming
\[ \newcommand{\Ell}{\mathcal{L}} \newcommand{\jump}{\mathcal{J}} \newcommand{\Var}{\mathrm{Var}} \newcommand{\Cov}{\mathrm{Cov}} \newcommand{\lmax}{\ell_\text{max}} \newcommand{\lmin}{\ell_\text{min}} \newcommand{\def}{\stackrel{\mathrm{def}}{=}} \]
The R package dpseg performs piecewise linear
segmentation of 2-dimensional data by a dynamic programming algorithm
(Bellman 1961). It was developed for time
series data, dissection of bacterial growth phases, and for genome-wide
read-count data from next generation sequencing.
print, plot and predict
methods allow quick evaluation of the results. Parameter scanning
routines (estimateP, scanP) help to evaluate
the best choice of parameters for a given problem.
The package and its documentation are also intended to serve as a
tutorial on the concepts of linear regression, dynamic programming and
the segmentation problem. The movie function animates the
progress of the algorithm through the data. Generic implementations of
dynamic programming routines allow to test alternative segmentation
criteria.
Theory
Recursion
The problem is to find break-points in 2-dimensional ordered data \(\{(x_i,y_i), i=1,...,n\}\), eg., a time series. This can be formulated as a dynamic programming recursion:
\[ \begin{equation} S_j = \max_{i\le j} (S_{i-\jump} + \text{score}(i,j)) - P\;, (\#eq:recur) \end{equation} \]
where the scoring function \(\text{score}(i,j)\) quantifies how well a segment between points \(i\) and \(j\) is defined.
The break-point penalty term \(P\) sets bounds on segment lengths and should be chosen close to the optimal value of the scoring function (Section @ref(selectp)). Higher \(P\) will yield longer segments, and especially for non-monotonic data \(P\) lower than the optimal value for the scoring function can work better.
The binary jump parameter \(\jump \in \{0,1\}\) determines whether the break-point \(i\) is both, the end of the previous and start of the current segment (\(\jump=0\)), or the previous segments ends at \(i-1\) (\(\jump=1\)). The latter choice allows discontinuous jumps between adjacent segments (Fig. @ref(fig:jumps)).
Scoring Functions
Segmentation into linear segments can be achieved by using a
goodness-of-fit measure. Linear models are often evaluated by the
coefficient of determination \(R^2\)
(“R-squared”), and we can use this directly as a scoring function
(type="r2" in dpseg):
\[ \begin{equation} \text{score}(i,j) = R^2 - 1 = \frac{\sum (\hat y_i - \bar y)^2}{\sum (y_i - \bar y)^2}-1\;, (\#eq:scorer2) \end{equation} \]
where \(\hat y_i\) is the linear
model and \(\bar y\) the mean of the
original data, see Section @ref(appii) and equation @ref(eq:r2) for
details. Its square root, the Pearson correlation can also be used as
scoring function (type="cor"). Subtraction of 1 simply
aligns different scoring functions in dpseg at \(\text{score} \le 0\) and thereby allows the
use of a consistent default penalty parameter of \(P=0\).
\(R^2\) depends on the slope (eq.
@ref(eq:r2)) and will score poorly for segments without slope (Fig.
@ref(fig:lines) and @ref(fig:custom), Section @ref(lines)). The negative
variance of the residuals \(r_i=y_i - \hat
y_i\) between data and the regression line does not depend on the
slope (eq. @ref(eq:var)) and is for many cases the better choice
and the default in dpseg
(type="var"):
\[ \begin{equation} \text{score}(i,j) = -\Var(r) = - \frac{1}{n-1} \sum r_i^2\;, (\#eq:score) \end{equation} \]
See Section @ref(scoring) for a short discussion on the choice of the built-in scoring functions. Section @ref(lines) discusses the handling of straight horizontal or vertical lines (with zero variance) in the data. And Sections @ref(matrix) and @ref(generic) introduce custom scoring options.
Incremental calculation of the variances (Section @ref(appii)) allows
for a computationally efficient implementation (Section @ref(benchmark))
of the recursion in dpseg.
Backtracing
During calculation of \(S_j\) the position \(i_\text{max}<j\) that yielded the maximal score is stored as a vector. The segmentation can be easily retrieved by starting at the last position \(j=n\), the end of the last segment, looking up its \(i_\text{max}\), the start of the segment, and proceed from \(j=i_\text{max} - \jump\) (the end of the next segment). If jumps were allowed (\(\jump=1\)) the previous segment ended one position ahead of the next segment at \(i-1\), and this must be accounted for in backtracing.
Segment Length Restrictions
Depending on the scoring function, short segments may not be meaningful, eg. goodness-of-fit measures for linear regression such as the variance of residuals are 0 for segments of length \(\ell=2\) and an “optimal” segmentation would split the data in pairs with perfect “fits”. Thus, we can restrict score function maximization for only \(i\le(j-\lmin+1)\).
When handling data sets much larger then the expected segment length, one can also define a maximal segment length, and thereby save memory and computation time by only considering \(i\ge(j-\lmax +1)\).
Combining both restrictions, and with \(\Ell_x=\ell_x+1\) the recursion becomes:
\[ \begin{equation} S_j = \max_{\substack{i\le j-\Ell_{\min}\\i\ge j-\Ell_{\max}}} (S_{i-\jump} + \text{score}(i,j)) - P\;. (\#eq:recurl) \end{equation} \]
Usage
Basic
The main function dpseg takes \(x\) and \(y\) vectors of equal length as input and
returns an object of class dpseg. This results object can
be inspected by print and plot methods. A
predict method returns \(y\) values and specified novel \(x\) values at the straight line
segments.
The results object is a simple list (S3 object) with the
list item segments as the main result: a table (data frame)
that lists the start and end x-values of the segments, the start and end
indices in the data vectors, the linear regression coefficients and
goodness-of-fit measures for the segments (intercept, slope, R-squared,
variance of residuals).
library(dpseg)
type <- "var" # use the (default) scoring, -var(residuals(lm(y~x)))
jumps <- FALSE # allow discrete jumps between segments?
P <- 1e-4 # break-point penalty, use higher P for longer segments
# get example data `oddata` - bacterial growth measured as optical density OD
x <- oddata$Time
y <- log(oddata[,"A2"]) # note: exponential growth -> log(y) is linear
# NOTE: the scoring function results are stored as a matrix for re-use below
segs <- dpseg(x=x, y=y, jumps=jumps, P=P, type=type, store.matrix=TRUE)## using 'recursion_linreg_c' with type 'var'
## calculating recursion for 229 datapoints##
## Dynamic programming-based segmentation of 229 xy data points:
##
## x1 x2 start end intercept slope r2 var
## 1 1.022778 1.362778 1 3 0.7194953 -4.383977 0.9909440 0.005076021
## 2 1.362778 1.703194 3 5 -9.3479269 3.016111 0.9626144 0.010235471
## 3 1.703194 2.213611 5 8 -6.2212774 1.153347 0.9766701 0.001532996
## ...
## x1 x2 start end intercept slope r2 var
## 7 15.68903 17.91181 87 100 -3.4426960 0.18899698 0.9940106 1.101408e-04
## 8 17.91181 21.32958 100 120 -0.3825874 0.01895679 0.8638382 6.374447e-05
## 9 21.32958 39.94000 120 229 0.4767462 -0.01989881 0.9919736 9.501867e-05
##
## Found: 9 segments.
## Parameters:
## type: var; minl: 3; maxl: 229; P: 1e-04; jumps: 0
Segmentation of a growth curve (E. coli in M9 minimal medium)
by dpseg. Vertical lines indicate segment starts (dashed
red) and ends (solid black). The predict method returns the
piecewise linear model (dashed yellow).
Exploratory & Educational Movies
For both, educational purposes or detailed evaluation of novel
scoring functions (Section @ref(generic)) or parameters, the
movie function allows to visualize the performance of the
dynamic programming as it progresses through the data:
# use arguments frames and res to control file size
movie(segs, format="gif", file.name="movie", path="pkg/vignettes/figures",
frames=seq(1,length(x),3), delay=.3,res=50)This will plot each step of the algorithm and visualize the
development of the scoring function. For this function to work, the
argument store.matrix=TRUE must be set in the call to
dpseg.
Animation of the progress of the algorithm through the data (gray circles and right y-axis). The black vertical line is the current position \(j\), and the black circles are the values of the scoring function \(\text{score}(i,j)\) for all \(i<j\). The blue line was the \(i_\text{max}\) where the maximal value of the recursion \(S_j\) was found. The thin red line indicates \(j-\lmin\). Colors indicate the final segmentation after backtracing.
Fine-tuning Segment Length: The Penalty Parameter \(P\)
The minimal and maximal segment length parameters \(\lmin\) and \(\lmax\) (arguments minl and
maxl of dpseg) are the easiest way to restrict
the algorithm (and memory usage) to certain segment lengths. However,
this overrules potentially better segmentations with lower variance.
A more meaningful way to restrict segment lengths for given data is
to explicitly allow higher variance of segments. This can be achieved
with the break-point penalty parameter \(P\) (argument P). This
parameter can be directly used to tune segmentation. It can be chosen in
the order of magnitude of the tolerated variances. A higher \(P\) will allow higher variance of the
individual segments and will yield longer segments. \(P<0\) will reward break-points and yield
more shorter segments.
To choose an optimal \(P\) for a
given application, the package offers two functions:
estimateP and scanP. estimateP
makes use of the excellent performance of base R’s
smooth.spline implementation and reports the variance of
the residuals; argument plot=TRUE allows to evaluate
results:
## using 'recursion_linreg_c' with type 'var'
## calculating recursion for 229 datapoints
The reported value is a good starting point. For our example data,
values an order of magnitude lower than this estimate, \(P\)=estimateP(x=x,y=y)/10
achieved a satisfying segmentation, but this will strongly depend on the
type of data and noise around expected segments.
Alternative estimators can be easily defined, eg.
estimateP simply calls smooth.spline:
The convenience function scanP calculates segmentations
for a vector of \(P\) values and
returns (plots) the resulting numbers of segments and the median of
segment variances of residuals. Based on these values one can select a
value of \(P\) that splits the data
into a reasonable number of segments with acceptable variance of
residuals:
scanP is more efficient than scanning over
dpseg calls since it calculates the scoring function matrix
only once and uses this matrix for scans over \(P\) (Section @ref(matrix)).
## NOTE: dpseg is slower for many segments!
sp <- scanP(x=x, y=y, P=seq(-.01,.1,length.out=50), plot=TRUE)## running dpseg 50 times: ..................................................
A higher \(P\) will yield fewer and longer segments. \(P\) should be chosen close to the optimal value of the scoring function.
Scoring Functions: \(-\mathrm{Var}(r)\) vs. \(R^2\)
The performance of the different built-in scoring functions depends on the type of data and the goal of the user.
The default scoring function, the negative variance of residuals
(type="var", \(-\Var(r)\)), is independent of the slope of
segments (eq. @ref(eq:var)) and is recommended in most cases.
Only when looking explicitly for segmentation of zig-zag like data
with only increases and decreases, the coefficient of determination
(type="r2", \(R^2\)) or
its square root, Pearson correlation (type="cor"), may be
better choices. In dpseg, these are not used in the sense
of a goodness-of-fit measure but in their mathematical meaning as a
measure of (linear) correlation: how well can \(x\) explain variation in \(y\)? Lines with low slope score worse than
steeper lines (see slope \(b_1\) in eq.
@ref(eq:r2)). This behaviour is exemplified in the Sections
@ref(example), @ref(lines), and @ref(generic). Figure
@ref(fig:rcppdynprogdata) provides an example where the solution is
stable over a broader range of penalty \(P\) when scoring by the Pearson
correlation.
Data Example: \(P<0\) and Correlation-based Scoring.
Using the non-monotonic, sine-based example
data from the RcppDynProg R package (Mount and Zumel 2019), we can first use
scanP to test different penalties for different scoring
functions. Here, dpseg yields the intended segmentation
with the default scoring function (type="var") and a
negative penalty, while the Pearson correlation
(type="cor") worked well with default penalty \(P=0\) and appears to have a broader range
of penalties that achieve the intented split into 7 segments:
## example from https://winvector.github.io/RcppDynProg/articles/SegmentationL.html
set.seed(2018)
d <- data.frame(x = 0.05*(1:(300))) # ordered in x
d$y_real <- sin((0.3*d$x)^2)
d$y <- d$y_real + 0.25*rnorm(length(d$y_real))
par(mfrow=c(2,2))
scanP(x=d$x, y=d$y, P=seq(-0.005,0.005,length.out=50), verb=0) # note: no messages by verb=0
scanP(x=d$x, y=d$y, P=seq(-0.05,0.05,length.out=50), type="cor", verb=0)
plot(dpseg(x=d$x, y=d$y, P=-.0025, verb=0))
plot(dpseg(x=d$x, y=d$y, type="cor", verb=0))
Segmentation of the test data from the RcppDynProg package with the default scoring function, negative variance of residuals (left), and with Pearson correlation (right).
Vertical and Horizontal Lines with Zero Variance.
Experimental data may contain short horizontal or vertical data stretches, e.g., as artefacts of low measurement sensitivity. For such segments the calculation of the scoring function involves division by 0 and undefined values.
For a horizontal straight line with \(\Var(y)=0\) (top panel in Fig.
@ref(fig:lines)), i.e., a short segment of \(x\) values with equal \(y\) values, the \(R^2\) and Pearson correlation measures are
undefined (eq. @ref(eq:r2)). dpseg will set them to 0,
their minimal value, consistent with their general behaviour to favor
sloped and ignore horizontal segments. The variance of residuals can be
calculated (eq. @ref(eq:var)) as \(\Var(r)=0\) and the segments can be
detected with the default scoring function (top left panel in Fig.
@ref(fig:lines)).
For a vertical line with \(\Var(x)=0\) (lower panel in Fig.
@ref(fig:lines)), i.e., different \(y\)
values at equal \(x\), all measures are
undefined (eq. @ref(eq:r2) and @ref(eq:var)). dpseg will
set the scoring function to \(-\infty\), effectively ignoring them in
segmentation. The data points will be part of adjacent or spanning
segments. If such cases are present in the data between intended
segments, it may be advisable to use jumps=TRUE to allow
for split segments and discrete jumps between adjacent segments (see
Section @ref(jumps)).
For both cases, dpseg will issue a summarized warning if
it was relevant for the used scoring function, and providing the counts
of occurrences of such cases during the recursion:
y <- c(1:5,rep(5,10),6:10) # several equal y values
x <- 1:length(y)
par(mfrow=c(2,2))
plot(dpseg(x=x, y=y, type="var", verb=0)) # Var(y)=0, horizontal line
plot(dpseg(x=x, y=y, type="r2", verb=0))## Warning in recursion(x = x, y = y, maxl = maxl, jumps = jumps, P = P, minl =
## minl, : Syy==0: equal y-values or numeric cancellation in segment, score was
## set to -1 (r2=cor=0). This occurred 45 times.## Warning in recursion(x = x, y = y, maxl = maxl, jumps = jumps, P = P, minl =
## minl, : Sxx==0: equal x-values or numeric cancellation in segment, score was
## set to -infinity. This occurred 45 times.## Warning in recursion(x = x, y = y, maxl = maxl, jumps = jumps, P = P, minl =
## minl, : Sxx==0: equal x-values or numeric cancellation in segment, score was
## set to -infinity. This occurred 45 times.
Effect of straight vertical (top) or horizontal (bottom) lines.
Discrete Jumps between Segments
Per default, dpseg assumes that adjacent segments are
linked, that is: the last data point of a segment is also the first data
point of the next adjacent segment. Setting the argument
jumps=TRUE allows to obtain segments that do not share
their start and end points (see \(\jump\) in eq. @ref(eq:recur)). This can be
useful if the data contains discrete jumps between adjacent segments
(Fig. @ref(fig:jumps)).
x <- c(1:10)
y <- c(1:4,7:12) # discrete jump between two lines
par(mfcol=c(1,2))
plot(dpseg(x=x, y=y, type="var", jumps=FALSE, verb=0))
plot(dpseg(x=x, y=y, type="var", jumps=TRUE, verb=0))
Handling discrete jumps in data: the default behaviour adds a third
segment across the discrete jump between segments (left panel). Allowing
for such jumps by setting argument jumps=TRUE correctly
identifies the jump and the two adjacent segments (right panel).
Pre-Calculated Scoring Matrix
When the scoring function can be pre-calculated, a simple look-up in
the triangular matrix (banded by \(\lmin\) and \(\lmax\)) suffices for the recursion. This
option is available in dpseg by passing a scoring matrix
\(\mathbf{S}_{ij}\) as argument
y (instead of a numeric vector).
For example, we can use the scoring matrix generated above (Fig.
@ref(fig:dpsegdemo), stored due to argument
store.matrix=TRUE) and test different parameters \(P\), \(\jump\), \(\lmin'>\lmin\) and \(\lmax'<\lmax\) more efficiently
(Section @ref(benchmark)). This approach is also used in the
scanP function (Section @ref(selectp)).
## use the scoring matrix from a previous run for generic recursion,
## with store.matrix=TRUE, and test different parameters.
segm <- dpseg(y=segs$SCR, jumps=jumps, P=2*P, minl=5, maxl=50)## setup use with pre-calculated scoring matrix
## calculating recursion for 229 datapoints##
## Dynamic programming-based segmentation of 229 xy data points:
##
## x1 x2 start end
## 1 1 37 1 37
## 2 37 46 37 46
## 3 46 87 46 87
## ...
## x1 x2 start end
## 5 100 131 100 131
## 6 131 180 131 180
## 7 180 229 180 229
##
## Found: 7 segments.
## Parameters:
## type: matrix; minl: 5; maxl: 50; P: 2e-04; jumps: 0Note, that in this case the algorithm does not use or know of the
original \(x,y\) data and the result
therefore contains only the segment break-point indices in the original
data vectors. The predict and plot functions
will not work. We can use the convenience function addLm to
add the original \(x,y\) data and
linear regression coefficients (via lm) for the calculated
segments:
## add lm-based regression coefficients and original x/y data
segm <- addLm(segm, x=oddata$Time, y=log(oddata[,"A2"]))
plot(segm)
Compared to the first run (Fig. @ref(fig:dpsegdemo)) both, the higher \(\lmin=5\) and higher \(P=0.0002\), contributed to get fewer segments, while the lower \(\lmax=50\) split the last segment in two.
Custom Scoring Functions
Alternative scoring functions can be tested easily by either
providing a scoring matrix \(\mathbf{S}_{ij}\) as described above, or by
providing the score(i,j) function directly. The latter can
be achieved by defining a score function with the signature as in the
following example, testing to use the coefficient of determination \(R^2\) (R-squared) for segmentation:
Since our scoring function doesn’t provide linear regression
parameters, we can use the argument add.lm=TRUE to add
intercept and slope data via base R’s lm, required for the
predict and plot methods.
score_rsq <- function(i, j, x, y,...) summary(lm(y[i:j]~x[i:j]))$r.squared
segn <- dpseg(x=x, y=y, jumps=jumps, P=.99, scoref=score_rsq, add.lm=TRUE)
plot(segn) 
Note, that as above a meaningful penalty term \(P\) should be close to the optimal value of the scoring function, here \(R^2=1\). Further note, that the \(R^2\) measure depends on the slope and misses a short almost horizontal segment around \(x=20\) that is detected by the \(-\mathrm{Var}(r)\) default scoring function.
Benchmarking
The implementation of equation @ref(eq:recurl) is straightforward.
Section @ref(appi) shows a fully functional implementation in about 30
lines of code in R. However, this is highly inefficient. The theoretical
complexity is quadratic, ie. \(\mathcal{O}(n^2)\), and even with segment
length restrictions a linear regression with base R’s lm
function is performed \((n-1)*(\lmax-\lmin+1)\) times.
A more efficient implementation calculates the linear regression
parameters incrementally (eq. @ref(eq:score) and Section @ref(appii))
while looping through the data. Benchmarking of different
implementations, using a growth curve of a culture of Escherichia
coli cells, showed that this incremental implementation was about 2
orders of magnitude faster then the generic implementation, and another
2 orders of magnitude faster when implemented in C++ via
the Rcpp package (Fig. @ref(fig:benchmark)). The latter
approach is thus the default recursion used in dpseg.
Without incremental regression, using a pre-calculated scoring matrix
as input to dpseg, the recursion was ca. 25% faster. This
allows to scan over parameters \(P\),
\(\jump\), \(\lmin\) and \(\lmax\) more efficiently.
The function piecewise_linear of the related package
RcppDynProg (Mount and Zumel
2019) (version 0.1.3, without weights) was only slightly faster
than incremental calculation in base R and ca. 100x slower than the
default implementation of dpseg.
Benchmarking of R & Rcpp implementations.
Dynamic Programming in base R
The implementation of the recursion @ref(eq:recurl), the scoring
function @ref(eq:score) and the backtrace is
straightforward:
## RECURSION
recursion <- function(x, y, maxl, jumps=0, P=0, minl=3, S0=1) {
N = length(x)
S = numeric(N) # init to 0
imax = numeric(N)
S[1] = -P
for ( j in 2:N) {
si = rep(-Inf, maxl-minl+1)
irng = (j-maxl):(j-minl) +1
irng = irng[irng>0]
for ( i in irng ) {
idx = i-(j-maxl)
sij = ifelse(i==1&jumps==1, S0, S[i-jumps])
si[idx] = sij + score(x, y, i, j) - P
}
S[j] = max(si)
idx = which.max(si)
imax[j] = idx + (j-maxl)
}
imax
}
## SCORING FUNCTION
score <- function(x, y, k, l) -var(residuals(lm(y[k:l]~x[k:l])))
## backtracing
backtrace <- function(imax, jumps=0) {
end = length(imax) # end of last segment
ends = end
while( end>1 ) {
end = imax[end] - jumps
ends = c(end, ends)
}
ends[1] <- ends[1] + jumps # note: start of first segment
ends
}We construct a simple test case, with three linear segments without jumps, and added noise:
# simple test case
k1=1
k2=.05
k3=-.5
y1 <- k1*1:5
y2 <- k2*1:5 + k1*5
y3 <- k3*1:5 + k2*5 + k1*5
set.seed(1)
ym <- c(y1, y2, y3)
nsd <- .25 # noise, standard deviation
y <- ym + rnorm(length(ym), 0, nsd) # add noise
x <- 1:length(y)
## run recursion
JUMPS = 0
imax = recursion(x, y, maxl=length(x), jumps=JUMPS, P=0, minl=3, S0=1)
## backtrace
ends = backtrace(imax, jumps=JUMPS)
print(ends)## [1] 1 5 10 15## plot
plot(x,y)
lines(x,ym)
legend("bottom", title="slopes:", legend=paste(k1,k2,k3,sep=", "), bty="n")
abline(v=ends)
Correction segmentation (horizontal lines) for a primitive test case.
Incremental Linear Regression
Summarized values such as the mean can not be calculated
incrementally, ie. while looping through the data \(i=1,...,n\). We are looking for a method to
calculate linear regression parameters incrementally for an efficient
implementation of the dynamic programming recursions (eq. @ref(eq:recur)
and @ref(eq:recurl)).
Least-Squares Method
Consider \(n\) measured data pairs \(\{(x_i,y_i), i=1,...,n\}\), with the dependent variable \(y_i\) (eg. a measured value) and independent variable \(x_i\) (eg. the time of measurement), for which we suspect a linear relation with intercept \(\beta_0\) and slope \(\beta_1\), and with added random measurement errors, in regression analysis denoted the residuals \(r_i\):
\[\begin{align} y_i &= \beta_0 + \beta_1 x_i + r_i \end{align}\]
The goal is to find a straight line, denoted the regression line, that best describes this linear relation:
\[\begin{align} \hat{y}_i &= b_0 + b_1 x_i\;. \end{align}\]
The gold standard approach to find the regression line is to minimize the sum of squares of the residuals: \(\min\limits_{\beta_0,\beta_1} \sum r_i^2\). We treat the data as constants and look for parameters \(b_0\) and \(b_1\), estimators of the real parameters \(\beta_0\) and \(\beta_1\), that minimize the deviation of the data from this line.
\[ \begin{equation} f(\beta_0, \beta_1) = \sum_{i=1}^n r_i^2 = \sum_{i=1}^n (y_i-\beta_0-\beta_1 x_i)^2 (\#eq:beta) \end{equation} \]
Note that for clarity we shorten the sum symbol from now on as: \(\sum = \sum_{i=1}^n\). It is important not to mix such estimators with the real parameters, since we do not know how accurate our estimation will be. The real parameters \(\beta_0\) and \(\beta_1\) reflect the actual physical process that underlies the data we measured.
To minimize the residuals, we get the partial derivatives and set them to 0.
\[ \begin{align} \frac{\partial f}{\partial \beta_0} \vert_{b_0} &= \sum 2\cdot (y_i-b_0-\beta_1 x_i)\cdot 1 &=0\\ &= 2 \sum y_i - 2 n b_0 - 2 \beta_1 \sum x_i &=0\\ b_0 &= \frac{1}{n}\sum y_i - \beta_1 \frac{1}{n} \sum x_i\\\hline \frac{\partial f}{\partial \beta_1} \vert_{b_1} &= \sum 2\cdot (y_i-\beta_0-b_1 x_i)\cdot x_i &=0\\ &= \sum x_i y_i - \beta_0 \sum x_i - b_1 \sum x_i^2 &=0 \end{align} \]
Combining both minimization criteria, with \(b_0=\beta_0\) and \(b_1=\beta_1\):
\[ \begin{align} \sum x_i y_i - \left(\frac{1}{n}\sum y_i - b_1 \frac{1}{n} \sum x_i\right) \sum x_i - b_1 \sum x_i^2 =0\\ \sum x_i y_i - \frac{1}{n}\sum y_i \sum x_i + b_1 \frac{1}{n} \left(\sum x_i\right)^2 - b_1 \sum x_i^2 =0\\ \sum x_i y_i - \frac{1}{n}\sum y_i \sum x_i = b_1 \left(\sum x_i^2 - \frac{1}{n} \left(\sum x_i\right)^2\right)\\ b_1 = \frac{\sum x_i y_i - \frac{1}{n}\sum y_i \sum x_i}{\sum x_i^2 - \frac{1}{n} \left(\sum x_i\right)^2} \end{align} \]
Introducing the arithmetic mean, \(\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i\), and using the centered form of the variance, we obtain the well known equation for linear regression:
\[ \begin{align} b_0 &= \bar y - b_1 \bar x\\ b_1 &= \frac{\sum(x_i-\bar x )(y_i-\bar y)}{\sum(x_i-\bar x)^2}\;. (\#eq:emin) \end{align} \]
This can be understood intuitively, especially the centered form. It contains the sum of the squared differences of the \(x_i\) data points from their arithmetic mean \(\bar x\), as well as a similar expression with the products of \(x\) and \(y\) data. These “sums of squares (products)” (\(SQ\)) are a measure of the spread of the data around their mean, known as variance (covariance).
Required Differentiation Rules:
- \((f(x)+g(x))' = f'(x) + g'(x)\),
- \((f(x)^2)' = 2 \cdot f(x) \cdot f'(x)\).
Required Summation Rules:
- \(\sum_{i=1}^n (a_i+b_i) = \sum_{i=1}^n a_i + \sum_{i=1}^n b_i\),
- \(\sum_{i=1}^n(c a_i) = c \sum_{i=1}^n a_i\),
- \(\sum_{i=1}^n c = n c\),
- also good to know, no rule for \(\sum_{i=1}^n (a_i b_i)\).
Incremental Calculation
The last step to obtain equation @ref(eq:emin) is known in German as Verschiebungssatz von Steiner (Steiner translation theorem), in French as théorème de König-Huygens. In English this transformation has no special name and the two forms are just known as the centered (left) and expanded (right) forms of the variance:
\[\begin{equation} \sum(x_i-\bar x)^2= \sum x_i^2 - n \bar x^2 = \sum x_i^2 - \frac{1}{n}(\sum x_i)^2\;. \end{equation}\]
We can apply this to all sum of squares required to calculate various statistical measures of the data:
\[ \begin{align} S_{xx} \def & \sum(x_i - \bar x)^2 &=& \sum x_i^2 - \frac{1}{n} (\sum x_i)^2\\ S_{yy} \def & \sum(y_i - \bar y)^2 &=& \sum y_i^2 - \frac{1}{n} (\sum y_i)^2\\ S_{xy} \def & \sum(y_i - \bar y)(x_i - \bar x) &=& \sum y_i x_i - \frac{1}{n} \sum y_i\sum x_i (\#eq:sos) \end{align} \]
Their expanded forms consist of additions of simple sums of the data, and will thus allow to calculate the sum of squares incrementally by adding up the sum expressions \(\sum x_i^2\), \(\sum x_i\), \(\sum y_i^2\), \(\sum y_i\), and \(\sum x_i y_i\).
An important caveat is that the expanded forms can become 0 for large \(\bar x^2\), known as “catastrophic cancellation”.
Useful Definitions
For completeness and context, we provide some standard definitions in statistics literature.
Variance & Covariance
Normalizing the sum of squares by the number of data points provides the variance, \(\Var(x)\):
\[ \begin{align} &\; \tilde s_x^2 =& \frac{1}{n} S_{xx}\\ \Var(x) \def &\; s_x^2 =& \frac{1}{n-1}S_{xx} \end{align} \]
The intuitive normalization by all data points \(n\) is often denoted the “empirical variance”, in German “mittleres Abweichungsquadrat”. Statistics packages usually use the second version, normalized by \(n-1\), the “degrees of freedom”, and denoted the “theoretical variance” or “inductive variance”, in German Stichprobenvarianz. This corrected version accounts for the fact that the deviation of the last value \(x_n\) from the mean is already defined by the first 1 to \(n-1\) values. It does not further contribute to the spread of the data. Correcting \(\tilde s^2\) by multiplication with \(\frac{n}{n-1}\) is in German textbooks sometimes called Bessel-Korrektur.
Again a similar concept exists combining \(x\) and \(y\) data, and this is called co-variance, \(\Cov(x,y)\):
\[ \Cov(x,y) \def s_{xy}^2 = \frac{1}{n-1} S_{xy}\;. \]
Going back to our residual error minimization in equation @ref(eq:emin), we see that the slope of our regression line is given by the ratio of the data co-variance over the variance of the \(x\) data, and the normalization terms are canceled out:
\[ b_1 = \frac{\Cov(x,y)}{\Var(x)}\;. \]
Correlation and R-squared
The fraction of variance in \(y\) explained by a linear dependence on \(x\) is often denoted as R-squared (\(R^2\); see below: Coefficient of Determination) and is the square of the Pearson correlation \(r\). The Pearson correlation is the covariance of x and y, divided by the square root of the product of the variances in x and y:
\[ r = \frac{\Cov(x,y)}{\Var(x)\Var(y)}\;. \]
Standard Deviation & Standard Error
Above symbols \(s^2\) refer to the sum of squares or products and accordingly have squared or two different units. To directly compare the spread of the data with the data itself, eg. the mean of the data, we simple take the (positive) square root of the variances to obtain the well known standard deviation:
\[ s = \sqrt{s^2} \;. \]
The standard deviation is a very useful measure in conjunction with the mean, if we a are interested how wide the population data are spread around a mean, consider eg. body height distributions. Ie. if the thing we measure actually varies.
If we are more interested in the “precision of the mean” value of our measurements, ie. if we are sure that the thing we measure does not vary, eg. the mass of an atom, and most variation of measurement values just comes from errors of our measurement device, we further normalize the standard deviation to obtain the standard error:
\[ e = \frac{s}{\sqrt n}\;. \]
In short, if we expect actual variation of the measured phenomenon, the standard deviation is the measure of choice, while standard error is an estimate of the precision of a measurement device, eg. chemical or optical probes.
Incremental Calculation of Scoring Functions
Let’s keep these definitions in mind but return to linear regression. The aim was to obtain expressions that we can calculate incrementally. We have found a regression line that minimizes the sum of squares of the residuals, the distance of actual measured values \(y\) from the regression line \(\hat y_i\).
Coefficient of Determination: \(R^2\)
To judge how well the regression line describes our data we can define a new term that quantifies the fraction of the spread of the data that can be explained by the regression line. Again “sum of squares” measures are used. The “Sum of sQuares Explained” (\(SQE\)) describes the spread of our prognosed data \(\hat y_i\) around the mean of the original data \(\bar y\):
\[\begin{equation} SQE \def \sum (\hat y_i - \bar y)^2\;, \end{equation}\]
and the “Sum of sQuares Total” (\(SQT\)) describes the total spread of the data around its mean:
\[\begin{equation} SQT \def \sum (y_i - \bar y)^2 = S_{yy}\;. \end{equation}\]
Their ratio is the fraction of the total data spread that we can explain by our regression line, often denoted R-squared:
\[ R^2 \def \frac{SQE}{SQT}\;, \]
which reaches \(R^2\rightarrow 1\) for a perfect fit. We have already introduced \(SQT\) as \(S_{yy}\) in equation @ref(eq:sos). To calculate \(SQE\) we replace the defining terms by our regression results:
\[ \begin{align} SQE = \sum (\hat y_i - \bar y)^2 &= \sum(b_0 + b_1 x_i - (b_0 + b_1 \bar x))^2\\ &= \sum(b_1(x_i - \bar x))^2\\ &= b_1^2 \sum (x_i - \bar x)^2\\ &= \left(\frac{S_{xy}}{S_{xx}}\right)^2S_{xx} = \frac{S_{xy}^2}{S_{xx}} \end{align} \]
and get:
\[ \begin{equation} R^2 = \frac{SQE}{SQT} = \frac{S_{xy}^2}{S_{xx}S_{yy}} = b_1 \frac{S_{xy}}{S_{yy}} = \frac{\Cov(x,y)}{\Var(x)}\cdot\frac{\Cov(x,y)}{\Var(y)}\;, (\#eq:r2) \end{equation} \]
for which we can obtain all values incrementally via the expanded forms of the sum of squares \(S_{xy}\), \(S_{xx}\) and \(S_{xy}\) in equation @ref(eq:sos).
Notably, R-squared is also the squared version of the Pearson correlation \(r\):
\[ | r | = \sqrt{R^2} = \frac{| S_{xy} |}{\sqrt{S_{xx}S_{yy}}} \;, \]
and both, R-squared (type="r2") and Pearson correlation
(type="cor") can be used as optional scoring functions of
dpseg, where -1 is subtracted (eq. @ref(eq:scorer2)) to
allow a consistent default penalty parameter \(P=0\).
Variance of Residuals: \(\mathrm{Var}(r)\)
The R-squared value (\(R^2\)) reflects a direct linear dependence of \(y\) on \(x\) values, and depends on the slope \(b_1\ne0\). \(R^2\) actually measures correlation between \(x\) and \(y\), or how well changes in \(y\) depend on changes in \(x\). A horizontal line has no correlation, \(y\) does not change with \(x\). When segmenting data into linear parts, \(R^2\)-based measures ignore regions without change in \(y\) at \(b_1\approx 0\), where data spreads around a mean value \(\bar y= b_0\). This can be seen in Figures @ref(fig:lines) and @ref(fig:custom) where a short horizontal segment of the data is ignored when using \(R^2\) as scoring functions.
Minimization of the variance of the residuals is independent of the slope and is thus a better optimization criterium for such cases:
\[ \Var(r) = s_r^2 = \frac{1}{n-1} \sum (r_i-\bar r)^2= \frac{1}{n-1} \sum r_i^2\;, \]
where \(\sum r_i = \bar r=0\) follows from the condition of the minimization \(\frac{\partial f}{\partial \beta_0} \vert_{b_0}=0\) of equation @ref(eq:beta).
Minimization of the “Sum of sQuares of the Residuals” (\(SQR\)) was the initial optimization criterium (eq. @ref(eq:beta)):
\[ SQR \def \sum r_i^2 = \sum(y_i- \hat y_i)^2\;, \]
and an expression for incremental calculation is obtained by partitioning the residual sum of squares (Quadratsummenzerlegung):
\[ \begin{align} \sum(y_i- \bar y_i)^2 &= \sum(y_i- \bar y_i + \hat y_i - \hat y_i)^2\\ &= \sum((\hat y_i -\bar y)+(y_i- \hat y_i))^2\\ &= \sum((\hat y_i -\bar y)+ r_i)^2\\ &= \sum((\hat y_i - \bar y)^2 + r_i^2 + 2 r_i \hat y_i - 2 r_i \bar y)\\ &= \sum(\hat y_i - \bar y)^2 + \sum r_i^2 + 2 \sum r_i \hat y_i - 2 \bar y \sum r_i\\ \sum(y_i- \bar y_i)^2 &= \sum(\hat y_i - \bar y)^2 + \sum r_i^2 \\ SQT &= SQE + SQR\;, \end{align} \]
again with \(\sum r_i = 0\).
In other words the total spread of the data is the sum of the spread explained by our regression line \(\hat y\) and the spread that remains in the un-explained residuals. We can obtain the latter as:
\[ \begin{align} \sum r_i^2 &= \sum(y_i- \bar y_i)^2 - \sum(\hat y_i - \bar y)^2 = SQT - SQE\;, \end{align} \]
and the variance of the residuals as:
\[ \begin{equation} \Var(r) = s_r^2 = \frac{1}{n-1} \sum r_i^2 = \frac{1}{n-1}\left(S_{yy} - \frac{S_{xy}^2}{S_{xx}}\right) = \Var(y) - \frac{\Cov(x,y)^2}{\Var(x)}\,, (\#eq:var) \end{equation} \]
which allows incremental calculation from the expanded forms of the sum of squares \(S_{xy}\), \(S_{xx}\) and \(S_{xy}\) in equation @ref(eq:sos).
The negative variance of the residuals, \(-\Var(r)\), is used as the default scoring
function (type="var", eq. @ref(eq:score)) of
dpseg.
Special Cases: \(\mathrm{Var}=0\).
Vertical segments in the data with \(\Var(x)=0\), i.e., multiple \(y\) values at a single \(x\) value, lead to division by 0 for all
scoring functions (eq. @ref(eq:r2) and @ref(eq:var)). These cases are
detected by dpseg, the scoring functions are set to \(-\infty\), and a warning issued. The data
points will be part of adjacent or spanning segments (bottom panels in
Fig. @ref(lines)).
Horizontal segments in the data with \(\Var(y)=0\) and slope \(b_1=0\), i.e., equal \(y\) values at adjacent \(x\), lead to division by 0 in the
correlation-based scoring functions (eq. @ref(eq:r2)). Here,
dpseg will set \(R^2=0\)
and issue a warning if correlation-based scoring is used
(type="r2" and type="cor"). The scoring
functions will evaluate to their minimum. This is consistent with the
general behaviour of correlation-based scoring functions to favor sloped
(\(|b_1|>0\)) over flat (\(b_1\approx 0\)) segments. The default
scoring function based on the variance of residuals (eq. @ref(eq:var))
is not affected and will detect such segments (top left panel in Fig.
@ref(lines)).
Both cases are exemplified in Section @ref(lines) and Figure @ref(lines).