Alaska datasets

remotePARTS ships with a data object. This data set contains NDVI values derived from NASA’s MODIS satellite for the US State of Alaska. The object is ndvi_AK10000 and, due to package size limitations, this data set is a random sampling of 10,000 pixels from the full Alaska data set. This is just for the sake of this vignette: remotePARTS can easily handle the full map (and much larger maps).

For this vignette, we will also create a smaller 3000 pixel subsample ndvi_AK3000 for demonstrative purposes:

data("ndvi_AK10000")
ndvi_AK3000 <- ndvi_AK10000[seq_len(3000),] # first 3000 pixels from the 10K

The ndvi_AK10000 object is a data.frame with 37 columns: lng and lat are longitude and latitude, respectively; AR_coef and CLS_coef are pre-calculated coefficient estimates of the temporal ndvi trends, from pixel-level time series analyses (AR REML and conditional least squares, respectively). These coefficient estimates have been standardized by the mean ndvi value for the pixel over that time period; land is a factor representing land cover classes; and the remaining 32 columns – of the form ndviYYYY – contain the NDVI values from 1982 to 2013. These data already have rare land classes (those that occur in less than 2% of pixels) removed and contain no missing values. remotePARTS does not fully support missing data, so it is recommended that missing observations are removed in pre-processing.

str(ndvi_AK10000)
## 'data.frame':    10000 obs. of  37 variables:
##  $ lng     : num  -151 -145 -144 -163 -145 ...
##  $ lat     : num  63.3 64.7 62.7 67.7 65.8 ...
##  $ AR_coef : num  0.01071 0.003586 0.002488 0.000373 0.002505 ...
##  $ CLS_coef: num  0.00529 0.001537 0.002276 0.000476 0.001514 ...
##  $ land    : Factor w/ 10 levels "Evergr needle",..: 6 7 7 6 7 7 6 6 7 7 ...
##  $ ndvi1982: num  1.88 7.68 8.46 6.85 10.25 ...
##  $ ndvi1983: num  1.74 8.14 7.99 6.97 10.35 ...
##  $ ndvi1984: num  1.96 8.47 8.04 6.67 10.29 ...
##  $ ndvi1985: num  1.74 7.72 7.96 6.9 10.02 ...
##  $ ndvi1986: num  1.92 8.31 8.64 6.43 10.39 ...
##  $ ndvi1987: num  2.04 8.89 8.4 7 10.47 ...
##  $ ndvi1988: num  2.13 8.5 8.32 7.12 10.05 ...
##  $ ndvi1989: num  1.93 8.99 8.79 6.24 10.16 ...
##  $ ndvi1990: num  1.73 8.66 8.45 7.46 10.31 ...
##  $ ndvi1991: num  1.99 8.74 8.34 6.86 10.28 ...
##  $ ndvi1992: num  1.74 7.99 7.94 6.53 9.77 ...
##  $ ndvi1993: num  1.98 8.62 7.93 6.97 10.35 ...
##  $ ndvi1994: num  2.04 8.59 8.26 7.15 10.25 ...
##  $ ndvi1995: num  1.9 8.1 8.55 7.08 10.18 ...
##  $ ndvi1996: num  2.09 8.61 8.23 7.08 10.33 ...
##  $ ndvi1997: num  1.81 8.77 8.99 7.24 10.23 ...
##  $ ndvi1998: num  1.95 7.9 8.57 7.45 9.99 ...
##  $ ndvi1999: num  1.75 8.14 8.61 6.75 10.04 ...
##  $ ndvi2000: num  1.71 7.5 8.37 6.7 9.92 ...
##  $ ndvi2001: num  1.57 7.76 8.29 6.66 9.69 ...
##  $ ndvi2002: num  1.72 7.83 8.62 7.09 9.76 ...
##  $ ndvi2003: num  1.95 7.94 8.35 6.87 9.76 ...
##  $ ndvi2004: num  2.2 8.37 8.83 7.3 11.74 ...
##  $ ndvi2005: num  2.18 8.45 8.63 7.18 11.62 ...
##  $ ndvi2006: num  2.06 8.32 8.36 6.74 11.59 ...
##  $ ndvi2007: num  2.39 8.53 8.63 7.47 11.91 ...
##  $ ndvi2008: num  2.36 8.6 8.41 7 11.8 ...
##  $ ndvi2009: num  2.35 10.02 9.27 6.59 11.62 ...
##  $ ndvi2010: num  2.95 11.09 9.26 7.09 11.46 ...
##  $ ndvi2011: num  2.5 8.9 8.94 7.05 12.15 ...
##  $ ndvi2012: num  2.71 8.71 8.39 6.21 10.97 ...
##  $ ndvi2013: num  2.54 9 8.95 6.93 10.38 ...

For this demonstration, we are interested in asking the following questions:

Is NDVI in Alaska increasing over time?
Are Alaska’s NDVI time trends associated with land cover classes? and
Do Alaska’s NDVI time trends differ with latitude?

The figure below shows a temporal cross-section of these data for 1982, 1998, and 2013.

tibble(ndvi_AK10000) |> 
  pivot_longer(cols = c("ndvi1982", "ndvi1998", "ndvi2013")) |> 
  ggplot(aes(x = lng, y = lat, col = value, fill = value)) + 
  geom_tile() +
  facet_wrap(~ gsub("ndvi", "", name), ncol = 3) +
  scale_color_viridis_c(option = "magma") +
  scale_fill_viridis_c(option = "magma") +
  labs(x = "Longitude", y = "Latitude", col = "ndvi", fill = "ndvi")

The following figure shows how Alaska’s three primary land cover classes are distributed.

ndvi_AK10000 |>
ggplot(aes(x = lng, y = lat, col = land, fill = land)) + 
  geom_tile() +
  scale_color_viridis_d(direction = -1, end = .9) +
  scale_fill_viridis_d(direction = -1, end = .9) +
  labs(y = "Latitude", x = "Longitude", col = "Land cover", 
       fill = "Land cover")

Use help("ndvi_AK10000") to see documentation for these data.

Model

When using remotePARTS, the data are assumed to follow the general stochastic process of the form

\[y(t) = X(t)\beta + \varepsilon(t)\] where

$y(t)$ is a response variable of interest at time $t$
$\beta$ is a vector of coefficients for the predictor variables $X(t)$ on $y(t)$
$\varepsilon(t)$ is an temporal autoregressive process: $\varepsilon(t) = \rho \varepsilon(t - 1) + \delta(t)$
at any time step, $\delta(t)$ is spatially autocorrelated according to covariance matrix $\Sigma$: $\delta(t) \sim N(0, \Sigma)$
there is no temporal dependence in $\delta(t)$ (i.e., $\delta(t_i)$ is independent of $\delta(t_j)$ except when $t_i$ = $t_j$)

Time-series analysis

The first step in a typical remotePARTS workflow is to obtain pixel-level estimates of time-series coefficients. In our example, we are interested in estimating the time trends in NDVI for each pixel $i$, represented by $\beta_1$ in the regression model

\[y_i(t) = \beta_0 + \beta_1 t + \varepsilon_i(t)\]

where the random errors $\varepsilon_i(t)$ follow an AR(1) process:

\[\varepsilon_i(t) = b\varepsilon_i(t - 1) + \delta_i(t)\] \[\delta_i(t) \sim N(0 , \sigma)\]

We will use the function fitAR_map() to estimate $\beta_1$, which fits pixel-level AR(1) models to a data frame of pixels and estimates coefficients using restricted maximum likelihood (REML). To do so, we must extract only our NDVI columns as the matrix Y. We’ll do this by matching all column names containing “ndvi” and slicing the data.frame:

Y <- ndvi_AK3000 |> select(contains("ndvi")) |> as.matrix()

We also need a 2-column coordinate matrix coords:

# coords <- as.matrix(ndvi_AK3000[, c("lng", "lat")])
coords <- ndvi_AK3000 |> select("lng", "lat") |> as.matrix()

Y and coords are then passed to fitAR_map() with default settings:

ARfit <- fitAR_map(Y = Y, coords = coords)

Coefficient estimates can be obtained from ARfit with coefficients(). The first column is the estimate of $\beta_0$, $\hat{\beta_0}$, and the second is $\hat{\beta_1}$.

coefficients(ARfit) |> head()
##      (Intercept)            t
## [1,]    1.703446  0.021923615
## [2,]    7.980618  0.030464243
## [3,]    8.146784  0.021130056
## [4,]    6.883592  0.002582853
## [5,]   10.081968  0.026466877
## [6,]    7.628119 -0.010395964

These time-series analyses calculate the time trend in the raw NDVI data. In most situations it makes sense to ask if there are time trends in the relative NDVI values, that is, changes in NDVI relative to the mean value of NDVI in a pixel. Scaling the trend in NDVI relative to the mean gives assessments of the proportional change in NDVI. These trends in proportional NDVI are calculated be dividing $\hat{\beta_1}$ by the mean. The values of the trend coefficients are contained in ARfit$coefficients, and since the coefficients for the trend are in the column of the coefficients matrix named t, the scaling is performed as

ARfit$coefficients[, "t"] <- ARfit$coefficients[,"t"]/rowMeans(Y)

These scaled values of the time trend are then stored in the ndvi_AK3000 data frame.

ndvi_AK3000$AR_coef <- coefficients(ARfit)[, "t"]

Below is an image of the estimated coefficients (pre-calculated and scaled) for the full ndvi_AK10000. From this, it appears that northern latitudes may be greening faster than more southern latitudes.

ndvi_AK10000 %>% 
  filter(AR_coef <= 0.06) |> 
  ggplot(aes(x = lng, y = lat, col = AR_coef, fill = AR_coef)) + 
  geom_tile() +
  scale_color_viridis_c(option = "magma") +
  scale_fill_viridis_c(option = "magma") +
  guides(fill = "none") + 
  labs(
    y = "Latitude", x = "Longitude", col = expression(beta[1]), 
    fill = expression(beta[1])
  )

fitAR_map and its conditional least-squares counterpart, fitCLS_map, are wrappers for the functions fitAR and fitCLS which conduct individual time series analysis. If the user wants, these can be applied on a pixel-by-pixel basis to the data to allow greater flexibility. Both AR REML and CLS methods account for temporal autocorrelation in the time series. See function documentation for more details (i.e., help("fitAR_map")).

Spatial relationships

Now that we’ve reduced the temporal component of each time series to a single value (i.e., estimates of $\beta_1$) while accounting for temporal autocorrelation, we can focus on the spatial component of our problem.

Distance

The first step is to calculate the distances among pixels as a distance matrix D. Here, we’ll calculate relative distances with distm_scaled() from our coordinate matrix.

D <- distm_scaled(coords)

distm_scaled() scales distances across the spatial domain so that the greatest distance between two pixels is 1. Because distm_scaled() uses geosphere::distm(), it treats coordinates as degrees longitude and latitude on a sphere and calculates distances accordingly.

Covariance

Next, we will estimate the expected correlation between the random errors of our spatial response variable (estimates of $\beta_1$) based on their distances. To do so, we need a spatial covariance function. In this example, we will use an exponential covariance: $V = \exp\big(\frac{-D}{r}\big)$ where $r$ is a parameter that dictates the range of spatial autocorrelation. The function covar_exp() implements exponential covariance.

The below figure is a semivariogram of covar_exp for two values of r.

ggplot() +
  xlim(0, 1) +
  geom_function(
    fun = covar_exp, args = list(r = 0.05), aes(col = "0.05"), linewidth = 1
  ) +
  geom_function(
    fun = covar_exp, args = list(r = 0.1), aes(col = "0.1"), linewidth = 1
  ) +
  geom_function(
    fun = covar_exp, args = list(r = 0.2), aes(col = "0.2"), linewidth = 1
  ) +
  labs(x = "distance", y = "covar_exp(distance, r)", col = "r") +
  theme_bw() + 
  theme(
    legend.position = "inside", legend.position.inside = c(1, 1),
    legend.justification = c(1, 1), legend.background = element_blank()
  ) +
  scale_color_viridis_d(end = 0.95)

If we know the range parameter $r$, we can calculate the covariance V from distances D with covar_exp():

r <- 0.1
V <- covar_exp(D, r)

V represents the correlation among points if all variation is accounted for. However, it is safest to assume that there is some additional source of unexplained and unmeasured variance (i.e., a nugget $\eta$).

Here is a semivariogram, with and without a nugget component for covar_exp:

ggplot() +
  xlim(0, 1) +
  geom_function(
    fun = function(x, n)(1-n)*covar_exp(x, 0.2), 
    args = list(n = 0), aes(col = "0"), linewidth = 1
  ) +
  geom_function(
    fun = function(x, n)(1-n)*covar_exp(x, 0.2), 
    args = list(n = 0.3), aes(col = "0.3"), linewidth = 1
  ) +
    geom_function(
    fun = function(x, n)(1-n)*covar_exp(x, 0.2), 
    args = list(n = 0.5), aes(col = "0.5"), linewidth = 1
  ) +
  labs(x = "distance", y = "covar_exp(distance, r = 0.2)", col = expression(eta)) +
  theme_bw() + 
  theme(
    legend.position = "inside", legend.position.inside = c(1, 1),
    legend.justification = c(1, 1), legend.background = element_blank()
  ) +
  scale_color_viridis_d(end = 0.95)

We assume that the covariance structure among pixels is given by $\Sigma = \eta I + (1-\eta)V$ where $I$ is the identity matrix:

nugget <- 0.2 
I <- diag(nrow(V)) # identity matrix
Sigma <- nugget*I + (1-nugget)*V

See help("covar_exp") for a description of the all covariance functions provided by remotePARTS and for more information regarding their use.

GLS: 3000-pixel subset

To test spatial hypotheses with remotePARTS, we use a generalized least-squares regression model (GLS):

\[\theta = X\alpha_{gls} + \gamma \] where

$\theta$ is a vector of response values
$\alpha_{gls}$ is a vector of the effects of predictor variables $X$ on $\theta$
the error term $\gamma$ is spatially autocorrelated according to $\Sigma_\gamma$: $\gamma \sim N(0, \Sigma_\gamma)$

$\theta$ will usually be a regression parameter. For example, if we’re interested in understanding trends in NDVI over time, we would use the pixel-level regression coefficient for the effect of time on NDVI (i.e., $\theta = \hat\beta$).

Known parameters

If the parameters that govern the spatial autocorrelation $\Sigma_\gamma$ are known, a GLS can be fit with fitGLS(). Here, we will fit the GLS by providing (i) a model formula, (ii) a data source, (iii) our V matrix, which was pre-calculated with a spatial parameter $r = 0.1$ and (iv) a nugget of $\eta = 0.2$.

The specific task in this examples is to estimate the effect of land cover class on our time trend. Because land is a factor, we’ll also specify a no-intercept model to make interpretation clearer.

The GLS fitting process requires an inversion of V. This means that even with only the 3000-pixel subset, it will take a few minutes to finish the computations on most computers (but see the ncores and parallel arguments).

GLS.0 <- fitGLS(
  formula = AR_coef ~ 0 + land, data = ndvi_AK3000, V = V, nugget = nugget
)

fitGLS adds the nugget to V internally. If we wanted to do this ourselves, we could pass the covariance matrix Sigma which already contains a nugget component and then set the nugget argument of fitGLS to 0:

# equivalently:
fitGLS(
  formula = AR_coef ~ 0 + land, data = ndvi_AK3000, V = Sigma, nugget = 0
)

The estimates for our land class effects can be extracted with coefficients().

coefficients(GLS.0)
## landShrubland   landSavanna landGrassland 
##  0.0008646565  0.0002554862  0.0011294576

The full model fit object looks like this:

GLS.0
## 
## Call:
## fitGLS(formula = AR_coef ~ 0 + land, data = ndvi_AK3000, V = V, 
##     nugget = nugget)
## 
## t-tests:
##                     Est t.stat pval.t
## landShrubland 0.0008647 0.5065 0.6126
## landSavanna   0.0002555 0.1493 0.8813
## landGrassland 0.0011295 0.6648 0.5062
## 
## F-test:
##                model df_F    SSE       MSE logLik Fstat  pval_F
## 1 AR_coef ~ 0 + land    2 0.1128 3.764e-05  12808 4.461 0.01163
## 2        AR_coef ~ 1 2997 0.1131 3.775e-05  12802    NA      NA

Although none of the three land cover classes show a statistically significant time trend in (proportional) NDVI, the F-test is significant. This means that including the model including land cover explains the data better than than the intercept-only model: overall, land cover explains significant variation in temporal trends of NDVI.

Parameter estimation

In practice, we rarely know the values of the parameters that govern spatial autocorrelation (e.g., the range and nugget) in advance. Therefore, these parameters will need to be estimated for most data.

Spatial parameters

The spatial parameters of a covariance function (e.g., covar_exp) can be estimated from residuals of pixel-level time-series models (see Ives et al. RSE, 2021). Although we conducted the time-series analyses as though each pixel was independent (with fitAR_map()), they are, in fact, dependent. Specifically, the correlation of the residuals from the pixel-level analyses is roughly proportional to the spatial autocorrelation of the residuals of the spatial model, $\gamma$, if all of the variation in $\gamma$ is due to the spatiotemporal variation produced by $\varepsilon_i(t)$. Therefore, we can estimate the range parameter for V as $V_{ij} \approx \text{cor}\big(\varepsilon_i(t), \varepsilon_j(t)\big)$

The function fitCor() performs the estimation of spatial parameters. We will pass to this function (i) the time-series residuals for our map, extracted from the time-series analysis output object with residuals(), (ii) the coordinate matrix coords, (iii) the covariance function covar_exp(), and (iv) a list specifying that we should start the search for the optimal range parameter at 0.1. For this example, we will also specify fit.n = 3000, which ensures that all 3000 pixels are used to estimate spatial parameters.

corfit <- fitCor(
  resids = residuals(ARfit), coords = coords, covar_FUN = "covar_exp", 
  start = list(range = 0.1), fit.n = 3000
)
(range.opt = corfit$spcor)
##     range 
## 0.1083729

By default, fitCor() uses distm_scaled() to calculate distances from the coordinate matrix, but any function that returns a distance matrix can be specified with the distm_FUN argument. It is important to scale the parameter values appropriately, accounting for your distances. For example, if we instead use distm_km() to calculate distance in km instead of relative distances, we would need to scale our starting range parameter by the maximum distance in km of our map:

max.dist <- max(distm_km(coords))
corfit.km <- fitCor(
  resids = residuals(ARfit), coords = coords, covar_FUN = "covar_exp",
  start = list(range = max.dist*0.1), distm_FUN = "distm_km", fit.n = 3000
)

Depending on the covariance function used, not all parameters will need scaling. For example, covar_exppow() is an exponential-power covariance function and takes a range and shape parameter, but only the range parameter should scale with distance. See ?fitCor() for more details.

After we’ve obtained our range parameter estimate, we can use it to re-calculate the V matrix:

V.opt <- covar_exp(D, range.opt)

Nugget

Similar to finding the optimal spatial parameters, the nugget can be estimated by selecting a nugget that maximizes the likelihood of the GLS given the data. fitGLS() will find this maximum-likelihood nugget when nugget = NA is specified. This type of optimization requires fitting multiple GLS models, which means it will be much slower than our call to fitGLS() with a known nugget.

In addition to our original arguments, we’ll also explicitly set no.F = FALSE so that F-tests are calculated. For the F-tests, the default reduced model is the intercept-only model, although it is also possible to specify alternative models as a formula in the formula0 option.

GLS.opt <- fitGLS(
  formula = AR_coef ~ 0 + land, data = ndvi_AK3000, V = V.opt, 
  nugget = NA, no.F = FALSE
)
(nug.opt = GLS.opt$nugget)
## [1] 0.1342314
coefficients(GLS.opt)
## landShrubland   landSavanna landGrassland 
##  0.0009201230  0.0003899361  0.0011467324

Let’s compare our GLS from earlier with this one with optimized parameters:

(mod_compare <- tribble(
  ~mod, ~range, ~nugget, ~logLik, ~MSE,
  "GLS.0", r, GLS.0$nugget, GLS.0$logLik, GLS.0$MSE,
  "GLS.opt", range.opt, GLS.opt$nugget, GLS.opt$logLik, GLS.opt$MSE
))
## # A tibble: 2 × 5
##   mod     range nugget logLik       MSE
##   <chr>   <dbl>  <dbl>  <dbl>     <dbl>
## 1 GLS.0   0.1    0.2   12808. 0.0000376
## 2 GLS.opt 0.108  0.134 12809. 0.0000500

In this example, logLik for GLS.opt is not functionally different than logLik for GLS.0. This indicates that using the values of range = 0.1 and nugget = 0.2 gives a similar likelihood than the optimal model when range value is calculated from covar_exp(), GLS.opt. The key difference between these models is that the MSE is higher for GLS.opt, indicative of the increased uncertainty.

Simultaneous parameter estimation

It is also possible to simultaneously estimate spatial parameters and the nugget without using the time-series residuals. This is done by finding the set of parameters (e.g., range and nugget) that maximizes the likelihood of a GLS given the data. This task is computationally slower than optimizing nugget alone with fitGLS() and therefore will take some time to run.

fitopt <- fitGLS_opt(
  formula = AR_coef ~ 0 + land, data = ndvi_AK3000, 
  coords = ndvi_AK3000[, c("lng", "lat")], 
  covar_FUN = "covar_exp", 
  start = c(range = .1, nugget = .2),
  method = "BFGS", # use BFGS algorightm (see ?stats::optim())
  control = list(reltol = 1e-5)  # lower the convergence tolerance (see ?stats::optim()) 
)
fitopt$opt$par
#      range     nugget 
# 0.02497874 0.17914929 
fitopt$GLS$logLik
# [1] 12824.77
fitopt$GLS$MSE
# [1] 2.475972e-05

Because fitGLS_opt() does not require time series residuals, it is possible to use fitGLS_opt() for statistical problems involving only spatial variables. In other words, rather than $\theta$ being limited to spatially distributed trend coefficients, it can also be any spatial variable.

When time-series residuals are available, we recommend estimating spatial parameters with fitCor() and fitGLS(), rather than fitGLS_opt(). The former typically has better statistical performance than the latter. See ?fitGLS_opt() for more information about this function and its use.

Hypothesis testing

Ultimately, the purpose of the tools provided by remotePARTS is map-level hypotheses testing. In this example, we will test 3 hypotheses using 3 different GLS models.

Intercept-only model

If we want to test the hypothesis that “there was a trend in Alaska NDVI
from 1982-2013”, we can fit an intercept-only GLS model:

(GLS.int <- fitGLS(
  AR_coef ~ 1, data = ndvi_AK3000, V = V.opt, nugget = nug.opt, no.F = TRUE
))
## 
## Call:
## fitGLS(formula = AR_coef ~ 1, data = ndvi_AK3000, V = V.opt, 
##     nugget = nug.opt, no.F = TRUE)
## 
## t-tests:
##                  Est t.stat pval.t
## (Intercept) 0.000972 0.4564 0.6481
## 
## Model statistics:
##      SSE       MSE logLik
## 1 0.1503 5.011e-05  12806

We can see from the t-test that the intercept is not statistically different from zero: there is no map-level temporal trend in NDVI across the entire data set.

land cover effects

If we want to test the hypothesis that “trends in Alaskan NDVI differ by land- cover class”, we can use GLS.opt from earlier:

GLS.opt <- fitGLS(
  formula = AR_coef ~ 0 + land, data = ndvi_AK3000, V = V.opt, 
  nugget = NA, no.F = FALSE
)

GLS.opt
## 
## Call:
## fitGLS(formula = AR_coef ~ 0 + land, data = ndvi_AK3000, V = V.opt, 
##     nugget = NA, no.F = FALSE)
## 
## t-tests:
##                     Est t.stat pval.t
## landShrubland 0.0009201 0.4306 0.6668
## landSavanna   0.0003899 0.1822 0.8554
## landGrassland 0.0011467 0.5385 0.5903
## 
## F-test:
##                model df_F    SSE       MSE logLik Fstat  pval_F
## 1 AR_coef ~ 0 + land    2 0.1499 5.003e-05  12809 3.263 0.03841
## 2        AR_coef ~ 1 2997 0.1503 5.014e-05  12805    NA      NA

The t-tests show that the trend in NDVI, for all land cover classes, was not statistically different from zero: NDVI did not exhibit a trend in any land cover class. The F-test (ANOVA table), however, provides evidence that land class collectively explain some variation in NDVI trends.

Latitude effects

Finally, to test the hypothesis that “temporal trends in NDVI differ with latitude”, we can regress the AR coefficient on latitude:

(GLS.lat <- fitGLS(
  AR_coef ~ 1 + lat, data = ndvi_AK3000, V = V.opt, nugget = nug.opt, 
  no.F = FALSE
))
## 
## Call:
## fitGLS(formula = AR_coef ~ 1 + lat, data = ndvi_AK3000, V = V.opt, 
##     nugget = nug.opt, no.F = FALSE)
## 
## t-tests:
##                    Est   t.stat pval.t
## (Intercept) -8.617e-04 -0.04362 0.9652
## lat          2.986e-05  0.09337 0.9256
## 
## F-test:
##               model df_F    SSE       MSE logLik    Fstat pval_F
## 1 AR_coef ~ 1 + lat    1 0.1503 5.012e-05  12806 0.008719 0.9256
## 2       AR_coef ~ 1 2998 0.1503 5.012e-05  12806       NA     NA

The t-tests show that temporal trends in NDVI did not differ with latitude. The p-value from the F-test is equivalent to that of the t-test p-value for effect of latitude because they are testing the same hypothesis.

Conclusions (ndvi_AK3000)

We can see from these hypothesis tests that, at least among the 3000-pixel sub-sample of Alaska, the answer to all three questions that we posed is no: there is no statistical evidence for a greening trend in Alaska, nor strong differences among land cover classes or latitude.

Large data sets

Until now, we have limited our analyses to the 3000-pixel subset of Alaska, ndvi_AK3000. Calls to fitGLS() involve inverting V, and the computational complexity scales with $N^3$ where $N$ is the number of pixels in the map. We have used the data set ndvi_AK3000 up to now because the computation time for the analyses is reasonable. Even with only 3000 pixels, we must handle distance and covariance matrices that each contain $3,000 \times 3,000 = 9,000,000$ elements. This is approaching the upper size limit for obtaining relatively fast results on most personal consumer hardware.

By contrast, the covariance matrix for the larger ndvi_AK10000 data set would have $10,000 \times 10,000 = 100,000,000$ elements which creates a computationally infeasible problem for a typical computer.

For these reasons, fitGLS_paritition may be the most useful function in the remotePARTS package. This function can perform the GLS analysis on the full ndvi_AK10000 data set. In fact, ndvi_AK10000 is quite small in comparison to many remote sensing data sets, any of which could be analyzed with fitGLS_partition().

Partitioned GLS

fitGLS_parition() conducts GLS analyses on partitions of the data and then combines the results from the partitions to give overall statistical results. Specifically, this process (1) breaks the data into smaller and more manageable pieces (partitions), (2) conducts GLS on each partition, (3) calculates cross-partition statistics from pairs of partitions, and (4) summarizes the results with statistical tests that account for correlations among partitions. We will use the full ndvi_AK10000 data set to demonstrate fitGLS_parition().

We have already performed the time-series analyses on the full data set. These are in the AR_coef column of ndvi_AK10000.

data("ndvi_AK10000")

Step (1) is to divide pixels into partitions, which is done with the function sample_parition(). Passing the number of pixels in our map and the argument partsize = 1500 will result in a partition matrix with 1,500 rows and 6 columns. Columns of the resulting partition matrix pm each contain a random sample of 1,500 pixels. Each of these 6 samples (partitions) are non-overlapping, containing no repeats. Setting npart = NA will automatically give the maximum number of partitions possible (i.e., 6).

pm <- sample_partitions(npix = nrow(ndvi_AK10000), partsize = 1500, npart = NA)
dim(pm)
## [1] 1500    6

Once we have our partition matrix, fitGLS_parititon() performs steps (2)-(4) of the analyses. The input is similar to fitGLS. For this example, we specify (i) a formula, (ii) the data as a data.frame (AK10000), (iii) the partition matrix (pm), (iv) the covariance function (covar_FUN), (v) a list of spatial parameters (covar.pars) including our optimized range parameter, and a nugget. If nugget is specified, this value is used for the calculations, while if nugget = NA (the default) it is estimated for each partition separately.

Although the compute time is much faster than if we needed to invert the full covariance matrix V, this example still takes a long time to fit (but scales roughly linearly with number of partitions). Therefore, we have saved the output of this code as an R object partGLS.ndviAK so that you can look at its output without having to execute the function.

The model was fit with this code:

partGLS_ndviAK <- fitGLS_partition(
  formula = AR_coef ~ 0 + land, data = AK10000, 
  partmat = pm, covar_FUN = "covar_exp", 
  covar.pars = list(range = range.opt),
  nugget = nug.opt, ncores = 4
)

and can be loaded with

data(partGLS_ndviAK)

Here are the t-tests, that show that land cover class does not significantly affect NDVI trend:

partGLS_ndviAK$overall$t.test
## $p.t
##                        Est          SE    t.stat    pval.t
## landShrubland 0.0013094359 0.001899589 0.6893260 0.4906359
## landSavanna   0.0004379666 0.001902064 0.2302586 0.8178960
## landGrassland 0.0016210956 0.001894683 0.8556027 0.3922404
## 
## $covar_coef
##              [,1]         [,2]         [,3]
## [1,] 3.623978e-06 3.569499e-06 3.561642e-06
## [2,] 3.566697e-06 3.633439e-06 3.537596e-06
## [3,] 3.562415e-06 3.540957e-06 3.605232e-06

It is highly recommended that users read the full documentation (help("fitGLS_parition")).

Here is the p-value for the chisqr test of the partitioned GLS

partGLS_ndviAK$overall$pval.chisqr
## pval.chisqr 
##       1e-06 
## attr(,"rankdef.MSR")
## [1] 0

This again, indicates that the model which includes land cover classes better explains variation in NDVI trends than the intercept-only model. The p-value is much lower than the p-value from the F-test conducted by fitGLS(). This is likely due to outliers in the data, which should be removed before conducting any real analysis. One simple way to filter potential trend outliers would be to remove any pixels whose time-series coefficient standard errors are unusually large (e.g., SE > 4*mean(SE)).

Conclusions

Our partitioned GLS has returned the same conclusions as our previous GLS analysis. No map-level trend in NDVI was found within any of the land cover classes.

This was only one of our three hypotheses tested earlier. The remaining two can easily be tested with fitGLS_partition(), as they were with fitGLS, by changing the formula argument accordingly.

Alaska

NDVI analysis (remotePARTS)

Introduction