fdakma (version 1.1.1)

kma.similarity: Similarity/dissimilarity index between two functions

Description

kma.similarity computes a similarity/dissimilarity measure between two functions $f$ and $g$. Users can choose among different types of measures.

Usage

kma.similarity(x.f = NULL, y0.f = NULL, y1.f = NULL,x.g = NULL, y0.g = NULL, y1.g = NULL, similarity.method, unif.grid = TRUE)

Arguments

x.f

vector length.f: abscissa grid where function $f$ and his first derivatives $f'$ is evaluated. length.f: numbrt of abscissa values where $f$ is evaluated. x.f must always be provided.

y0.f

vector length.f or matrix length.f X d: evaluations of function $f$ on the abscissa grid x.f. length.f: number of abscissa values where $f$ is evaluated. d (only if $f$ and $g$ are multidimensional

y1.f

vector length.f or matrix length.f X d: evaluations of $f$ first derivative, i.e., $f'$, on the abscissa grid x.f. Default value of y1.f is NULL. The vector y1.f must be provid

x.g

vector length.g: abscissa grid where function $g$ and his first derivatives $g'$ is evaluated. length.g: numbrt of abscissa values where $g$ is evaluated. x.g must always be provided.

y0.g

vector length.g or matrix length.g X d: evaluations of function $g$ on the abscissa grid x.g. length.g: number of abscissa values where $g$ is evaluated. d (only if $f$ and $g$ are multidimension

y1.g

vector length.g or matrix length.g X d: evaluations of $g$ first derivative, i.e., $g'$, on the abscissa grid x.g. Default value is of y1.g NULL. The vector y1.g must be provid

similarity.method

character: similarity/dissimilarity between $f$ and $g$. Possible choices are: 'd0.pearson', 'd1.pearson', 'd0.L2', 'd1.L2','d0.L2.centered', 'd1.L2.centered'. Default value is

unif.grid

boolean: if equal to TRUE the similarity measure is computed over an uniform grid built in the intersection domain of the two functions, that is an additional discretization is performed. If equal to FALSE the additional discreti

Value

• scalar: similarity/dissimilarity measure between the two functions $f$ and $g$ computed via the similarity/dissimilarity measure specified.

Details

We report the list of the currently available similarities/dissimilarities. Note that all norms and inner products are computed over $D$, that is the intersection of the domains of $f$ and $g$. $\overline{f}$ and $\overline{g}$ denote the mean value, respectively, of functions $f$ and $g$.

1. 'd0.pearson': this similarity measure is the cosine of the angle between the two functions $f$ and $g$.

$$\frac{_{L^2}}{\|{f}\|_{L^2} \|{g}\|_{L^2}}$$ 2. 'd1.pearson': this similarity measure is the cosine of the angle between the two function derivatives $f'$ and $g'$.

$$\frac{_{L^2}}{\|{f'}\|_{L^2} \|{g'}\|_{L^2}}$$

3. 'd0.L2': this dissimilarity measure is the L2 distance of the two functions $f$ and $g$ normalized by the length of the common domain $D$.

$$\frac{\|{f-g}\|_{L^2}}{|D|}$$

4. 'd1.L2': this dissimilarity measure is the L2 distance of the two function first derivatives $f'$ and $g'$ normalized by the length of the common domain $D$.

$$\frac{\|{f'-g'}\|_{L^2}}{|D|}$$

5. 'd0.L2.centered': this dissimilarity measure is the L2 distance of $f-\overline{f}$ and $g-\overline{g}$ normalized by the length of the common domain $D$.

$$\frac{\|{(f-\overline{f})-(g-\overline{g})}\|_{L^2}}{|D|}$$

6. 'd1.L2.centered': this dissimilarity measure is the L2 distance of $f'-\overline{f'}$ and $g'-\overline{g'}$ normalized by the length of the common domain $D$.

$$\frac{\|{(f'-\overline{f'})-(g'-\overline{g'})}\|_{L^2}}{|D|}$$

For multidimensional functions, if similarity.method='d0.pearson' or 'd1.pearson' the similarity/dissimilarity measure is computed via the average of the indexes in all directions. The coherence properties specified in Sangalli et al. (2010) implies that if similarity.method is set to 'd0.L2', 'd1.L2', 'd0.L2.centered' or 'd1.L2.centered', value of warping.method must be 'shift' or 'NOalignment'. If similarity.method is set to 'd0.pearson' or 'd1.pearson' all values for warping.method are allowed.

References

Sangalli, L.M., Secchi, P., Vantini, S., Vitelli, V., 2010. "K-mean alignment for curve clustering", Computational Statistics and Data Analysis, 54, 1219-1233, 2010.

See Also

kma, kma.compare, kma.show.results, fdakma, kma.data

Examples

data(kma.data)

x.f <- kma.data$x # abscissas of f and f'
x.g <- kma.data$x # abscissas of g and g'

y0.f <- kma.data$y0[1,] # evaluations of f on the abscissa grid x.f
y1.f <- kma.data$y1[1,] # evaluations of f' on the abscissa grid x.f
y0.g <- kma.data$y0[3,] # evaluations of g on the abscissa grid x.g
y1.g <- kma.data$y1[3,] # evaluations of g' on the abscissa grid x.g


# Plot of the two functions f and g
  plot(t(x.g),t(y0.g), type='l', xlab='x', ylab='y',, col='red')
  points(t(x.f),t(y0.f), type='l')
  title ('f and g')
  legend('bottomleft', legend=c('f','g'), 
         col=c('black','red'), lty=c(1,1), cex = 0.5)

# Example: 'd0.pearson'
kma.similarity (x.f=x.f, y0.f=y0.f, x.g=x.g,  y0.g=y0.g, similarity.method='d0.pearson')

# Example: 'd0.L2'
kma.similarity (x.f=x.f, y0.f=y0.f, x.g=x.g,  y0.g=y0.g, similarity.method='d0.L2')

# Plot of the two function first derivatives f' and g'
  plot(t(x.g),t(y1.g), type='l', xlab='x', ylab='y', col='red')
  points(t(x.f),t(y1.f), type='l')
  title ("f' and  g'")
  legend('bottomleft', legend=c("f'","g'"), 
         col=c('black','red'), lty=c(1,1), cex = 0.5)

# Example: 'd1.pearson'
kma.similarity (x.f=x.f, y1.f=y1.f, x.g=x.g,  y1.g=y1.g, similarity.method='d1.pearson')

# Example: 'd1.L2'
kma.similarity (x.f=x.f, y1.f=y1.f, x.g=x.g,  y1.g=y1.g, similarity.method='d1.L2')