Chapter 4 Matrix Models I: Intro to MPMs, and Raw (Empirical) MPMs

An ounce of algebra is worth a ton of verbal argument.

— J.B.S. Haldane

Matrix projection models (MPMs) are representations of the dynamics of a population across all life history stages deemed relevant, across the most relevant time interval (typically one year, but sometimes different, and assumed to be consistent within each analysis). In the simplest MPMs, matrix elements (\(a_{kj}\)) show either the probability of transition for an individual in stage \(j\) at occasion t (along the columns), to stage \(k\) at occasion t+1 (along the rows), or the mean rate of production of offspring into stage \(k\) at occasion t+1 (along the rows) by adult individuals in reproductive stage \(j\) in occasion t (along the columns). Conceptually, each individual is in a particular stage in the instance of monitoring or observation, and then either dies or completes a transition in the interval between that observation occasion and the next. Death is not an explicit life stage and so is not modeled as such, instead becoming a potential endpoint of each transition.

Figure 4.1: MPM for Cypripedium candidum, a North American herbaceous plant

Figure 4.1 shows a simple matrix that assumes five stages in the Cypripedium candidum life cycle. These stages are dormant seed (S), juvenile (J, comprised of protocorms and seedlings), non-flowering but sprouting adult (V), non-sprouting adult (D), and flowering adult (F). In equation 4.1, we project a column vector of numbers of individuals alive in each stage in time 1 (e.g., \(n _{1,S}\) is the number of individuals in stage \(S\) in time 1) a single time step forward using standard matrix multiplication with the projection matrix. The product is another column vector showing the predicted numbers of individuals alive in each stage in the next occasion (e.g., \(n _{2,S}\) is the number of individuals in stage \(S\) in time 2).

\[\begin{equation} \left[\begin{array} {rrrrr} n _{2,S} \\ n _{2,J} \\ n _{2,V} \\ n _{2,D} \\ n _{2,F} \\ \end{array}\right] = \left[\begin{array} {ccccc} S_{SS} & 0 & 0 & 0 & F _{SF} \\ S_{JS} & S_{JJ} & 0 & 0 & F _{JF} \\ 0 & S_{VJ} & S_{VV} & S_{VD} & S _{VF} \\ 0 & S_{DJ} & S_{DV} & S_{DD} & S _{DF} \\ 0 & 0 & S_{FV} & S_{FD} & S _{FF} \\ \end{array}\right] \left[\begin{array} {rrrrr} n _{1,S} \\ n _{1,J} \\ n _{1,V} \\ n _{1,D} \\ n _{1,F} \\ \end{array}\right] \tag{4.1} \end{equation}\]

Given this, the total population size in any occasion is the sum of the numbers of individuals in all stages in that occasion, or in other words the sum of the elements in the column vector. The true discrete population growth rate (\(\lambda\)) is calculated as the ratio of the true total population size in occasion t+1 to the true total population size in occasion t. Naturally, projections are estimates, so the estimated population growth rate (\(\hat{\lambda}\)) can be calculated as either the ratio of projected population size in time t+1 to projected population size in time t, or as the dominant eigenvalue of the projection matrix. In the latter case, estimated \(\hat{\lambda}\) is actually the expected population growth rate assuming that the population is at stable stage equilibrium, and so may differ from the true \(\lambda\) depending on conditions.

How does one estimate the elements composing the matrix? First of all, we note that there are two primary sorts of elements - survival-transition elements, and fecundity elements. The former are probabilities denoting the chance of transitioning from one stage in time t to another in stage t+1, and the latter are rates denoting the number of offspring produced per reproductive individual in time t that survive to time t+1 (this is the pre-breeding model as described in Chapter 2, while fecundity in the post-breeding case would equal the survival of the reproductive parent from time t to time t+1 multiplied by the number of offspring produced). In the raw MPM (also called an empirical MPM), which is the first and still dominant kind of MPM, survival-transition elements are estimated as

\[\begin{equation} a _{kj} = \frac{n _{kj}}{\sum_{i=1}^{d} n _{ij}} \tag{4.2} \end{equation}\]

Here, we can refer to matrix elements as \(a_{kj}\), where the element represents the rate at which individuals transition to stage \(k\) in occasion t+1 after having been stage \(j\) in occasion t. Then, \(n _{kj}\) is the number of individuals making this transition, and \(d\) is the number of stages plus death in the life history model (\(d=m+1\), where \(m\) is the number of life history stages). So the denominator represents the total number of individuals alive in stage \(j\) in time t regardless of whether they are alive or dead in time t+1. For example, if 100 individuals are alive in stage D in year 2010, and 20 of these are alive in stage D in year 2011, as are a further 40 in stage V and a further 25 in stage F, then the associated transition probabilities are 0.20, 0.40, and 0.25, respectively. The overall survival probability for stage D is the sum of these transitions, or 0.85, with 15% of individuals alive in 2010 dying by the time of the monitoring occasion in 2011. Fecundity in the pre-breeding model can be estimated as

\[\begin{equation} a _{kj} = s_{k} \frac{\sum_{i=1}^{n_{.j}} fec_{i}}{ n _{.j}} \tag{4.3} \end{equation}\]

Here, \(s_{k}\) is the survival of newborns to the first-year stage from time t to time t+1, \(fec_{i}\) is the actual fecundity of individual \(i\) in time t, \(n_{.j}\) is the number of reproductive adults in stage \(j\) at time t, and the right-hand term, \(\frac{\sum_{i=1}^{n_{.j}} fec_{i}}{ n _{.j}}\), is the average offspring production per reproductive adult.

4.1 Developing raw MPMs with lefko3

In package lefko3, the primary function to create basic raw MPMs is rlefko2(). In the code below, we use this function to create a series of raw matrices for our Cypripedium candidum dataset. In our call to rlefko2(), we need to provide R at the very least with the dataset that it will use to estimate the matrix with (data = cypraw_v1), the stageframe identifying and describing all stages in the life history model (stageframe = cypframe_raw), and the names of the variables coding for stage in times t+1 and t (stages = c("stage3", "stage2")). Our call also includes a statement that we wish to estimate matrices for all occasions possible (year = "all", which also requires the identification of the monitoring time variable, yearcol = "year2"). We will also specify that we wish matrices to be estimated for all three patches (patch = "all", which also requires the identification of the patch variable, patchcol = "patchid"). This function assumes default variable names for most of the status variables in hfv datasets, and since we are using size variables other than the defaults, we need to supply those names (size = c("size3added", "size2added")). Finally, we will supply our supplement table (supplement = cypsupp2_raw), and note the name of the variable coding for individual identity (indivcol = "individ").

cypmatrix2rp <- rlefko2(data = cypraw_v1, stageframe = cypframe_raw,
  year = "all", patch = "all", stages = c("stage3", "stage2"),
  size = c("size3added", "size2added"), supplement = cypsupp2_raw, 
  yearcol = "year2", patchcol = "patchid", indivcol = "individ")

cypmatrix2rp
> $A
> $A$`1`
>       [,1] [,2] [,3] [,4]      [,5] [,6]      [,7]  [,8]         [,9] [,10]
>  [1,] 0.08  0.0  0.0 0.00 0.0000000    0 0.0000000 500.0 1666.6666667     0
>  [2,] 0.10  0.0  0.0 0.00 0.0000000    0 0.0000000 500.0 1666.6666667     0
>  [3,] 0.00  0.1  0.0 0.00 0.0000000    0 0.0000000   0.0    0.0000000     0
>  [4,] 0.00  0.0  0.1 0.00 0.0000000    0 0.0000000   0.0    0.0000000     0
>  [5,] 0.00  0.0  0.0 0.05 0.0500000    0 0.0000000   0.0    0.0000000     0
>  [6,] 0.00  0.0  0.0 0.00 0.0000000    0 0.0000000   0.0    0.0000000     0
>  [7,] 0.00  0.0  0.0 0.00 0.4454545    0 0.6363636   0.2    0.0000000     0
>  [8,] 0.00  0.0  0.0 0.00 0.1909091    0 0.2727273   0.6    0.6666667     1
>  [9,] 0.00  0.0  0.0 0.00 0.0000000    0 0.0000000   0.2    0.3333333     0
> [10,] 0.00  0.0  0.0 0.00 0.0000000    0 0.0000000   0.0    0.0000000     0
> [11,] 0.00  0.0  0.0 0.00 0.0000000    0 0.0000000   0.0    0.0000000     0
>       [,11]
>  [1,]     0
>  [2,]     0
>  [3,]     0
>  [4,]     0
>  [5,]     0
>  [6,]     0
>  [7,]     0
>  [8,]     0
>  [9,]     0
> [10,]     0
> [11,]     0
> 
> $A$`2`
>       [,1] [,2] [,3] [,4]   [,5] [,6]    [,7]         [,8] [,9] [,10] [,11]
>  [1,] 0.08  0.0  0.0 0.00 0.0000    0 312.500 1111.1111111 1250     0     0
>  [2,] 0.10  0.0  0.0 0.00 0.0000    0 312.500 1111.1111111 1250     0     0
>  [3,] 0.00  0.1  0.0 0.00 0.0000    0   0.000    0.0000000    0     0     0
>  [4,] 0.00  0.0  0.1 0.00 0.0000    0   0.000    0.0000000    0     0     0
>  [5,] 0.00  0.0  0.0 0.05 0.0500    0   0.000    0.0000000    0     0     0
>  [6,] 0.00  0.0  0.0 0.00 0.0875    0   0.125    0.0000000    0     0     0
>  [7,] 0.00  0.0  0.0 0.00 0.5250    0   0.750    0.3333333    0     0     0
>  [8,] 0.00  0.0  0.0 0.00 0.0000    0   0.000    0.4444444    0     0     0
>  [9,] 0.00  0.0  0.0 0.00 0.0000    0   0.000    0.2222222    1     0     0
> [10,] 0.00  0.0  0.0 0.00 0.0000    0   0.000    0.0000000    0     0     0
> [11,] 0.00  0.0  0.0 0.00 0.0000    0   0.000    0.0000000    0     0     0
> 
> $A$`3`
>       [,1] [,2] [,3] [,4]      [,5] [,6]      [,7] [,8] [,9] [,10] [,11]
>  [1,] 0.08  0.0  0.0 0.00 0.0000000    0 0.0000000    0 0.00     0     0
>  [2,] 0.10  0.0  0.0 0.00 0.0000000    0 0.0000000    0 0.00     0     0
>  [3,] 0.00  0.1  0.0 0.00 0.0000000    0 0.0000000    0 0.00     0     0
>  [4,] 0.00  0.0  0.1 0.00 0.0000000    0 0.0000000    0 0.00     0     0
>  [5,] 0.00  0.0  0.0 0.05 0.0500000    0 0.0000000    0 0.00     0     0
>  [6,] 0.00  0.0  0.0 0.00 0.0000000    0 0.0000000    0 0.00     0     0
>  [7,] 0.00  0.0  0.0 0.00 0.4666667    0 0.6666667    0 0.00     0     0
>  [8,] 0.00  0.0  0.0 0.00 0.2333333    1 0.3333333    1 0.25     0     0
>  [9,] 0.00  0.0  0.0 0.00 0.0000000    0 0.0000000    0 0.75     0     0
> [10,] 0.00  0.0  0.0 0.00 0.0000000    0 0.0000000    0 0.00     0     0
> [11,] 0.00  0.0  0.0 0.00 0.0000000    0 0.0000000    0 0.00     0     0
> 
> $A$`4`
>       [,1] [,2] [,3] [,4]      [,5] [,6]      [,7]      [,8]      [,9] [,10]
>  [1,] 0.08  0.0  0.0 0.00 0.0000000    0 0.0000000 0.0000000 0.0000000     0
>  [2,] 0.10  0.0  0.0 0.00 0.0000000    0 0.0000000 0.0000000 0.0000000     0
>  [3,] 0.00  0.1  0.0 0.00 0.0000000    0 0.0000000 0.0000000 0.0000000     0
>  [4,] 0.00  0.0  0.1 0.00 0.0000000    0 0.0000000 0.0000000 0.0000000     0
>  [5,] 0.00  0.0  0.0 0.05 0.0500000    0 0.0000000 0.0000000 0.0000000     0
>  [6,] 0.00  0.0  0.0 0.00 0.1166667    0 0.1666667 0.2222222 0.0000000     0
>  [7,] 0.00  0.0  0.0 0.00 0.3500000    0 0.5000000 0.4444444 0.3333333     0
>  [8,] 0.00  0.0  0.0 0.00 0.0000000    0 0.0000000 0.3333333 0.3333333     0
>  [9,] 0.00  0.0  0.0 0.00 0.0000000    0 0.0000000 0.0000000 0.3333333     0
> [10,] 0.00  0.0  0.0 0.00 0.0000000    0 0.0000000 0.0000000 0.0000000     0
> [11,] 0.00  0.0  0.0 0.00 0.0000000    0 0.0000000 0.0000000 0.0000000     0
>       [,11]
>  [1,]     0
>  [2,]     0
>  [3,]     0
>  [4,]     0
>  [5,]     0
>  [6,]     0
>  [7,]     0
>  [8,]     0
>  [9,]     0
> [10,]     0
> [11,]     0
> 
> $A$`5`
>       [,1] [,2] [,3] [,4]      [,5]      [,6]      [,7]  [,8] [,9] [,10] [,11]
>  [1,] 0.08  0.0  0.0 0.00 0.0000000 0.0000000 0.0000000 500.0 5000     0     0
>  [2,] 0.10  0.0  0.0 0.00 0.0000000 0.0000000 0.0000000 500.0 5000     0     0
>  [3,] 0.00  0.1  0.0 0.00 0.0000000 0.0000000 0.0000000   0.0    0     0     0
>  [4,] 0.00  0.0  0.1 0.00 0.0000000 0.0000000 0.0000000   0.0    0     0     0
>  [5,] 0.00  0.0  0.0 0.05 0.0500000 0.0000000 0.0000000   0.0    0     0     0
>  [6,] 0.00  0.0  0.0 0.00 0.0000000 0.0000000 0.0000000   0.0    0     0     0
>  [7,] 0.00  0.0  0.0 0.00 0.3111111 0.3333333 0.4444444   0.0    0     0     0
>  [8,] 0.00  0.0  0.0 0.00 0.3111111 0.6666667 0.4444444   0.6    0     0     0
>  [9,] 0.00  0.0  0.0 0.00 0.0000000 0.0000000 0.0000000   0.0    0     0     0
> [10,] 0.00  0.0  0.0 0.00 0.0000000 0.0000000 0.0000000   0.0    1     0     0
> [11,] 0.00  0.0  0.0 0.00 0.0000000 0.0000000 0.0000000   0.0    0     0     0
> 
> $A$`6`
>       [,1] [,2] [,3] [,4]       [,5] [,6]      [,7]        [,8]   [,9]   [,10]
>  [1,] 0.08  0.0  0.0 0.00 0.00000000    0 0.0000000 166.6666667 625.00 1875.00
>  [2,] 0.10  0.0  0.0 0.00 0.00000000    0 0.0000000 166.6666667 625.00 1875.00
>  [3,] 0.00  0.1  0.0 0.00 0.00000000    0 0.0000000   0.0000000   0.00    0.00
>  [4,] 0.00  0.0  0.1 0.00 0.00000000    0 0.0000000   0.0000000   0.00    0.00
>  [5,] 0.00  0.0  0.0 0.05 0.05000000    0 0.0000000   0.0000000   0.00    0.00
>  [6,] 0.00  0.0  0.0 0.00 0.07777778    0 0.1111111   0.0000000   0.00    0.00
>  [7,] 0.00  0.0  0.0 0.00 0.46666667    0 0.6666667   0.2666667   0.00    0.00
>  [8,] 0.00  0.0  0.0 0.00 0.07777778    0 0.1111111   0.6000000   0.50    0.00
>  [9,] 0.00  0.0  0.0 0.00 0.00000000    0 0.0000000   0.1333333   0.25    0.25
> [10,] 0.00  0.0  0.0 0.00 0.00000000    0 0.1111111   0.0000000   0.25    0.50
> [11,] 0.00  0.0  0.0 0.00 0.00000000    0 0.0000000   0.0000000   0.00    0.25
>       [,11]
>  [1,]     0
>  [2,]     0
>  [3,]     0
>  [4,]     0
>  [5,]     0
>  [6,]     0
>  [7,]     0
>  [8,]     0
>  [9,]     0
> [10,]     0
> [11,]     0
> 
> $A$`7`
>       [,1] [,2] [,3] [,4]       [,5] [,6]       [,7]         [,8]    [,9]
>  [1,] 0.08  0.0  0.0 0.00 0.00000000    0 0.00000000 1.666667e+03 4375.00
>  [2,] 0.10  0.0  0.0 0.00 0.00000000    0 0.00000000 1.666667e+03 4375.00
>  [3,] 0.00  0.1  0.0 0.00 0.00000000    0 0.00000000 0.000000e+00    0.00
>  [4,] 0.00  0.0  0.1 0.00 0.00000000    0 0.00000000 0.000000e+00    0.00
>  [5,] 0.00  0.0  0.0 0.05 0.05000000    0 0.00000000 0.000000e+00    0.00
>  [6,] 0.00  0.0  0.0 0.00 0.06363636    0 0.09090909 8.333333e-02    0.00
>  [7,] 0.00  0.0  0.0 0.00 0.31818182    0 0.45454545 2.500000e-01    0.25
>  [8,] 0.00  0.0  0.0 0.00 0.25454545    1 0.36363636 5.833333e-01    0.50
>  [9,] 0.00  0.0  0.0 0.00 0.00000000    0 0.00000000 0.000000e+00    0.25
> [10,] 0.00  0.0  0.0 0.00 0.00000000    0 0.00000000 0.000000e+00    0.00
> [11,] 0.00  0.0  0.0 0.00 0.00000000    0 0.00000000 0.000000e+00    0.00
>         [,10] [,11]
>  [1,] 5000.00 17500
>  [2,] 5000.00 17500
>  [3,]    0.00     0
>  [4,]    0.00     0
>  [5,]    0.00     0
>  [6,]    0.00     0
>  [7,]    0.00     0
>  [8,]    0.50     0
>  [9,]    0.25     0
> [10,]    0.25     0
> [11,]    0.00     1
> 
> $A$`8`
>       [,1] [,2] [,3] [,4]      [,5] [,6]        [,7]     [,8]   [,9] [,10]
>  [1,] 0.08  0.0  0.0 0.00 0.0000000    0 277.7777778 781.2500 6250.0  5000
>  [2,] 0.10  0.0  0.0 0.00 0.0000000    0 277.7777778 781.2500 6250.0  5000
>  [3,] 0.00  0.1  0.0 0.00 0.0000000    0   0.0000000   0.0000    0.0     0
>  [4,] 0.00  0.0  0.1 0.00 0.0000000    0   0.0000000   0.0000    0.0     0
>  [5,] 0.00  0.0  0.0 0.05 0.0500000    0   0.0000000   0.0000    0.0     0
>  [6,] 0.00  0.0  0.0 0.00 0.0000000    0   0.0000000   0.1875    0.0     0
>  [7,] 0.00  0.0  0.0 0.00 0.2333333    1   0.3333333   0.3125    0.0     0
>  [8,] 0.00  0.0  0.0 0.00 0.2333333    0   0.3333333   0.3750    0.5     0
>  [9,] 0.00  0.0  0.0 0.00 0.0000000    0   0.1111111   0.0625    0.0     0
> [10,] 0.00  0.0  0.0 0.00 0.0000000    0   0.0000000   0.0625    0.5     1
> [11,] 0.00  0.0  0.0 0.00 0.0000000    0   0.0000000   0.0000    0.0     0
>       [,11]
>  [1,] 10000
>  [2,] 10000
>  [3,]     0
>  [4,]     0
>  [5,]     0
>  [6,]     0
>  [7,]     0
>  [8,]     0
>  [9,]     0
> [10,]     1
> [11,]     0
> 
> $A$`9`
>       [,1] [,2] [,3] [,4]      [,5]      [,6]      [,7]  [,8] [,9]   [,10]
>  [1,] 0.08  0.0  0.0 0.00 0.0000000 0.0000000 0.0000000 250.0  0.0 1250.00
>  [2,] 0.10  0.0  0.0 0.00 0.0000000 0.0000000 0.0000000 250.0  0.0 1250.00
>  [3,] 0.00  0.1  0.0 0.00 0.0000000 0.0000000 0.0000000   0.0  0.0    0.00
>  [4,] 0.00  0.0  0.1 0.00 0.0000000 0.0000000 0.0000000   0.0  0.0    0.00
>  [5,] 0.00  0.0  0.0 0.05 0.0500000 0.0000000 0.0000000   0.0  0.0    0.00
>  [6,] 0.00  0.0  0.0 0.00 0.1272727 0.3333333 0.1818182   0.0  0.0    0.00
>  [7,] 0.00  0.0  0.0 0.00 0.1909091 0.3333333 0.2727273   0.2  0.0    0.00
>  [8,] 0.00  0.0  0.0 0.00 0.3181818 0.0000000 0.4545455   0.2  0.5    0.00
>  [9,] 0.00  0.0  0.0 0.00 0.0000000 0.0000000 0.0000000   0.4  0.5    0.50
> [10,] 0.00  0.0  0.0 0.00 0.0000000 0.3333333 0.0000000   0.0  0.0    0.25
> [11,] 0.00  0.0  0.0 0.00 0.0000000 0.0000000 0.0000000   0.0  0.0    0.25
>       [,11]
>  [1,]     0
>  [2,]     0
>  [3,]     0
>  [4,]     0
>  [5,]     0
>  [6,]     0
>  [7,]     0
>  [8,]     0
>  [9,]     0
> [10,]     0
> [11,]     0
> 
> $A$`10`
>       [,1] [,2] [,3] [,4] [,5] [,6]      [,7]  [,8]        [,9] [,10] [,11]
>  [1,] 0.08  0.0  0.0 0.00 0.00    0 0.0000000 0.000 714.2857143  1250  2500
>  [2,] 0.10  0.0  0.0 0.00 0.00    0 0.0000000 0.000 714.2857143  1250  2500
>  [3,] 0.00  0.1  0.0 0.00 0.00    0 0.0000000 0.000   0.0000000     0     0
>  [4,] 0.00  0.0  0.1 0.00 0.00    0 0.0000000 0.000   0.0000000     0     0
>  [5,] 0.00  0.0  0.0 0.05 0.05    0 0.0000000 0.000   0.0000000     0     0
>  [6,] 0.00  0.0  0.0 0.00 0.00    0 0.0000000 0.000   0.0000000     0     0
>  [7,] 0.00  0.0  0.0 0.00 0.40    1 0.5714286 0.125   0.0000000     0     0
>  [8,] 0.00  0.0  0.0 0.00 0.20    0 0.2857143 0.750   0.0000000     0     0
>  [9,] 0.00  0.0  0.0 0.00 0.00    0 0.0000000 0.125   0.7142857     0     0
> [10,] 0.00  0.0  0.0 0.00 0.00    0 0.0000000 0.000   0.2857143     1     0
> [11,] 0.00  0.0  0.0 0.00 0.00    0 0.0000000 0.000   0.0000000     0     1
> 
> $A$`11`
>       [,1] [,2] [,3] [,4]      [,5] [,6]      [,7]   [,8]        [,9] [,10]
>  [1,] 0.08  0.0  0.0 0.00 0.0000000    0 0.0000000 625.00 833.3333333     0
>  [2,] 0.10  0.0  0.0 0.00 0.0000000    0 0.0000000 625.00 833.3333333     0
>  [3,] 0.00  0.1  0.0 0.00 0.0000000    0 0.0000000   0.00   0.0000000     0
>  [4,] 0.00  0.0  0.1 0.00 0.0000000    0 0.0000000   0.00   0.0000000     0
>  [5,] 0.00  0.0  0.0 0.05 0.0500000    0 0.0000000   0.00   0.0000000     0
>  [6,] 0.00  0.0  0.0 0.00 0.0000000    0 0.0000000   0.00   0.0000000     0
>  [7,] 0.00  0.0  0.0 0.00 0.4666667    0 0.6666667   0.25   0.0000000     0
>  [8,] 0.00  0.0  0.0 0.00 0.2333333    0 0.3333333   0.50   0.3333333     0
>  [9,] 0.00  0.0  0.0 0.00 0.0000000    0 0.0000000   0.25   0.6666667     0
> [10,] 0.00  0.0  0.0 0.00 0.0000000    0 0.0000000   0.00   0.0000000     1
> [11,] 0.00  0.0  0.0 0.00 0.0000000    0 0.0000000   0.00   0.0000000     0
>       [,11]
>  [1,]  7500
>  [2,]  7500
>  [3,]     0
>  [4,]     0
>  [5,]     0
>  [6,]     0
>  [7,]     0
>  [8,]     0
>  [9,]     0
> [10,]     0
> [11,]     1
> 
> $A$`12`
>       [,1] [,2] [,3] [,4]  [,5] [,6] [,7]      [,8]         [,9] [,10] [,11]
>  [1,] 0.08  0.0  0.0 0.00 0.000    0 0.00 0.0000000 4166.6666667  1250  5000
>  [2,] 0.10  0.0  0.0 0.00 0.000    0 0.00 0.0000000 4166.6666667  1250  5000
>  [3,] 0.00  0.1  0.0 0.00 0.000    0 0.00 0.0000000    0.0000000     0     0
>  [4,] 0.00  0.0  0.1 0.00 0.000    0 0.00 0.0000000    0.0000000     0     0
>  [5,] 0.00  0.0  0.0 0.05 0.050    0 0.00 0.0000000    0.0000000     0     0
>  [6,] 0.00  0.0  0.0 0.00 0.000    0 0.00 0.0000000    0.0000000     0     0
>  [7,] 0.00  0.0  0.0 0.00 0.525    0 0.75 0.1666667    0.0000000     0     0
>  [8,] 0.00  0.0  0.0 0.00 0.000    0 0.00 0.8333333    0.6666667     0     0
>  [9,] 0.00  0.0  0.0 0.00 0.000    0 0.00 0.0000000    0.3333333     1     0
> [10,] 0.00  0.0  0.0 0.00 0.000    0 0.00 0.0000000    0.0000000     0     1
> [11,] 0.00  0.0  0.0 0.00 0.000    0 0.00 0.0000000    0.0000000     0     0
> 
> $A$`13`
>       [,1] [,2] [,3] [,4]  [,5] [,6] [,7]         [,8]         [,9] [,10] [,11]
>  [1,] 0.08  0.0  0.0 0.00 0.000    0 0.00 1071.4285714 2500.0000000  2500     0
>  [2,] 0.10  0.0  0.0 0.00 0.000    0 0.00 1071.4285714 2500.0000000  2500     0
>  [3,] 0.00  0.1  0.0 0.00 0.000    0 0.00    0.0000000    0.0000000     0     0
>  [4,] 0.00  0.0  0.1 0.00 0.000    0 0.00    0.0000000    0.0000000     0     0
>  [5,] 0.00  0.0  0.0 0.05 0.050    0 0.00    0.0000000    0.0000000     0     0
>  [6,] 0.00  0.0  0.0 0.00 0.000    0 0.00    0.0000000    0.0000000     0     0
>  [7,] 0.00  0.0  0.0 0.00 0.350    0 0.50    0.0000000    0.3333333     0     0
>  [8,] 0.00  0.0  0.0 0.00 0.175    0 0.25    0.5714286    0.3333333     0     0
>  [9,] 0.00  0.0  0.0 0.00 0.000    0 0.00    0.2857143    0.3333333     0     0
> [10,] 0.00  0.0  0.0 0.00 0.000    0 0.00    0.1428571    0.0000000     1     0
> [11,] 0.00  0.0  0.0 0.00 0.000    0 0.00    0.0000000    0.0000000     0     0
> 
> $A$`14`
>       [,1] [,2] [,3] [,4]      [,5] [,6]      [,7]        [,8]      [,9] [,10]
>  [1,] 0.08  0.0  0.0 0.00 0.0000000    0 0.0000000 416.6666667 0.0000000   0.0
>  [2,] 0.10  0.0  0.0 0.00 0.0000000    0 0.0000000 416.6666667 0.0000000   0.0
>  [3,] 0.00  0.1  0.0 0.00 0.0000000    0 0.0000000   0.0000000 0.0000000   0.0
>  [4,] 0.00  0.0  0.1 0.00 0.0000000    0 0.0000000   0.0000000 0.0000000   0.0
>  [5,] 0.00  0.0  0.0 0.05 0.0500000    0 0.0000000   0.0000000 0.0000000   0.0
>  [6,] 0.00  0.0  0.0 0.00 0.2333333    0 0.3333333   0.1666667 0.0000000   0.0
>  [7,] 0.00  0.0  0.0 0.00 0.2333333    0 0.3333333   0.1666667 0.3333333   0.0
>  [8,] 0.00  0.0  0.0 0.00 0.0000000    0 0.0000000   0.3333333 0.3333333   0.5
>  [9,] 0.00  0.0  0.0 0.00 0.0000000    0 0.3333333   0.3333333 0.3333333   0.0
> [10,] 0.00  0.0  0.0 0.00 0.0000000    0 0.0000000   0.0000000 0.0000000   0.0
> [11,] 0.00  0.0  0.0 0.00 0.0000000    0 0.0000000   0.0000000 0.0000000   0.5
>       [,11]
>  [1,]     0
>  [2,]     0
>  [3,]     0
>  [4,]     0
>  [5,]     0
>  [6,]     0
>  [7,]     0
>  [8,]     0
>  [9,]     0
> [10,]     0
> [11,]     0
> 
> $A$`15`
>       [,1] [,2] [,3] [,4]  [,5] [,6] [,7] [,8] [,9] [,10] [,11]
>  [1,] 0.08  0.0  0.0 0.00 0.000  0.0 0.00 0.00  625     0     0
>  [2,] 0.10  0.0  0.0 0.00 0.000  0.0 0.00 0.00  625     0     0
>  [3,] 0.00  0.1  0.0 0.00 0.000  0.0 0.00 0.00    0     0     0
>  [4,] 0.00  0.0  0.1 0.00 0.000  0.0 0.00 0.00    0     0     0
>  [5,] 0.00  0.0  0.0 0.05 0.050  0.0 0.00 0.00    0     0     0
>  [6,] 0.00  0.0  0.0 0.00 0.000  0.0 0.00 0.00    0     0     0
>  [7,] 0.00  0.0  0.0 0.00 0.175  0.5 0.25 0.00    0     0     0
>  [8,] 0.00  0.0  0.0 0.00 0.525  0.5 0.75 0.50    0     0     0
>  [9,] 0.00  0.0  0.0 0.00 0.000  0.0 0.00 0.25    1     0     0
> [10,] 0.00  0.0  0.0 0.00 0.000  0.0 0.00 0.25    0     0     0
> [11,] 0.00  0.0  0.0 0.00 0.000  0.0 0.00 0.00    0     0     1
> 
> 
> $U
> $U$`1`
>       [,1] [,2] [,3] [,4]      [,5] [,6]      [,7] [,8]      [,9] [,10] [,11]
>  [1,] 0.08  0.0  0.0 0.00 0.0000000    0 0.0000000  0.0 0.0000000     0     0
>  [2,] 0.10  0.0  0.0 0.00 0.0000000    0 0.0000000  0.0 0.0000000     0     0
>  [3,] 0.00  0.1  0.0 0.00 0.0000000    0 0.0000000  0.0 0.0000000     0     0
>  [4,] 0.00  0.0  0.1 0.00 0.0000000    0 0.0000000  0.0 0.0000000     0     0
>  [5,] 0.00  0.0  0.0 0.05 0.0500000    0 0.0000000  0.0 0.0000000     0     0
>  [6,] 0.00  0.0  0.0 0.00 0.0000000    0 0.0000000  0.0 0.0000000     0     0
>  [7,] 0.00  0.0  0.0 0.00 0.4454545    0 0.6363636  0.2 0.0000000     0     0
>  [8,] 0.00  0.0  0.0 0.00 0.1909091    0 0.2727273  0.6 0.6666667     1     0
>  [9,] 0.00  0.0  0.0 0.00 0.0000000    0 0.0000000  0.2 0.3333333     0     0
> [10,] 0.00  0.0  0.0 0.00 0.0000000    0 0.0000000  0.0 0.0000000     0     0
> [11,] 0.00  0.0  0.0 0.00 0.0000000    0 0.0000000  0.0 0.0000000     0     0
> 
> $U$`2`
>       [,1] [,2] [,3] [,4]   [,5] [,6]  [,7]      [,8] [,9] [,10] [,11]
>  [1,] 0.08  0.0  0.0 0.00 0.0000    0 0.000 0.0000000    0     0     0
>  [2,] 0.10  0.0  0.0 0.00 0.0000    0 0.000 0.0000000    0     0     0
>  [3,] 0.00  0.1  0.0 0.00 0.0000    0 0.000 0.0000000    0     0     0
>  [4,] 0.00  0.0  0.1 0.00 0.0000    0 0.000 0.0000000    0     0     0
>  [5,] 0.00  0.0  0.0 0.05 0.0500    0 0.000 0.0000000    0     0     0
>  [6,] 0.00  0.0  0.0 0.00 0.0875    0 0.125 0.0000000    0     0     0
>  [7,] 0.00  0.0  0.0 0.00 0.5250    0 0.750 0.3333333    0     0     0
>  [8,] 0.00  0.0  0.0 0.00 0.0000    0 0.000 0.4444444    0     0     0
>  [9,] 0.00  0.0  0.0 0.00 0.0000    0 0.000 0.2222222    1     0     0
> [10,] 0.00  0.0  0.0 0.00 0.0000    0 0.000 0.0000000    0     0     0
> [11,] 0.00  0.0  0.0 0.00 0.0000    0 0.000 0.0000000    0     0     0
> 
> $U$`3`
>       [,1] [,2] [,3] [,4]      [,5] [,6]      [,7] [,8] [,9] [,10] [,11]
>  [1,] 0.08  0.0  0.0 0.00 0.0000000    0 0.0000000    0 0.00     0     0
>  [2,] 0.10  0.0  0.0 0.00 0.0000000    0 0.0000000    0 0.00     0     0
>  [3,] 0.00  0.1  0.0 0.00 0.0000000    0 0.0000000    0 0.00     0     0
>  [4,] 0.00  0.0  0.1 0.00 0.0000000    0 0.0000000    0 0.00     0     0
>  [5,] 0.00  0.0  0.0 0.05 0.0500000    0 0.0000000    0 0.00     0     0
>  [6,] 0.00  0.0  0.0 0.00 0.0000000    0 0.0000000    0 0.00     0     0
>  [7,] 0.00  0.0  0.0 0.00 0.4666667    0 0.6666667    0 0.00     0     0
>  [8,] 0.00  0.0  0.0 0.00 0.2333333    1 0.3333333    1 0.25     0     0
>  [9,] 0.00  0.0  0.0 0.00 0.0000000    0 0.0000000    0 0.75     0     0
> [10,] 0.00  0.0  0.0 0.00 0.0000000    0 0.0000000    0 0.00     0     0
> [11,] 0.00  0.0  0.0 0.00 0.0000000    0 0.0000000    0 0.00     0     0
> 
> $U$`4`
>       [,1] [,2] [,3] [,4]      [,5] [,6]      [,7]      [,8]      [,9] [,10]
>  [1,] 0.08  0.0  0.0 0.00 0.0000000    0 0.0000000 0.0000000 0.0000000     0
>  [2,] 0.10  0.0  0.0 0.00 0.0000000    0 0.0000000 0.0000000 0.0000000     0
>  [3,] 0.00  0.1  0.0 0.00 0.0000000    0 0.0000000 0.0000000 0.0000000     0
>  [4,] 0.00  0.0  0.1 0.00 0.0000000    0 0.0000000 0.0000000 0.0000000     0
>  [5,] 0.00  0.0  0.0 0.05 0.0500000    0 0.0000000 0.0000000 0.0000000     0
>  [6,] 0.00  0.0  0.0 0.00 0.1166667    0 0.1666667 0.2222222 0.0000000     0
>  [7,] 0.00  0.0  0.0 0.00 0.3500000    0 0.5000000 0.4444444 0.3333333     0
>  [8,] 0.00  0.0  0.0 0.00 0.0000000    0 0.0000000 0.3333333 0.3333333     0
>  [9,] 0.00  0.0  0.0 0.00 0.0000000    0 0.0000000 0.0000000 0.3333333     0
> [10,] 0.00  0.0  0.0 0.00 0.0000000    0 0.0000000 0.0000000 0.0000000     0
> [11,] 0.00  0.0  0.0 0.00 0.0000000    0 0.0000000 0.0000000 0.0000000     0
>       [,11]
>  [1,]     0
>  [2,]     0
>  [3,]     0
>  [4,]     0
>  [5,]     0
>  [6,]     0
>  [7,]     0
>  [8,]     0
>  [9,]     0
> [10,]     0
> [11,]     0
> 
> $U$`5`
>       [,1] [,2] [,3] [,4]      [,5]      [,6]      [,7] [,8] [,9] [,10] [,11]
>  [1,] 0.08  0.0  0.0 0.00 0.0000000 0.0000000 0.0000000  0.0    0     0     0
>  [2,] 0.10  0.0  0.0 0.00 0.0000000 0.0000000 0.0000000  0.0    0     0     0
>  [3,] 0.00  0.1  0.0 0.00 0.0000000 0.0000000 0.0000000  0.0    0     0     0
>  [4,] 0.00  0.0  0.1 0.00 0.0000000 0.0000000 0.0000000  0.0    0     0     0
>  [5,] 0.00  0.0  0.0 0.05 0.0500000 0.0000000 0.0000000  0.0    0     0     0
>  [6,] 0.00  0.0  0.0 0.00 0.0000000 0.0000000 0.0000000  0.0    0     0     0
>  [7,] 0.00  0.0  0.0 0.00 0.3111111 0.3333333 0.4444444  0.0    0     0     0
>  [8,] 0.00  0.0  0.0 0.00 0.3111111 0.6666667 0.4444444  0.6    0     0     0
>  [9,] 0.00  0.0  0.0 0.00 0.0000000 0.0000000 0.0000000  0.0    0     0     0
> [10,] 0.00  0.0  0.0 0.00 0.0000000 0.0000000 0.0000000  0.0    1     0     0
> [11,] 0.00  0.0  0.0 0.00 0.0000000 0.0000000 0.0000000  0.0    0     0     0
> 
> $U$`6`
>       [,1] [,2] [,3] [,4]       [,5] [,6]      [,7]      [,8] [,9] [,10] [,11]
>  [1,] 0.08  0.0  0.0 0.00 0.00000000    0 0.0000000 0.0000000 0.00  0.00     0
>  [2,] 0.10  0.0  0.0 0.00 0.00000000    0 0.0000000 0.0000000 0.00  0.00     0
>  [3,] 0.00  0.1  0.0 0.00 0.00000000    0 0.0000000 0.0000000 0.00  0.00     0
>  [4,] 0.00  0.0  0.1 0.00 0.00000000    0 0.0000000 0.0000000 0.00  0.00     0
>  [5,] 0.00  0.0  0.0 0.05 0.05000000    0 0.0000000 0.0000000 0.00  0.00     0
>  [6,] 0.00  0.0  0.0 0.00 0.07777778    0 0.1111111 0.0000000 0.00  0.00     0
>  [7,] 0.00  0.0  0.0 0.00 0.46666667    0 0.6666667 0.2666667 0.00  0.00     0
>  [8,] 0.00  0.0  0.0 0.00 0.07777778    0 0.1111111 0.6000000 0.50  0.00     0
>  [9,] 0.00  0.0  0.0 0.00 0.00000000    0 0.0000000 0.1333333 0.25  0.25     0
> [10,] 0.00  0.0  0.0 0.00 0.00000000    0 0.1111111 0.0000000 0.25  0.50     0
> [11,] 0.00  0.0  0.0 0.00 0.00000000    0 0.0000000 0.0000000 0.00  0.25     0
> 
> $U$`7`
>       [,1] [,2] [,3] [,4]       [,5] [,6]       [,7]       [,8] [,9] [,10]
>  [1,] 0.08  0.0  0.0 0.00 0.00000000    0 0.00000000 0.00000000 0.00  0.00
>  [2,] 0.10  0.0  0.0 0.00 0.00000000    0 0.00000000 0.00000000 0.00  0.00
>  [3,] 0.00  0.1  0.0 0.00 0.00000000    0 0.00000000 0.00000000 0.00  0.00
>  [4,] 0.00  0.0  0.1 0.00 0.00000000    0 0.00000000 0.00000000 0.00  0.00
>  [5,] 0.00  0.0  0.0 0.05 0.05000000    0 0.00000000 0.00000000 0.00  0.00
>  [6,] 0.00  0.0  0.0 0.00 0.06363636    0 0.09090909 0.08333333 0.00  0.00
>  [7,] 0.00  0.0  0.0 0.00 0.31818182    0 0.45454545 0.25000000 0.25  0.00
>  [8,] 0.00  0.0  0.0 0.00 0.25454545    1 0.36363636 0.58333333 0.50  0.50
>  [9,] 0.00  0.0  0.0 0.00 0.00000000    0 0.00000000 0.00000000 0.25  0.25
> [10,] 0.00  0.0  0.0 0.00 0.00000000    0 0.00000000 0.00000000 0.00  0.25
> [11,] 0.00  0.0  0.0 0.00 0.00000000    0 0.00000000 0.00000000 0.00  0.00
>       [,11]
>  [1,]     0
>  [2,]     0
>  [3,]     0
>  [4,]     0
>  [5,]     0
>  [6,]     0
>  [7,]     0
>  [8,]     0
>  [9,]     0
> [10,]     0
> [11,]     1
> 
> $U$`8`
>       [,1] [,2] [,3] [,4]      [,5] [,6]      [,7]   [,8] [,9] [,10] [,11]
>  [1,] 0.08  0.0  0.0 0.00 0.0000000    0 0.0000000 0.0000  0.0     0     0
>  [2,] 0.10  0.0  0.0 0.00 0.0000000    0 0.0000000 0.0000  0.0     0     0
>  [3,] 0.00  0.1  0.0 0.00 0.0000000    0 0.0000000 0.0000  0.0     0     0
>  [4,] 0.00  0.0  0.1 0.00 0.0000000    0 0.0000000 0.0000  0.0     0     0
>  [5,] 0.00  0.0  0.0 0.05 0.0500000    0 0.0000000 0.0000  0.0     0     0
>  [6,] 0.00  0.0  0.0 0.00 0.0000000    0 0.0000000 0.1875  0.0     0     0
>  [7,] 0.00  0.0  0.0 0.00 0.2333333    1 0.3333333 0.3125  0.0     0     0
>  [8,] 0.00  0.0  0.0 0.00 0.2333333    0 0.3333333 0.3750  0.5     0     0
>  [9,] 0.00  0.0  0.0 0.00 0.0000000    0 0.1111111 0.0625  0.0     0     0
> [10,] 0.00  0.0  0.0 0.00 0.0000000    0 0.0000000 0.0625  0.5     1     1
> [11,] 0.00  0.0  0.0 0.00 0.0000000    0 0.0000000 0.0000  0.0     0     0
> 
> $U$`9`
>       [,1] [,2] [,3] [,4]      [,5]      [,6]      [,7] [,8] [,9] [,10] [,11]
>  [1,] 0.08  0.0  0.0 0.00 0.0000000 0.0000000 0.0000000  0.0  0.0  0.00     0
>  [2,] 0.10  0.0  0.0 0.00 0.0000000 0.0000000 0.0000000  0.0  0.0  0.00     0
>  [3,] 0.00  0.1  0.0 0.00 0.0000000 0.0000000 0.0000000  0.0  0.0  0.00     0
>  [4,] 0.00  0.0  0.1 0.00 0.0000000 0.0000000 0.0000000  0.0  0.0  0.00     0
>  [5,] 0.00  0.0  0.0 0.05 0.0500000 0.0000000 0.0000000  0.0  0.0  0.00     0
>  [6,] 0.00  0.0  0.0 0.00 0.1272727 0.3333333 0.1818182  0.0  0.0  0.00     0
>  [7,] 0.00  0.0  0.0 0.00 0.1909091 0.3333333 0.2727273  0.2  0.0  0.00     0
>  [8,] 0.00  0.0  0.0 0.00 0.3181818 0.0000000 0.4545455  0.2  0.5  0.00     0
>  [9,] 0.00  0.0  0.0 0.00 0.0000000 0.0000000 0.0000000  0.4  0.5  0.50     0
> [10,] 0.00  0.0  0.0 0.00 0.0000000 0.3333333 0.0000000  0.0  0.0  0.25     0
> [11,] 0.00  0.0  0.0 0.00 0.0000000 0.0000000 0.0000000  0.0  0.0  0.25     0
> 
> $U$`10`
>       [,1] [,2] [,3] [,4] [,5] [,6]      [,7]  [,8]      [,9] [,10] [,11]
>  [1,] 0.08  0.0  0.0 0.00 0.00    0 0.0000000 0.000 0.0000000     0     0
>  [2,] 0.10  0.0  0.0 0.00 0.00    0 0.0000000 0.000 0.0000000     0     0
>  [3,] 0.00  0.1  0.0 0.00 0.00    0 0.0000000 0.000 0.0000000     0     0
>  [4,] 0.00  0.0  0.1 0.00 0.00    0 0.0000000 0.000 0.0000000     0     0
>  [5,] 0.00  0.0  0.0 0.05 0.05    0 0.0000000 0.000 0.0000000     0     0
>  [6,] 0.00  0.0  0.0 0.00 0.00    0 0.0000000 0.000 0.0000000     0     0
>  [7,] 0.00  0.0  0.0 0.00 0.40    1 0.5714286 0.125 0.0000000     0     0
>  [8,] 0.00  0.0  0.0 0.00 0.20    0 0.2857143 0.750 0.0000000     0     0
>  [9,] 0.00  0.0  0.0 0.00 0.00    0 0.0000000 0.125 0.7142857     0     0
> [10,] 0.00  0.0  0.0 0.00 0.00    0 0.0000000 0.000 0.2857143     1     0
> [11,] 0.00  0.0  0.0 0.00 0.00    0 0.0000000 0.000 0.0000000     0     1
> 
> $U$`11`
>       [,1] [,2] [,3] [,4]      [,5] [,6]      [,7] [,8]      [,9] [,10] [,11]
>  [1,] 0.08  0.0  0.0 0.00 0.0000000    0 0.0000000 0.00 0.0000000     0     0
>  [2,] 0.10  0.0  0.0 0.00 0.0000000    0 0.0000000 0.00 0.0000000     0     0
>  [3,] 0.00  0.1  0.0 0.00 0.0000000    0 0.0000000 0.00 0.0000000     0     0
>  [4,] 0.00  0.0  0.1 0.00 0.0000000    0 0.0000000 0.00 0.0000000     0     0
>  [5,] 0.00  0.0  0.0 0.05 0.0500000    0 0.0000000 0.00 0.0000000     0     0
>  [6,] 0.00  0.0  0.0 0.00 0.0000000    0 0.0000000 0.00 0.0000000     0     0
>  [7,] 0.00  0.0  0.0 0.00 0.4666667    0 0.6666667 0.25 0.0000000     0     0
>  [8,] 0.00  0.0  0.0 0.00 0.2333333    0 0.3333333 0.50 0.3333333     0     0
>  [9,] 0.00  0.0  0.0 0.00 0.0000000    0 0.0000000 0.25 0.6666667     0     0
> [10,] 0.00  0.0  0.0 0.00 0.0000000    0 0.0000000 0.00 0.0000000     1     0
> [11,] 0.00  0.0  0.0 0.00 0.0000000    0 0.0000000 0.00 0.0000000     0     1
> 
> $U$`12`
>       [,1] [,2] [,3] [,4]  [,5] [,6] [,7]      [,8]      [,9] [,10] [,11]
>  [1,] 0.08  0.0  0.0 0.00 0.000    0 0.00 0.0000000 0.0000000     0     0
>  [2,] 0.10  0.0  0.0 0.00 0.000    0 0.00 0.0000000 0.0000000     0     0
>  [3,] 0.00  0.1  0.0 0.00 0.000    0 0.00 0.0000000 0.0000000     0     0
>  [4,] 0.00  0.0  0.1 0.00 0.000    0 0.00 0.0000000 0.0000000     0     0
>  [5,] 0.00  0.0  0.0 0.05 0.050    0 0.00 0.0000000 0.0000000     0     0
>  [6,] 0.00  0.0  0.0 0.00 0.000    0 0.00 0.0000000 0.0000000     0     0
>  [7,] 0.00  0.0  0.0 0.00 0.525    0 0.75 0.1666667 0.0000000     0     0
>  [8,] 0.00  0.0  0.0 0.00 0.000    0 0.00 0.8333333 0.6666667     0     0
>  [9,] 0.00  0.0  0.0 0.00 0.000    0 0.00 0.0000000 0.3333333     1     0
> [10,] 0.00  0.0  0.0 0.00 0.000    0 0.00 0.0000000 0.0000000     0     1
> [11,] 0.00  0.0  0.0 0.00 0.000    0 0.00 0.0000000 0.0000000     0     0
> 
> $U$`13`
>       [,1] [,2] [,3] [,4]  [,5] [,6] [,7]      [,8]      [,9] [,10] [,11]
>  [1,] 0.08  0.0  0.0 0.00 0.000    0 0.00 0.0000000 0.0000000     0     0
>  [2,] 0.10  0.0  0.0 0.00 0.000    0 0.00 0.0000000 0.0000000     0     0
>  [3,] 0.00  0.1  0.0 0.00 0.000    0 0.00 0.0000000 0.0000000     0     0
>  [4,] 0.00  0.0  0.1 0.00 0.000    0 0.00 0.0000000 0.0000000     0     0
>  [5,] 0.00  0.0  0.0 0.05 0.050    0 0.00 0.0000000 0.0000000     0     0
>  [6,] 0.00  0.0  0.0 0.00 0.000    0 0.00 0.0000000 0.0000000     0     0
>  [7,] 0.00  0.0  0.0 0.00 0.350    0 0.50 0.0000000 0.3333333     0     0
>  [8,] 0.00  0.0  0.0 0.00 0.175    0 0.25 0.5714286 0.3333333     0     0
>  [9,] 0.00  0.0  0.0 0.00 0.000    0 0.00 0.2857143 0.3333333     0     0
> [10,] 0.00  0.0  0.0 0.00 0.000    0 0.00 0.1428571 0.0000000     1     0
> [11,] 0.00  0.0  0.0 0.00 0.000    0 0.00 0.0000000 0.0000000     0     0
> 
> $U$`14`
>       [,1] [,2] [,3] [,4]      [,5] [,6]      [,7]      [,8]      [,9] [,10]
>  [1,] 0.08  0.0  0.0 0.00 0.0000000    0 0.0000000 0.0000000 0.0000000   0.0
>  [2,] 0.10  0.0  0.0 0.00 0.0000000    0 0.0000000 0.0000000 0.0000000   0.0
>  [3,] 0.00  0.1  0.0 0.00 0.0000000    0 0.0000000 0.0000000 0.0000000   0.0
>  [4,] 0.00  0.0  0.1 0.00 0.0000000    0 0.0000000 0.0000000 0.0000000   0.0
>  [5,] 0.00  0.0  0.0 0.05 0.0500000    0 0.0000000 0.0000000 0.0000000   0.0
>  [6,] 0.00  0.0  0.0 0.00 0.2333333    0 0.3333333 0.1666667 0.0000000   0.0
>  [7,] 0.00  0.0  0.0 0.00 0.2333333    0 0.3333333 0.1666667 0.3333333   0.0
>  [8,] 0.00  0.0  0.0 0.00 0.0000000    0 0.0000000 0.3333333 0.3333333   0.5
>  [9,] 0.00  0.0  0.0 0.00 0.0000000    0 0.3333333 0.3333333 0.3333333   0.0
> [10,] 0.00  0.0  0.0 0.00 0.0000000    0 0.0000000 0.0000000 0.0000000   0.0
> [11,] 0.00  0.0  0.0 0.00 0.0000000    0 0.0000000 0.0000000 0.0000000   0.5
>       [,11]
>  [1,]     0
>  [2,]     0
>  [3,]     0
>  [4,]     0
>  [5,]     0
>  [6,]     0
>  [7,]     0
>  [8,]     0
>  [9,]     0
> [10,]     0
> [11,]     0
> 
> $U$`15`
>       [,1] [,2] [,3] [,4]  [,5] [,6] [,7] [,8] [,9] [,10] [,11]
>  [1,] 0.08  0.0  0.0 0.00 0.000  0.0 0.00 0.00    0     0     0
>  [2,] 0.10  0.0  0.0 0.00 0.000  0.0 0.00 0.00    0     0     0
>  [3,] 0.00  0.1  0.0 0.00 0.000  0.0 0.00 0.00    0     0     0
>  [4,] 0.00  0.0  0.1 0.00 0.000  0.0 0.00 0.00    0     0     0
>  [5,] 0.00  0.0  0.0 0.05 0.050  0.0 0.00 0.00    0     0     0
>  [6,] 0.00  0.0  0.0 0.00 0.000  0.0 0.00 0.00    0     0     0
>  [7,] 0.00  0.0  0.0 0.00 0.175  0.5 0.25 0.00    0     0     0
>  [8,] 0.00  0.0  0.0 0.00 0.525  0.5 0.75 0.50    0     0     0
>  [9,] 0.00  0.0  0.0 0.00 0.000  0.0 0.00 0.25    1     0     0
> [10,] 0.00  0.0  0.0 0.00 0.000  0.0 0.00 0.25    0     0     0
> [11,] 0.00  0.0  0.0 0.00 0.000  0.0 0.00 0.00    0     0     1
> 
> 
> $F
> $F$`1`
>       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]     [,9] [,10] [,11]
>  [1,]    0    0    0    0    0    0    0  500 1666.667     0     0
>  [2,]    0    0    0    0    0    0    0  500 1666.667     0     0
>  [3,]    0    0    0    0    0    0    0    0    0.000     0     0
>  [4,]    0    0    0    0    0    0    0    0    0.000     0     0
>  [5,]    0    0    0    0    0    0    0    0    0.000     0     0
>  [6,]    0    0    0    0    0    0    0    0    0.000     0     0
>  [7,]    0    0    0    0    0    0    0    0    0.000     0     0
>  [8,]    0    0    0    0    0    0    0    0    0.000     0     0
>  [9,]    0    0    0    0    0    0    0    0    0.000     0     0
> [10,]    0    0    0    0    0    0    0    0    0.000     0     0
> [11,]    0    0    0    0    0    0    0    0    0.000     0     0
> 
> $F$`2`
>       [,1] [,2] [,3] [,4] [,5] [,6]  [,7]     [,8] [,9] [,10] [,11]
>  [1,]    0    0    0    0    0    0 312.5 1111.111 1250     0     0
>  [2,]    0    0    0    0    0    0 312.5 1111.111 1250     0     0
>  [3,]    0    0    0    0    0    0   0.0    0.000    0     0     0
>  [4,]    0    0    0    0    0    0   0.0    0.000    0     0     0
>  [5,]    0    0    0    0    0    0   0.0    0.000    0     0     0
>  [6,]    0    0    0    0    0    0   0.0    0.000    0     0     0
>  [7,]    0    0    0    0    0    0   0.0    0.000    0     0     0
>  [8,]    0    0    0    0    0    0   0.0    0.000    0     0     0
>  [9,]    0    0    0    0    0    0   0.0    0.000    0     0     0
> [10,]    0    0    0    0    0    0   0.0    0.000    0     0     0
> [11,]    0    0    0    0    0    0   0.0    0.000    0     0     0
> 
> $F$`3`
>       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
>  [1,]    0    0    0    0    0    0    0    0    0     0     0
>  [2,]    0    0    0    0    0    0    0    0    0     0     0
>  [3,]    0    0    0    0    0    0    0    0    0     0     0
>  [4,]    0    0    0    0    0    0    0    0    0     0     0
>  [5,]    0    0    0    0    0    0    0    0    0     0     0
>  [6,]    0    0    0    0    0    0    0    0    0     0     0
>  [7,]    0    0    0    0    0    0    0    0    0     0     0
>  [8,]    0    0    0    0    0    0    0    0    0     0     0
>  [9,]    0    0    0    0    0    0    0    0    0     0     0
> [10,]    0    0    0    0    0    0    0    0    0     0     0
> [11,]    0    0    0    0    0    0    0    0    0     0     0
> 
> $F$`4`
>       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
>  [1,]    0    0    0    0    0    0    0    0    0     0     0
>  [2,]    0    0    0    0    0    0    0    0    0     0     0
>  [3,]    0    0    0    0    0    0    0    0    0     0     0
>  [4,]    0    0    0    0    0    0    0    0    0     0     0
>  [5,]    0    0    0    0    0    0    0    0    0     0     0
>  [6,]    0    0    0    0    0    0    0    0    0     0     0
>  [7,]    0    0    0    0    0    0    0    0    0     0     0
>  [8,]    0    0    0    0    0    0    0    0    0     0     0
>  [9,]    0    0    0    0    0    0    0    0    0     0     0
> [10,]    0    0    0    0    0    0    0    0    0     0     0
> [11,]    0    0    0    0    0    0    0    0    0     0     0
> 
> $F$`5`
>       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
>  [1,]    0    0    0    0    0    0    0  500 5000     0     0
>  [2,]    0    0    0    0    0    0    0  500 5000     0     0
>  [3,]    0    0    0    0    0    0    0    0    0     0     0
>  [4,]    0    0    0    0    0    0    0    0    0     0     0
>  [5,]    0    0    0    0    0    0    0    0    0     0     0
>  [6,]    0    0    0    0    0    0    0    0    0     0     0
>  [7,]    0    0    0    0    0    0    0    0    0     0     0
>  [8,]    0    0    0    0    0    0    0    0    0     0     0
>  [9,]    0    0    0    0    0    0    0    0    0     0     0
> [10,]    0    0    0    0    0    0    0    0    0     0     0
> [11,]    0    0    0    0    0    0    0    0    0     0     0
> 
> $F$`6`
>       [,1] [,2] [,3] [,4] [,5] [,6] [,7]     [,8] [,9] [,10] [,11]
>  [1,]    0    0    0    0    0    0    0 166.6667  625  1875     0
>  [2,]    0    0    0    0    0    0    0 166.6667  625  1875     0
>  [3,]    0    0    0    0    0    0    0   0.0000    0     0     0
>  [4,]    0    0    0    0    0    0    0   0.0000    0     0     0
>  [5,]    0    0    0    0    0    0    0   0.0000    0     0     0
>  [6,]    0    0    0    0    0    0    0   0.0000    0     0     0
>  [7,]    0    0    0    0    0    0    0   0.0000    0     0     0
>  [8,]    0    0    0    0    0    0    0   0.0000    0     0     0
>  [9,]    0    0    0    0    0    0    0   0.0000    0     0     0
> [10,]    0    0    0    0    0    0    0   0.0000    0     0     0
> [11,]    0    0    0    0    0    0    0   0.0000    0     0     0
> 
> $F$`7`
>       [,1] [,2] [,3] [,4] [,5] [,6] [,7]     [,8] [,9] [,10] [,11]
>  [1,]    0    0    0    0    0    0    0 1666.667 4375  5000 17500
>  [2,]    0    0    0    0    0    0    0 1666.667 4375  5000 17500
>  [3,]    0    0    0    0    0    0    0    0.000    0     0     0
>  [4,]    0    0    0    0    0    0    0    0.000    0     0     0
>  [5,]    0    0    0    0    0    0    0    0.000    0     0     0
>  [6,]    0    0    0    0    0    0    0    0.000    0     0     0
>  [7,]    0    0    0    0    0    0    0    0.000    0     0     0
>  [8,]    0    0    0    0    0    0    0    0.000    0     0     0
>  [9,]    0    0    0    0    0    0    0    0.000    0     0     0
> [10,]    0    0    0    0    0    0    0    0.000    0     0     0
> [11,]    0    0    0    0    0    0    0    0.000    0     0     0
> 
> $F$`8`
>       [,1] [,2] [,3] [,4] [,5] [,6]     [,7]   [,8] [,9] [,10] [,11]
>  [1,]    0    0    0    0    0    0 277.7778 781.25 6250  5000 10000
>  [2,]    0    0    0    0    0    0 277.7778 781.25 6250  5000 10000
>  [3,]    0    0    0    0    0    0   0.0000   0.00    0     0     0
>  [4,]    0    0    0    0    0    0   0.0000   0.00    0     0     0
>  [5,]    0    0    0    0    0    0   0.0000   0.00    0     0     0
>  [6,]    0    0    0    0    0    0   0.0000   0.00    0     0     0
>  [7,]    0    0    0    0    0    0   0.0000   0.00    0     0     0
>  [8,]    0    0    0    0    0    0   0.0000   0.00    0     0     0
>  [9,]    0    0    0    0    0    0   0.0000   0.00    0     0     0
> [10,]    0    0    0    0    0    0   0.0000   0.00    0     0     0
> [11,]    0    0    0    0    0    0   0.0000   0.00    0     0     0
> 
> $F$`9`
>       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
>  [1,]    0    0    0    0    0    0    0  250    0  1250     0
>  [2,]    0    0    0    0    0    0    0  250    0  1250     0
>  [3,]    0    0    0    0    0    0    0    0    0     0     0
>  [4,]    0    0    0    0    0    0    0    0    0     0     0
>  [5,]    0    0    0    0    0    0    0    0    0     0     0
>  [6,]    0    0    0    0    0    0    0    0    0     0     0
>  [7,]    0    0    0    0    0    0    0    0    0     0     0
>  [8,]    0    0    0    0    0    0    0    0    0     0     0
>  [9,]    0    0    0    0    0    0    0    0    0     0     0
> [10,]    0    0    0    0    0    0    0    0    0     0     0
> [11,]    0    0    0    0    0    0    0    0    0     0     0
> 
> $F$`10`
>       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]     [,9] [,10] [,11]
>  [1,]    0    0    0    0    0    0    0    0 714.2857  1250  2500
>  [2,]    0    0    0    0    0    0    0    0 714.2857  1250  2500
>  [3,]    0    0    0    0    0    0    0    0   0.0000     0     0
>  [4,]    0    0    0    0    0    0    0    0   0.0000     0     0
>  [5,]    0    0    0    0    0    0    0    0   0.0000     0     0
>  [6,]    0    0    0    0    0    0    0    0   0.0000     0     0
>  [7,]    0    0    0    0    0    0    0    0   0.0000     0     0
>  [8,]    0    0    0    0    0    0    0    0   0.0000     0     0
>  [9,]    0    0    0    0    0    0    0    0   0.0000     0     0
> [10,]    0    0    0    0    0    0    0    0   0.0000     0     0
> [11,]    0    0    0    0    0    0    0    0   0.0000     0     0
> 
> $F$`11`
>       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]     [,9] [,10] [,11]
>  [1,]    0    0    0    0    0    0    0  625 833.3333     0  7500
>  [2,]    0    0    0    0    0    0    0  625 833.3333     0  7500
>  [3,]    0    0    0    0    0    0    0    0   0.0000     0     0
>  [4,]    0    0    0    0    0    0    0    0   0.0000     0     0
>  [5,]    0    0    0    0    0    0    0    0   0.0000     0     0
>  [6,]    0    0    0    0    0    0    0    0   0.0000     0     0
>  [7,]    0    0    0    0    0    0    0    0   0.0000     0     0
>  [8,]    0    0    0    0    0    0    0    0   0.0000     0     0
>  [9,]    0    0    0    0    0    0    0    0   0.0000     0     0
> [10,]    0    0    0    0    0    0    0    0   0.0000     0     0
> [11,]    0    0    0    0    0    0    0    0   0.0000     0     0
> 
> $F$`12`
>       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]     [,9] [,10] [,11]
>  [1,]    0    0    0    0    0    0    0    0 4166.667  1250  5000
>  [2,]    0    0    0    0    0    0    0    0 4166.667  1250  5000
>  [3,]    0    0    0    0    0    0    0    0    0.000     0     0
>  [4,]    0    0    0    0    0    0    0    0    0.000     0     0
>  [5,]    0    0    0    0    0    0    0    0    0.000     0     0
>  [6,]    0    0    0    0    0    0    0    0    0.000     0     0
>  [7,]    0    0    0    0    0    0    0    0    0.000     0     0
>  [8,]    0    0    0    0    0    0    0    0    0.000     0     0
>  [9,]    0    0    0    0    0    0    0    0    0.000     0     0
> [10,]    0    0    0    0    0    0    0    0    0.000     0     0
> [11,]    0    0    0    0    0    0    0    0    0.000     0     0
> 
> $F$`13`
>       [,1] [,2] [,3] [,4] [,5] [,6] [,7]     [,8] [,9] [,10] [,11]
>  [1,]    0    0    0    0    0    0    0 1071.429 2500  2500     0
>  [2,]    0    0    0    0    0    0    0 1071.429 2500  2500     0
>  [3,]    0    0    0    0    0    0    0    0.000    0     0     0
>  [4,]    0    0    0    0    0    0    0    0.000    0     0     0
>  [5,]    0    0    0    0    0    0    0    0.000    0     0     0
>  [6,]    0    0    0    0    0    0    0    0.000    0     0     0
>  [7,]    0    0    0    0    0    0    0    0.000    0     0     0
>  [8,]    0    0    0    0    0    0    0    0.000    0     0     0
>  [9,]    0    0    0    0    0    0    0    0.000    0     0     0
> [10,]    0    0    0    0    0    0    0    0.000    0     0     0
> [11,]    0    0    0    0    0    0    0    0.000    0     0     0
> 
> $F$`14`
>       [,1] [,2] [,3] [,4] [,5] [,6] [,7]     [,8] [,9] [,10] [,11]
>  [1,]    0    0    0    0    0    0    0 416.6667    0     0     0
>  [2,]    0    0    0    0    0    0    0 416.6667    0     0     0
>  [3,]    0    0    0    0    0    0    0   0.0000    0     0     0
>  [4,]    0    0    0    0    0    0    0   0.0000    0     0     0
>  [5,]    0    0    0    0    0    0    0   0.0000    0     0     0
>  [6,]    0    0    0    0    0    0    0   0.0000    0     0     0
>  [7,]    0    0    0    0    0    0    0   0.0000    0     0     0
>  [8,]    0    0    0    0    0    0    0   0.0000    0     0     0
>  [9,]    0    0    0    0    0    0    0   0.0000    0     0     0
> [10,]    0    0    0    0    0    0    0   0.0000    0     0     0
> [11,]    0    0    0    0    0    0    0   0.0000    0     0     0
> 
> $F$`15`
>       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
>  [1,]    0    0    0    0    0    0    0    0  625     0     0
>  [2,]    0    0    0    0    0    0    0    0  625     0     0
>  [3,]    0    0    0    0    0    0    0    0    0     0     0
>  [4,]    0    0    0    0    0    0    0    0    0     0     0
>  [5,]    0    0    0    0    0    0    0    0    0     0     0
>  [6,]    0    0    0    0    0    0    0    0    0     0     0
>  [7,]    0    0    0    0    0    0    0    0    0     0     0
>  [8,]    0    0    0    0    0    0    0    0    0     0     0
>  [9,]    0    0    0    0    0    0    0    0    0     0     0
> [10,]    0    0    0    0    0    0    0    0    0     0     0
> [11,]    0    0    0    0    0    0    0    0    0     0     0
> 
> 
> $hstages
> [1] NA
> 
> $agestages
> [1] NA
> 
> $ahstages
>    stage_id stage original_size original_size_b original_size_c min_age max_age
> 1         1    SD           0.0              NA              NA       0      NA
> 2         2    P1           0.0              NA              NA       0      NA
> 3         3    P2           0.0              NA              NA       0      NA
> 4         4    P3           0.0              NA              NA       0      NA
> 5         5    SL           0.0              NA              NA       0      NA
> 6         6     D           0.0              NA              NA       0      NA
> 7         7   XSm           1.0              NA              NA       0      NA
> 8         8    Sm           3.0              NA              NA       0      NA
> 9         9    Md           6.0              NA              NA       0      NA
> 10       10    Lg          11.0              NA              NA       0      NA
> 11       11   XLg          19.5              NA              NA       0      NA
>    repstatus obsstatus propstatus immstatus matstatus entrystage indataset
> 1          0         0          1         0         0          1         0
> 2          0         0          0         1         0          1         0
> 3          0         0          0         1         0          0         0
> 4          0         0          0         1         0          0         0
> 5          0         0          0         1         0          0         0
> 6          0         0          0         0         1          0         1
> 7          1         1          0         0         1          0         1
> 8          1         1          0         0         1          0         1
> 9          1         1          0         0         1          0         1
> 10         1         1          0         0         1          0         1
> 11         1         1          0         0         1          0         1
>    binhalfwidth_raw sizebin_min sizebin_max sizebin_center sizebin_width
> 1               0.0         0.0         0.0            0.0             0
> 2               0.0         0.0         0.0            0.0             0
> 3               0.0         0.0         0.0            0.0             0
> 4               0.0         0.0         0.0            0.0             0
> 5               0.0         0.0         0.0            0.0             0
> 6               0.5        -0.5         0.5            0.0             1
> 7               0.5         0.5         1.5            1.0             1
> 8               1.5         1.5         4.5            3.0             3
> 9               1.5         4.5         7.5            6.0             3
> 10              3.5         7.5        14.5           11.0             7
> 11              5.0        14.5        24.5           19.5            10
>    binhalfwidthb_raw sizebinb_min sizebinb_max sizebinb_center sizebinb_width
> 1                 NA           NA           NA              NA             NA
> 2                 NA           NA           NA              NA             NA
> 3                 NA           NA           NA              NA             NA
> 4                 NA           NA           NA              NA             NA
> 5                 NA           NA           NA              NA             NA
> 6                 NA           NA           NA              NA             NA
> 7                 NA           NA           NA              NA             NA
> 8                 NA           NA           NA              NA             NA
> 9                 NA           NA           NA              NA             NA
> 10                NA           NA           NA              NA             NA
> 11                NA           NA           NA              NA             NA
>    binhalfwidthc_raw sizebinc_min sizebinc_max sizebinc_center sizebinc_width
> 1                 NA           NA           NA              NA             NA
> 2                 NA           NA           NA              NA             NA
> 3                 NA           NA           NA              NA             NA
> 4                 NA           NA           NA              NA             NA
> 5                 NA           NA           NA              NA             NA
> 6                 NA           NA           NA              NA             NA
> 7                 NA           NA           NA              NA             NA
> 8                 NA           NA           NA              NA             NA
> 9                 NA           NA           NA              NA             NA
> 10                NA           NA           NA              NA             NA
> 11                NA           NA           NA              NA             NA
>    group                       comments alive almostborn
> 1      0                   Dormant seed     1          0
> 2      0               1st yr protocorm     1          0
> 3      0               2nd yr protocorm     1          0
> 4      0               3rd yr protocorm     1          0
> 5      0                       Seedling     1          0
> 6      0                  Dormant adult     1          0
> 7      0    Extra small adult (1 shoot)     1          0
> 8      0       Small adult (2-4 shoots)     1          0
> 9      0      Medium adult (5-7 shoots)     1          0
> 10     0      Large adult (8-14 shoots)     1          0
> 11     0 Extra large adult (>14 shoots)     1          0
> 
> $labels
>    pop patch year2
> 1    1     A  2004
> 2    1     A  2005
> 3    1     A  2006
> 4    1     A  2007
> 5    1     A  2008
> 6    1     B  2004
> 7    1     B  2005
> 8    1     B  2006
> 9    1     B  2007
> 10   1     B  2008
> 11   1     C  2004
> 12   1     C  2005
> 13   1     C  2006
> 14   1     C  2007
> 15   1     C  2008
> 
> $matrixqc
> [1] 266  70  15
> 
> $dataqc
> [1]  74 320
> 
> attr(,"class")
> [1] "lefkoMat"

The output from this analysis is a lefkoMat object, which is described in detail in section 1.5.1. In brief, this object is a list with the following elements:

  1. A: a list of full population projection matrices, in order of population, patch, and occasion in time t.
  2. U: a list of matrices showing only survival-transition elements, in the same order as A.
  3. F: a list of matrices showing only fecundity elements, in the same order as A.
  4. hstages: a data frame showing the order of paired stages (given if matrices are historical, otherwise NA).
  5. agestages: a data frame showing the order of age-stages (if matrices are age-by-stage, otherwise NA).
  6. ahstages: the stageframe used in analysis, with stages reordered and edited as they occur in the matrix.
  7. labels: a table showing the order of matrices, according to population, patch, and occasion in time t.
  8. matrixqc: a short vector used in summary statements to describe the overall quality of each matrix.
  9. dataqc: a short vector used in summary statements to describe key sampling aspects of the dataset (only in raw MPMs).
  10. modelqc: a short vector used in summary statements to describe the vital rate models (only in function-based MPMs).

Calling particular values and elements within lefkoMat objects is not complicated, once you know the structure. Figure 4.2 illustrates how to call a particular element from one of the A matrices, for example.

Figure 4.2: Organization of a lefkoMat object, and how to call a specific element

Objects of class lefkoMat have their own summary() statements, which we can use to understand more about them.

summary(cypmatrix2rp)
> 
> This ahistorical lefkoMat object contains 15 matrices.
> 
> Each matrix is square with 11 rows and columns, and a total of 121 elements.
> A total of 266 survival transitions were estimated, with 17.733 per matrix.
> A total of 70 fecundity transitions were estimated, with 4.667 per matrix.
> This lefkoMat object covers 1 population, 3 patches, and 5 time steps.
> 
> The dataset contains a total of 74 unique individuals and 320 unique transitions.
> 
> Survival probability sum check (each matrix represented by column in order):
>          [,1]  [,2]  [,3]  [,4]  [,5]  [,6]  [,7]  [,8]  [,9] [,10] [,11] [,12]
> Min.    0.000 0.000 0.000 0.000 0.000 0.000 0.050 0.050 0.000 0.050 0.000 0.000
> 1st Qu. 0.075 0.025 0.075 0.025 0.075 0.075 0.140 0.140 0.100 0.140 0.100 0.100
> Median  0.180 0.100 0.180 0.100 0.180 0.180 0.909 0.778 0.686 0.857 0.750 0.575
> Mean    0.457 0.361 0.471 0.328 0.417 0.464 0.631 0.611 0.530 0.631 0.562 0.523
> 3rd Qu. 0.955 0.769 1.000 0.592 0.781 1.000 1.000 1.000 0.955 1.000 1.000 1.000
> Max.    1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
>         [,13] [,14] [,15]
> Min.    0.000 0.000 0.000
> 1st Qu. 0.075 0.075 0.100
> Median  0.180 0.180 0.750
> Mean    0.432 0.450 0.562
> 3rd Qu. 0.875 1.000 1.000
> Max.    1.000 1.000 1.000

We start off learning that 15 matrices were estimated, and we learn the dimensionality of those matrices. Next we see how many elements were actually estimated, both overall and per matrix (actually the number of non-zero elements), followed by the number of populations, patches or subpopulations, and time steps covered by the MPM. Then we see the number of individuals and transitions the matrices are based on. It is typical for population ecologists to consider the total number of transitions in a dataset as a measure of the statistical power of a matrix, but the number of individuals used is just as important because each transition that an individual experiences is dependent on the other transitions that it also experiences. The final portion of the summary shows us the range of survival probabilities of stages in the matrices, where the survival probabilities are calculated as column sums of each U matrix. Since there are 15 matrices, there are 15 survival summaries. It is important to check to see that no stage survives outside the realm of possibility (i.e. no probability should be greater than 1.0 or lower than 0.0). Unusual stage survival probabilities will result in a warning as a part of the summary() output.

The input for the rlefko2() function includes patch = "all" and year = "all", but can be set to focus on any set of patches / subpopulations or years included within the data. Package lefko3 includes a great deal of flexibility here, and can estimate many matrices covering all of the populations, patches, and years occurring in a specific dataset. For example, if we had wished to skip the patch divisions within the population and instead estimate only annual matrices at the population level, then we could have eliminated the patch and patchcol options altogether from the input, as below.

cypmatrix2r <- rlefko2(data = cypraw_v1, stageframe = cypframe_raw,
  year = "all", stages = c("stage3", "stage2"),
  size = c("size3added", "size2added"), supplement = cypsupp2_raw, 
  yearcol = "year2", patchcol = "patchid", indivcol = "individ")

summary(cypmatrix2r)
> 
> This ahistorical lefkoMat object contains 5 matrices.
> 
> Each matrix is square with 11 rows and columns, and a total of 121 elements.
> A total of 120 survival transitions were estimated, with 24 per matrix.
> A total of 40 fecundity transitions were estimated, with 8 per matrix.
> This lefkoMat object covers 1 population, 1 patch, and 5 time steps.
> 
> The dataset contains a total of 74 unique individuals and 320 unique transitions.
> 
> Survival probability sum check (each matrix represented by column in order):
>          [,1]  [,2]  [,3]  [,4]  [,5]
> Min.    0.000 0.050 0.050 0.000 0.050
> 1st Qu. 0.100 0.140 0.140 0.100 0.140
> Median  0.689 0.870 0.864 0.610 0.882
> Mean    0.552 0.629 0.629 0.528 0.627
> 3rd Qu. 1.000 1.000 1.000 0.960 1.000
> Max.    1.000 1.000 1.000 1.000 1.000

Notice what happened here. The first call to rlefko2() yielded 15 matrices, because there are three patches and a total of six years of data, yielding five transitions between years (also referred to as time steps or periods). So, there are \(3 \times 5 = 15\) matrices. But in the second call, we no longer recognize patches and so have only estimated one set of five matrices covering the whole population. We can also focus in on specific patches and specific sets of years, setting the options appropriately.

Let’s try another exercise in this vein. This time, we will estimate matrices at the population level, but only for the years 2006 and 2007. This will result in six matrices, since there are three patches and two monitoring occasions (note that focusing on years 2006 and 2007 means that we will only build matrices in which these are the years in time t).

cypmatrix2rp67 <- rlefko2(data = cypraw_v1, stageframe = cypframe_raw,
  year = c(2006, 2007), patch = "all", stages = c("stage3", "stage2"),
  size = c("size3added", "size2added"), supplement = cypsupp2_raw, 
  yearcol = "year2", patchcol = "patchid", indivcol = "individ")

summary(cypmatrix2rp67)
> 
> This ahistorical lefkoMat object contains 6 matrices.
> 
> Each matrix is square with 11 rows and columns, and a total of 121 elements.
> A total of 111 survival transitions were estimated, with 18.5 per matrix.
> A total of 22 fecundity transitions were estimated, with 3.667 per matrix.
> This lefkoMat object covers 1 population, 3 patches, and 2 time steps.
> 
> The dataset contains a total of 74 unique individuals and 320 unique transitions.
> 
> Survival probability sum check (each matrix represented by column in order):
>          [,1]  [,2]  [,3]  [,4]  [,5]  [,6]
> Min.    0.000 0.000 0.050 0.000 0.000 0.000
> 1st Qu. 0.075 0.025 0.140 0.100 0.075 0.075
> Median  0.180 0.100 0.778 0.686 0.180 0.180
> Mean    0.471 0.328 0.611 0.530 0.432 0.450
> 3rd Qu. 1.000 0.592 1.000 0.955 0.875 1.000
> Max.    1.000 1.000 1.000 1.000 1.000 1.000

Finally, we will build matrices only for the years 2006 and 2007 in patch A. We expect only two matrices.

cypmatrix2rA67 <- rlefko2(data = cypraw_v1, stageframe = cypframe_raw,
  year = c(2006, 2007), patch = "A", stages = c("stage3", "stage2"),
  size = c("size3added", "size2added"), supplement = cypsupp2_raw, 
  yearcol = "year2", patchcol = "patchid", indivcol = "individ")

summary(cypmatrix2rA67)
> 
> This ahistorical lefkoMat object contains 2 matrices.
> 
> Each matrix is square with 11 rows and columns, and a total of 121 elements.
> A total of 30 survival transitions were estimated, with 15 per matrix.
> A total of 0 fecundity transitions were estimated, with 0 per matrix.
> This lefkoMat object covers 1 population, 1 patch, and 2 time steps.
> 
> The dataset contains a total of 74 unique individuals and 320 unique transitions.
> 
> Survival probability sum check (each matrix represented by column in order):
>          [,1]  [,2]
> Min.    0.000 0.000
> 1st Qu. 0.075 0.025
> Median  0.180 0.100
> Mean    0.471 0.328
> 3rd Qu. 1.000 0.592
> Max.    1.000 1.000

This output is also interesting because it showcases one of the strengths and weaknesses of the raw MPM approach. We see that no individuals produced any offspring in patch A over these two time steps, so no fecundity terms were estimated. While this is a very accurate result, it can lead to problems if we think of the MPM as reflecting the underlying biology of the species. In this case, fecundity is missing most likely because there were too few individuals in this patch at these times, and so the luck of the draw left us with some biologically estimable elements not being estimated.

4.2 Ahistorical vs. historical matrix models

The matrices that we have built so far are all examples of raw ahistorical MPMs (ahMPMs). These are projection models in which the future stage of an individual is dependent only on its current stage, and not on previous stages. In other words, individual history is not incorporated into ahMPMs. It may seem odd that individual history is not incorporated into the matrices given that demographic datasets are composed of records of individual histories spanning several or even many monitoring occasions. However, construction of an ahMPM breaks up these individual histories into pairs of consecutive stages across time, with each stage-pair treated as independent of every other pair of consecutive stages. For example, if an individual is in stage A in occasion 1, stage B in occasion 2, stage E in occasion 3, and dead in occasion 4, then its individual history is broken up into A-B, B-E, and E-Dead, with each stage pair assumed to be independent. The resulting projection matrix would be the same if these transitions originated from different individuals or the same one - their order and relationships do not have any impacts in ahMPM construction.

The independence of consecutive stage-pairs reflects a central assumption in ahMPM analysis: the stage of an individual in the next occasion is influenced only by its current stage. Conceptually, if an organism’s stage in the next occasion is entirely determined by its current stage, then its previous states do not influence these transitions. Thus, standard ahMPMs are two dimensional and reflect only the current and next immediate stage of individuals, given by the columns and rows, respectively. This is ultimately an extension of the iid assumption in statistics - that the states of individuals are independent and originate from identically-distributed random variables. This assumption helped make matrix projection models relatively easy to estimate even before the advent of personal computers.

The first MPMs were produced in the 1940s, with Leslie’s introduction of the age-structured model (Leslie 1945; Caswell 2001). We have never done a meta-analysis of all of these studies, but nonetheless it is safe to say that studies considering individual history are rare. The typical MPM study involves only ahMPMs, and so assumes independence of stage transitions across time even from the same individual. In fact, at the time of writing, we are aware of only five examples of studies breaking these assumptions and using a historical approach, in which some degree of individual history is incorporated into matrix estimation and analysis (Ehrlén 2000; Shefferson et al. 2014, 2017, 2018; deVries & Caswell 2018).

The historical MPM (hMPM) is an extension of the matrix projection model that incorporates information on one previous occasion into the determination of vital rates. Thus, the expected survival-transition probability of an individual in stage \(j\) at occasion t to stage \(k\) at occasion t+1 depends not only on its stage in occasion t but also on its stage in occasion t-1. Population ecologists considering this problem analytically might be inclined to add an extra dimension to the matrix to deal with this, thus creating a 3d array or cube. However, this is mathematically intractable, with many analyses becoming impossible (deVries & Caswell 2018). Instead, we utilize the approach developed by Ehrlén (2000), in which rows and columns represent life history stages paired in consecutive occasions. Thus, columns now represent the From pair of stages (stages in occasions t-1 and t), and rows now represent the To pair of stages (stages in occasions t and t+1), as in figure 4.3. This model is equivalent to the second-order model proposed by deVries & Caswell (2018), although the latter parameterizes history in transitions involving individuals just born differently (we explain the difference further in this vignette).

Figure 4.3: Historical MPM for Cypripedium candidum, a North American herbaceous plant

Historical MPMs can be projected forward in the same way that ahMPMs can. In figure 4.4 we see an example of a projection one time step forward using the hMPM shown in figure 4.3.

Figure 4.4: Historical MPM projection in Ehrlén format

Here, \(n _{1,SS}\) is the number of individuals that were in stage \(S\) in both occasions 0 and 1, and \(n _{2,DF}\) is the number of individuals that were in stage \(D\) in occasion 2 and stage \(F\) in occasion 1.

Ahistorical matrices and historical matrices can be derived from the same datasets provided that individual fates are noted for three consecutive times each. We can refer to matrix elements as \(a _{kjl}\), where the element represents the rate at which individuals transition to stage \(k\) in occasion t+1 after having been in stage \(j\) in occasion t and in stage \(l\) in occasion t-1. If \(n _{kjl}\) is the number of individuals making this transition, \(n _{.jl}\) is the total number of individuals in stage \(j\) in occasion t and stage \(l\) in occasion t-1 regardless of stage or even status as alive or dead in occasion t+1, \(m\) is the number of stages in the life history model, and \(d\) is the number of stages plus death in the life history model (so that \(d=m+1\)), then we have:

\[\begin{equation} a _{kjl} = \frac{n _{kjl}}{n _{.jl}} = \frac{n _{kjl}}{\sum_{i=1}^{d} n _{ijl}} \tag{4.4} \end{equation}\]

Note that although we can use the actual numbers of individuals making historical transitions to estimate ahistorical MPMs, we cannot use the matrix elements in a historical MPM to calculate the associated ahistorical MPM elements, nor can we use the elements in an ahistorical MPM to calculate the elements in a historical MPM. This is due to the fact that historical transitions must be weighted by the numbers of individuals in each previous stage in occasions t and t-1 (\(n _{.jl}\)) to produce the proper ahistorical transition rates. Historical matrix elements contain no information about these weights. Thus, we have the following relationships between ahistorical survival-transition terms and the historical datasets:

\[\begin{equation} a _{kj} = \frac{\sum_{l=1}^{m} n _{kjl}}{\sum_{l=1}^{m} \sum_{i=1}^{d} n _{ijl}} \tag{4.5} \end{equation}\]

Historical MPMs normally require a much larger number of elements to be parameterized than ahMPMs do. Figure 4.3 illustrates this issue. Typically, a historical matrix has dimensions equal to the number of stages squared (although this may sometimes be reduced, if certain conditions are met). However, most of the increase in matrix elements is in structural zeros. In an ahistorical matrix, the only true zeros will be those elements corresponding to biologically impossible transitions, or those elements without any individuals taking the respective transition in a given time step. However, most elements in a historical MPM are structural zeros, because transition elements in hMPMs are only estimable if stage at occasion t is equal in the column and row stage pairs. For example, the transition probability between stage A in occasion t-1 and stage B in occasion t (column stage-pair), to stage C in occasion t and stage D in occasion t+1 (row stage-pair) equals 0, because an organism cannot be in both stages B and C in occasion t. Increasing the number of stages in the life history model causes these structural zeros to increase at a faster rate than the rate at which the number of truly estimable elements increases. In fact, if there are \(m\) stages in a life history, yielding \(m ^{2}\) elements in an ahistorical matrix, then although there will be \(m ^{4}\) elements in the historical matrix, only \(m^{3}\) will be potentially estimable while \((m-1) m^{3}\) will be structural zeros (we say potentially because some of the logically possible transitions may still be biologically impossible, or may have no data to yield values other than 0).

4.2.1 Historical models in lefko3

Let’s build our first historical MPM with the Cypripedium candidum data. As before, we will build our MPM to cover all years and all patches.

cypmatrix3rp <- rlefko3(data = cypraw_v1, stageframe = cypframe_raw,
  year = "all", patch = "all", stages = c("stage3", "stage2", "stage1"), 
  size = c("size3added", "size2added", "size1added"), supplement = cypsupp3_raw, 
  yearcol = "year2", patchcol = "patchid", indivcol = "individ")

summary(cypmatrix3rp)
> 
> This historical lefkoMat object contains 12 matrices.
> 
> Each matrix is square with 121 rows and columns, and a total of 14641 elements.
> A total of 516 survival transitions were estimated, with 43 per matrix.
> A total of 70 fecundity transitions were estimated, with 5.833 per matrix.
> This lefkoMat object covers 1 population, 3 patches, and 4 time steps.
> 
> The dataset contains a total of 74 unique individuals and 320 unique transitions.
> 
> Survival probability sum check (each matrix represented by column in order):
>          [,1]   [,2]   [,3]  [,4]  [,5]  [,6] [,7]  [,8]  [,9]  [,10] [,11]
> Min.    0.000 0.0000 0.0000 0.000 0.000 0.000 0.00 0.000 0.000 0.0000 0.000
> 1st Qu. 0.000 0.0000 0.0000 0.000 0.000 0.000 0.00 0.000 0.000 0.0000 0.000
> Median  0.000 0.0000 0.0000 0.000 0.000 0.000 0.00 0.000 0.000 0.0000 0.000
> Mean    0.107 0.0945 0.0851 0.101 0.158 0.158 0.14 0.169 0.119 0.0851 0.119
> 3rd Qu. 0.000 0.0000 0.0000 0.000 0.100 0.100 0.05 0.100 0.000 0.0000 0.000
> Max.    1.000 1.0000 1.0000 1.000 1.000 1.000 1.00 1.000 1.000 1.0000 1.000
>         [,12]
> Min.    0.000
> 1st Qu. 0.000
> Median  0.000
> Mean    0.144
> 3rd Qu. 0.000
> Max.    1.000

Quickly scanning this output shows a number of differences relative to the ahMPM. First, there are three fewer matrices here than in the ahistorical case. There are three patches that we are estimating matrices for, and six years of data for each patch, leading to five possible ahistorical time steps and 15 possible ahistorical matrices. Since historical matrices require three years of transition data, only four historical transitions are possible per patch, leading to 12 total historical matrices. Second, the dimensionality of the matrices is the square of the dimensions of the ahistorical matrices. This leads to vastly more matrix elements within each matrix, although it turns out that most of these matrix elements are structural zeros because they reflect impossible transitions. Indeed, in this case, although there are 14,641 elements in each matrix, on average only 48.83 are actually estimated to values greater than 0.

Let’s look at the first matrix, corresponding to the transition from 2004 and 2005 to 2006 in the first patch (patch A, according to the labels element). Because this is a huge matrix, we will only look at the top corner, followed by a section toward the middle. Particularly note the sparseness - most elements are zeros, because most transitions are actually impossible. These matrices may also be exported to Microsoft Excel, Apple Numbers, or another spreadsheet program to look over in greater detail.

cypmatrix3rp$A[[1]][1:20,1:10]
>       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
>  [1,] 0.08  0.0    0    0    0    0    0    0    0     0
>  [2,] 0.10  0.0    0    0    0    0    0    0    0     0
>  [3,] 0.00  0.0    0    0    0    0    0    0    0     0
>  [4,] 0.00  0.0    0    0    0    0    0    0    0     0
>  [5,] 0.00  0.0    0    0    0    0    0    0    0     0
>  [6,] 0.00  0.0    0    0    0    0    0    0    0     0
>  [7,] 0.00  0.0    0    0    0    0    0    0    0     0
>  [8,] 0.00  0.0    0    0    0    0    0    0    0     0
>  [9,] 0.00  0.0    0    0    0    0    0    0    0     0
> [10,] 0.00  0.0    0    0    0    0    0    0    0     0
> [11,] 0.00  0.0    0    0    0    0    0    0    0     0
> [12,] 0.00  0.0    0    0    0    0    0    0    0     0
> [13,] 0.00  0.0    0    0    0    0    0    0    0     0
> [14,] 0.00  0.1    0    0    0    0    0    0    0     0
> [15,] 0.00  0.0    0    0    0    0    0    0    0     0
> [16,] 0.00  0.0    0    0    0    0    0    0    0     0
> [17,] 0.00  0.0    0    0    0    0    0    0    0     0
> [18,] 0.00  0.0    0    0    0    0    0    0    0     0
> [19,] 0.00  0.0    0    0    0    0    0    0    0     0
> [20,] 0.00  0.0    0    0    0    0    0    0    0     0
print(cypmatrix3rp$A[[1]][66:85,73:81], digits = 3)
>          [,1]     [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
>  [1,]   0.000    0.000    0    0    0    0    0    0    0
>  [2,] 357.143    0.000    0    0    0    0    0    0    0
>  [3,] 357.143    0.000    0    0    0    0    0    0    0
>  [4,]   0.000    0.000    0    0    0    0    0    0    0
>  [5,]   0.000    0.000    0    0    0    0    0    0    0
>  [6,]   0.000    0.000    0    0    0    0    0    0    0
>  [7,]   0.143    0.000    0    0    0    0    0    0    0
>  [8,]   0.714    0.000    0    0    0    0    0    0    0
>  [9,]   0.000    0.000    0    0    0    0    0    0    0
> [10,]   0.000    0.000    0    0    0    0    0    0    0
> [11,]   0.000    0.000    0    0    0    0    0    0    0
> [12,]   0.000    0.000    0    0    0    0    0    0    0
> [13,]   0.000 1666.667    0    0    0    0    0    0    0
> [14,]   0.000 1666.667    0    0    0    0    0    0    0
> [15,]   0.000    0.000    0    0    0    0    0    0    0
> [16,]   0.000    0.000    0    0    0    0    0    0    0
> [17,]   0.000    0.000    0    0    0    0    0    0    0
> [18,]   0.000    0.000    0    0    0    0    0    0    0
> [19,]   0.000    0.667    0    0    0    0    0    0    0
> [20,]   0.000    0.333    0    0    0    0    0    0    0

As before, we can create matrices at the full population level by withholding the patch option, as below.

cypmatrix3r <- rlefko3(data = cypraw_v1, stageframe = cypframe_raw,
  year = "all", stages = c("stage3", "stage2", "stage1"), 
  size = c("size3added", "size2added", "size1added"), supplement = cypsupp3_raw, 
  yearcol = "year2", patchcol = "patchid", indivcol = "individ")

summary(cypmatrix3r)
> 
> This historical lefkoMat object contains 4 matrices.
> 
> Each matrix is square with 121 rows and columns, and a total of 14641 elements.
> A total of 242 survival transitions were estimated, with 60.5 per matrix.
> A total of 54 fecundity transitions were estimated, with 13.5 per matrix.
> This lefkoMat object covers 1 population, 1 patch, and 4 time steps.
> 
> The dataset contains a total of 74 unique individuals and 320 unique transitions.
> 
> Survival probability sum check (each matrix represented by column in order):
>          [,1]  [,2]  [,3]  [,4]
> Min.    0.000 0.000 0.000 0.000
> 1st Qu. 0.000 0.000 0.000 0.000
> Median  0.000 0.000 0.000 0.000
> Mean    0.173 0.179 0.166 0.198
> 3rd Qu. 0.100 0.100 0.100 0.100
> Max.    1.000 1.000 1.000 1.000

We now have only four matrices, corresponding to the full population in each year at time t. Next, let’s focus only on years 2006 and 2007 in all patches, as before.

cypmatrix3rp67 <- rlefko3(data = cypraw_v1, stageframe = cypframe_raw,
  year = c(2006, 2007), patch = "all", stages = c("stage3", "stage2", "stage1"), 
  size = c("size3added", "size2added", "size1added"), supplement = cypsupp3_raw, 
  yearcol = "year2", patchcol = "patchid", indivcol = "individ")

summary(cypmatrix3rp67)
> 
> This historical lefkoMat object contains 6 matrices.
> 
> Each matrix is square with 121 rows and columns, and a total of 14641 elements.
> A total of 260 survival transitions were estimated, with 43.333 per matrix.
> A total of 30 fecundity transitions were estimated, with 5 per matrix.
> This lefkoMat object covers 1 population, 3 patches, and 2 time steps.
> 
> The dataset contains a total of 74 unique individuals and 320 unique transitions.
> 
> Survival probability sum check (each matrix represented by column in order):
>           [,1]   [,2]  [,3] [,4]   [,5]  [,6]
> Min.    0.0000 0.0000 0.000 0.00 0.0000 0.000
> 1st Qu. 0.0000 0.0000 0.000 0.00 0.0000 0.000
> Median  0.0000 0.0000 0.000 0.00 0.0000 0.000
> Mean    0.0945 0.0851 0.158 0.14 0.0851 0.119
> 3rd Qu. 0.0000 0.0000 0.100 0.05 0.0000 0.000
> Max.    1.0000 1.0000 1.000 1.00 1.0000 1.000

We find six matrices, corresponding to the years 2006 and 2007 for each of the three patches. And now let’s focus in on years 2006 and 2007 in the first patch, patch A.

cypmatrix3rA67 <- rlefko3(data = cypraw_v1, stageframe = cypframe_raw,
  year = c(2006, 2007), patch = "A", stages = c("stage3", "stage2", "stage1"), 
  size = c("size3added", "size2added", "size1added"), supplement = cypsupp3_raw, 
  yearcol = "year2", patchcol = "patchid", indivcol = "individ")

summary(cypmatrix3rA67)
> 
> This historical lefkoMat object contains 2 matrices.
> 
> Each matrix is square with 121 rows and columns, and a total of 14641 elements.
> A total of 78 survival transitions were estimated, with 39 per matrix.
> A total of 0 fecundity transitions were estimated, with 0 per matrix.
> This lefkoMat object covers 1 population, 1 patch, and 2 time steps.
> 
> The dataset contains a total of 74 unique individuals and 320 unique transitions.
> 
> Survival probability sum check (each matrix represented by column in order):
>           [,1]   [,2]
> Min.    0.0000 0.0000
> 1st Qu. 0.0000 0.0000
> Median  0.0000 0.0000
> Mean    0.0945 0.0851
> 3rd Qu. 0.0000 0.0000
> Max.    1.0000 1.0000

As before, we find no fecundity terms estimated, as no individuals in patch A were reproductive in these two times.

4.3 Different parameterizations of the historical MPM

deVries & Caswell (2018) detail different approaches to the development of hMPMs, differing in how state or stage in occasion t-1 is incorporated into the matrix. Full prior stage dependence models deal with history by incorporating prior condition as the exact stage of an organism in occasion t-1, yielding a 2d matrix showing stage pairs in occasion t-1 and t along the columns of the matrix, and stage pairs in occasion t and t+1 along the rows of the matrix. Prior condition models deal with history by making the transition from stage t to stage t+1 a function of stage at occasion t and condition in occasion t-1. Prior condition can be determined in the same way that current stage is determined (e.g. size classification), or a different measure of condition can be used, such as growth (i.e. the change in size between occasions t-1 and t).

One key feature proposed by deVries & Caswell (2018) is the addition of a stage to account for the prior status of newborn individuals. This reflects a different interpretation from Ehrlén Ehrlén (2000) of the historical transition. In Ehrlén (2000), all matrix elements simply reflect the order of the events in a single transition, including both fecundity and survival-transition events. However, in deVries & Caswell (2018), these matrix elements must also reflect the history of specific individuals. In the latter case, the fact that a newborn in occasion t did not exist in occasion t-1 requires that a new prior stage is constructed and used in the matrix to account for that individual’s prior lack of existence. This new stage only exists in the prior occasion, and is only used for newborns in the first survival transition from birth. So, for example, in a historical MPM where the first stage is the newborn stage and only the third stage is reproductive, we add a fourth stage to the prior portion of the row and column. Thus, we may start with the following hMPM in Ehrlén (2000) format:

Figure 4.5: Example historical MPM projection in Ehrlén format

This hMPM becomes as follows in deVries & Caswell (2018) format, where the prior “pre-born” stage is shown as \(P\).

Figure 4.6: Example historical MPM projection in deVries format

Package lefko3 generally implements the full prior stage dependence model approach, particularly in the development of raw matrices. However, in principle, function-based matrices developed with lefko3 can be seen as falling within the prior condition model approach where prior condition is determined in the same way as current stage. We also implement both the Ehrlén (2000) format and the deVries & Caswell (2018) format, and use the former as the default. We leave it to the user to decide which approach to use, and point out only that the literature suggests that maternal condition can have long-term effects on offspring survival, for example through maternal care and epigenetic influences (Descamps et al. 2008; Beamonte-Barrientos et al. 2010; Dogra & Dani 2019). Thus, retaining maternal state as the prior condition of a newborn makes some logical sense and has a basis in the literature.

Let’s build a de Vries format hMPM for the Cypripedium candidum dataset.

cypmatrix3rp_dev <- rlefko3(data = cypraw_v1, stageframe = cypframe_raw,
  year = "all", patch = "all", stages = c("stage3", "stage2", "stage1"), 
  size = c("size3added", "size2added", "size1added"), supplement = cypsupp3_raw, 
  yearcol = "year2", patchcol = "patchid", indivcol = "individ",
  format = "deVries")

summary(cypmatrix3rp_dev)
> 
> This historical lefkoMat object contains 12 matrices.
> 
> Each matrix is square with 132 rows and columns, and a total of 17424 elements.
> A total of 516 survival transitions were estimated, with 43 per matrix.
> A total of 70 fecundity transitions were estimated, with 5.833 per matrix.
> This lefkoMat object covers 1 population, 3 patches, and 4 time steps.
> 
> The dataset contains a total of 74 unique individuals and 320 unique transitions.
> 
> Survival probability sum check (each matrix represented by column in order):
>           [,1]   [,2]  [,3]   [,4]  [,5]   [,6]  [,7]   [,8]  [,9] [,10] [,11]
> Min.    0.0000 0.0000 0.000 0.0000 0.000 0.0000 0.000 0.0000 0.000 0.000 0.000
> 1st Qu. 0.0000 0.0000 0.000 0.0000 0.000 0.0000 0.000 0.0000 0.000 0.000 0.000
> Median  0.0000 0.0000 0.000 0.0000 0.000 0.0000 0.000 0.0000 0.000 0.000 0.000
> Mean    0.0981 0.0866 0.078 0.0923 0.144 0.1445 0.128 0.1548 0.109 0.078 0.109
> 3rd Qu. 0.0000 0.0000 0.000 0.0000 0.000 0.0125 0.000 0.0125 0.000 0.000 0.000
> Max.    1.0000 1.0000 1.000 1.0000 1.000 1.0000 1.000 1.0000 1.000 1.000 1.000
>         [,12]
> Min.    0.000
> 1st Qu. 0.000
> Median  0.000
> Mean    0.132
> 3rd Qu. 0.000
> Max.    1.000

Our new matrices have 11 more rows and columns, and 2,783 more elements. To see what is different, compare the hstages element in each on your own, and you will see the inclusion of some new stage pairs with the AlmostBorn stage. It is also possible for some transitions to be split, occasionally yielding more transitions than were estimated in the Ehrlén format case.

4.4 Matrix reduction

One important issue that arises in hMPM estimation is matrix dimensionality reduction. Occasionally, historical matrices are produced in which stage-pairs exist that are not associated with any transitions. This is particularly the case with raw hMPMs, since these are quite data-hungry MPMs. The result is that sometimes all of the rows and columns associated with a particular stage pair are completely full of zeros. In these situations, matrices can be reduced by eliminating the stage-pair altogether. The result is a set of smaller matrices, which is useful considering that historical MPMs are generally large and can take up a great deal of memory. We can tell R to develop reduced MPMs by using the reduce = TRUE option in the matrix generating function that we are using. Note that this option is available in ALL matrix generating functions, even for those creating ahistorical MPMs, and that rows and columns will be reduced only if ALL matrices within the lefkoMat object being produced have empty rows and columns associated with a particular stage or stage-pair.

cypmatrix3rp_red <- rlefko3(data = cypraw_v1, stageframe = cypframe_raw,
  year = "all", patch = "all", stages = c("stage3", "stage2", "stage1"), 
  size = c("size3added", "size2added", "size1added"), supplement = cypsupp3_raw, 
  yearcol = "year2", patchcol = "patchid", indivcol = "individ",
  reduce = TRUE)

summary(cypmatrix3rp_red)
> 
> This historical lefkoMat object contains 12 matrices.
> 
> Each matrix is square with 43 rows and columns, and a total of 1849 elements.
> A total of 516 survival transitions were estimated, with 43 per matrix.
> A total of 70 fecundity transitions were estimated, with 5.833 per matrix.
> This lefkoMat object covers 1 population, 3 patches, and 4 time steps.
> 
> The dataset contains a total of 74 unique individuals and 320 unique transitions.
> 
> Survival probability sum check (each matrix represented by column in order):
>          [,1]  [,2]  [,3]  [,4]  [,5]  [,6]  [,7]  [,8]  [,9] [,10] [,11] [,12]
> Min.    0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
> 1st Qu. 0.000 0.000 0.000 0.000 0.025 0.075 0.000 0.075 0.000 0.000 0.000 0.000
> Median  0.100 0.100 0.100 0.100 0.200 0.200 0.200 0.200 0.100 0.100 0.100 0.180
> Mean    0.301 0.266 0.239 0.283 0.443 0.443 0.393 0.475 0.336 0.239 0.336 0.405
> 3rd Qu. 0.650 0.200 0.200 0.475 1.000 1.000 1.000 1.000 0.875 0.200 0.875 1.000
> Max.    1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

Our new hMPM has matrices with 43 rows and columns, while the original unreduced hMPM has matrices with 121 rows and columns. This is a reduction of 78 rows and columns, with an overall reduction in the size of the matrix by \(14641-1849 = 12792\) elements. This should speed up later calculations greatly.

4.5 Arithmetic mean matrices

Now that we have created our MPMs, we might wish to create element-wise arithmetic mean matrices to aid inference and further analysis. For example, we might be interested in developing patch-level means and an overall population mean, but one in which the element means weight each patch and each year equally. For this purpose, we can use the lmean() function. Let’s take a look at the element-wise mean raw ahMPM first.

cyp2rp_mean <- lmean(cypmatrix2rp)

cyp2rp_mean
> $A
> $A[[1]]
>       [,1] [,2] [,3] [,4]       [,5]       [,6]        [,7]         [,8]
>  [1,] 0.08  0.0  0.0 0.00 0.00000000 0.00000000 62.50000000 422.22222222
>  [2,] 0.10  0.0  0.0 0.00 0.00000000 0.00000000 62.50000000 422.22222222
>  [3,] 0.00  0.1  0.0 0.00 0.00000000 0.00000000  0.00000000   0.00000000
>  [4,] 0.00  0.0  0.1 0.00 0.00000000 0.00000000  0.00000000   0.00000000
>  [5,] 0.00  0.0  0.0 0.05 0.05000000 0.00000000  0.00000000   0.00000000
>  [6,] 0.00  0.0  0.0 0.00 0.04083333 0.00000000  0.05833333   0.04444444
>  [7,] 0.00  0.0  0.0 0.00 0.41964646 0.06666667  0.59949495   0.19555556
>  [8,] 0.00  0.0  0.0 0.00 0.14707071 0.33333333  0.21010101   0.59555556
>  [9,] 0.00  0.0  0.0 0.00 0.00000000 0.00000000  0.00000000   0.08444444
> [10,] 0.00  0.0  0.0 0.00 0.00000000 0.00000000  0.00000000   0.00000000
> [11,] 0.00  0.0  0.0 0.00 0.00000000 0.00000000  0.00000000   0.00000000
>               [,9] [,10] [,11]
>  [1,] 1.583333e+03   0.0     0
>  [2,] 1.583333e+03   0.0     0
>  [3,] 0.000000e+00   0.0     0
>  [4,] 0.000000e+00   0.0     0
>  [5,] 0.000000e+00   0.0     0
>  [6,] 0.000000e+00   0.0     0
>  [7,] 6.666667e-02   0.0     0
>  [8,] 2.500000e-01   0.2     0
>  [9,] 4.833333e-01   0.0     0
> [10,] 2.000000e-01   0.0     0
> [11,] 0.000000e+00   0.0     0
> 
> $A[[2]]
>       [,1] [,2] [,3] [,4]       [,5]       [,6]        [,7]         [,8]
>  [1,] 0.08  0.0  0.0 0.00 0.00000000 0.00000000 55.55555556 572.91666667
>  [2,] 0.10  0.0  0.0 0.00 0.00000000 0.00000000 55.55555556 572.91666667
>  [3,] 0.00  0.1  0.0 0.00 0.00000000 0.00000000  0.00000000   0.00000000
>  [4,] 0.00  0.0  0.1 0.00 0.00000000 0.00000000  0.00000000   0.00000000
>  [5,] 0.00  0.0  0.0 0.05 0.05000000 0.00000000  0.00000000   0.00000000
>  [6,] 0.00  0.0  0.0 0.00 0.05373737 0.06666667  0.07676768   0.05416667
>  [7,] 0.00  0.0  0.0 0.00 0.32181818 0.46666667  0.45974026   0.23083333
>  [8,] 0.00  0.0  0.0 0.00 0.21676768 0.20000000  0.30966811   0.50166667
>  [9,] 0.00  0.0  0.0 0.00 0.00000000 0.00000000  0.02222222   0.14416667
> [10,] 0.00  0.0  0.0 0.00 0.00000000 0.06666667  0.02222222   0.01250000
> [11,] 0.00  0.0  0.0 0.00 0.00000000 0.00000000  0.00000000   0.00000000
>               [,9]  [,10] [,11]
>  [1,] 2392.8571429 2875.0 6e+03
>  [2,] 2392.8571429 2875.0 6e+03
>  [3,]    0.0000000    0.0 0e+00
>  [4,]    0.0000000    0.0 0e+00
>  [5,]    0.0000000    0.0 0e+00
>  [6,]    0.0000000    0.0 0e+00
>  [7,]    0.0500000    0.0 0e+00
>  [8,]    0.4000000    0.1 0e+00
>  [9,]    0.3428571    0.2 0e+00
> [10,]    0.2071429    0.6 2e-01
> [11,]    0.0000000    0.1 4e-01
> 
> $A[[3]]
>       [,1] [,2] [,3] [,4]       [,5] [,6]       [,7]         [,8]         [,9]
>  [1,] 0.08  0.0  0.0 0.00 0.00000000  0.0 0.00000000 422.61904762 1625.0000000
>  [2,] 0.10  0.0  0.0 0.00 0.00000000  0.0 0.00000000 422.61904762 1625.0000000
>  [3,] 0.00  0.1  0.0 0.00 0.00000000  0.0 0.00000000   0.00000000    0.0000000
>  [4,] 0.00  0.0  0.1 0.00 0.00000000  0.0 0.00000000   0.00000000    0.0000000
>  [5,] 0.00  0.0  0.0 0.05 0.05000000  0.0 0.00000000   0.00000000    0.0000000
>  [6,] 0.00  0.0  0.0 0.00 0.04666667  0.0 0.06666667   0.03333333    0.0000000
>  [7,] 0.00  0.0  0.0 0.00 0.35000000  0.1 0.50000000   0.11666667    0.1333333
>  [8,] 0.00  0.0  0.0 0.00 0.18666667  0.1 0.26666667   0.54761905    0.3333333
>  [9,] 0.00  0.0  0.0 0.00 0.00000000  0.0 0.06666667   0.22380952    0.5333333
> [10,] 0.00  0.0  0.0 0.00 0.00000000  0.0 0.00000000   0.07857143    0.0000000
> [11,] 0.00  0.0  0.0 0.00 0.00000000  0.0 0.00000000   0.00000000    0.0000000
>       [,10]  [,11]
>  [1,] 750.0 2500.0
>  [2,] 750.0 2500.0
>  [3,]   0.0    0.0
>  [4,]   0.0    0.0
>  [5,]   0.0    0.0
>  [6,]   0.0    0.0
>  [7,]   0.0    0.0
>  [8,]   0.1    0.0
>  [9,]   0.2    0.0
> [10,]   0.4    0.2
> [11,]   0.1    0.4
> 
> $A[[4]]
>       [,1] [,2] [,3] [,4]       [,5]       [,6]         [,7]         [,8]
>  [1,] 0.08  0.0  0.0 0.00 0.00000000 0.00000000 39.351851852 472.58597884
>  [2,] 0.10  0.0  0.0 0.00 0.00000000 0.00000000 39.351851852 472.58597884
>  [3,] 0.00  0.1  0.0 0.00 0.00000000 0.00000000  0.000000000   0.00000000
>  [4,] 0.00  0.0  0.1 0.00 0.00000000 0.00000000  0.000000000   0.00000000
>  [5,] 0.00  0.0  0.0 0.05 0.05000000 0.00000000  0.000000000   0.00000000
>  [6,] 0.00  0.0  0.0 0.00 0.04707912 0.02222222  0.067255892   0.04398148
>  [7,] 0.00  0.0  0.0 0.00 0.36382155 0.21111111  0.519745070   0.18101852
>  [8,] 0.00  0.0  0.0 0.00 0.18350168 0.21111111  0.262145262   0.54828042
>  [9,] 0.00  0.0  0.0 0.00 0.00000000 0.00000000  0.029629630   0.15080688
> [10,] 0.00  0.0  0.0 0.00 0.00000000 0.02222222  0.007407407   0.03035714
> [11,] 0.00  0.0  0.0 0.00 0.00000000 0.00000000  0.000000000   0.00000000
>               [,9]        [,10]        [,11]
>  [1,] 1.867063e+03 1.208333e+03 2833.3333333
>  [2,] 1.867063e+03 1.208333e+03 2833.3333333
>  [3,] 0.000000e+00 0.000000e+00    0.0000000
>  [4,] 0.000000e+00 0.000000e+00    0.0000000
>  [5,] 0.000000e+00 0.000000e+00    0.0000000
>  [6,] 0.000000e+00 0.000000e+00    0.0000000
>  [7,] 8.333333e-02 0.000000e+00    0.0000000
>  [8,] 3.277778e-01 1.333333e-01    0.0000000
>  [9,] 4.531746e-01 1.333333e-01    0.0000000
> [10,] 1.357143e-01 3.333333e-01    0.1333333
> [11,] 0.000000e+00 6.666667e-02    0.2666667
> 
> 
> $U
> $U[[1]]
>       [,1] [,2] [,3] [,4]       [,5]       [,6]       [,7]       [,8]
>  [1,] 0.08  0.0  0.0 0.00 0.00000000 0.00000000 0.00000000 0.00000000
>  [2,] 0.10  0.0  0.0 0.00 0.00000000 0.00000000 0.00000000 0.00000000
>  [3,] 0.00  0.1  0.0 0.00 0.00000000 0.00000000 0.00000000 0.00000000
>  [4,] 0.00  0.0  0.1 0.00 0.00000000 0.00000000 0.00000000 0.00000000
>  [5,] 0.00  0.0  0.0 0.05 0.05000000 0.00000000 0.00000000 0.00000000
>  [6,] 0.00  0.0  0.0 0.00 0.04083333 0.00000000 0.05833333 0.04444444
>  [7,] 0.00  0.0  0.0 0.00 0.41964646 0.06666667 0.59949495 0.19555556
>  [8,] 0.00  0.0  0.0 0.00 0.14707071 0.33333333 0.21010101 0.59555556
>  [9,] 0.00  0.0  0.0 0.00 0.00000000 0.00000000 0.00000000 0.08444444
> [10,] 0.00  0.0  0.0 0.00 0.00000000 0.00000000 0.00000000 0.00000000
> [11,] 0.00  0.0  0.0 0.00 0.00000000 0.00000000 0.00000000 0.00000000
>             [,9] [,10] [,11]
>  [1,] 0.00000000   0.0     0
>  [2,] 0.00000000   0.0     0
>  [3,] 0.00000000   0.0     0
>  [4,] 0.00000000   0.0     0
>  [5,] 0.00000000   0.0     0
>  [6,] 0.00000000   0.0     0
>  [7,] 0.06666667   0.0     0
>  [8,] 0.25000000   0.2     0
>  [9,] 0.48333333   0.0     0
> [10,] 0.20000000   0.0     0
> [11,] 0.00000000   0.0     0
> 
> $U[[2]]
>       [,1] [,2] [,3] [,4]       [,5]       [,6]       [,7]       [,8]      [,9]
>  [1,] 0.08  0.0  0.0 0.00 0.00000000 0.00000000 0.00000000 0.00000000 0.0000000
>  [2,] 0.10  0.0  0.0 0.00 0.00000000 0.00000000 0.00000000 0.00000000 0.0000000
>  [3,] 0.00  0.1  0.0 0.00 0.00000000 0.00000000 0.00000000 0.00000000 0.0000000
>  [4,] 0.00  0.0  0.1 0.00 0.00000000 0.00000000 0.00000000 0.00000000 0.0000000
>  [5,] 0.00  0.0  0.0 0.05 0.05000000 0.00000000 0.00000000 0.00000000 0.0000000
>  [6,] 0.00  0.0  0.0 0.00 0.05373737 0.06666667 0.07676768 0.05416667 0.0000000
>  [7,] 0.00  0.0  0.0 0.00 0.32181818 0.46666667 0.45974026 0.23083333 0.0500000
>  [8,] 0.00  0.0  0.0 0.00 0.21676768 0.20000000 0.30966811 0.50166667 0.4000000
>  [9,] 0.00  0.0  0.0 0.00 0.00000000 0.00000000 0.02222222 0.14416667 0.3428571
> [10,] 0.00  0.0  0.0 0.00 0.00000000 0.06666667 0.02222222 0.01250000 0.2071429
> [11,] 0.00  0.0  0.0 0.00 0.00000000 0.00000000 0.00000000 0.00000000 0.0000000
>       [,10] [,11]
>  [1,]   0.0   0.0
>  [2,]   0.0   0.0
>  [3,]   0.0   0.0
>  [4,]   0.0   0.0
>  [5,]   0.0   0.0
>  [6,]   0.0   0.0
>  [7,]   0.0   0.0
>  [8,]   0.1   0.0
>  [9,]   0.2   0.0
> [10,]   0.6   0.2
> [11,]   0.1   0.4
> 
> $U[[3]]
>       [,1] [,2] [,3] [,4]       [,5] [,6]       [,7]       [,8]      [,9] [,10]
>  [1,] 0.08  0.0  0.0 0.00 0.00000000  0.0 0.00000000 0.00000000 0.0000000   0.0
>  [2,] 0.10  0.0  0.0 0.00 0.00000000  0.0 0.00000000 0.00000000 0.0000000   0.0
>  [3,] 0.00  0.1  0.0 0.00 0.00000000  0.0 0.00000000 0.00000000 0.0000000   0.0
>  [4,] 0.00  0.0  0.1 0.00 0.00000000  0.0 0.00000000 0.00000000 0.0000000   0.0
>  [5,] 0.00  0.0  0.0 0.05 0.05000000  0.0 0.00000000 0.00000000 0.0000000   0.0
>  [6,] 0.00  0.0  0.0 0.00 0.04666667  0.0 0.06666667 0.03333333 0.0000000   0.0
>  [7,] 0.00  0.0  0.0 0.00 0.35000000  0.1 0.50000000 0.11666667 0.1333333   0.0
>  [8,] 0.00  0.0  0.0 0.00 0.18666667  0.1 0.26666667 0.54761905 0.3333333   0.1
>  [9,] 0.00  0.0  0.0 0.00 0.00000000  0.0 0.06666667 0.22380952 0.5333333   0.2
> [10,] 0.00  0.0  0.0 0.00 0.00000000  0.0 0.00000000 0.07857143 0.0000000   0.4
> [11,] 0.00  0.0  0.0 0.00 0.00000000  0.0 0.00000000 0.00000000 0.0000000   0.1
>       [,11]
>  [1,]   0.0
>  [2,]   0.0
>  [3,]   0.0
>  [4,]   0.0
>  [5,]   0.0
>  [6,]   0.0
>  [7,]   0.0
>  [8,]   0.0
>  [9,]   0.0
> [10,]   0.2
> [11,]   0.4
> 
> $U[[4]]
>       [,1] [,2] [,3] [,4]       [,5]       [,6]        [,7]       [,8]
>  [1,] 0.08  0.0  0.0 0.00 0.00000000 0.00000000 0.000000000 0.00000000
>  [2,] 0.10  0.0  0.0 0.00 0.00000000 0.00000000 0.000000000 0.00000000
>  [3,] 0.00  0.1  0.0 0.00 0.00000000 0.00000000 0.000000000 0.00000000
>  [4,] 0.00  0.0  0.1 0.00 0.00000000 0.00000000 0.000000000 0.00000000
>  [5,] 0.00  0.0  0.0 0.05 0.05000000 0.00000000 0.000000000 0.00000000
>  [6,] 0.00  0.0  0.0 0.00 0.04707912 0.02222222 0.067255892 0.04398148
>  [7,] 0.00  0.0  0.0 0.00 0.36382155 0.21111111 0.519745070 0.18101852
>  [8,] 0.00  0.0  0.0 0.00 0.18350168 0.21111111 0.262145262 0.54828042
>  [9,] 0.00  0.0  0.0 0.00 0.00000000 0.00000000 0.029629630 0.15080688
> [10,] 0.00  0.0  0.0 0.00 0.00000000 0.02222222 0.007407407 0.03035714
> [11,] 0.00  0.0  0.0 0.00 0.00000000 0.00000000 0.000000000 0.00000000
>             [,9]      [,10]     [,11]
>  [1,] 0.00000000 0.00000000 0.0000000
>  [2,] 0.00000000 0.00000000 0.0000000
>  [3,] 0.00000000 0.00000000 0.0000000
>  [4,] 0.00000000 0.00000000 0.0000000
>  [5,] 0.00000000 0.00000000 0.0000000
>  [6,] 0.00000000 0.00000000 0.0000000
>  [7,] 0.08333333 0.00000000 0.0000000
>  [8,] 0.32777778 0.13333333 0.0000000
>  [9,] 0.45317460 0.13333333 0.0000000
> [10,] 0.13571429 0.33333333 0.1333333
> [11,] 0.00000000 0.06666667 0.2666667
> 
> 
> $F
> $F[[1]]
>       [,1] [,2] [,3] [,4] [,5] [,6] [,7]     [,8]     [,9] [,10] [,11]
>  [1,]    0    0    0    0    0    0 62.5 422.2222 1583.333     0     0
>  [2,]    0    0    0    0    0    0 62.5 422.2222 1583.333     0     0
>  [3,]    0    0    0    0    0    0  0.0   0.0000    0.000     0     0
>  [4,]    0    0    0    0    0    0  0.0   0.0000    0.000     0     0
>  [5,]    0    0    0    0    0    0  0.0   0.0000    0.000     0     0
>  [6,]    0    0    0    0    0    0  0.0   0.0000    0.000     0     0
>  [7,]    0    0    0    0    0    0  0.0   0.0000    0.000     0     0
>  [8,]    0    0    0    0    0    0  0.0   0.0000    0.000     0     0
>  [9,]    0    0    0    0    0    0  0.0   0.0000    0.000     0     0
> [10,]    0    0    0    0    0    0  0.0   0.0000    0.000     0     0
> [11,]    0    0    0    0    0    0  0.0   0.0000    0.000     0     0
> 
> $F[[2]]
>       [,1] [,2] [,3] [,4] [,5] [,6]     [,7]     [,8]     [,9] [,10] [,11]
>  [1,]    0    0    0    0    0    0 55.55556 572.9167 2392.857  2875  6000
>  [2,]    0    0    0    0    0    0 55.55556 572.9167 2392.857  2875  6000
>  [3,]    0    0    0    0    0    0  0.00000   0.0000    0.000     0     0
>  [4,]    0    0    0    0    0    0  0.00000   0.0000    0.000     0     0
>  [5,]    0    0    0    0    0    0  0.00000   0.0000    0.000     0     0
>  [6,]    0    0    0    0    0    0  0.00000   0.0000    0.000     0     0
>  [7,]    0    0    0    0    0    0  0.00000   0.0000    0.000     0     0
>  [8,]    0    0    0    0    0    0  0.00000   0.0000    0.000     0     0
>  [9,]    0    0    0    0    0    0  0.00000   0.0000    0.000     0     0
> [10,]    0    0    0    0    0    0  0.00000   0.0000    0.000     0     0
> [11,]    0    0    0    0    0    0  0.00000   0.0000    0.000     0     0
> 
> $F[[3]]
>       [,1] [,2] [,3] [,4] [,5] [,6] [,7]    [,8] [,9] [,10] [,11]
>  [1,]    0    0    0    0    0    0    0 422.619 1625   750  2500
>  [2,]    0    0    0    0    0    0    0 422.619 1625   750  2500
>  [3,]    0    0    0    0    0    0    0   0.000    0     0     0
>  [4,]    0    0    0    0    0    0    0   0.000    0     0     0
>  [5,]    0    0    0    0    0    0    0   0.000    0     0     0
>  [6,]    0    0    0    0    0    0    0   0.000    0     0     0
>  [7,]    0    0    0    0    0    0    0   0.000    0     0     0
>  [8,]    0    0    0    0    0    0    0   0.000    0     0     0
>  [9,]    0    0    0    0    0    0    0   0.000    0     0     0
> [10,]    0    0    0    0    0    0    0   0.000    0     0     0
> [11,]    0    0    0    0    0    0    0   0.000    0     0     0
> 
> $F[[4]]
>       [,1] [,2] [,3] [,4] [,5] [,6]     [,7]    [,8]     [,9]    [,10]    [,11]
>  [1,]    0    0    0    0    0    0 39.35185 472.586 1867.063 1208.333 2833.333
>  [2,]    0    0    0    0    0    0 39.35185 472.586 1867.063 1208.333 2833.333
>  [3,]    0    0    0    0    0    0  0.00000   0.000    0.000    0.000    0.000
>  [4,]    0    0    0    0    0    0  0.00000   0.000    0.000    0.000    0.000
>  [5,]    0    0    0    0    0    0  0.00000   0.000    0.000    0.000    0.000
>  [6,]    0    0    0    0    0    0  0.00000   0.000    0.000    0.000    0.000
>  [7,]    0    0    0    0    0    0  0.00000   0.000    0.000    0.000    0.000
>  [8,]    0    0    0    0    0    0  0.00000   0.000    0.000    0.000    0.000
>  [9,]    0    0    0    0    0    0  0.00000   0.000    0.000    0.000    0.000
> [10,]    0    0    0    0    0    0  0.00000   0.000    0.000    0.000    0.000
> [11,]    0    0    0    0    0    0  0.00000   0.000    0.000    0.000    0.000
> 
> 
> $hstages
> [1] NA
> 
> $agestages
>   NA
> 1 NA
> 
> $ahstages
>    stage_id stage original_size original_size_b original_size_c min_age max_age
> 1         1    SD           0.0              NA              NA       0      NA
> 2         2    P1           0.0              NA              NA       0      NA
> 3         3    P2           0.0              NA              NA       0      NA
> 4         4    P3           0.0              NA              NA       0      NA
> 5         5    SL           0.0              NA              NA       0      NA
> 6         6     D           0.0              NA              NA       0      NA
> 7         7   XSm           1.0              NA              NA       0      NA
> 8         8    Sm           3.0              NA              NA       0      NA
> 9         9    Md           6.0              NA              NA       0      NA
> 10       10    Lg          11.0              NA              NA       0      NA
> 11       11   XLg          19.5              NA              NA       0      NA
>    repstatus obsstatus propstatus immstatus matstatus entrystage indataset
> 1          0         0          1         0         0          1         0
> 2          0         0          0         1         0          1         0
> 3          0         0          0         1         0          0         0
> 4          0         0          0         1         0          0         0
> 5          0         0          0         1         0          0         0
> 6          0         0          0         0         1          0         1
> 7          1         1          0         0         1          0         1
> 8          1         1          0         0         1          0         1
> 9          1         1          0         0         1          0         1
> 10         1         1          0         0         1          0         1
> 11         1         1          0         0         1          0         1
>    binhalfwidth_raw sizebin_min sizebin_max sizebin_center sizebin_width
> 1               0.0         0.0         0.0            0.0             0
> 2               0.0         0.0         0.0            0.0             0
> 3               0.0         0.0         0.0            0.0             0
> 4               0.0         0.0         0.0            0.0             0
> 5               0.0         0.0         0.0            0.0             0
> 6               0.5        -0.5         0.5            0.0             1
> 7               0.5         0.5         1.5            1.0             1
> 8               1.5         1.5         4.5            3.0             3
> 9               1.5         4.5         7.5            6.0             3
> 10              3.5         7.5        14.5           11.0             7
> 11              5.0        14.5        24.5           19.5            10
>    binhalfwidthb_raw sizebinb_min sizebinb_max sizebinb_center sizebinb_width
> 1                 NA           NA           NA              NA             NA
> 2                 NA           NA           NA              NA             NA
> 3                 NA           NA           NA              NA             NA
> 4                 NA           NA           NA              NA             NA
> 5                 NA           NA           NA              NA             NA
> 6                 NA           NA           NA              NA             NA
> 7                 NA           NA           NA              NA             NA
> 8                 NA           NA           NA              NA             NA
> 9                 NA           NA           NA              NA             NA
> 10                NA           NA           NA              NA             NA
> 11                NA           NA           NA              NA             NA
>    binhalfwidthc_raw sizebinc_min sizebinc_max sizebinc_center sizebinc_width
> 1                 NA           NA           NA              NA             NA
> 2                 NA           NA           NA              NA             NA
> 3                 NA           NA           NA              NA             NA
> 4                 NA           NA           NA              NA             NA
> 5                 NA           NA           NA              NA             NA
> 6                 NA           NA           NA              NA             NA
> 7                 NA           NA           NA              NA             NA
> 8                 NA           NA           NA              NA             NA
> 9                 NA           NA           NA              NA             NA
> 10                NA           NA           NA              NA             NA
> 11                NA           NA           NA              NA             NA
>    group                       comments alive almostborn
> 1      0                   Dormant seed     1          0
> 2      0               1st yr protocorm     1          0
> 3      0               2nd yr protocorm     1          0
> 4      0               3rd yr protocorm     1          0
> 5      0                       Seedling     1          0
> 6      0                  Dormant adult     1          0
> 7      0    Extra small adult (1 shoot)     1          0
> 8      0       Small adult (2-4 shoots)     1          0
> 9      0      Medium adult (5-7 shoots)     1          0
> 10     0      Large adult (8-14 shoots)     1          0
> 11     0 Extra large adult (>14 shoots)     1          0
> 
> $labels
>   pop patch
> 1   1     A
> 2   1     B
> 3   1     C
> 4   1     0
> 
> $matrixqc
> [1] 118  34   4
> 
> $dataqc
> [1]  74 320
> 
> attr(,"class")
> [1] "lefkoMat"

A quick scan through our output shows that we have four matrices. The labels element shows us their order. There is no time term in the labels element, because all matrices are temporal means. Instead, we see that the first three matrices are the patch-level means for patches A, B, and C. This is followed by the overall population mean matrix, listed as patch 0. It also pays to look at the summary.

summary(cyp2rp_mean)
> 
> This ahistorical lefkoMat object contains 4 matrices.
> 
> Each matrix is square with 11 rows and columns, and a total of 121 elements.
> A total of 118 survival transitions were estimated, with 29.5 per matrix.
> A total of 34 fecundity transitions were estimated, with 8.5 per matrix.
> This lefkoMat object covers 1 population, 4 patches, and 0 time steps.
> 
> The dataset contains a total of 74 unique individuals and 320 unique transitions.
> 
> Survival probability sum check (each matrix represented by column in order):
>          [,1]  [,2]  [,3]  [,4]
> Min.    0.000 0.050 0.050 0.050
> 1st Qu. 0.100 0.140 0.140 0.140
> Median  0.200 0.642 0.600 0.467
> Mean    0.407 0.573 0.506 0.495
> 3rd Qu. 0.763 0.917 0.850 0.776
> Max.    1.000 1.000 1.000 1.000

We now have four matrices, and we see that these matrices have slightly more elements estimated, on average, than in the raw ahMPM. This is because some of the zeros in the original raw ahMPM were zeros only because of a sampling issue - no individuals actually transitioned through a particular transition in a particular year, but may have transitioned in other years, yielding higher numbers of non-zero elements in the arithmetic mean matrices than in the original raw matrices. Unfortunately these zeros will nonetheless drag these mean element values down artificially, potentially impacting our analyses.

Now let’s create the final sets of arithmetic mean matrices for the hMPMs.

cyp3rp_mean <- lmean(cypmatrix3rp)

summary(cyp3rp_mean)
> 
> This historical lefkoMat object contains 4 matrices.
> 
> Each matrix is square with 121 rows and columns, and a total of 14641 elements.
> A total of 309 survival transitions were estimated, with 77.25 per matrix.
> A total of 74 fecundity transitions were estimated, with 18.5 per matrix.
> This lefkoMat object covers 1 population, 4 patches, and 0 time steps.
> 
> The dataset contains a total of 74 unique individuals and 320 unique transitions.
> 
> Survival probability sum check (each matrix represented by column in order):
>           [,1]  [,2]  [,3]  [,4]
> Min.    0.0000 0.000 0.000 0.000
> 1st Qu. 0.0000 0.000 0.000 0.000
> Median  0.0000 0.000 0.000 0.000
> Mean    0.0968 0.156 0.117 0.123
> 3rd Qu. 0.0500 0.200 0.100 0.167
> Max.    1.0000 1.000 1.000 0.991

We see many more elements, and many more estimated elements, the latter reflecting the fact that our estimated transitions now reflect different starting stages in time t-1.

Let’s now look at function-based MPMs in the next chapter.

4.6 Points to remember

  1. Raw, or empirical, MPMs are created by calculating probability transitions directly from the dataset as proportions of individuals actually transitioning to each stage in the next time out of all individuals alive at some time. Fecundity is calculated in a number of ways, but often involves direct calculation based either on actual offspring produced or counts / measures of fecundity proxies, such as flowers.
  2. Ahistorical MPMs include transitions from each stage at time t to each stage in time t+1. Historical MPMs include transitions from each stage in times t and t-1, to each stage in time t+1. Therefore, the latter include information on one time of individual history, while the former do not.
  3. Historical MPMs are sparse and vastly larger than their ahistorical counterparts.
  4. Historical MPMs exist in two formats - Ehrlén and deVries. The latter treats survival transitions from a reproductive step in time t-1 as originating from a prior unborn stage for the newborn, while the former does not separate maternal fecundity from the fate of the newborn.
  5. Use functions rlefko2() and rlefko3() to produce raw ahistorical and historical matrices, respectively.

References

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Caswell, H. (2001). Matrix population models: Construction, analysis, and interpretation. Second edition. Sinauer Associates, Inc., Sunderland, Massachusetts, USA.
Descamps, S., Boutin, S., Berteaux, D. & Gaillard, J.-M. (2008). Age-specific variation in survival, reproductive success and offspring quality in red squirrels: Evidence of senescence. Oikos, 117, 1406–1416.
deVries, C. & Caswell, H. (2018). Demography when history matters: Construction and analysis of second-order matrix population models. Theoretical Ecology, 11, 129–140.
Dogra, H. & Dani, K.G.S. (2019). Defining features of age-specific fertility and seed quality in senescing indeterminate annuals. American Journal of Botany, 106, 604–610.
Ehrlén, J. (2000). The dynamics of plant populations: Does the history of individuals matter? Ecology, 81, 1675–1684.
Leslie, P.H. (1945). On the use of matrices in certain population mathematics. Biometrika, 33, 183–212.
Shefferson, R.P., Kull, T., Hutchings, M.J., Selosse, M.-A., Jacquemyn, H., Kellett, K.M., et al. (2018). Drivers of vegetative dormancy across herbaceous perennial plant species. Ecology Letters, 21, 724–733.
Shefferson, R.P., Mizuta, R. & Hutchings, M.J. (2017). Predicting evolution in response to climate change: The example of sprouting probability in three dormancy-prone orchid species. Royal Society Open Science, 4, 160647.
Shefferson, R.P., Warren II, R.J. & Pulliam, H.R. (2014). Life history costs make perfect sprouting maladaptive in two herbaceous perennials. Journal of Ecology, 102, 1318–1328.