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Spectral signals and the hodrick-prescott filter

08 July, 2011 - Dan Hammer

The Hodrick-Prescott filter is commonly used to parse cyclical behavior from an economic time-series. Instead, we applied the H-P filter to smooth time-series data from the Moderate Resolution Imaging Spectroradiometer (MODIS) sensor aboard NASA's Terra and Aqua satellites. The MODIS sensors observe the earth's surface every 1 to 2 days, collecting data that can be used for earth systems modeling. We use the MODIS data to estimate forest clearing activity in the tropics. The H-P filter was originally built for business cycles; but we applied it to natural cycles.

The FORMA project is a Center for Global Development initiative to provide information on deforestation each month. We -- the FORMA development team -- are interested in the detection of steep, extra-seasonal drops in the "greenness" of vegetation. We use these abrupt changes to help identify where and when forest clearing activity is occurring. The Normalized Difference Vegetation Index (NDVI) is closely correlated with the health and density of terrestrial vegetation. The NDVI data stream, however, is subject to all sorts of problems, includign idiosyncratic error due to (among other things) persistent cloud cover. We are experimenting with different methods to smooth the time-series over anomalous observations, but we've found that the H-P filter works pretty well.

The graph below displays the a pre-conditioned NDVI time-series for a 1km x 1km tract of forested land in Indonesia between February 2000 and December 2010 [in blue]. We have already removed and interpolated "unreliable" values, as indicated by an associated MODIS measurement of the error from cloud cover (among other things). The H-P filter for this NDVI time-series is also plotted [in red], with a smoothing parameter λ = 50. The λ parameter reflects the filter's sensitivity to short-term variation in the NDVI. As λ increases, the filtered curve approaches the OLS regression line -- which is not at all sensitive to short-term fluctuations in the time-series. As λ decreases, the filter approaches the original time-series; it is absolutely sensitive to short-term variation.

There is no standard value for the smoothing parameter. For quarterly economic data, Hodrick and Prescott suggest a λ value of 1600. We apply our change detection algorithms to the filtered NDVI series associated with many values of λ in order to identify the point at which there is a persistent and significant decline in the vegetation index. For this particular pixel, displayed above, this point is around mid-2008.

We are currently porting the entire FORMA system to Clojure, a very elegant Lisp that allows us to efficiently implement our algorithms on a cloud computation platform. We therefore don't have to choose a λ value a priori but instead apply thousands of smoothing parameters to the data, incorporating only the derived information that best represents the signals from forest clearing activity.

I wasn't able to find a pre-written H-P filter in Clojure, so I wrote my own. I am relatively new to Clojure; and so, please feel free to tear this code apart.

;; Hodrick-Prescott filter

(use '(incanter core charts))

(defn insert-at
  "insert list [b] into list [a] at index [idx]."
  [idx a b]
  (let [opened (split-at idx a)]
    (concat (first opened) b (second opened))))

(defn insert-into-zeros
  "insert vector [v] into a vector of zeros of total length [len]
  at index [idx]."
  [idx len v]
  (insert-at idx (repeat (- len (count v)) 0) v))

(defn hp-mat
  "create the matrix of coefficients from the minimization problem
  required to parse the trend component from a time-series of le-
  gth [T], which has to be greater than or equal to 9 periods."
  {:pre [(>= T 9)]
   :post [(= [T T] (dim %))]}
  (let [first-row  (insert-into-zeros 0 T [1 -2 1])
        second-row (insert-into-zeros 0 T [-2 5 -4 1])
        inner (for [x (range (inc (- T 5)))]
                   (insert-into-zeros x T [1 -4 6 -4 1]))]
      (concat [first-row]
             [(reverse second-row)]
             [(reverse first-row)]))))

(defn hp-filter
  "return a smoothed time-series, given the HP filter parameter."
  [ts lambda]
  (let [T (count ts)
        trend-cond (solve 
                       (mult lambda (hp-mat T))
                       (identity-matrix T)))]
    (mmult trend-cond ts)))
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