1. Dendroband data as time series
2. Naive forecasting methods and issues
3. Proposed modeling approach
4. TODO’s

Pictured: One particular litu (082422)

Same data as in Cam’s July 31 preso:

Raw dendroband measurements. Call these \(D_t^{obs}\)

Take diffs to compute growth. What do you observe?

Observations about data:

• Exhibits both seasonal pattern and trend
• Very noisy
• Issue: Negative growth?
• Issue: Irregular intervals of measurement

Seasonal pattern (blue) + increasing trend (red)

Say we have observed measurements \(y_1, \ldots, y_t\). Here are some naive forecasts of \(y_{t+1}\):

1. Naive: \(y_{t+1}\) = last value \(y_{t}\)
2. Mean: \(y_{t+1}\) = \(\overline{y}\)
3. Seasonal naive: \(y_{t+1}\) = \(y_{\text{one seasonal period earlier}}\)

Assuming seasonal period of 1 year we have (1) forecast and (2) prediction intervals of uncertainty

(Zoomed-in to 2018-2020) Recall however irregular intervals of measurements

I cheated by back-filling in missing values:

A general class of models used to estimate latent i.e. unobservable variables

• Spatial data: Markov random fields
• Time series: Hidden Markov Model
  • Hidden: Latent variables
  • Markov (property): Current observations depend on previous ones

\[ \begin{eqnarray*} \text{Data model: } D_t^{obs} &\sim& \text{Normal}(D_t, \tau_{obs})\\ \text{Process model: }D_{t+1} &\sim& \text{Normal}(\beta_0 + \beta_1D_{t}, \tau_{pro})\\ &=& \beta_0 + \beta_1D_{t} + \text{Normal}(0, \tau_{pro})\\ \end{eqnarray*} \] where

1. \(D_t^{obs}\) are observed diameters
2. \(D_t, D_{t+1}\) are unobservable latent “true” diameters
3. \(\tau_{obs}\) dictates variation of measurement error from dendroband + caliper
4. \(\tau_{pro}\) dictates variation of error not captured by model i.e. residual

1. Separate observation errors (don’t propagate) from process errors (do propagate)
2. Missing data
3. Prediction/forecasting: Future observations as missing data
4. Data fusion (at the end)

Missing data: Two gaps of missing data in red were imputed.

Predictions of future observations in red:

1. Minimally viable model
2. Next iteration

Based on Clark (2007)

1. Lognormal model for growth because of noisy & negative observed values
2. Hierarchical process model for \(D_t\) and \(\tau_{pro}\):
   1. Random effects (things to account for): tag, site, species
   2. Fixed effects (things we’re interested in): shared effect of year (climate)
3. More emphasis on explicit forecasts/predictions

Next iteration: Bayesian methods allow for data fusion disparate data sources. Ex: different data that have

1. Different observed data types
2. Different time scales
3. Different error structures

1. Diameter censuses
   1. Diameters (same as dendrobands)
   2. Every 5 years
   3. Errors from tape/calipers
2. Ring width from tree coring
   1. Increments, not diameters
   2. Time scale?
   3. Errors from dendrochronology