rand: Support arbitrary normal distributions

3 files changed, 90 insertions, 14 deletions

diff --git a/lib/stdlib/doc/src/rand.xml b/lib/stdlib/doc/src/rand.xml
index 8745e16908..3a5d18dc35 100644
--- a/lib/stdlib/doc/src/rand.xml
+++ b/lib/stdlib/doc/src/rand.xml
@@ -119,6 +119,11 @@ S0 = rand:seed_s(exsplus),
    <pre>
 {SND0, S2} = rand:normal_s(S1),</pre>
 
+   <p>Create a normal deviate with mean -3 and variance 0.5:</p>
+
+   <pre>
+{ND0, S3} = rand:normal_s(-3, 0.5, S2),</pre>
+
     <note>
       <p>This random number generator is not cryptographically
         strong. If a strong cryptographic random number generator is
@@ -201,6 +206,15 @@ S0 = rand:seed_s(exsplus),
     </func>
 
     <func>
+      <name name="normal" arity="2"/>
+      <fsummary>Return a normal distributed random float.</fsummary>
+      <desc>
+        <p>Returns a normal N(Mean, Variance) deviate float
+          and updates the state in the process dictionary.</p>
+      </desc>
+    </func>
+
+    <func>
       <name name="normal_s" arity="1"/>
       <fsummary>Return a standard normal distributed random float.</fsummary>
       <desc>
@@ -211,6 +225,15 @@ S0 = rand:seed_s(exsplus),
     </func>
 
     <func>
+      <name name="normal_s" arity="3"/>
+      <fsummary>Return a normal distributed random float.</fsummary>
+      <desc>
+        <p>Returns, for a specified state, a normal N(Mean, Variance)
+          deviate float and a new state.</p>
+      </desc>
+    </func>
+
+    <func>
       <name name="seed" arity="1"/>
       <fsummary>Seed random number generator.</fsummary>
       <desc>
diff --git a/lib/stdlib/src/rand.erl b/lib/stdlib/src/rand.erl
index 1f457b9e0e..cef83da287 100644
--- a/lib/stdlib/src/rand.erl
+++ b/lib/stdlib/src/rand.erl
@@ -28,7 +28,7 @@
 	 export_seed/0, export_seed_s/1,
          uniform/0, uniform/1, uniform_s/1, uniform_s/2,
          jump/0, jump/1,
-	 normal/0, normal_s/1
+	     normal/0, normal/2, normal_s/1, normal_s/3
 	]).
 
 -compile({inline, [exs64_next/1, exsplus_next/1,
@@ -178,6 +178,13 @@ normal() ->
     _ = seed_put(Seed),
     X.
 
+%% normal/2: returns a random float with N(μ, σ²) normal distribution
+%% updating the state in the process dictionary.
+
+-spec normal(Mean :: number(), Variance :: number()) -> float().
+normal(Mean, Variance) ->
+    Mean + (math:sqrt(Variance) * normal()).
+
 %% normal_s/1: returns a random float with standard normal distribution
 %% The Ziggurat Method for generating random variables - Marsaglia and Tsang
 %% Paper and reference code: http://www.jstatsoft.org/v05/i08/
@@ -198,6 +205,13 @@ normal_s(State0) ->
 	false -> normal_s(Idx, Sign, -X, State)
     end.
 
+%% normal_s/3: returns a random float with normal N(μ, σ²) distribution
+
+-spec normal_s(Mean :: number(), Variance :: number(), state()) -> {float(), NewS :: state()}.
+normal_s(Mean, Variance, State0) when Variance > 0 ->
+    {X, State} = normal_s(State0),
+    {Mean + (math:sqrt(Variance) * X), State}.
+
 %% =====================================================================
 %% Internal functions
 
diff --git a/lib/stdlib/test/rand_SUITE.erl b/lib/stdlib/test/rand_SUITE.erl
index fe5eaccda5..cb97d27992 100644
--- a/lib/stdlib/test/rand_SUITE.erl
+++ b/lib/stdlib/test/rand_SUITE.erl
@@ -27,6 +27,7 @@
 -export([interval_int/1, interval_float/1, seed/1,
          api_eq/1, reference/1,
 	 basic_stats_uniform_1/1, basic_stats_uniform_2/1,
+	 basic_stats_standard_normal/1,
 	 basic_stats_normal/1,
 	 plugin/1, measure/1,
 	 reference_jump_state/1, reference_jump_procdict/1]).
@@ -52,7 +53,8 @@ all() ->
 
 groups() ->
     [{basic_stats, [parallel],
-      [basic_stats_uniform_1, basic_stats_uniform_2, basic_stats_normal]},
+      [basic_stats_uniform_1, basic_stats_uniform_2,
+       basic_stats_standard_normal, basic_stats_normal]},
      {reference_jump, [parallel],
       [reference_jump_state, reference_jump_procdict]}].
 
@@ -294,12 +296,35 @@ basic_stats_uniform_2(Config) when is_list(Config) ->
      || Alg <- algs()],
     ok.
 
-basic_stats_normal(Config) when is_list(Config) ->
+basic_stats_standard_normal(Config) when is_list(Config) ->
     ct:timetrap({minutes,6}), %% valgrind needs a lot of time
-    io:format("Testing normal~n",[]),
-    [basic_normal_1(?LOOP, rand:seed_s(Alg), 0, 0) || Alg <- algs()],
+    io:format("Testing standard normal~n",[]),
+    IntendedMean = 0,
+    IntendedVariance = 1,
+    [basic_normal_1(?LOOP, IntendedMean, IntendedVariance,
+                    rand:seed_s(Alg), 0, 0)
+     || Alg <- algs()],
     ok.
 
+basic_stats_normal(Config) when is_list(Config) ->
+    IntendedMeans = [-1.0e6, -50, -math:pi(), -math:exp(-1),
+                     0.12345678, math:exp(1), 100, 1.0e6],
+    IntendedVariances = [1.0e-6, math:exp(-1), 1, math:pi(), 1.0e6],
+    IntendedMeanVariancePairs =
+        [{Mean, Variance} || Mean <- IntendedMeans,
+                             Variance <- IntendedVariances],
+
+    ct:timetrap({minutes, 6 * length(IntendedMeanVariancePairs)}), %% valgrind needs a lot of time
+    lists:foreach(
+      fun ({IntendedMean, IntendedVariance}) ->
+              io:format("Testing normal(~.2f, ~.2f)~n",
+                        [float(IntendedMean), float(IntendedVariance)]),
+              [basic_normal_1(?LOOP, IntendedMean, IntendedVariance,
+                              rand:seed_s(Alg), 0, 0)
+               || Alg <- algs()]
+      end,
+      IntendedMeanVariancePairs).
+
 basic_uniform_1(N, S0, Sum, A0) when N > 0 ->
     {X,S} = rand:uniform_s(S0),
     I = trunc(X*100),
@@ -339,19 +364,33 @@ basic_uniform_2(0, {#{type:=Alg}, _}, Sum, A) ->
     abs(?LOOP div 100 - Max) < 1000 orelse ct:fail({max, Alg, Max}),
     ok.
 
-basic_normal_1(N, S0, Sum, Sq) when N > 0 ->
-    {X,S} = rand:normal_s(S0),
-    basic_normal_1(N-1, S, X+Sum, X*X+Sq);
-basic_normal_1(0, {#{type:=Alg}, _}, Sum, SumSq) ->
-    Mean = Sum / ?LOOP,
-    StdDev =  math:sqrt((SumSq - (Sum*Sum/?LOOP))/(?LOOP - 1)),
-    io:format("~.10w: Average: ~7.4f StdDev ~6.4f~n", [Alg, Mean, StdDev]),
+basic_normal_1(N, IntendedMean, IntendedVariance, S0, StandardSum, StandardSq) when N > 0 ->
+    {X,S} = normal_s(IntendedMean, IntendedVariance, S0),
+    % We now shape X into a standard normal distribution (in case it wasn't already)
+    % in order to minimise the accumulated error on Sum / SumSq;
+    % otherwise said error would prevent us of making a fair judgment on
+    % the overall distribution when targeting large means and variances.
+    StandardX = (X - IntendedMean) / math:sqrt(IntendedVariance),
+    basic_normal_1(N-1, IntendedMean, IntendedVariance, S,
+                   StandardX+StandardSum, StandardX*StandardX+StandardSq);
+basic_normal_1(0, _IntendedMean, _IntendedVariance, {#{type:=Alg}, _}, StandardSum, StandardSumSq) ->
+    StandardMean = StandardSum / ?LOOP,
+    StandardVariance = (StandardSumSq - (StandardSum*StandardSum/?LOOP))/(?LOOP - 1),
+    StandardStdDev =  math:sqrt(StandardVariance),
+    io:format("~.10w: Standardised Average: ~7.4f, Standardised StdDev ~6.4f~n",
+              [Alg, StandardMean, StandardStdDev]),
     %% Verify that the basic statistics are ok
     %% be gentle we don't want to see to many failing tests
-    abs(Mean) < 0.005 orelse ct:fail({average, Alg, Mean}),
-    abs(StdDev - 1.0) < 0.005 orelse ct:fail({stddev, Alg, StdDev}),
+    abs(StandardMean) < 0.005 orelse ct:fail({average, Alg, StandardMean}),
+    abs(StandardStdDev - 1.0) < 0.005 orelse ct:fail({stddev, Alg, StandardStdDev}),
     ok.
 
+normal_s(Mean, Variance, State0) when Mean == 0, Variance == 1 ->
+    % Make sure we're also testing the standard normal interface
+    rand:normal_s(State0);
+normal_s(Mean, Variance, State0) ->
+    rand:normal_s(Mean, Variance, State0).
+
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 %% Test that the user can write algorithms.